CONFERENCE PROCEEDINGS
International Student Conference
“Science and Progress”
German-Russian
Interdisciplinary
Science Center
St. Petersburg – Peterhof
November, 14-18
2011
Organizing committee
Prof. Dr. S.F. Bureiko, Prof. Dr. A.M. Shikin, E.I. Spirin,
E.V. Serova, Dr. A.G. Rybkin, A.A. Popova, Dean of Faculty Physics, SPSU
Coordinator of G-RISC, SPSU
Dean-assistant of Faculty of Physics, SPSU
Head of Academic Mobility Department, SPSU
G-RISC office, SPSU
G-RISC office, SPSU
Program Committee
Prof. Dr. E. Rühl, Prof. Dr. C. Laubschat, Prof. Dr. A.M. Shikin, Prof. Dr. Yu.S. Tver’yanovich, Prof. Dr. V.N. Troyan, Coordinator of G-RISC, FU Berlin
Faculty of Physics, TU Dresden
Coordinator of G-RISC, SPSU
Faculty of Chemistry, SPSU
Faculty of Physics, SPSU
Contacts
Faculty of Physics Saint-Petersburg State University
Ulyanovskaya ul. 3,
Peterhof, St. Petersburg, Russia
198504
Tel. +7 (812) 428-46-56,
Fax. +7 (812) 428-46-55
E-mail: g-risc@phys.spbu.ru
Website: www.phys.spbu.ru/grisc
3
Heads of sections
A. Chemistry – Prof. Dr. Yu.S. Tver’yanovich, Faculty of Chemistry, SPSU
B. Geo- and Astrophysics – Prof. Dr. V.N. Troyan, Faculty of Physics, SPSU,
Dr. V.G. Nagnibeda, Faculty of Mathematics and Mechanics, SPSU
C. Mathematics and Mechanics
– Prof. Dr. V. Reitmann,
Faculty of Mathematics and Mechanics, SPSU
D. Solid State Physics – Prof. Dr. A.P. Baraban,
Faculty of Physics, SPSU
E. Applied Physics
– Prof. Dr. A.S. Chirtsov,
Faculty of Physics, SPSU
F. Optics and Spectroscopy
– Prof. Dr. Yu.V. Chizhov,
Prof. Dr. N.A. Timofeev,
Faculty of Physics, SPSU
G. Theoretical, Mathematical and Computational Physics
– Prof. Dr. Yu.M. Pis’mak,
Faculty of Physics, SPSU
H. Biophysics
– Prof. Dr. N.V. Tsvetkov,
Faculty of Physics, SPSU
I. Resonance Phenomena in Condenced Matter
– Prof. Dr. V.I. Chizhik,
Faculty of Physics, SPSU
4
A. Chemistry
Solid-contact ion-selective electrodes with ion-to-electron
transducer layer composed of nanostructured materials
Ivanova Nataliya
inm251284@mail.ru
Scientific supervisor: Prof. Dr. Mikhelson K.N., Department of
Physical Chemistry, Chemical Faculty, Saint-Petersburg State
University
Introduction
Ion-selective electrodes (ISEs): potentiometric sensors of ions, comprise routinely used analytical tool for essay of various analytes of clinical, industrial, and
environmental relevance. However ISEs of the conventional design, containing
internal aqueous solution and internal reference electrode, don’t fit modern requirements. The so-called solid-contact ISEs (SC-ISEs) – without internal filling
– would allow for easier miniaturization, for planar technology of manufacturing,
and, eventually, for better quality combined with lower production cost. The fundamental problem with SC-ISEs refers to ion-to-electron conductivity transduction
at a stable and reproducible potential difference at the interface between the ionically conducting sensor layer (membrane) and electronically conducting substrate.
This problem has been successfully solved for electrodes with glass and crystalline
membranes. Between a glass membrane containing an alkali metal oxide and the
lead they place a tin alloy doped with the respective alkali metal, thus utilizing the
first-kind electrode concept. The second-kind electrode concept is implemented
in SC-ISEs with crystalline membranes normally containing Ag2S: the inner side
of the membrane is vacuum-sputtered with Ag, and then the lead is soldered to
this silver layer.
None of these approaches can be used for SC-ISEs with solvent polymeric
membranes containing ionophores: neutral or charged species selectively binding
ions, and in this way ensuring a selective potentiometric response of the sensor.
Conducting polymers appear the most promising ion-to-electron transducers for this
kind of sensors [1]. So far, however, the long-term stability and the piece-to-piece
reproducibility of the potentials of ionophore-based SC-ISEs does not fit practical
needs, and remains well below that of conventional ISEs.
In this work we try graphenes, nanostructured polymeric composite of Cu(I),
and also hexacyanoferrates as active components of ion-to-electron transducer
layer in SC-ISEs.
Materials and methods
The electronically conducting substrate was always graphite encapsulated in
a PVC body. The transducer layers were formed by drop-casting solutions of the
respective materials on top of the substrate. As active components of ion-to-electron
transducer layer we used nanostructured materials such as electroactive conjugated
polymer: polymeric complex of Cu(I) with bichynolyl-containing polyamidoacid
6
(as 4.5 % solution in N-metylpyrrolidone – N-MP) – PAC-2, graphemes (as 2.5%
solution in dimethylformamide - DMFA), both kindly provided by the Institute
of Macromolecular compounds RAS, St.Petersburg, and fullerene – C60 (kindly
provided by St.PSU). Optionally, the layers were doped with dispersed carbon
black (CB) and/or RedOx pair: K3Fe(CN)6/ K4Fe(CN)6. For the transducer layer
compositions see Table 1.
Table 1.
Electrode
Composition of the transducer layer
1
2,5 % solution of graphenes in DMFA
2
4,5 % solution of polymeric complex of copper Cu(I) PAC-2 in N-MP
3
No transducer layer (the so-called coated-wire electrode - CWE)
4
200 µl saturated solution of salts mixture K3[Fe(CN)6]+
+K4[Fe(CN)6]·3H2O+300 mg PVC+1,7 ml THF
5
Same as 4, but dispersed in an ultrasonic bath
6
Conventional ISEs with inner solution KCl 10-2 M based on valinomycin
7
300 mg PVC+1,7 ml THF+150µl saturated solution of salts mixture
K3[Fe(CN)6]+ K4[Fe(CN)6]·3H2O, prepared by 1 M aqueous solution
KCl
8
300 mg PVC+150µl DOP+1,7 ml THF+150µl saturated solution of
salts mixture K3[Fe(CN)6]+ K4[Fe(CN)6]·3H2O, prepared by 1 M aqueous solution KCl
9
Dry mixture of salts K3[Fe(CN)6]+ K4[Fe(CN)6]·3H2O with suspension
of carbon black in the ratio 1:1
10
100 µl saturated solution of salts mixture K3[Fe(CN)6]+
K4[Fe(CN)6]·3H2O+ 100µl 2,5 % solution of graphenes in
DMFA+300 mg PVC+1,7 ml THF
11
100 µl saturated solution of salts mixture K3[Fe(CN)6]+
K4[Fe(CN)6]·3H2O+ 100µl 2,5 % solution of graphenes in
DMFA+600 mf mixture of PVC:carbon black=1:1+ 2 ml THF
12
300 mg PVC+1,7 ml THF+100µl saturated solution of salts mixture K3[Fe(CN)6]+ K4[Fe(CN)6]·3H2O+200 mg fullerenes- C60
K+-selective membranes contained poly(vinylchloride) (PVC) and bis(2ethylhexyl)phthalate (DOP) (1:3) doped with 0.03 M potassium tetrakis(p-Clphenyl)borate (KClTPB). The membranes were formed by drop-casting the
membrane cocktail: the aforementioned substances dissolved in tetrahydrofuran
(THF) on top of the transducer layer. Conventional ISE with internal aqueous
7
solution (0.01 M KCl) and Ag/AgCl internal reference electrode was used for
the back-to-back comparison with SC-ISEs.
Zero-current potentiometric measurements were accompanied by chronopotentiometry (ChP) and electrochemical impedance spectroscopy (EIS). The reference
electrode was always saturated Ag/AgCl electrode, as counter electrode for ChP
and EIS measurements we used glassy carbon rod.
Results and discussion
Electrodes 1 and 2 showed steady positive drift of the potentials, while electrodes 4, 5, 8, 7 and 12 showed negative drift. Only electrodes 3, 9, 10 and 11
showed relatively stable potential readings over time, although worse than conventional ISE 6, see Fig. 1.
Fig. 1. Drift of the potential in control 0.1 M KCl solution for SC-ISEs 3, 9, 10,
11 and conventional ISE 6.
Electrode 10, although showing a relatively stable potential in the control solution 0.1 M KCl, slowly by slowly lost response in solutions below 10-3 M - see
Fig. 2, while electrodes 1, 2, 3, 6, 7, 8 and 9 retained Nernstian response down to
10-5 M KCl – see Fig. 3.
Fig. 2. Behaviour of electrode 9 retaining Nernstain response down to 10-5 M KCl
over 6 months of observation.
8
Fig. 3. Behavior of electrode 10 over time: gradual degradation of the response.
The potential drift may be caused by slow Red-Ox reaction responsible for the
ion-to-electron conductivity transduction. This transduction can be studied by impedance and chronopotentiometric measurements. For selected electrodes, appeared
most promising we carried out EIS ansd ChP measurements. SC-ISEs 1, 2, and 12
showed only bulk impedance, while SC-ISE 11 showed depressed semicircle, most
likely a superposition of a bulk and a slow charge-transfer process, Fig. 4.
Fig. 4. EIS curves for SC-ISEs 1, 2, 11, 12 – curves 1, 2, 3 and 4, respectively.
Chronopotentiometric data
support the latter conclusion: polarization then plotted vs. square
root of time is almost linear for
ISEs 1, 2 and significantly nonlinear for ISEs 11, 12 (Fig. 5)
suggesting slow charge-transfer in
the latter electrodes. Indeed, SCISEs 12, and especially 11 show
relatively large charge-transfer
resistance.
Besides insufficient stability
of the ion-to-electron transduction, the additional reason for the Fig. 5. Chronopotentiometric measurements:
degradation in sensor response can polarization vs. square root of time.
9
be the existence of water layer between the membrane and the electrode substrate,
which behaves unintentionally as an extremely non-buffered electrolyte reservoir
[2]. Indeed, some SC-ISEs, in particularly electrode 9 exhibit large potential drifts
when 0.1 M KCl is replaced with 0.1 M NaCl, suggesting the existence of thin
water layer between the membrane and the transducer layer, see Fig. 6.
Fig. 6. Response of SC-K +-SEs upon replacement of 0.1 M KCl with 0.1 M NaCl
and back.
Conclusions
The insufficient stability of SC-ISEs under study is caused, most likely, both by
relatively slow RedOx process at the transducer layer, and the existence of water
layer beneath the membrane.
Profs. A. Yakimanski and M. Goikhman, Institute of Macromolecular compounds RAS, St. Petersburg, are greatly acknowledged for providing with Cu(I)
-polymeric nanocomposite.
References
1. Ivaska A. // Electroanalysis, 1991, 3, 247.
2. Fibbioli M., Morf W.E., Badertscher M., N.F. de Rooij, Pretsch E. //
Electroanalysis, 2000, 12, 1286.
10
Using of semiconductor oxide films for detection of
volatile organic compounds in gases
Lopatnikov Artem
artlopatnikov@gmail.com
Scientific supervisor: Prof. Dr. Povarov V.G., Department of
Analytical Chemistry, Faculty of Chemistry, Saint-Petersburg
State University
Introduction
To create a gas-sensitive films that are used in chemical sensors, the semiconductors based on transitive metal oxides are widely used [1-3].
The principle of detection is based on the phenomenon of heterogeneous catalytical reactions occurring on the surface of the solid and gas. The character of
transformations of organic compounds depends on the nature of the substance and
the catalyst used, as well as the conditions of catalytic reaction. At low temperatures
only physical adsorption is observed, while temperature increases, the growing role
is played by the forces of chemical interaction between the adsorbed molecules and
surface atoms of solid-state or adsorbed oxygen, located on its surface [4].
The presence of volatile organic compounds leads to its adsorption on a surface
of film with subsequent oxidation. The mechanism of oxidation is accompanied
by formation of the number of ionic and ion-radical forms, which are temporarily
increases the conductivity of film [5]. Thus, these changes can be captured and
treated as an analytical signal. Fig. 1 shows a schematic diagram of this detection
method
Fig.1. Working principle and
circuit scheme of gas-sensitive
sensor analyzer.
1 - substrate, 2 - oxide film,
3 - silver contact, 4 - flow of
gas-oxidant in a mixture with
volatile organic compounds,
5 - variable resistance, 6 - signal detection, 7 - constant voltage source
During the current research work we used the films of tin dioxide modified with
oxide of copper and pentoxide of vanadium. Tin dioxide film provides electrical
conductivity over a wide temperature range, while copper and vanadium oxides
are typical catalysts for oxidation processes.
Results and Discussion
The films of mixed metal oxides were deposited on the edge of a rod of alum
silicate. Through contacts rods are connected with the electrical circuit and a vari11
able resistor, which is simultaneously connected to the ADC. The detector response
received without additional amplification. During the experiment the detection
system was heated at a required temperature range. The detector is located inside
the main channel of the aluminum thermostat of serial gas chromatograph.
For researching of the films activity the mixtures of test compounds with air
were injected into the chromatograph evaporator.
Heating of the film was carried out in steps. Fig. 2 shows analytical signal of
ethanol vapors depending on the temperature. The peaks have the usual "chromatographic" shape, their width decreases with increasing temperature, and the height
increases up to a temperature 375 oC.
Fig. 2. The peaks of ethanol (three for each temperature) on the film of (SnO2
+ 3% CuO). The feed rate of air through the hollow column - 80 ml/min. The first
peak at 350oC was registered against the background of random noise in the measuring circuit.
To determine the sensitivity
of the detector the dependence
of the peak area on the amount
of substance was investigated. It turned out that for all
compounds under consideration
the calibration dependences are
generally nonlinear. However,
in the range from 1 to 5 mkg of
substance there is a linear plot.
Fig 3. shows a calibration curve
for n-hexane. The reproducibility of the analytical signal
Fig. 3. The dependence of the peak areas on the was determined by repeated
injection of 1 ml mixture of nmass of n-hexane.
12
hexane and air at a concentration of 2 mg / ml in evaporator. It turned out, the mean
relative deviation of the peak area based on 25 measurements was about 2%.
Dependences obtained for tin dioxide films modified with vanadium pentaoxide
are generally similar for ones obtained for films with the addition of copper dioxide.
Differences in the behavior of the two films of different composition appear more
completely in the comparison of analytical signals obtained simultaneously with
respect to the same substance. For this purpose, we placed two sensors of different composition into a standard heating block of gas chromatograph. The films
deposited on the edge of cylindrical rods made of alum silicate with two throughholes in parallel, located along the main axis of the cylinder. Two metal wires with
silver edges were passed through the holes and slightly protruding above plane of
the film-covered end of the rod, were served as the contacts. For each of the film
we used the electrical measuring circuit, described previously.
During use of two sensors on the gas chromatograph was equipped with a set
packed column with a phase 5% SE-30 on inertone, which provided the first real
chromatograms, one of which is shown in Fig. 4.
Most of the compounds give
responses of standard form,
herewith ethoxyethan exhibits
different behavior. Oxygen of
ethoxyethan has the ability to
capture electrons. But after sorption and a corresponding reduction in electrical conductivity
subsequent oxidation begins to
liberate electrons into the measuring circuit. As a result, the large
positive deviation is replaced by Fig. 4. Typical view of analytical signal.
a slight negative.
A graphical dependence on
each other’s signals from two
detectors of different composition in relation to the same
compound seems to be the most
informative. Each compound is
conformed to a curve of the special form, which make possible
its identification. Fig. 5 shows
the trajectories of the signals
from a number of substances,
which are obtained with a combination of films with different Fig. 5. A graphical dependence on each other’s
composition.
signals from two detectors of different composition in relation to the same compound.
13
This phenomenon can be used as a means of identification. Due to the fact, that
the analytical signal, obtained this way, corresponds to only one aspect of the proceeding on the surface of the membrane process - the concentration of free charge
carriers in a given time, one can conclude that this method yields the identification
of a limited number of compounds.
Conclusions
We can make the following conclusions:
1. The sensitivity of the detector is about 1 microgram.
2. There is no need for a signal amplifier.
3. The reason for the nonlinear dependence of the height of the response on the
concentration is a saturation of the surface and blocking the active centers.
4. The linear dependence of the peak area of concentration allows for quantitative analysis of the separated peaks.
5. Individuality of oxidation process strongly affects the shape of a peak.
6. The constructed model of the detector can be installed on a gas chromatograph, using its standard components.
7. Installing the detector on a capillary column chromatogram will receive
high-resolution analysis in the mode of temperature programming.
8. Simultaneous usage of several films with different oxidation catalysts can
be used as a detector electronic tongue.
1.
2.
3.
4.
5.
References
Ahlers S. et al. // Sensors and Actuators B 107 (2005) 587-599.
Zhang W.-M. et al. // Sensors and Actuators B 123 (2007) 454-460.
Kim K.-W. et al. //Sensors and Actuators B 123 (2007) 318-324.
Yamazoe N. // Sensors and Actuators B 108 (2005) 2-14.
Gouma P.I. // Rev. Adv. Mater. Sci. 5 (2003) 147-154.
14
Digital spectrographic analysis of human biological
fluids for determination of microelements
Savinov Sergey
s.sergei.s@mail.ru
Scientific supervisor: Prof. Drobyshev A.I., Department of
Analytical Chemistry, Faculty of Chemistry, Saint Petersburg
State University
Introduction
Chemical elements being presented in the human body can be divided into
two groups, namely, essential and toxic [1]. Toxic elements (xenobiotics), which
penetrate into the body usually from the environment, even in low concentrations
damage not only health but also human life [2]. Significant increase or decrease in
the concentration of essential (vital) elements can also lead to disorder of normal
body functioning [3]. In this regard, very important, promising and called-for
trend in modern clinical diagnostics is a biomonitoring of chemical elements in
human organs and tissues, as well as in biological fluids [4]. To implement this
direction highly sensitive, multi-element, high-performance and available methods of analysis of bio-organic samples are required. The aim of this study was to
develop a technique of direct (without sample treatment) atomic emission digital
spectrographic analysis of human biological fluids (as an example, saliva) with
spectrum excitation of dried residue of a sample from end of carbon electrode in
a.c. arc for determining the concentration of microelements in them.
Materials and methods
To create an a.c. arc generator IVS-28 was used. The decomposition of
emission of arc plasma in a spectrum was performed by spectral device MFS-8.
Control of the spectral plant, including the registration of the spectrum with a
linear photodiode detector MAES-10, as well as photometry of spectral lines and
analytical information processing carried out by computer software "Atom 3.2"
(VMK "Optoelektronika", Novosibirsk). Carbon electrodes: upper, sharpened to
a cone, and bottom, machined to a diameter of 3 mm with a hole on its end, were
used. Drops of liquid samples and auxiliary solutions were applied to the bottom
electrode by microsyringe MSh-10. For preparation of calibration solutions deionized water, metals, their oxides or salts, concentrated hydrochloric acid, nitric acid,
sulfuric acid and hydrogen peroxide were used.
Results and discussion
Development of technique of microelement determination in liquid samples
Preparing of electrodes for an application on their end of liquid samples included
cleaning of the end zones from possible contaminations presented in coal rods by
firing of electrodes in a.c. arc during 20 seconds at a current strength 15 A. Then on the
15
end of the electrode with a hole a thin polymer film was created by applying and
drying of 20 ml of solution of polystyrene in toluene (0.3%) to prevent a penetration of the sample depth into the surface layer of the electrode (during evaporation
of drops of samples).
As an easy ionized addition agent which is known [5] to inhibit the influence of
sample matrix components on the plasma temperature, determining the efficiency
of atomic excitation, we used sodium chloride applied as 10 ml drop of an aqueous
solution (15 g/l) on the end of the electrode.
3,00
Intensity, relative units
2,50
2,00
Al 309,3 nm
1,50
Cu 327,5 nm
Mg 279,6 nm
1,00
Zn 213,9 nm
0,50
0,00
13
14
15
16
17
Current strenght, А
18
19
Fig. 1. Dependence of the intensities of analytical lines on the current strength
of a.c. arc.
0,90
0,80
RSD, relative units
0,70
0,60
0,50
Al 309,3 nm
0,40
Cu 327,5 nm
0,30
Mg 279,6 nm
0,20
Zn 213,9 nm
0,10
0,00
13
14
15
16
17
Current strenght, А
18
19
Fig. 2. Dependence of relative standard deviations of the intensities of analytical
lines on the current strength of a.c. arc.
An important point in the development of technique of biological fluids
analysis for microelement determination is to optimize the conditions of spectrum
excitation in order to obtain maximum value of the analytical signal and the least
error of its measurement. With a view of it, experiments in which the spectra of
the samples had been excited at different current strength of a.c. arc were made.
On the grounds of the obtained dependences of the intensities of analytical lines
16
(Fig. 1) and their relative standard deviations (Fig. 2) on the current strength we
chose 17 A as the optimal value.
Verification of the developed technique
To confirm the universality of the developed technique comparative measurements of the intensities of spectral lines when spectrum exciting of the source
sample of saliva, as well as samples treated with concentrated nitric acid and
concentrated hydrogen peroxide - the most commonly used reagents for the treatment of biological objects to destruct an organic matrix [6, 7] – were carried out.
As can be seen in Fig. 3, the relative difference of intensities does not exceed 10%.
Thus, developed technique allows to determine microelements in saliva samples
without preliminary mineralization.
20
18
IIntensity, relative units
16
14
Nitric acid treatment
12
10
Hydrogen peroxide
treatment
8
Without treatment
6
4
2
0
Zn
Mn
Al
P
Ti
Fig. 3. The intensities of analytical lines of elements by different ways of treatment of same sample.
Determination of microelements in human saliva
The developed technique has been used by us for determination of microelements in samples of saliva of Saint Petersburg residents. Based on obtained results
we calculated average values of concentrations, as well as intervals of their variations by 58 analyzed samples. The following table shows our data, as well as from
other papers. Although there is spread of values of mean contents of most elements
obtained by different authors, however, they fall into intervals estimated by us.
Element
Current work
[8]
[9]
[10]
[11]
Mg
10 (3,6-71)
6,76
Si
2,1 (0,5-18)
5,36
Zn
0,28 (0,01-1,3)
0,17
1,3
0,26
Fe
0,18 (0,01-1,7)
0,44
Ti
0,086 (0,001-1,1)
0,758
Cr
0,050 (0,010-0,12)
0,05
0,03
Mn
0,049 (0,009-0,67)
0,042
0,025
0,003
Cu
0,010 (0,001-0,72)
0,07
0,05
0,005
0,02
Co
0,008 (0,001-0,036)
0,003
Ag
0,001 (0,0001-0,077)
Table 1. Average content of several elements (with intervals of their variations)
in saliva, mg/l.
17
Conclusion
Thus, we developed a technique of atomic emission digital spectrographic
analysis of biological fluids with spectrum excitation of dried residue of a sample
from the end of carbon electrode in the a.c. arc with a relative error of microelement
determination 10 - 20%. This technique was used to analyze samples of saliva.
Findings on the average contents of microelements and intervals of their variations
are in good agreement with other researchers data.
References
1. Avtsyn A.P. et al. Microelementoses of man. Etiology, classification, organopathology (In Russian). - Moscow: Medicine, 1991, - 496 pp.
2. Chandramouli K. et al. // Archives of Disease in Childhood. V. 94, pp. 844–848
(2009).
3. Takser L. et al. // Environmental Research. V. 95, pp. 119-125 (2004).
4. Kakkar P., Jaffery F.N. // Environmental Toxicology and Pharmacology. V. 19,
pp. 335–349 (2005).
5. Terek T., Mika J., Gegush E. Emission spectral analysis (In Russian). V.1 Moscow: Mir, 1982, - 286 pp.
6. Chiappin S. et al. // Clinica Chimica Acta. V. 383, pp. 30–40 (2007).
7. Burguera J.L., Burguera M. // Spectrochimica Acta. Part B. V. 64, pp. 451-458
(2009).
8. Baranovskaya, I.A. (In Russian) // Kazan Medical Journal. V. 90, pp. 87-89
(2009).
9. Notova S.V., Ordzhonikidze G.Z., Nigmatullina Y.F. (In Russian) // Journal of
Orenburg State University. V. 6, pp. 146-147 (2003).
10.Watanabe K. et al. // Journal of Trace elements in Medicine and Biology. V.23,
pp. 93-99 (2009).
11.Wang D., Dua X., Zheng W. // Toxicology Letters. V. 176, pp. 40–47 (2008).
18
Synthesis of condensed imidazole derivatives with a
Nodal nitrogen atom - pyrido[1,2-a]benzimidazoles
Sokolov Alexandr
morose@mail.ru
Scientific supervisor: Dr. Begunov R.S., Department of Organic
and Biological Chemistry, Faculty of Biology and Ecology,
Yaroslavl Demidov State University
Introduction
At the present time there is a demand for a number of new chemical structures,
the scope of which also emerged recently. Such young science as molecular biology,
genetic engineering, medical genetics need to provide specialized knowledge-based
reagents for research in their fields.
Opportunities to implement these requirements relate to the development in
recent years new methodologies for the synthesis of chemicals and the development
of physico-chemical methods of analysis and mathematical (quantum mechanical)
modeling of chemical processes.
One group of compounds, which are high demand in contemporary applied and
experimental sciences, are condensed imidazole derivatives containing a bridging
nitrogen atom. For example, to such structures belong pyrido[1,2-a] benzimidazoles,
which are bioisosteric analogues of nitrogenous bases of DNA and exhibit a high
biological activity [1-2], and furthermore, due to the system of conjugated bonds
have fluorescent properties. Compounds with imidazole fragment, despite the wide
spread in nature, not cost-effective to obtain from natural raw materials. In the
literature described many techniques for their synthesis [3-4], but the drawbacks,
such as the use of expensive reagents, high temperatures for the process, the low
selectivity and low yield of final products do not allow to use well-known methodology for the creation of industrial production technologies of these products.
Based on analysis of more than 240 literary sources, it was concluded that the
most promising method of synthesis of substituted pyrido[1,2-a]benzimidazoles is
reductive cyclization of pyridinium salts. This method allows to realizes the process
selectively, with a high yield of target products. It is based on the increased activity
of pyridine and, in particular, its derivatives - N-oxides and quaternary salts with a
full positive charge on the endocyclic nitrogen atom to the aromatic nucleophilic
substitution reactions by α-positions of the ring (Fig. 1).
However, an obstacle to widespread use of this method of synthesis is insufficient knowledge about the laws of the process of reductive cyclization. Therefore
were investigated the main factors determining the reaction and the possibility for
further chemical modification of the resulting products.
19
Fig. 1. Scheme of reductive cyclization of pyridinium salts.
1 a) R=R1=H, b) R=R1=CH3, c) R=H, R1=CH3;
2 =CH, a) R2=H, R3=CF3, b) R2=H, R3=CN, c) R2=H, R3=COOH,
d) R2=H, R3=CONH2, e) R2=R3=CN; f) R2=H, R3=NO2 g) X=N, R2=R3=H;
3,5 Х=CH; 4,6 Х=N.
Results and Discussion
In this paper we studied the influence of the nature of the reducing agent,
solvent, temperature on the direction of intramolecular reductive amination of
salts of 1-(2-nitro-(het) aryl) pyridinium on example of 7-trifluoromethylphenyl
pyridinium chloride.
For the experimental investigation of the influence of nature of reducing agent
on the direction of the reaction, we used agents that work in alkaline or in acidic
medium. In the application of Na2S (pH> 7) occur addition of OH ¯-ion to α-position
of the pyridine ring of salt 3a, followed by opening of the pyridine ring and the
formation of an aldehyde derivative 7 (Fig. 2).
Fig. 2. Scheme of reduction of pyridinium salts in alkaline medium.
In the future, as the reducing agents
we used metal chlorides of variable
oxidation states (TiCl3, SnCl2), working
in an acidic environment. Reduction
of 3a with chloride titanium (III)
shown in Fig.3 leads to the formation
of the product, defined by NMR and
mass spectrometry as 1-(2-amino-4(trifluoromethyl) phenyl) pyridinium
Fig. 3. Scheme of reduction of pyridini- chloride (8).
um salts with chloride titanium (III).
20
Thus, in the course of the reaction occurs reduction of the nitro group to amino
group, without the formation of condensed tricyclic structures.
Intramolecular cyclization with the formation of product 3a -7-(trifluoromethyl)
pyrido[1,2 –a]benzimidazole (5a) (Fig. 4) is realized with application of chloride
of tin (II) in acidic aqueous-alcoholic medium (3% HCl).
Fig. 4. Scheme of reductive cyclization of pyridinium salts with chloride tin (II).
Influence of solvent nature on the course of the reductive amination 3a was
investigated on a number of alcohols, the results are presented in Table 1.
As can be seen from the table that Table 1. The yield of the reductive aminathe best conditions for the synthesis tion of 1-(2-nitro-4-(trifluoromethyl) phenyl)
of pyrido[1,2 –a]benzimidazoles pyridinium chloride (3a) [C = 0.16 mmol/l,
(5) is an acidic aqueous-alcoholic t = 500C].
medium, and the nature of the alco- №
Solvent
Yield 5a, %
hol has no significant effect on the 1
81.5
Propan-2-ol
yield of 5a.
96.3
2 Propan-2-ol(H2O, HCl)
In addition to the medium, tem- 3
91.7
Ethanol (H O, HCl)
perature is important factor in the 4 Methanol (H2 O, HCl)
96.0
2
reduction of nitro compounds and
16.3
5
Water (HCl)
nucleophilic substitution reactions.
Found (Table 2) that the temperature
rise of reduction of 1-(2-nitro-4- Table 2. The yield of products of the reductive
(trifluoromethyl) phenyl) pyridini- amination of 1-(2-nitro-4-(trifluoromethyl)
um chloride (3a) to 50 °C increases phenyl) pyridinium chloride
the yield of product 5a, a further [C = 0.16 mmol / l propane-2-ol-H2O].
increase of temperature of the reacYield, %
0
tion mixture leads to decrease in the № Temperature, С
5а
8
yield of target products, which may 1
0
82.9
8.9
explain the occurrence of alternative 2
10
84.8
6.1
chemical processes.
3
20
91.0
Thus, the experimentally were
4
30
93.2
selected the most suitable conditions
5
40
96.3
for the process of intramolecular
6
50
97
cyclization of salts 3: medium 7
60
96.8
propan-2-ol-water-HCl, t = 50 °C.
8
70
95.9
Based on experimental data and
quantum-mechanical simulation was set key stage of reduction of the nitro group,
which determines way of the process of reductive cyclization of pyridinium salts
(Fig. 1) and was proposed the mechanism of intramolecular reductive cyclization.
21
This makes it possible to use the developed technique for synthesis a variety
of pyrido[1,2- α]benzimidazoles (Fig. 1).
Compounds with fused tricyclic core are interesting as intercalators of DNA.
At the same time, most intercalary activity should have pyrido[1,2-a] benzimidazoles containing NH2-, C(O)NH2-groups, because they increase the affinity of the
compounds to the nucleic acid molecules. So were synthesized amino derivatives
of pyrido[1,2-a]benzimidazoles based on reaction of electrophilic substitution nitration and reduction reactions (Fig. 5).
Fig. 5. Scheme of reductive cyclization of pyridinium salts and futher modification
- nitration and reduction. a) R=CF3; b) R=NH2
The reactions of electrophilic substitution in the 7-R-pyrido[1,2-a]benzimidazole proceed under relatively mild conditions (nitration - 3 h at 70° C), which
allows to obtain the final products of high purity and good yields 81-94%.
Reduction of compound 9 was carried out with chlorides of metals of variable oxidation state, with the best results were observed when using titanium chloride (III).
36-substituted pyrido[1,2-α]benzimidazoles, which were obtained in our
research have been investigated as drugs, with mechanism of action is based on
intercalation into DNA and inhibition of replication of target cells. Among them,
the greatest biological activity have structures containing amino, amido, or both
amino and amido groups. Such drugs have high intercalary activity, which exceeding the known DNA intercalators in a few times [5].
Conclusion
a) Developed an effective method for the synthesis of tricyclic condensed
compounds based on reductive cyclization of pyridinium salts.
b) Investigated main factors determining direction of reductive cyclization and
proposed its mechanism.
c) Obtained variety of pyrido[1,2-α]benzimidazoles, which have high biological activity and can be used in genetics research and pharmacology as DNA intercalators.
References
1. Rida S.M., El-Hawash S.A.M. Fahmy H.T.Y. // Arch. Pharm. Res.,V. 29, No.
10, рр. 826-833 (2006).
2. Dupuy M., Pinguet F., Chavignon O., Chezal J.-M., Teulade J.-C., Chapat J.-P.,
Blache Y. // Chem. Pharm. Bull.,V. 49, No. 9, рр. 1061-1065 (2001).
3. Tereshchenko A.D., Tolmachev A.A., Tverdokhlebov A.V. // Synthesis. No.
3, рp. 373-376 (2004).
4. O'Shaughnessy J., Aldabbagh F. // Synthesis. No. 7, pp. 1069-1076 (2005).
5. Ryzvanovich G.A., Begunov R.S., Rachinskaya O.A., Muravenko O.V., Sokolov
A.A. // Chem.-pharm. Journal, V. 45, № 3, p. 13-15 (2011).
22
С. Mathematics and Mechanics
Algebraic approximation of global attractors of discrete
dynamical systems
Malykh Artem
art.malykh@gmail.com
Scientific supervisor: Prof. Dr. Volker Reitmann, Department of
Applied Cybernetics, Faculty of Mathematics and Mechanics,
Saint-Petersburg State University
1. Introduction
One of the typical goals related to dynamical systems is the approximation of
global B-attractors. Many results in this area were shown by Foias and Temam [1, 2].
In their papers they have shown how to construct the approximation of an attractor
with any given precision for a large class of continuous-time analytic dynamical
systems. The approximation is done with the help of algebraic sets.
In the first part of the present paper we repeat the definition of algebraic sets,
describe the reasons for their use for the approximation of attractors and give a
theorem for discrete-time dynamical systems similar to one which was shown by
Foias and Temam [1, 2].
In the second part of the paper we will describe Whitney stratifications of
algebraic sets and its connection with global B-attractors.
2. Basic tools of global attractors and algebraic sets
Consider the dynamical system with discrete time:
( ϕt
, ( M , ρ)), {}
( t ∈T )
(1)
where (M, ρ) is a metric space, T∈{Z,Z+}; for all t∈𝕋 the map φt:M→M is continuous and for all t, s∈T we have φt+s=φt◦φs. Let us recall the definition of a global
B-attracting set and of a B-attractor [3].
Definition 1: A set XÌM is called global B-attracting for (1) if dist(φt(B), X)
→0 for t→∞ and for all bounded sets BÌM. Here for all Y, ZÌM the function
dist(Y, Z)=supu∈Y infv∈Z ρ(u,v) is the Hausdorff semi-distance.
Definition 2: A set X is called global B-attractor for (1) if
1. X is a bounded and closed set;
2. φt(X)=X for all t∈T;
3. X is a B-attracting set for (1).
In this paper we consider only dynamical systems where M=n.
It is a rather often used approach to do the approximation of an attractor with the
help of manifolds, however some attractors have not-integer Hausdorff dimension,
i.e. have fractal structure. Therefore they cannot be approximated by manifolds
with any given precision. That is why we use for approximation algebraic sets
which are not necessary manifolds. Here is the definition:
24
Definition 3: Consider the following system of equations:
 P1 (u1 , …, un ) = 0 
 P2 (u1 , …, un ) = 0

…

 Pm (u1 , …, un ) = 0

Where m, n∈N, P, P1,…,Pm are polynomials defined on n. Then the set of its
solutions is called an algebraic set.
Let us provide an example of algebraic set which is not a manifold. Consider
the simple algebraic set:
V1 = { (u1 , u2 )| (u1 − u2 )(u1 + u2 ) = 0} Fig. 1.
As we can see in Fig. 1, in the one-dimensional neighborhood of the point (0, 0)
this set is not homeomorphic to an open ball in , so it is not a manifold.
3. Algebraic approximation for a class of discrete-time dynamical systems
Let us consider the system generated by the iteration
ut +1 = − Aut − R (ut ), t ∈Z (2)
where A:n→n is a linear symmetric operator. Let the system (2) has a global
B-attractor X. Suppose that R: n→n is an analytic mapping inside some open
set DÌn : XÌD. Define the map F(u) := -Aut – R(ut). Let F be an invertible map
and F-1 be an analytic map inside D. Let D contain the point b such that F-1, R are
given by convergent Taylor series in the open ball Br(b): XÌBr(b).
Before stating our approximation theorem we give some auxiliary notation.
Let f : n→ be given by the convergent Taylor series in the neighborhood of the
point b. Then
l
∂ k1 ...∂ kn f (b)
f l ,b (u ) := ∑
(u1 − b1 ) k1 ...(un − bn ) kn k1 ,..., kn = 0 (k1 ...kn ) !
is a finite part of series. For the map f : n →m, f(u)=(f1(u),…,fm(u)) given by
convergent Taylor series in the neighborhood of the point b we have similar notation:
f l ,b = ( f1l ,b , …, f m l ,b )
Now we can give the statement of the approximation theorem.
25
Theorem: Suppose we have a system of the form (2). Let λ1,…,λn be the eigenvalues of A and suppose that |λ1|≥ |λ2| ≥ … ≥ |λn| and |λn| < 1. Let m be the minimal
index such that |λm+1|<1 and Qm be the projector on the linear subspace of n
generated by the eigenvectors corresponding to λm+1, …, λn. Define for any L, M,
N∈N, u ∈ n the sum
L
k −1
J
(u ) = ∑ (−1)k (Q A) R F − k (u ) L,M , N
m
k =1
N ,b
(
)
M ,b
Then for any ε > 0 there exist L, M, N ∈N such that for any u0 ∈ X
Q u −J
(u ) < ε m 0
L,M , N
0
and the set
 L0, M , N = u|J L , M , N (u ) − Qm u = 0 is an algebraic set.
Proof: Here we will give only a sketch of the proof. First, we represent an
arbitrary point of the attractor u0 as an infinite sum depending on A, R, Qm, F-1 and
u0. Then we prove that a finite part of this sum KL(u0) can approximate u0 with any
given precision and that the norm of the difference between u0 and KL(u0) can be
estimated uniformly for any point of the attractor. Then we represent R and F-1
by finite parts of their Taylor series around b (RN,b and F-1M,b, respectively). Now
we substitute these representations into KLand get the sum JL,M,N. Then we show
that JL,M,N(u0) can approximate KL(u0) with any given precision and that the norm
of the difference between KL(u0) and JL,M,N(u0) can be estimated uniformly for any
point of the attractor.
The function JL,M,N is a sum of polynomial functions. Therefore the set
 L0, M , N = u ∈  n |J L , M , N (u ) − Qm u = 0
{
is an algebraic set.
}
{
}
4. Whitney stratification
Besides the ability to approximate attractors with any given precision, algebraic
sets have another useful property related to the analysis of a global B-attractor. Any
algebraic set has a Whitney stratification. Informally speaking, a stratification is a
presentation of a manifold as an union of disjoint manifolds. The Whitney stratification requires some additional properties of tangent spaces from these manifolds.
The definition from [4] is following.
A stratification of a set SÌn is a partition of S into submanifolds {Sj} of n
such that the family of strata {Sj} is locally finite at each point of S. Let us denote
as Gn,m (0 ≤ m≤ n) the set of all linear m-dimensional subspaces of n. This set has
the structure of a C∞ smooth manifold with dim(Gn,m = m(n-m)). The manifold Gn,m
is called Grassman manifold.
Definition 4. A stratification of a set S is called Whitney stratification if each
pair of strata Si, Sj; i ≠ j satisfies the following condition:
If
{ pk }∞k =1 , {qk }∞k =1
26
are sequences of points in Si and Sj , respectively, both converging to a point p of
Si, if the sequence of tangent spaces
{Tqk S j }∞k =1 converges to a subspace L in Gn,m, where m = dim(Sj), and if the sequence
 ∞ { pk , qk }k =1
of lines containing 0 and qk-pk converges to a line lÌn in G1,l, then lÌL.
Let us consider an example of a Whitney stratification of a simple algebraic set.
Example 1: Consider the algebraic set
(
)
S = {(u1 , u2 ) ∈  2 | u1 u1 − u22 = 0} = {u1 = 0}  {u1 = u22 }
Let us write S in the form S = 5i =1 Si
S2
S3
S4
, where
{(0,u ) ∈  |u > 0},
= {(0, u ) ∈  |u < 0},
= {(u , u ) ∈  |u = u , u > 0},
= {(u , u ) ∈  |u = −(u ), u > 0},
S1 =
2
2
2
1
1
2
2
2
2
S5 = {(0, 0)}.
2
2
2
1
1
1/ 2
2
2
1/ 2
2
2
Fig. 2.
It is easy to see that {Si}, i=1,…,5 is a Whitney stratification of S. One can
find the detailed explanation in [5].
Suppose we have a Whitney stratification of a set S into strata {Si}, i = 1,...,N,
N∈N. And suppose that m = max(dim(Si)), i=1,…,N. Our main interest in the
analysis of an attractor are strata with dimension lower than m (let us call them
low-dimension strata). Our suggestion is the following: if a low-dimension stratum
appears near some region in the Whitney stratification of all approximating algebraic
set, then in this region we have the fractal structure of the B-attractor.
5. Conclusion
We described the concept of algebraic sets and the reason why they are useful
in the analysis of global B-attractors. Also, a theorem with explicit form of approximating algebraic sets is given. In the last part we have briefly described the
concept of stratification, its possible connection with approximating sets and the
direction for further investigation.
References
1. Foias C. and Temam, R. // Phys. D. V. 32. P. 163–182. (1988).
2. Foias C., Temam R. // SIAM J. Math. Anal. V. 25. No 5, P. 1269–1302. (1994).
3. Boichenko V.A., Leonov G.A., Reitmann V., Dimension Theory for ODE.
Wiesbaden: Vieweg-Teubner Verlag. (2005).
4. Whitney H. // Trans. Am. Math. Soc. V. 36. P. 63-89. (1934).
5. Leonov G.A., Malykh A.E., Reitmann V. // Proc. Conf. PHYSCON, 2009.
27
Taken’s time delay embedding theorem for dynamical
systems on infinite-dimensional manifolds
Popov Sergey
serg.pobeda@gmail.com
Scientific supervisor: Prof., Dr. V. Reitmann, Department of
Applied Cybernetics, Faculty of Mathematics and Mechanics,
Saint-Petersburg State University
1. Introduction
In 1981 Takens proved a theorem that allows the phase space of dynamical
systems evolving on smooth finite-dimensional manifolds to be reconstructed from
an appropriate time series. Later this result was generalized by Robinson [1] to the
case of dynamical systems on an arbitrary Hilbert space.
In this paper dynamical systems defined on infinite-dimensional manifolds
are considered. We provide a certain generalization of Robinson’s result for such
dynamical systems. Moreover, unlike Robinson’s paper where the results were
obtained by an embedding theorem for infinite-dimensional Hilbert spaces due to
Hunt & Kaloshin [2], our result makes use of an embedding theorem for infinitedimensional manifolds due to Okon [3]. We apply also the Taken's embedding
theory to the investigation of the dynamical system arising from the microwave
heating problem [4]. Some approximation of the fractal dimension of the invariant set is given.
2. Time series analysis
Let M be a Cr - smooth n-dimensional manifold. Let’s consider a dynamical
system on M generated by a diffeomorphism φ: M→ M. The motion corresponding
to an initial point u∈M is denoted by φt(u), t≥0.
In a typical experiment [5], the phase space of this dynamical system is unknown. We try to infer properties of the system by taking measurements. Since
each state of the dynamical system is uniquely specified by a point u in the phase
space, a measured quantity is a function from phase space to the real numbers
h: M→. Let τ be the length of the interval between the measurements. Then we
get the sequence of observations:
z0 = h (u ), .., zi = h(ϕ iτ (u )), i ∈1,.., N 0 .
Let d∈N be an arbitrary number. Then we get the vectors
ξi := (zi , zi +1 , …, zi + d −1 ) ∈  d , i = 0,1, …, N 0 − d + 1 .
This sequence of vectors
{ξi }iN=00− d +1 is called a time series. The question arises: how to reconstruct the dynamics of
the initial system by this time series? To answer this question let’s construct an
embedding function which is defined as follows:
Φ (u ) := (h(u ), h(ϕ1 (u )), ..., h(ϕ d −1 (u ))), u ∈ M . ϕ,h
28
The next theorem shows how to choose the number d in this function in order
to get an embedding from M to d.
Theorem 1 [5]: Let M be a compact Cr - smooth n-dimensional manifold. Let
d∈N such that d ≥2n+1. Then the set (φ,h) of pairs for which the embedding function Фφ,h(u) is an embedding is open and dense in the space
Diff r ( M ) × C r ( M , ) for r ≥ 1 Here Diff r(M) is the linear space of Cr- diffeomorphisms on M and Cr(M,)
is the linear space of Cr - smooth maps from M to .
Note that the inequality d ≥2n+1 arises from Whitney's embedding theorem.
This theorem states that the set of embeddings from the n-dimensional manifold M
to d is an open and dence set in the space of Cr - smooth maps from M to d.
Now let us consider Robinson’s theorem [1]. This theorem is a modification of
Taken’s theorem to the case of an arbitrary Hilbert space. The formulation of this
theorem uses the property of ‘prevalence’. This concept, which generalizes the
notion of ‘almost every’ from finite-dimensional spaces to infinite-dimensional
spaces, was introduced be Hunt, Sauer & Yorke [6].
Definition 1: A Borel subset S of a normed linear space V is prevalent if there
is a finite-dimensional subspace E of V such that for each ν∈V, ν+e belongs to S
for almost every e∈E (with respect to a Borel measure μ).
The fact that S is prevalent means that if we start at any point in the ambient
space V and explore along the finite-dimensional space of directions specified by
E, then almost every point encountered will lie in S.
Now let us consider the notion of the thickness exponent. If X is a subset of
a Banach space V, then the thickness exponent of X in V, τ(X;V), is a measure of
how well X can be approximated by linear subspaces of V.
More formally (see [1]), denote by εV(X,n) the minimum distance between X
and any n - dimensional linear subspace of V. Then
−log(n) .
τ (X ;V ) = lim
.
n →∞ log( ε ( X , n))
V
This formula shows that if εV(X,n) ~n-1/τ then τ is the thickness exponent of X . Now
we can formulate Robinson’s theorem.
Theorem 2 [1]: Let H be a Hilbert space and S be a compact set whose fractal
dimension (see [7]) satisfies dimf(S)<d, d∈N, and which has the thickness exponent
τ(S,H). Choose a natural number k>(2+τ(S,H))d, and suppose further that S is an
invariant set for a Lipschitz map φ:H→H, such that:
1. The set Γ of points in S such that φ(x)=x satisfies dimf(Γ)<1/2, and
2. S contains no periodic orbits of φ of periods 2,...,k.
Then a prevalent set of Lipschitz maps h:H→ make the embedding
Фφ,h:H→k one-to-one on S.
Note that unlike Taken’s theorem where the Whitney embedding theorem
was used, Robinsons’s theorem makes use of an embedding theorem for finitedimensional sets due to Hunt & Kaloshin [2].
29
3. Numerical results
The fractal dimension plays an important role in the embedding theory. In this
section some numerical results of estimating the fractal dimension are presented.
Let us consider the one-dimensional two-phase microwave heating process (see
[4]) which is defined in the following way:
1

ε( x) wtt = (
wx ) x − σ ( x, θ) wt ,

µ (x )


A(θ)t = θ xx + σ (x, θ)wt2 ,

w (0, t ) = f1 (t ), w (1, t ) = f 2 (t ),
(1)


θ (0, t ) = 0, θ (1, t ) = 0,

θ x (0, t ) = 0, θ x (1, t ) = 0,

 w x, 0 = 0, w x, 0 = e x , θ x, 0 = θ ( x)
(
)
) 0( ) ( ) 0
t (

Here x∈(0,1), t∈[0,T), θ(x,t) is the temperature of the material, ε(x) is the
electric permittivity, μ(x) is the magnetic permeability, σ(x,θ) is the electrical
conductivity, A(θ) is the operator of entalphy which describes the two-phase nature
of the process.
We can consider an approximating problem for (1), hence we have finitedimensional approximation of the given process. Let X be an approximation of
the phase space of this process. In order to estimate the fractal dimension of X we
can estimate the correlation dimension of X.
The correlation dimension is defined as follows:
lnC (ε) dimcor (X ) = lim
ε→+0 ln( ε )
where C(x) is the correlation integral defined by the following formula:
1 N
C (ε ) = lim 2 ∑ H ε− || xi − x j ||
(2)
N →∞ N
i , j =1
(
)
where xi are N vectors from X, ||·|| is the distance on X and H(x) is the Heaviside
function:
 1, x ≥ 0
H (x ) = 
0, x < 0.
Let θ0(t) be the solution of the approximating problem for (1) at a fixed point
x0∈(0,1) and at the time moment t. Let τ>0 be a time delay. Then we can consider
an embedding phase space of dimension m, which consists of the points
.
y mj = {θ0 (τj ), θ0 (τ ( j + 1)), …, θ0 (τ ( j + m − 1))}, j = 1, 2, …, n = N − m + 1 Here N is a sufficiently large natural number. Then we can calculate the correlation
integral (2) by the following formula:
1 n
Cm , n (ε ) = 2 ∑ H (ε− || y mj − ykm ||)
n j , k =1
This formula depends on the parameters m,n and ε.
30
Fig. 1. The estimation of the correlation dimension.
In the Fig.1 the connection between the estimates of the fractal dimension and
the parameter m is shown. At small values of m the value dimcor increases. Starting
from some m0 the increasing stops. The value dimcor corresponding to this m0 is the
estimate of the correlation dimension of the phase space.
4. Generalization to the case of an infinite-dimensional manifolds
Robinson’s theorem holds on an arbitrary Hilbert space. But many dynamical
systems (for example the dynamical system arising from the Sine-Gordon equation),
arising in real life, act on infinite-dimensional manifolds. We can try to generalize
the result of Robinson to such infinite-dimensional manifolds. To do this we have
to choose an appropriate embedding theorem like the theorems by Whitney and
Hunt & Kaloshin.
Such an appropriate theorem is proved by Okon [3] Using this theorem the
following result [8] was obtained.
References
1. Robinson J.C. // Nonlinearity № 18 pp. 2135-2143 (2005).
2. Hunt B.R. and Kaloshin V.Yu. // Nonlinearity № 12, pp. 1263-1275 (1999).
3. Okon T. // Arch. Math. № 78 pp. 36-42 (2002).
4. Manoranjan R.V., Yin H.M. // Contin. and Discrete and Dynamical Systems,
Serie A, № 15 pp. 1155-1168 (2006).
5. Takens F. // Lecture Notes in Mathematics № 898 (1981).
6. Hunt B.R., Sauer T., and Yorke J. // Bull. Amer. Math. Soc. № 27, pp. 217-238
(1993).
7. Boichenko V.A., Leonov G.A. and Reitmann V. Dimension Theory for Ordinary
Differential Equation. - Teubner, 2005.
8. Popov S.A. // Preliminary version (2012).
31
A two-phase problem arising from a microwave heating
process in nonhomogeneous material
Serkova Nadezhda
seizen@yandex.ru
Scientific supervisor: Prof. Dr. Reitmann V., Department of
Applied Cybernetics, Faculty of Mathematics and Mechanics,
Saint-Petersburg State University
Introduction
In the present work process of local heating of material by microwave radiation is considered. The mathematical model of this process consists of Maxwell’s
equations and the heat transfer equation. It also takes into account that the material
can be in two phases – liquid or solid. We consider the one-dimensional arising
from a microwave heating problem in special the case of nonhomogeneity with
respect to Maxwell's equations.
Studies on the effect of microwaves on the material are of great importance to the
issues of industrial and medical applications. In particular, the effects of microwave
radiation are used to treat cancer. The so called hyperthermia procedure is applied
in combination with other types of treatment and increases their effectiveness [7].
In this case, the mathematical model of the material is considered as biological
tissue, which is characterized by specific features, such as heterogeneity.
In the first part of the paper we formulate the heating problem, define the
concept of a weak solution and provide the theorem of the existence of a weak
solution for described problem.
In the second part we model the problem using a numerical method to investigate
dependencies of temperature profile form on values of electric permittivity and
magnetic permeability at different intervals with respect to Maxwell's equations.
One-dimensional heating problem
Let us consider the initial-boundary problem that describes a microwave heating process in one-dimensional space:
The first equation is Maxwell’s equation; it is derived from two Maxwell’s equations in one-dimensional space. The second equation is the heat transfer equation,
(3)-(5) are the boundary conditions, (6) are initial conditions.
32
Here θ(x,t) is the temperature of the material, ε(x) is the electric permittivity, μ(x) is the magnetic permeability, σ(x, θ(x,t)) is the conductivity. A(θ) is the
nonlinear operator of enthalpy which shows the two-phase nature of the process
and takes the form
where m denotes the melting temperature of the material.
Since we consider the case of heterogeneity with respect to Maxwell’s equations, electromagnetic properties vary depending on the interval. So the electric
permittivity and the magnetic permeability and take a piecewise constant form:
Here a is the boundary of the layers.
Existence of a weak solution
In connection with problem (1)-(6) one can explore various issues, one being
the existence of the solution.
Let us give the definition of weak solution.
Definition. A couple of functions
mentioned system if
is a weak solution of the
is satisfied for any ψ∈H1(0,T,H1(0,1)) and
is satisfied for any η∈H1(0,T,H1(0,1)). If we can take T=∞ such solution is called
global weak solution.
Here H1(0,T,H1(0,1)) is a special Sobolev space defined in [3]. The second
identity contains the operator of the enthalpy A, so that a solution defined in this
way is also called enthalpy solution.
The question of existence of a weak solution is solved by the following theorem,
which is the main result of the present paper.
33
Theorem. Let the following conditions hold for the system (1)-(6):
1) ε(x), μ(x) are piecewise constant function.
There are constants 0<r0, 0<R0 such that
2) There is a σ0>0 such that 0 ≤ σ(x,θ) ≤ σ0(1+θ) for all (x, θ)∈(0,1)×[0,∞)
3)There is an a0>0 such that Aθ≥ a for all θ∈, θ ≠ m
4)
Then the problem (1)-(6) has a global weak solution.
Numerical results
Under certain additional restrictions one can show that the system (1)-(6) has
a classical solution. We assume that these solutions are smooth enough and use an
explicit difference scheme as a numerical method to model this system. Through
the modeling we observe a changing of the temperature profile depending on different parameters.
The temperature profile in case of homogeneity is presented in Fig. 1. One can
Fig. 1. Typical form of the
temperature profile without
heterogeneity: “hot spot” at
the end of the interval.
see that at the end of the interval a so called “hot spot” appears. Solution norm goes
to infinity in the neighborhood of some point. It means that temperature grows
very fast at the end of the interval and the heat is localized in a bounded region of
the space. It is clear that the uncontrolled rise of temperature is dangerous when
the microwave radiation is used for medical purposes.
In our experiments, we consider a changing of the type of the graph and localization of the “hot spots” with respect to this initial situation when the electric permit34
tivity and magnetic permeability at different intervals with respect to Maxwell’s
equation and position of boundary of the layers were varied.
Figs. 2-4 illustrate the results of our experiments.
One can observe connections between variations of electromagnetic characteristics at the intervals with respect to Maxwell’s equation and heat distribution on
Fig. 2. Form of the temperature profile depending on parameters: growing of magnetic permeability at the second
interval leads to the decreasing of maximum temperature
of the “hot spot”.
Fig. 3. Form of the temperature profile depending on parameters: growing of
electric permittivity at the second interval leads to the appearance of a second
“hot spot” and the disappearance of the first one.
35
Fig. 4. Form of the temperature profile and
maximum temperature
depending on the location of the boundary of
layers: location of the
“second” “hot spot” is
before the boundary of
layers; maximum temperature on the second
“hot spot” depends on
its location.
the whole interval. The maximum temperature of the first observed “hot spot” can
be managed by varying the magnetic permeability. Changing of electric permittivity allows to obtain another “hot spot” instead of the first one. Its location and
maximum temperature depend on position of the boundary of layers.
References
1. Landau L.D., Lifshitz E.I., Electrodynamics of continuous environments (In
Russian) – Moscow: Nauka, 1992.
2. Yin H.M. // SIAM J. of Mathematical Analysis, 29, pp. 409-433 (2002).
3. Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N. Linear and quasilinear equations of second order and parabolic type(In Russian) – Moscow: Nauka, 1967.
4. Sun D., Manoranjan V.S., Yin H.M. // Discrete and Continuous Dynamical
Systems, Supplement, pp. 956-964 (2007).
5. Serkova N.D. // Diploma project, SPbSU (2011).
6. Kalinin Y.N., Reitmann V., Yumaguzin N.Y. // Discrete and Continuous
Dynamical Systems, Supplement, Vol.2 pp. 956-964 (2011).
7. Kumar S., Katiyar V.K. // Int. J. of Appl. Math and Mech. 3(3), pp. 1-17 (2007).
36
Lyapunov functions in upper Hausdorff dimension
estimates of cocycle attractors
Slepukhin Alexander
St_Sandro@mail.ru
Scientific supervisor: Prof. Dr. Reitmann V., Department of
Applied Cybernetics, Faculty of Mathematics and Mechanics,
Saint-Petersburg State University
1. Introduction
General upper estimates of the Hausdorff dimension of attractors of dynamical systems have been derived for the first time by A. Douady and J. Oesterlé in
[1]. Later these results were generalized by other authors [2, 3]. For the first time
Lyapunov functions have been introduced into the estimates of Hausdorff dimension
by V.A. Boichenko and G.A. Leonov in [4]. The investigation of nonautonomous
differential equations leads to the theory of cocycles and their attractors [5-8].
In a certain way one can consider random dynamical systems and the associated
random attractors. Elements of the Douady-Oesterlé theory of upper Hausdorff
dimension estimates for random attractors were developed by H. Crauel and
F. Flandoli in [9].
In this paper we state two theorems about upper Hausdorff dimension estimates
of cocycle attractors or invariant sets which include Lyapunov functions. These
results can be looked as generalization of the estimates for attractors or invariant
sets of autonomous systems [4, 10] to cocycle attractors.
2. Basic tools for cocycle theory
Let (Θ,ρθ) be a compact complete metric space. A base flow ({σt}t∈, (Θ,ρθ)) is
defined by a continuous mapping σ:×Θ→Θ, (t,θ)→ σt(θ) satisfying
1) σ 0 ()
⋅ = id Θ
2) σ t + s ()
⋅ = σ t ()
⋅  σ s ()
⋅ for all t, s∈
A cocycle over the base flow ({σt}t∈, (Θ,ρθ)) is defined by the pair
 t

n
 ϕ (θ, ⋅) t ∈R ,  ,|| ⋅ ||  ,where
θ∈Θ
} (
{
)
1) φt(θ, ·):n → n for all t∈, θ∈Θ
2) φ0(θ, ·) = idn , for all θ∈Θ
3) φt+s(θ, ·) = φt(σs(θ), φs(θ, ·)) for all t, s∈, θ∈Θ
In the sequel we shortly denote a cocycle
 t
 n
{
} , ( ,|| ⋅ ||)
 ϕ (θ, ⋅)
t ∈R
θ∈Θ
over the base flow ({σt}t∈, (Θ,ρθ)) by (φ, σ). The basics of the cocycle theory one
can find in [5]. If θ∈Θ→Z(θ)Ìn is a map, we call Ẑ = {Z(θ)}θ∈Θ a nonautonomous
set. The nonautonomous set Ẑ is said to be compact if all sets Z(θ)Ìn, θ∈Θ are
compact, and invariant for the cocycle (φ,σ) if
37
(
)
ϕ t (θ, Z (θ)) = Z σ t (θ) for all t∈ and θ∈Θ.
The set Ẑ = {Z(θ)}θ∈Θ is said to be globally B-pullback attracting for the cocycle (φ,σ) if
lim dist (ϕ t (σ ( − t ) (θ), B), Z (θ)) = 0 t →∞
for any θ∈Θ and any bounded B Ìn.
A nonautonomous set  = {A(θ)}θ∈Θ is called global B-pullback attractor for
the cocycle (φ,σ) if the set  is compact, invariant and globally B-pulback attracting for the cocycle.
3. Upper Hausdorff dimension estimates for cocycles
Let (M,ρ) be a metric space and ZÌM be an arbitrary subset of M. We denote
the Hausdorff dimension of Z by dimHZ (c.f. [10]).
Let L:n→n be a linear operator and let a1(L)≥...≥ an(L) denote [10] the singular
values of L. Let d∈[0,n] be an arbitrary number. It can be represented as d=d0+s,
where d0∈{0,1,...,n-1} and s∈(0,1]. Now we put
s
α L …α L α
L , for d ∈ 0, n ,

ω d (L) := 

1
( )
d0
( )
d 0 +1
1,
( )
( ]
for d = 0
and we call ωd(L) the singular value function of L of order d.
Suppose that (φ,σ) is a cocycle for which the maps φt(θ, ·):n → n are smooth
enough for all t∈, and θ∈Θ. Let us make the following assumptions:
(A1) The nonautonomous set Ẑ = {Z(θ)}θ∈Θ is a compact invariant set for the
cocycle (φ,σ).
(A2) For each θ∈Θ and t>0 let ∂2φt(θ, ·): n → n be the differential of φt(θ, ·)
with respect to the second argument, which has the following properties:
a) For each ε>0 and t>0 the function
|| ϕ t (θ, v ) − ϕ t (θ, u ) − ∂ 2 ϕ t (θ, u )(v − u ) || ηε (t , θ) := sup
|| v − u ||
v , u ∈Z (θ )
0 < ||v − u || ≤ε
is bounded on Θ and converges to zero as ε→0 for each fixed t>0.
b) For each t>0
sup sup || ∂ 2 ϕ t (θ, u ) ||op < ∞
θ∈Θ u ∈Z ( θ )
where ||L||op denotes the operator norm of L.
Theorem 1. Suppose that the assumptions (A1) and (A2) are satisfied and the
following conditions hold:
1) There exists a compact set K̃Ì n such that
U Z (θ) ⊂ K .
(1)
θ∈È
n
2)There exist a continuous function k:Θ×  →>0, a time τ>0 and a number
d∈(0,n] such that
k (σ τ (θ), ϕ τ (θ, u ))
sup
ω d (∂ 2 ϕ τ (θ, u )) < 1.
k (θ, u )
( θ , u ) ∈Θ × K
Then dimHZ(θ)≤d for each θ∈Θ.
38
Theorem 1 is presented and proved in [11]. A shorter announcement of these
result has been given in [12].
4. Cocycles generated by differential equations
Let us consider the nonautonomous ODE
u = f (t , u ) (2)
n
k
where f:× → is a C -smooth (k≥1) vector field. With respect to the vector
field (2) we introduce the hull of f given by
H ( f ) = {f (⋅ + t , ⋅), t ∈} ,
where the closure is taken in the compact-open topology. One can show that H(f)
is metrizable with a metric ρ. As a result we get the complete metric space (H(f),ρ)
on which a base flow called Bebutov flow [6] is given by the shift map
σ t ( f ) = f (⋅ + t , ⋅) for any f̃ ∈ H(f). We assume that H(f) is compact. A sufficient condition for this
is the almost-periodicity of f(t,u) with respect to t.
Suppose now that we have on Θ= H(f) the evaluation map given by
(θ, u ) ∈Θ ×  n → θ(0, u ). In particular we get for θ = f∈ H(f)
fˆ (t , u ) = f (0, u ) It follows that
fˆ σ t ( f ), u = f (t , u ) for all t∈ and u∈n.
Using this map we can associate to (2) the family of vector fields
(3)
u = fˆ (σ t (θ), u ), where θ∈H(f) is arbitrary. The given system (2) is included into (3) as special case.
Under the following additional assumptions on (2) one can show for system (3)
the existence of a cocycle over the base flow ({σt}t∈, (H(f),ρ)) (cf. [5]).
(A3) The map (t,u)∈ ×n→f(t,u) is continuous and satisfies a local Lipschitz
condition with respect to u.
(A4) There exist locally integrable functions p,q:→ such that
|| f (t , u ) ||≤ p (t ) || u ||2 + q (t ) for all (t,u)∈ ×n.
For a point (θ0,u0)∈Θ×n we denote by w(t,u0) the solution of the variational
equation along the orbit of the cocycle through (θ0,u0) i.e., the equation
w = ∂ 2 fˆ σ t (θ0 ), ϕ t (θ0 , u0 ) w (4)
(
)
(
)
with the initial condition w(0,w0)∈n. Then we have for t≥0
.
∂ 2 ϕ t (θ0 , u0 )w0 = w (t , w0 ) i.e. ∂ 2 ϕ t (θ0 , u0 )w0
is a solution of the variational equation (4).
Let λ1(θ,u)≥... ≥λn(θ,u) be the eigenvalues of the matrix
1 ˆ
T
∂ 2 f (θ, u ) + ∂ 2 fˆ (θ, u ) 
.
2
Theorem 2. Suppose that there exist a continuous function V:Θ×n→ for
which the generalized derivative
39
d
V σ t (θ), ϕ t (θ, u0 )
dt
exists along the given trajectory. Suppose further that there are a number d∈(0,n]
written as d=d0+s with d0∈ {0,1,..., n-1} and s∈(0,1] and a time τ>0 such that
ϕ τ (θ, Z (θ)) = Z σ τ (θ) for all θ∈Θ, the condition (1) is satisfied and
τ
t
t
t
t
∫[λ1 σ (θ), ϕ (θ, u0 ) + …+ λ d0 σ (θ), ϕ (θ, u0 ) +
(
)
(
0
(
)
(
)
(
)
+ sλ d0 +1 σ t (θ), ϕ t (θ, u0 ) +
)
d
V σ t (θ), ϕ t (θ, u0 ) ]dt < 0
dt
(
)
for all θ∈Θ and u0∈K̃. Then dimHZ(θ)≤d for all θ∈Θ.
Theorem 2 is also presented in [11,12].
5. Upper Hausdorff dimension estimate for invariant
set of nonautonomous Rössler System
We consider the nonautonomous Rössler system [13]
(5)
x = − y − z , y = x, z = −b (t ) z + a (t ) y − y 2 where the parameters are functions a,b:→>0 which we write as
a (t ) = a0 + a1 (t ), b (t ) = b0 + b1 (t ) Here a0 and b0 are positive constants; a1(·) and b1(·) are smooth functions satisfying the inequalities
a1 (t ) ≤ εa0 , b1 (t ) ≤ εb0 (6)
for all t∈ where ε∈(0,1) is a small parameter. Assume also, that there is an l>0
such that
b (t ) ≤ εl (7)
for all t∈ and the hull H(f) with f as right-hand side of (5) is compact in compactopen topology. A sufficient condition for this is the almost periodicity of a and b.
It follows that system (5) is a special type of system (2) for which the assumptions
of Wakeman's theorem are satisfied. Thus (5) generates a cocycle
 t

n
 ϕ (θ, ⋅) t ∈R ,  ,|| ⋅ || 
(
{
}
θ∈H ( f )
({ }
(
)
)
)
over the base flow σ t ∈ , (H ( f ), ρ)
We assume that for this cocycle there exist a compact set Ẑ={Z(θ)}θ∈H(f), which
satisfies (1) with a compact K̃, and a time τ>0 such that
ϕ τ (θ, Z (θ)) = Z σ τ (θ) for all θ∈H(f).
Instead of (5) we consider the family of systems
x = − y − z , y = x, z = −bθ (t ) z + aθ (t )( y − y 2 ), where for brevity we have written
a (t ) ≡ aˆ σ t (θ) , b (t ) ≡ bˆ σ t (θ) t
(
θ
(
)
θ
40
)
(
)
Our aim is to estimate from above the Hausdorff dimension of Ẑ with the help
of Theorem 2. To do so we have to check the inequality
d
(8)
λ1,θ (t , x, y, z ) + λ 2,θ (t , x, y, z ) + sλ 3,θ (t , x, y, z ) + Vθ (t , x, y, z ) < 0
dt
for all t∈[0,τ], all (x,y,z)∈K̃, and all θ∈H(f) in which λ(k,θ)(t,x,y,z)≡λk(σt(θ),
φt(θ,x,y,z)), k=1,2,3 are the eigenvalues of the symmetrized Jacobian matrix for the
right-hand side of (8) ordered with respect to their size as λ1,θ≥λ2,θ≥λ3,θ and
Vθ (t , x, y, z ) ≡ V σ t (θ), ϕ t (θ, x, y, z ) is a Lyapunov-type function defined for (x,y,z)∈ K̃, θ∈H(f) and t∈[0,τ] by the
relation
1
V σ t (θ), x, z := (1 − s )ξ (z − bθ (t ) x ) ,
2
where ξ is a varying parameter.
We calculate the eigenvalues λk,θ and the derivative dVθ/dt and substitute them
into (8). Direct computations with the use of (6), (7) and Theorem 2 finally give
the estimate
2bθ (t )
2 (1 − ε )b0
dim H Z (θ) ≤ 3 −
≤ 3−
2
bθ (t ) + hθ (t , x, y; ξ)
(1 + ε)b0 + (a0 + 2b0 ) + b02 + 1 + ε⋅ C
with some constant C which is calculated from the parameters of the system.
It is clear that if we turn back to the autonomous Rössler system, i.e. tend ε→0, we
will get the already known Hausdorff dimension estimate for a compact invariant
set of the Rössler system (cf. [10])
2b0
dimH K ≤ 3 −
.
b0 + (a0 + 2b0 ) 2 + b02 + 1
(
(
)
)
References
1. Douady A., Oesterlé J. // Comptes Rendus Acad. Science. A. V. 290. P. 11351138 (1980).
2. Smith R.A. // Proc. Royal Society Edinburg. V. 140 A. P. 235-259 (1986).
3. Temam R. Infinite-Dimensional Systems in Mechanics and Physics. New York
– Berlin: Springer (1988).
4. Boichenko V.A., Leonov G.A. // Acta Appl. Math. V. 26, P. 1-60 (1992).
5. Wakeman D.R. // J. Diff. Equations. V. 17. No 2, P. 259-295 (1975).
6. Bebutov M. // Moscow Univ. Math. Bulletin. V. 2, P. 1-52 (1941).
7. Kloeden P.E., Schmalfuss B. // Numerical Algorithms. V. 14. No 1-3. P. 141152 (1997).
8. Chepyzhov V.V., Vishik M.I. // J. Math. Pures et Appliquées. V. 73. P. 279-333 (1994).
9. Crauel H., Flandoli F. // J. Dyn. Diff. Equations. V. 10. P. 449-474 (1998).
10. Boichenko V.A., Leonov G.A., Reitmann V. Dimension Theory for ODE.
Wiesbaden: Vieweg-Teubner Verlag (2005).
11. Reitmann V., Slepukhin A.S. // Vestnik St-Petersburg Univ. Math. V. 44. No 4.
P. 292-300 (2011).
12. Leonov G.A., Reitmann V., Slepukhin A.S. // Doklady Mathematics. V. 84.
No 1. P. 1-4 (2011).
13. Rössler O.E. // Z. Naturforsch. A. V. 31. P. 1664-1670 (1976).
41
D. Solid State Physics
Intercalation of Al as a method of formation of
quasifreestanding graphene
Anna Popova, Alexander M. Shikin
popova.anna@bk.ru
Scientific supervisor: Prof. Dr. Shikin A.M., Solid State
Electronics Department, Faculty of Physics, Saint-Petersburg
State University
1. Introduction
It is well known that the unique electronic structure and physical-chemical
properties of graphene will appreciably changed under the interaction of graphene
with a substrate. This interaction leads to an energy shift of the electronic states in
the valence band, to an appearance of the local energy gap and, in some cases, to
a distortion of linear dispersion dependences of the π states of graphene [1-4] that
can be followed by a loss of unique characteristic for graphene [5, 6]. The present
work deals with investigation of a process of intercalation of Al underneath of a
graphene, synthesized by cracking of propylene (C3H6) on top of monocrystalline
substrate Ni(111), and analysis of changes of the electronic structure of graphene
during the Al intercalation. On the one hand, one of the main aims of this work was
a verification of the results of early published works [7, 8] about a possible energy
shift of the Dirac point under the interaction of graphene with a Al (due to a possible
charge transfer between Al and graphene). On the other hand, an intercalation of
only noble metals [3] underneath of graphene was intensively studied before (i.e.
metals with d-type of the valence band). It was showed that intercalation of metals
with d electrons in valence band underneath of graphene leads to a hybridization
of π states of graphene with d states of metal with the corresponding distortion of
dispersion dependences of π states of graphene in the crossing region of these states
and the formation of local energy gaps in the dispersion dependences of π states
of graphene [3]. The valence band of Al does not include d electrons, so it was
expected that intercalation of Al underneath of a graphene will lead to a blocking
of strong covalent interaction of graphene with Ni substrate without any breaks in
the dispersion dependences of π states in the valence band. Investigations of the
electronic structure were carried out by angle-resolved photoelectron spectroscopy
with the application of synchrotron radiation. Influence of different concentration
of the deposited and intercalated Al atoms on the result of intercalation process of
Al underneath of graphene/Ni(111) was studied.
2. Results and Discussion
Intercalation of Al is very complicated process with definite stages of the formation of system. First of all, we should illustrate all these stages of the intercalated
process up to stage of stable electronic structure typical for Al layer at the interface
underneath of graphene (see Fig. 1). So, the stages (a) and (b) in Fig. 1 correspond
44
to a synthesis of graphene by cracking of propylene on top of monocrystalline
substrate Ni(111). As a result a well-ordered graphene monolayer is formed [9].
The stage (c) in Fig. 1 is a deposition of Al on top of graphene with a subsequent
annealing of the system at the temperature of 400°C during 5 minutes (stage (d) in
Fig. 1). It was found that after annealing of the system the Al atoms intercalated
underneath a graphene dissolve mainly in the Ni layer with formation of surface
alloy enriched by Ni. In order to increase a relative concentration of Al in Ni-Al
alloy an additional deposition of Al is necessary (stage (e) in Fig. 1) with subsequent
annealing of the system at the same temperature ((f) in Fig. 1). Simultaneously with
a growth of the relative concentration of Al in underlying surface Ni-Al alloy and
a partial accumulation of Al takes place at the interface between graphene and the
Ni-Al layer. At certain concentration of intercalated Al (after the next deposition
of the Al on top of system (stage (g) in Fig. 1) and subsequent annealing of the
system at the same temperature (stage (h) in Fig. 1)) a formation of continuous
layer of Al at the interface under graphene takes place. In this case the electronic
structure of graphene becomes similar to the electronic structure of quasifreestand-
Al
Fig. 1. Schematic illustration of all stages of the experiment. (a) - monocrystalline substrate Ni(111); (b) –graphene synthesized by cracking of propylene on top
of Ni(111); (c) – Al deposition on graphene; (d) – annealing of the system; (e) –
extra deposition of Al; (f) – annealing of the system; (g) – third (extra) Al deposition; (h) – annealing of the system.
ing graphene on top of Al.
Changes of the valence band structure during the process of formation of the
system are shown in Fig. 2. The photoemission spectra were measured in the normal emission geometry with angle resolution ~1o. The spectrum for graphene on
Ni(111) (Fig. 2a and stage (b) in Fig. 1) is characterized by the binding energy of π
states of graphene in the Г point of the surface Brillouin zone of about 10 eV. While
for pyrolytic graphite binding energy of π states (with a weak interaction between
45
graphite layers) is about 8 eV (Fig. 2e) [10]. Peaks near the 0.2-0.5eV and 1.5 eV
in the spectrum for the system graphene/Ni(111) correspond to 3d states of Ni.
Annealing of the system with Al deposited on graphene (stage (d) in Fig. 1)
at the temperature 400°C does not lead to any visible changes of energy position
of π states of graphene in the valence band. Binding energy of π states is about
10 eV as it was before. But some tracks at the 8.55 eV of binding energy can be
distinguished. This stage is characterized by a solution of the intercalated Al metal
into Ni substrate or by a formation of the surface alloy with a stoichiometry near
the Ni3Al. We didn’t observe any shift of π states of graphene that can be evidence
of absence of blocking of the strong interaction of graphene with a substrate at this
stage. At the same time 3d states of Ni is weaken and shift toward higher binding
energies (BE’s becomes ~1.7 and 1.8 eV). According to early published works
[11, 12] alloying of Al with d metal Ni leads
to a shift of the Ni d band toward the higher
binding energies in parallel with decreasing
of the concentration of Ni in alloy. Therefore,
these changes of energy position of Ni d peaks
can be related to an alloying of Al with Ni (with
predominant concentration of Ni) during the
intercalation of Al at the first step.
After the additional Al deposition (see
stage (e) in Fig. 1) with subsequent annealing of the system (see stage (f) in Fig. 1) we
observe considerable changes in the valence
band spectra. Aluminum which is accumulated
at the interface between graphene and substrate
begins blocking strong covalent interaction of
graphene with Ni d states in some areas under
Fig. 2. Changes of the valence graphene monolayer. As a result an additional
band photoemission spectra at peak of π states appears at the binding energy
different stages of experiment (see ~8.55 eV. However the weak peak of π states
Fig. 1). The spectra are measured at the binding energy ~10 eV also remains in
in the normal emission geometry. the spectrum that leads to two-peak structure of
(a) - graphene synthesized on top the π states. This is evidence that there are difof Ni(111); (b) - annealing of the ferent areas underneath a graphene monolayer
system after the deposition of Al with weak and strong interaction of graphene
(stage (d)in Fig. 1); (c) – anneal- with a substrate. In regions where intercalated
ing of the system after the addi- Al atoms are located underneath of graphene
tional Al deposition (stage (f) in monolayer a blocking of strong interaction of
Fig. 1); (d) – annealing of the sys- graphene with a substrate takes place. In the
tem after Al deposition (stage (h) places where Al atoms are located underneath a
in Fig. 1); (e) – pyrolytic graph- graphene the binding energy of π states is about
ite (for comparison). Photon en- ~8.55 eV and in the places where Al atoms are
ergy is 60 eV.
not located at the interface – the binding energy
46
of π states is about ~10 eV (as it is in the case of strong interaction of graphene
with a Ni substrate).
Fig. 3. (a) – Series of photoelectron spectra, measured with angle resolution for
graphene after the intercalation of Al (stage (h) in Fig. 1); (b) - corresponding
dispersion dependences of π states of graphene measured in the ГК direction of
the surface Brillouin zone. Dispersion dependences are represented in the form of
dN/dE. Photon energy is 60 eV.
Additional Al deposition on top of system (stage (g) in Fig. 1) and subsequent
annealing of the system (stage (h) in Fig. 1) lead to further considerable changes in
electronic structure of the valence states of the system. When the concentration of
Al is enough for filling of the whole layer at the interface, only one peak of π states
is observed in the valence band spectrum at the binding energy ~ 8.55 eV (Fig. 2e).
Fig. 3 shows series of photoelectron spectra measured with angle resolution for
this system and corresponding dispersion dependences of π states of graphene measured in
the ГК direction of the Brillouin zone. Fig. 4
shows in details dispersion dependences of π
states of graphene near the Fermi level in the
region of the K point of the surface Brillouin
zone.
The top of lower cone of graphene is located at the energy of about 0.4 eV relative to
the Fermi level (Fig. 4). The lower edge of the
band of graphene π states in the region of the Г
point is located at the energy of 8.55 eV (Fig. 3)
that differs from case of graphene on top of
Ni(111). The shift of the energy position of π
states of graphene toward the Fermi level, in Fig. 4. Detailed dispersion depencomparison with graphene/Ni(111) [3, 9], can dences of π states of graphene in
testify to blocking of the strong interaction of the region of the K point of the surgraphene with a substrate after intercalation of face Brillouin zone.
47
Al. Detailed analysis of the dispersion dependences of π states near the Fermi level
in the region of the K point shows a some distortion of the dispersion dependences
in the region of the crossing of π states with the substrate-derived states located at
the energy of about ~1 eV due to covalent interaction with these states (Fig. 4).
Conclusions
Intercalation of Al underneath graphene synthesized on Ni(111) leads to the
“blocking” of strong interaction of graphene with a substrate and formation of
electronic structure characteristic for a quasifreestanding graphene. Dispersion
dependences of π states have a linear character in the region of the K point of the
surface Brillouin zone and the Dirac point is located near the Fermi level. But a
small energy gap is formed between occupied and unoccupied cones of π states near
the K point. A charge transfer from Al to atoms of graphene after intercalation of
Al is not observed in opposite to the prediction in work [7, 8]. At the initial stages
of Al intercalation alloying of Al with underlying Ni layer takes place. It makes the
process of intercalation of Al more complex in comparison with other metals.
References
1. Oshima C. and Nagashima A. // J. Phys.: Condens. Matter, 9, 1–20 (1997).
2. Tontegode A.Ya. // Progress in Surface Science, 38, 201 – 429 (1991).
3. Попова А.А., Шикин А.М., Марченко А.Г., Рыбкин А.Г., Вилков О.Ю.,
Макарова А.А., Варыхалов А.Ю., Rader O. // Физика твердого тела, 53, 12,
2409 (2011).
4. Varykhalov A., Scholz M.R., Kim T.K., Rader O. // Phys. Rev. B 82, 121101
(2010).
5. Benakker C.W. // Rev. Mod. Phys. 80, 1337 (2008).
6. Katsnelson M.I.,. Novoselov K.S, Geim A.K. // Nature Phys. 2, 620 (2006).
7. Giovannetti G., Khomyakov P.A., Brocks G., Karpan V.M., Brink J., Kelly P.J.
// Phys. Rev. Lett. 101, 026803 (2008).
8. Khomyakov P.A., Giovannetti G., Rusu P.C., Brocks G., Brink J., Kelly P.J. //
Phys. Rev. B 79, 195245 (2009).
9. Varykhalov A., Sanchez-Barriga J., Shikin A.M., Bismas C., Veskovo E., Rybkin
A., Marchenko D., Rader O. // Phys. Rev. Lett. 101, 157601 (2008).
10.Molodtsov S.L., Laubschat C., Richter M., Gantz T., Shikin A.M. // Phys. Rev.
B 53, 16621 (1996).
11.Morinaga M, Nasu S., Adachis H., Saito J., Yukawa N. // J. Phys.: Condens.
Matter 3, 6817 (1991).
12.P.A. Bennett, J.C. Fuggle, F.U. Hillebrecht // Phys. Rev. B 27, 2194 (1983).
48
Calculation of Sound Speed in Artificial Opal
Andrey Uskov
yskov@yandex.ru
Scientific supervisor: Dr. Borisov B.F., Department of Solid State
Physics, Faculty of Physics, Saint-Petersburg State University
Introduction
Microcellular materials are widely used as catalysts, filters and adsorbents.
They also could be used to investigate properties of materials in confined geometry.
For example, it is possible to investigate properties of phase transitions in liquid
metals embedded in porous glasses and artificial opals by measuring dependency
of sound speed on temperature.
There are at least three methods allowing to determine phase transitions. They
are X-ray analysis, NMR, and acoustical methods. The first two methods have some
serious disadvantages. Each measurement takes several minutes, thus dynamics of
phase transitions could not be clarified. But the theory which could link elasticity
of embedded material and sound speed is still absent [1]. Thus, it is still impossible to link quantity of liquid in pores with sound speed. The aim of this work is
calculation of sound speed in artificial opal with empty pores and dependency of
sound speed on pore filling factor.
Elasticity of opals
The theory we have
developed takes into account internal structure
of artificial opals. They
consist of closely packed
silica spheres. Each sphere
consists of smaller silica
spheres of the second order.
The spheres of the second
order consist of the smaller
spheres of the third order
[1, 3].
Each silica sphere has
several neighbor spheres.
The summary force acting
on this sphere depends on
neighbor’s coordinates and Fig. 1. Internal structure of artificial opal [3].
elasticity of spheres. In general, the elasticity factors for different spheres could
differ.
49
Fig. 2. Scheme of interaction between silica spheres.
According to the second Newton’s law, motion of silica sphere could be de-
ma = ∑ cos α i ki ∆li
scribed by the following equation:
It could be shown that motion of sphere could be described by the following
differential equation:
ki a 2 cos 2 α i ∂ 2u
∂ 2u
=
∑ m ∂x 2
∂t 2
Where u is coordinate of the corresponding sphere along the wave vector. After
introduction of the porosity of the material:
m = (1 − γ ) ρ m a 3
We can calculate elasticity of the whole material and link it with sound speed
in material:
c=
∑ k cos
i
2
αi
a ρ m (1 − γ )
And finally we evaluate elasticity factors of each contact between two silica
spheres. Thus sound speed could be calculated as a function on compression factor
of silica spheres, what is equal to diameter of silica spheres substituted by distance
between to neighbor silica spheres and divided by two.


2π
c=
 6 ln  2 R + δ


 δ

3
2





c11
(1 − γ ) ρm
Mercury porometry simulation
Unfortunately, we have no possibility to measure compression factor directly,
but we had mercury porometry data for samples under investigation. We simulated
50
porometry for ideal opals with different compression factors and compared with
experimental results.
Fig. 3. Cross-sections of ideal opal without compression.
In order to calculate porometry, we simulated cross-sections of ideal opals.
Then, for each point the minimal distance to pore well was calculated. Actually,
this distance has similar physical sense to pore size for current point. But the
dependence of number of points on corresponding size doesn’t fit porometry data.
This data should be normalized in the following way: we should take into account
the fact that cross-section is two-dimensional and a single spherical pore makes
quite complicated contribution to this dependence. Contributions for spherical
pores are presented in the Fig. 4. Using this data, the dependence of volume of
pores on it’s radius was calculated for different cross-sections. The average value
between this dependencies has a physical sense of porometry. This data were
compared to experimental results.
Fig. 4. Dependence of point number on corresponding size.
51
Comparison results are presented in the Fig. 6. It could be seen that experimental
results fit theoretical ones for compression range about 5%. And the difference
between theoretical and experimental results could be explained by two effects.
The first one is dislocation of solica spheres in real sample. The second effect is
deformation of silica spheres.
Fig. 5. Dependence of volume of pores on radius. (relative to silica sphere size).
Fig. 6. Comparison of simulation results for different compression factors with
experimental ones.
52
&RPSDULVRQRIVLPXODWLRQUHVXOWVIRUGLIIHUHQWFRPSUHVVLRQIDFWRUV
LPHQWDORQHV
Fig. 7. Dependence of sound speed on compression factor.
)LJ'HSHQGHQFHRIVRXQGVSHHGRQFRPSUHVVLRQIDFWRU
speed fits quite good experimental
It could be seen that the calculated sound
results when compression factor is equal to calculated one. Another significant
FRXOGEHVHHQWKDWWKHFDOFXODWHGVRXQGVSHHGILWVTXLWHJRRGH[SHULP
result is prediction of sound speed dependence on filling factor. It is known that
VZKHQFRPSUHVVLRQIDFWRULVHTXDOWRFDOFXODWHGRQH$QRWKHUVLJQLI
compression factor rises when the sample is being filled, thus sound speed rises
too. Similar dependence was obtained experimentally.
LVSUHGLFWLRQRIVRXQGVSHHGGHSHQGHQFHRQILOOLQJIDFWRU,WLVNQRZQ
UHVVLRQIDFWRUULVHVZKHQWKHVDPSOHLVEHLQJILOOHGWKXVVRXQGVSHHG
References
LPLODUGHSHQGHQFHZDVREWDLQHGH[SHULPHQWDOO\
1. Borisov B.F.// XIX Session of Russian Acoustic Society, p. 111 (2006).
2. Borisov B.F.// XVI Session of Russian Acoustic Society, p. 15 (2003).
3. Gartvik A.V.// Doctoral thesis, St.Petersburg (2005).
5HIHUHQFHV
ULVRY%);,;6HVVLRQRI5XVVLDQ$FRXVWLF6RFLHW\S
ULVRY%);9,6HVVLRQRI5XVVLDQ$FRXVWLF6RFLHW\S
UWYLN$9'RFWRUDOWKHVLV6W3HWHUVEXUJ
53
Modification of spin and electronic structure of
graphene by intercalation of Bi
Evgeny Zhizhin
Evgeny_liquid@mail.ru
Scientific supervisors: Prof. Dr. Vladimirov G.G., Prof. Dr.
Shikin A.M., Department of Solid State Electronics, Faculty of
Physics, St. Petersburg State University
Introduction
Exploration of spin and electronic structure of various nanostructured systems
have attracted increasing interest in recent years, due to wide development of
Spintronics – a new branch of nanotechnology. Spintronics is based on exploiting
the 'spin' of the electron [1, 2].
By the normal conditions, value of the spin-orbit splitting of electronic states of
graphene is negligible [2], but the intercalation of Au atoms underneath graphene
monolayer leads to the effects induced of the substrate-induced spin-orbit splitting of the π-states of graphene [3, 4]. However, the interaction of the π-states of
graphene with the d - states of Au, in addition to the spin polarization of π states,
accompanied by the formation of discontinuities in the dispersion dependences
of the π- states [4]. The aim of our research was to investigate a possibility of
the intercalation atoms of another metal with high atomic number Bi underneath
graphene monolayer. Additionally, Bi is characterized by sp-type of the valence
band structure, i.e. by the lack of the d-electrons in the valence band. We present
in the current work the results of investigations of the features of the electronic
structure of graphene with intercalated Bi underneath and the features of the spin
structure of the formed system.
Experimental details
The experiment was carried out in the Helmholtz-Zentrum (BESSY II) at
the Russian-German beamline by photoelectron spectroscopy with angular and
spin resolution. As a result the dispersion dependences were measured with spin
resolution with using hemispherical energy analyzer SPECS "Phoibos 150".
Graphene monolayer on the surface of Ni(111) was formed by cracking of
propylene at heated surface of thin Ni(111) layer. The layer of Ni(111) with a
thickness of about 100Ǻ was deposited on an atomically clean surface of W(110).
Cleaning the surface of W(110) were produced by annealing in oxygen at a pressure
of 5.10-7 mbar and the temperature 1200°С followed a short heating to temperatures
of ∼2000°С in ultrahigh vacuum. Graphene was produced by a catalytic reaction
of cracking of propylene (С3H6) at the surface of Ni(111) during for 5 min at a
pressure of propylene 1.10-6 mbar and temperature of the sample 450°С. After that
the sample was annealed in ultrahigh vacuum and at 400-450°С. Intercalation of
Bi atoms underneath a graphene monolayer was produced by deposition of Bi on
54
the surface of a graphene monolayer with subsequent annealing of the system at a
temperature of ∼300-350°С. The pressure in
the research chamber during the experiment
was on the level of 1-2.10-10 mbarr.
Experimental results and discussion
Fig. 1 shows the series of the normal
emission photoelectron spectra for the various stages of formation of the system with
graphene synthesized on the surface of Ni(111)
with subsequent intercalation of Bi underneath a graphene. It is seen that a graphene
on Ni(111) is characterized by the binding
energy of π- states 1 eV that testify to strong
interaction of graphene with the Ni-substrate.
After the deposition of Bi on a top of graphene
monolayer (Fig. 1c) and following annealing
of the system at a temperature of ∼300-350°С
(Fig. 1d) π-state of graphene is shifted toward
the lower binding energy (~ 8,1 eV) that
indicates about a “blocking” of the strong
interaction between graphene and substrate
due to the intercalation of Bi.
Figs. 2 and 3 show the dispersion dependences of π states for graphene interca-
Fig. 2. The dispersion dependence of
the valence band electronic states for
graphene on Ni(111) in the ГК direction of the Brillouin zone synthesized
by cracking of propylene.
Fig. 1. The stages of the formation
of the MG/Bi/Ni(111) system. Series
of the normal emission photoelectron spectra: a) after deposition of
∼ 100 Å Ni on the surface of the
W(110), b) after the formation of
graphene by cracking of propylene,
c) after deposition of ∼ 1 ML Bi on
graphene, d) after intercalation of
Bi underneath graphene.
Fig. 3. The dispersion dependence of the
valence band electron states for a graphene after intercalation of Bi atoms underneath a graphene measured in the ГК
direction of the Brillouin zone.
55
lated of Bi underneath. From Fig. 2 we can see that the π-state of graphene for this
system doesn’t reach the Fermi level, and dispersion dependence of the π states of
graphene in the region of the K point of the Brillouin zone has parabolic character.
Compared to the quasi-free graphene dispersion relations the
π-state of graphene synthesized
on the surface Ni(111) are significantly shifted towards higher
binding energies approximately
on 2 eV. The upper edge of the
band of the π-states in the region
of the К point of the Brillouin
zone is located below the Fermi
level. An intercalation atom
of Bi underneath a graphene
monolayer leads to a substantial
change in the electronic structure of graphene as compared
to the electronic structure of
graphene on the surface of the
Fig .4. The dispersion dependence of the π states Ni(111).
of graphene after intercalation of Bi atoms meaAs a result of intercalasured in the region of the K point of the Brillouin tion atoms of Bi a significant
zone.
energy shift of the π-states of
graphene toward lower binding
energies is observed that testify
to a “blocking” of a strong interaction of graphene with the
substrate. Due to the Bi intercalation the electronic structure
of the valence band becomes
similar that characteristic for a
quasi-free graphene. Thereat,
practically no any distortions
in the dispersion dependences
of the π- states of graphene and
a formation of discontinuities
in the dispersion dependences
are observed.
In Fig. 4 the detailed disperFig. 5. Photoelectron spectrum with the spin res- sion dependence of the π- states
olution for π states of graphene measured in the of graphene in the region of the
ГК direction of the Brillouin zone after interca- K point of the Brillouin zone
lation of Bi.
is shown.
56
The band edge of the π-states reaches practically the Fermi level. Only small
energy gap is formed with a width of ~ 0,2 eV near the Fermi level.
Fig. 5 shows the corresponding photoelectron spectrum with the spin resolution
measured for π states of graphene in the direction ГК of the Brillouin zone after
intercalation of Bi. It is seen that intercalation of atoms
Bi underneath a graphene does not lead to any visible spin splitting of the π
states of graphene. It means that the effect of the substrate-induced spin-orbit
splitting of the π-states of graphene does not manifests itself after intercalation of
Bi despite its high atomic number.
Conclusions
As a result of the experiment one can make the following conclusions:
1. Intercalation of Bi underneath a graphene synthesized on Ni(111) leads to
blocking of the strong interaction of graphene with substrate and formation of
electronic structure similar that characteristic for quasi-free graphene with small
energy gap near the Fermi level.
2. Availability of only high atomic number Z of intercalated atoms (Bi) is
not sufficient for the effect of the induced spin-orbit splitting of the π-states of
graphene.
Acknowledgment. The experiment was carried out done in the HelmholtzZentrum (BESSY II) at the Russian-German beamline. The work was supported
in framework of G-RISC and Euler program.
References
1. Rashba E. I. // J. Supercond.— Vol. 15, no. 1 (2002).
2. Rybkin A.G. Electronic, energetic and spin structure of the thin layers of metals
induces by spin-orbit interaction // St. Petersburg – 2010.
3. Varykhalov A. // PRL, 101, 157601 (2008).
4. Popova A.A. // FTT, vol. 53, no. 12 (2011).
57
E. Applied Physics
Usage Pocket Comsol for the Numerical Nonstationary
Nonlocal Plasma Modeling
Burkova Zoya
goldenvirent@gmail.com
Scientific supervisor: Dr. Chirtsov A.S., Department of General
Physics, Faculty of Physics, Saint Petersburg State University
One of the most interesting kinds of plasma (ion-ion plasma) has been investigated for a long time [1]. The formation of ion-ion plasma is interesting to simulate
due to a high range of relationships of different kinds of plasma characteristics
can be received.
It’s easy to use a numerical simulation to simulate wide range of systems,
which is interesting for physics. Software package COMSOL is well adapted to
solve such problems [2]. There are a lot of modules in this software. One of them
is the plasma module.
The foundation of the COMSOL Multiphysics Plasma Module is the Drift
Diffusion interface which describes the transport of electrons in an electric field.
The Drift Diffusion interface solves a pair of reaction/convection/diffusion equations, one for the electron density and the other for the electron energy density.
The problem is that COMSOL can simulate only one task in one session. Because
of that were used package Wolfram Mathematica to compare the results.
External circuit
One of the simplest model has been investigated - DC glow discharges. Such
discharge in the low pressure regime has long been used for gas lasers and fluorescent lamps. DC discharges are attractive to study because the solution is time
independent.
The DC discharge consists of two electrodes, one powered (the anode) and one
grounded (the cathode). The electrons are emitted from the cathode surface and
then accelerated by the strong electric field, where they acquire enough energy to
initiate ionization. The positive column is coupled to an external circuit:
60
Domain equations
The electron density and mean electron energy are computed by solving a pair
of drift-diffusion equations for the electron density and mean electron energy.
Convection of electrons due to fluid motion is neglected.
∂
(1)
ne ) + ∇ ⋅  − ne (µ e ⋅ E ) − De ⋅ ∇ne  = Re
(
∂t
∂
(2)
(nε ) + ∇ ⋅  − nε (µ µ ⋅ E ) − Dµ ⋅∇nε  + E ⋅ Γ e = Rε ∂t
Γ e = − (µ e ⋅ E ) ne − De ⋅∇ne
(3)
and, ne denotes the electron density, Re is the electron rate expression, μe is the
electron mobility which is either a scalar or tensor, E is the electric field, and De
is the electron diffusivity, which is either a scalar or a tensor. The first term on the
right side of Eq. 1 represents migration of electrons due to an electric field. The
second term on the right side of Eq. 1 represents diffusion of electrons from regions
of high electron density to low electron density.
Here, ne is the electron energy density, Re is the energy loss/gain due to inelastic collisions, μe is the electron energy mobility, E is the electric field, and De is
the electron energy diffusivity. The subscript refers to electron energy. The third
term on the left side of Eq. 2 represents heating of the electrons due to an external
electric field. Note that this term can either be a heat source or a heat sink depending on whether the electrons are drifting in the same direction as the electric field
or not. For a Maxwellian electron energy distribution function, the following
relationships hold:
 5
De = µ eTe , µ ε =   µ e , Dε = µ εTe (4)
 3
Where Te is the electron “temperature”. So, given the electron mobility. From
which the electron diffusivity, energy mobility and energy diffusivity are computed.
Reaction
Formula
Type
1
2
3
4
5
6
7
e+Ar=>e+Ar
e+Ar=>e+Ars
e+Ars=>e+Ar
e+Ar=>2e+Ar+
e+Ars=>2e+Ar+
Ars+Ars=>e+Ar+ Ar+
Ars+Ar=>Ar+Ar
Elastic
Excitation
Superelastic
Ionization
Ionization
Penning ionization
Metastable quenching
ε(eV)
0
11,5
-11,5
15,8
4,2
-
Plasma chemistry
Argon is one of the simplest mechanisms to implement at low pressures. The
electronically excited states can be lumped into a single species which results in a
61
chemical mechanism consisting of only 3 species and 7 volumetric reactions and
2 surface reactions:
Volumetric reactions
Surface reactions
Reaction
Formula
Sticking coefficient
1
Ars=>Ar
1
2
Ar+=>Ar
1
Where: Ar-argon in ground state, Ars – argon in metastable state, Ar+- ionized
argon, e - electron.
Results
The electron density occurs along the axial length of the column. The electron
density peaks in the region between the cathode fall and positive column. This
region is sometimes referred to as Faraday dark space. The electron density also
decreases rapidly in the radial direction. This is caused by diffusive loss of electrons
to the outer walls of the column where they accumulate a surface charge. The build
Fig. 1. Surface plot of electron density of argon with metastable level.
Fig. 2. Surface plot of electron density of argon without metastable level.
62
up of negative charge leads to a positive potential in the center of the column with
respect to the walls. The electron density of argon with metastable level in Fig. 1 and
without in Fig. 2 are plotted along the axial length of the column.
The solution shows that the measurement of Argon without metastable level is
by a factor of ten lower then Argon with metastable level.
Fig. 3. Difference between data of Ars and Ar (2D).
Fig. 4. Difference between data of Ars and Ar (3D).
The software Wolfram Mathematica has been used to compare the results. Plot
data saved in a text file in COMSOL exported in Wolfram Mathematica. Difference
between the data of Ars and Ar is plotted in Figs. 3 and 4.
The results received in this work a good coordinated with theory and allows to
continue the investigation of the plasma behavior using program package COMSOL and Mathematica.
References
1. Кудрявцев А.А. // Письма в ЖТФ, 1996, т.22, вып.17, с. 11-14.
2. http:// www.comsol.com
63
Factorization of charge formfactors for clusterized light
nuclei in reactions e+16O and e+12C
Danilenko Valeria
valeriadanilenko@gmail.com
Scientific supervisor: Prof. Dr. Gridnev K.A., Department of
Nuclear Physics, Faculty of Physics, Saint-Petersburg State
University
Introduction
A Bose–Einstein condensate is a state of matter of a dilute gas of weakly interacting bosons confined in an external potential and cooled to temperatures very
near absolute zero (0 K or −273.15°C). Under such conditions, a large fraction
of the bosons occupy the lowest quantum state of the external potential, at which
point quantum effects become apparent on a macroscopic scale.
There are some cases in which one can use a model of binding alpha-particles
[1] in order to describe the structure of the nuclei. It can also useful for identifying
the properties of the elastic scattering of the electrons. We use a model of binding
alpha particles in out approach – the nucleus consists of interacting alpha-particles
and nuclear binding energy is a sum of alpha-particles pair interactions. We consider
the coordinates of alpha-particles to be frozen and calculate the positions of them
under the following requirements:
1. The configuration of alpha-particles corresponds to the minimum of the potential
energy.
2. The nucleus must have a nearly spherical form [2].
The process of formation of alpha-particles in nucleus is dynamical. They form a
crystal lattice in the nucleus and the interaction is counted per bond. The appropriate
interaction potential is the form of the one parameter Yukawa potential:
Vnucl (r ) = −V0
e − γr
,
r
where V0 is fitted through the experimental data and γ is inverse value of the
Compton wavelength of the neutral π-meson [2].
There is a conjecture that nuclei of the most widespread elements in the universe
might be viewed as a tight packing of alpha-particles. The binding energies within
such a framework show a good agreement with experimental values. The basis of
the model is the assumption that the binding energy can be expressed as a sum of
energies coming from interactions between alpha-particles and their self-energies.
So the nuclear binding energy can be written as (1)
EB = A0 (6 N α + nα ) + C , where A0 determinates the energy of interaction, Nα is the number of alpha-particles
in the nucleus, nα is the number of bonds between alpha-particles and C is the
Coulomb energy of the nucleus calculated as
64
C=
3e 2
Z ( Z − 1) A−1/ 3 , 5r0
where A is the mass number and r0=1.25 fm. The term with the number of bonds
stands for the interactions between alpha-particles [1].
Using of the model:
1. In general we allow nuclei to consist not only of alpha-particles but also
of neutrons not bound into alpha-particles, so that the model can be used for
predicting binding energies of nuclei with arbitrary mass number.
Generalization is done under the following requirements:
The dependence of binding energy on the number of neutron pairs at first
behaves linearly, but then acquires a slope, and there are no particular shell
effects – the separation energy of two neutrons is monotonically decreasing.
This could be explained: it is energetically favorable to place neutrons inside
different parts of alpha-nucleus, assuming that their bonding to alpha-particles
contributes more than bonding between neutron pairs themselves. This gives
initial liner behavior until the neutron pairs due to inflating the nucleus starting
to distort the alpha-bonds. This generalizes (1) as follows:
EB = A0 (6 N α + nα + βN 2 n ) + C ,
which brings in a linear term βN2n, but the number of bonds between alphaparticles has to be recalculated.
2. If we consider a tetrahedral structure, under very general assumption about
the alpha-particles potential one can calculate the location of alpha-particles
within such construction. Depending on their size, such constructions have a
different number of bonds per particle which brings to a picture similar to the
shell model of nuclei. If we consider the experimental value of a separation
energy of an alpha-particle as a a function of the number of alpha-particles
we will se that apart from peaks stemming from magical numbers, there are a
few nuclei with relatively high alpha-particles binding energy. These “alphamagical” numbers are 3, 7, 13…Similar magic numbers were obtained through
the analysis of clusters in molecular physics [1].
So we call the nuclei clusterized if we use the model of binding alpha-particles
for describing them.
In our research we considered folded formfactors for the clusterized light nuclei. At certain conditions the scattering amplitude can be expressed through the
formfactor. In case of electrons it is a charge formfactor. This formfactor in the
case of cluster structure can be factorized as two multipliers: one for distribution
of clusters in nuclei and the second for the cluster formfactor itself.
The cross-section is calculated using the following:
2
d σ  z1 z2 (e 2 ) 
1
2
2
 dσ 
=
F (q) = 
F (q)

2 
ϑ
 d Ω  res
d Ω  2mv 
sin 4
2
65
The factorization of the formfactor:
F (q ) = Fα ∗ ηα We consider 2 possible types of alpha-particles density distribution, namely
surface and volume distribution. In the first case the formfactor will be proportional
to the spherical Bessel function of order zero j0 in the second case it will be the
spherical Bessel function of order one j1 [3]
sin(kr )
η1α = j0 (kr ) =
kr
1
cos( kr )
ηα2 = j1 (kr ) =
sin(kr ) −
kr
(kr ) 2
As the formfactor for clusters itself we used:
Fα = exp(−αr 2 ) The calculations were made for 2 reactions on different energies:
Reaction
e+160
e+12C
e+12C
e+12C
e+12C
e+160
Energy, Mev
87,2
21,34
22,5
24,5
27,3
75
The obtained results of the angle distribution of the cross-sections are presented
in Fig. 1.
3,5E -14
Cross-section, mBrn/ster
3E -14
2,5E -14
87,2 Mev
2E -14
21,34 Mev
22,5 Mev
1,5E -14
24,5 Mev
1E -14
26,3 Mev
5E -15
75 Mev
0
0
20
40
60
80
100
Ang le , g ra d
Fig. 1. Angle distributions of the cross-sections in the system of center of mass.
The calculations show that the alpha-particle density distribution in the nuclei in
question is a surface distribution. The results were obtained with the formfactor 1.
66
Conclusion
We have used the factorized formfactors for clusterized light nuclei. Factorized
formfactors give us a possibility to scan nuclear surface and some intrinsic layers
of a nucleus. The obtained result shows that the alpha-particle density distribution
in the considered nuclei is a surface distribution.
References
1. Gridnev K.A., Torilov S.Yu., Gridnev D.K., Kartawenko V.G., Greiner W. //
Int. J. Mod. Phys. 14 (2005), 635.
2. Torilov S.Yu., Gridnev K.A., Greiner W. // Int. J. Mod. Phys. 16 (2007),
1757.
3. Arscen J.B., Weber H.J. Mathematical methods for physicists. - Academic
Press, 2005.
67
Study of interaction forces between constant magnet and
high-temperature superconductor
Marek Veronika
nika.marek@mail.ru
Scientific supervisor: Dr. Chirtsov A.S., Department of
General Physics - I, Faculty of Physics, Saint-Petersburg State
University
Introduction
The high-temperature superconductivity is interesting in terms of both theory
and experiment. Presently a great number of articles and overviews dedicated to
this phenomenon exist (i.e. [1, 2]). But a complete theory has not been already
created although the high-temperature superconductivity was discovered 25 years
ago. However this topic well demonstrates laws of electrodynamics that is why it
is traditionally studied in classical physics courses. A study is usually accompanied
by an experiment’s show but an implementation of this experiment is joined with a
number of problems (e.g. sizes of a superconductive ceramics and of a magnet are
small). Therefore an interactive demonstrations set and a video clip (composed of a
real experiment and an animation) about the mechanism of levitation of a constant
magnet over a high-temperature superconductor were created [3, 4] (Fig. 1).
(a)
(b)
(c)
Fig. 1. a – one of interactive models: magnet (current ring) + superconductor
(set of induced round currents); b – clip’s screenshot: a real experiment; c – clip’s
screenshot: a three-dimensional animated model of the system.
Experiment
During the video clip production, an existence of the space interval where the
magnet is in neutral equilibrium instead of the expected stable one was found. To
study this phenomenon a setup which allows the measurement of magnet’s altitude depending on the applied external force (electrically regulated) was thought
up. Systematical measurements of two types were implemented. In the first one
the applied force has been monotonically increased up to its maximum and then
likewise decreased to the value defined by own magnet’s weight (m = 200 mg).
68
In the second experiment the nonmonotonically varying force was used. Thereby
obtained plots (Fig. 2) show that hysteresis effects don’t exist in case of large
distances between the magnet and the superconductive ceramics. In the case of
twice smaller distance one can observe a strong hysteresis effect and an existence
a
b
Fig. 2: Appearance of hysteresis effects in case of monotonically (a) and nonmonotonically (b) change of the applied force.
of several magnet equilibrium positions. Moreover after the removal of the external force the magnet doesn’t return to its initial position and its levitation altitude
noticeably decreases.
Results and Discussion
To explain the observed phenomenon two possible models were considered.
The first one – the “abrupt collapse model” – supposes that when current flowing on the superconductor exceeds the critical value icr, the destruction of the
superconductive state in the corresponding areas of ceramics occurs. The model
of critical Bean state [5] supposes only a limitation of the current by the critical
value. It was assumed that, when the induced currents exceed the critical value, a
circle of disturbed superconductivity appears. Inside this circle the induced currents
turn either to zero or to the critical value (depending on the model used), that was
considered by adding the new compensative currents. This partial destruction of
superconductivity, which conserves after the magnet’s removal from the ceramics,
was considered as the cause of the magnet’s non-return to the initial state. By means
of these assumptions, the mentioned models and the image method, an analytical
description of the phenomena was worked out. This description allows in principle
the calculation of the critical current’s density from the experimentally obtained
plots, but actually it doesn’t take into account several features of the system as
the following: the finiteness of the superconductive ceramics, the nonsimpleconnectedness of its geometrical form and a breach of boundary conditions for
the magnet induction B under the additive currents.
Thus we have the additive currents which consider geometrical features of
the problem and ones which assure the boundary conditions. Thereby the Laplace
problem (which evidently can be solved only numerically) for a vector potential A
with mixed-boundary conditions was considered. Therefore the relaxation method
69
was applied, that with a glance of cylindrical symmetry of the problem leads to the
following connection between vector potential’s values in different points:
A ( r , φ, z ) ≈
≈
1
 A ( r − δ r , φ, z ) + A ( r − δ r , φ, z ) + 2 A ( r − δ r , φ, z ) + A ( r , φ, z − δ z ) + A ( r , φ, z + δ z ) 
6
In initial approximation A was supposed to be zero everywhere except the
layer near the part of ceramics with the additional effective currents. For this and
overlying layers the current has to satisfy Neiman’s boundary conditions:
∂Aϕ
4π '
−
= Br =
iϕ
c
∂z
Because of a slow convergence of the method in the area of superconductivity’s
destruction, the “cloud” computing was used. The thereby found vector potential
allows the calculation of the additional currents in those parts of ceramics which
preserve the superconductivity.
Fig. 3 represents an example of the total current calculation results. The distant
plot area corresponds to the central hole in the superconductive ceramics, the close
one - to the area out of ceramics. The central part for the first model (Fig. 3a) contains “dents” appeared because of the currents’ exceeding the critical value (that
leads to a broadening of the destruction superconductivity area). For the Bean model
(Fig. 3b) this is expressed in appearance of the “plateau” instead of “dents”.
Note that the influence of new superconductivity destruction zones appearing
during calculation of vector potential (and therefore currents’ distribution) is also
taken into account.
The lifting force acting on the magnet is defined by the current distribution and
can be calculated as a sum of interactions between the magnet and the total round
currents iƩ(R) on the ceramics’ surface:
∞
mR 2 H 3
F = ∫ iΣ ( R)
dR
5
/
2
H 2 + R2
0
(
)
a
b
Fig. 3. Results of the calculation of the total current on the superconductor’s surface for the “abrupt collapse model” (a) and the Bean model (b).
70
The area of disturbed superconductivity reaches its maximum size in a point of
maximum approaching the magnet to the ceramics which means the maximum of the
applied force. Then under decreasing of the applied force and under the repulsion
of induced surface currents, the magnet moves away from the superconductor but
reaches lesser altitude. That happens because of the superconductivity’s destruction which turns to zero a part of the induced currents in the area adjoining to the
central hole in the magnet.
Thus there are two causes seeking for compensating each other: an increasing of
the induced currents’ density and an appearance of the conductivity’s destruction. In
the real experiment it’s a levitation magnet’s altitude which assures the equilibrium
between the gravity force and the external applied force. In the numerical experiment, carried out to verify an applicability of the considered models, the role of
this adjustable parameter is played by the critical current value. In case of its right
choice, the calculated values of magnetic repulsion forces must be equal.
Conclusions
The computational modelling of the experiment showed, the “abrupt collapse
model” is more applicable for description of the lifting force hysteresis than the
Bean’s one. By means of above described method the critical current value (in
conventional units) was calculated in the network of the first model. Further improvements of the model could be made by more precise consideration of a magnet
field configuration or by creation of the hybrid model on the base of described
ones. Evidently it’s necessary to create much precise and perfect device for the
experiment’s implementation.
It should be added that above described effects was observed only on the one of
high-superconductive ceramics. On the others one can observe the trivial altitude
dependence on the applied force: under small deviations from magnet’s equilibrium
a
b
Fig. 4. Results of ceramics surface studies. a – the sample, demonstrating hysteresis effects, b – the trivial sample.
position weak hesitations appear, and no hysteresis effects are found out. That’s
why the comparative analysis of two superconductive samples has was carried out
on optic microscope. The results are represented in the Fig. 4. As one can notice,
a size of microgrits composing the “hysteresis” sample is much greater than one
of the trivial ceramics.
71
References
1. Delft D., Kes P. The discovery of superconductivity // Physics Today. September
2010, pp. 38-42.
2. Гинзбург В. Л., Ландау Л.Д. // ЖЭТФ, 1950, Т. 20, С. 1064.
3. Марек В.П., Чирцов А.С. Использование возможностей мультимедиа и
компьютерного моделирования для организации самостоятельной работы
студентов и их подготовки к работам физ. практикумов // В сб. «Материалы
Х Межд. Конф. «Физика в системе современного образования» (ФССО-09)»,
31 мая- 4 июня 2009 г., СПб, 2009, Т. 2, С. 193-195.
4. Марек В.П. Использование возможностей мультимедиа и компьютерного
моделирования для создания иллюстраций по курсу общей физики // В сб.
«Тезисы докладов молодежной научной конференции «Физика и прогресс»,
18-20 ноября 2009 г, Санкт-Петербург, Россия, С. 137.
5. Bean C.P. // Rev. Mod. Phys. 1964. V. 36. N 1. P. 31-36.
72
Usage of stereoscopic 3D-visualization technologies
Marek Veronika
nika.marek@mail.ru
Scientific supervisor: Dr. Chirtsov A.S., Department of
General Physics - I, Faculty of Physics, Saint-Petersburg State
University
Introduction
The three-dimensional (3D) computer models are greatly used in up-to-date
educational process, especially in scientific areas, by reason of their proximity to
real experiments. However these 3D objects are rather complicated to be apprehended by means of its plane 2D projections. This problem was partially solved
due to the possibility of rotating the model relative to the plane of the monitor. But
today another solution is proposed by 3D stereoscopic images and virtual reality
technologies which are becoming more and more popular at present [1]. Until
recently principal products in this area were dedicated to the creation of complex
expensive virtual reality systems. Currently a great number of simple rather cheap
methods of stereoscopic computer images making and also a lot of software for
its creation and demonstration exist. All of this actually allows using these new
technologies in education, but evidently only in those cases when it gives irrefutable advantages in comparison with traditional forms of studying.
Main methods of stereoscopic images creation are known since the XIX century:
for this it is necessary to project two pictures recorded by two spatially separated
observers on the retinas of both eyes. This procedure can be fulfilled in different
ways. One of the simplest and best-known is the anaglyphic method. It’s based on
the chromatic selection of images for each eye and uses glasses with color filters
for the images’ viewing. Essential disadvantages of this method are low quality of
the color rendering and easy fatigability of the onlooker. Therefore the anaglyphic
method isn’t suitable for the studying process but however it can be used for a
rapid assessment of the scene “voluminosity”.
More suitable for educational purposes is the polarization method. Here images
for stereomates are created by light beams with different polarizations; the division
of images is made with a help of polarization glasses.
Equipment
A setup for computer stereo visualizations was worked out (Fig. 1). It includes
two multimedia projectors ACER PD520 (based on DLP technology) on movable
stands, linear polarizers with perpendicular polarization planes and a screen. An
adjustment of projectors’ position was made by means of the calibrating images.
Note also that the screen has to save the polarization in the scattered light. Special
screens with metallic sputtering have this property but they rather expensive, that’s
73
a
b
Fig. 1. A principal scheme (a) and a photo (b)
of the developed equipment for the stereovisualisations.
why for this first test setup the screen was made
of means at hand with a mat aluminic foil on the
surface.
For viewing stereo images a polarized glasses with a perpendicular polarization for each eye was utilized. For stereo images and clips playback Stereoscopic
Player [2] was used.
Results
Let’s consider several three-dimensional objects obtained by a computer modelling which are actually complex for the two-dimensional demonstration. They
are for example a toroidal coil with a low number of turns or a problem about a
magnetic field in a system “grounded sphere + electric charge” which is solved by
the image method. In the first case, it’s better to observe the coil’s field at minimum
from two different angles, in the second one, several elements outside a screen’s
plane have to be eliminated for reducing image’s complexity. The usage of 3D technologies allows the solution of mentioned problems. It’s a collection of interactive
3D-models created in the special programs - “construction sets”, which was a base
for first experiments on the new technology [3]. The models the most difficult for
visualisation have been chosen, for instance “Beams motion in slightly misaligned
a
b
Fig. 2. Examples of the stereo images (anaglyphic method) based on results of the
computer modelling: a – “Beams motion in slightly misaligned resonator”, b –
“Compound objective”.
resonator”, “Magnetic field of a constant magnet in the presence of superconductor”, “Absolute optical system”, “Compound objective” etc (Fig. 2).
Another tested method for stereo demonstrations creation is implemented by
means of Autosdesk 3ds Max software [4].
74
a
b
Fig. 3. The vector model of one-electron atom in external magnetic field (fine and
hyperfine-structure splitting are taken into account): a – video clip’s frame,b –
anaglyphic variant.
Thereby a number of clips for the course
“Quantum theory of atomic and molecular spectra” [5] have been created (Fig. 3). These clips
visualize the vector model of an atom and different
approximations for atomic and molecular static
states computation. According to the simplest one,
an electron moves in the elliptical quasi-classical
orbit round of the nuclear and the orbit is oriented
orthogonally to the electron’s angular momentum l.
Then the spin-orbit coupling, the nuclear spin and
the external magnetic field (leads to Zeeman effect) are added.
The next step in stereo technologies usage is
the three-dimensional projection of the four-dimensional Minkowski space-time. An educational
video clip on this topic was worked out (by means
of Autodesk 3ds Max). It demonstrates the light
cone and also one- and two-sheet 4D-hyperboloids
whose����������������������������������������������
���������������������������������������������
lateral surface is set by the following relativistically invariant expressions:
c 2t 2 − r 2 = 0
c 2 t 2 − r 2 = ± A2
where A is a constant.
Fig. 4. Different variants of light cone interpretation: a – traditionally used 2D-projection; b, c –
anaglific 3D-projections; d – a frame from stereo
clip’s about Minkowski space-time structure.
75
a
b
c
d
The upper part of two-sheet 4D-hyperboloid corresponds to the space of 4Dvelocities. The sections of mentioned quadric surfaces by planes ct = constant in
the 4D-space are 3D-spheres.
The 3D projection of the 4D-dimensional Minkowski space-time is a set of
parallel 3D planes ct = const situated perpendicular to the time axis. Such presenta-
a
b
Fig. 5. Video clip’s frames: a – light cone plotting; b – 4D-velocity space.
tion is evidently better than a traditional image of the light cone (Figs. 4, 5). The
carried out video clip was recorded on virtual cameras imitating right and left eyes
of an observer that allowed create a stereo clip.
References
1. Андреев C.В., Денисов Е.Ю., Кириллов Н.Е. // Программные продукты и
системы, № 3, 2007, с. 37-40.
2. Stereoscopic Player [Электронный ресурс] / 3dtv.at website [сайт]. [2005 2011]. URL: http://www.3dtv.at/Index_en.aspx.
3. Колинько К.П., Никольский Д.Ю., Чирцов А.С. Многофункциональный
компьютерный учебник по фундаментальному курсу физики. Разделы:
“Движение частиц в силовых полях”, “Релятивистская динамика”,
“Геометрическая оптика” // В сборнике трудов IV Международной
конференции “Физика в системе современного образования”, Волгоград,
15-19 сентября 1997 г.
4. Autodesk 3ds Max Products. [Электронный ресурс] / Autodesk: [сайт].
[2010]. URL: http://usa.autodesk.com/3ds-max/.
5. Чирцов А.С. Конспект лекций по курсу «Атомные спектры» (ч. 1). – СПб:
«Соло», 2007.
76
Evaluation of the influence of readout cables in the CBM
Silicon Tracking System
Prokofyev Nikita
cryodrink@ya.ru
Scientific supervisor: Dr. Kondratyev V.P., Department of Nuclear
Physics, Faculty of Physics, Saint-Petersburg State University
Introduction
The Compressed Baryonic Matter (CBM) experiment is designed to explore
the QCD phase diagram in the region of high net-baryon densities. The Silicon
Tracking System (STS) is the central detector to perform charged-particle tracking
and high-resolution momentum measurement. It consists of eight planar stations
of about 3.5 m2 total active area. Each station combined from ladders - units of
10 sectors that are planned to be assembled in the lab and then mounted into the
experimental setup. A sector may consist of a individual double-sided silicon microstrip sensor or of two or three conjoined sensors. The thickness of every sensor is
300 μm. The signal is being read out via low-mass cables of up to 50 cm length in
order to keep the active area of the detector free of electronics. The cables consist of
micro-line-structured aluminum layers on polyimide carrier foils. The cables have a
multi-layer structure: two layers of aluminum wires, shielding, spacers and carriers
with a total thickness of 348 μm. At the first look, the thicknesses of cables and
sensors are of comparable values, and it is difficult to say a priori, how significant
the cable presence in the active area is. Even if cable material is more transparent
for particles than sensor material - silicon. However, in a tracking station several
layers of cables will be stacked and increase the material budget. To ensure that
cables are transparent enough and do not cause significant noise, simulations with
realistic STS models are being performed as described below.
Description of the simulation models
The first STS station is shown in Fig. 1 (top view). Each ten sectors by vertical
combined to ladder. Other stations contain more sensors, and different types of
ladders, but first station is the simplest to describe simulation models. Two models
were used as "starting points" to compare simulation and reconstruction results.
Both are standard STS geometries, containing only sensors. First is the model with
STS geometry sts_v11a.geo (v11a). Based on this geometry is the Realistic Cable
Model (RCM), that contains cables, described as 9-layer boxes with overall thickness of 348 μm. This model produced from the exact drawings. Detailed description of cables, including exact drawings and radiation lengths is available here [1].
Second model to refer results is the sts_v11b.geo (v11b). Based on this geometry
are the Silicon-Equivalent Model (SEM), and several models for reconstruction
performance studies (not taking into account the cables): with sensor thickness
400 μm each, 500, 600 and 800 μm. Significant difference between the RCM and
77
the SEM is that SEM have no special GEANT volumes for cables. Instead of this,
thicknesses of sensors were increased at the corresponding value.
Fig. 1. First STS station, view in the direction of the beam. This picture looks same
for all models described. Dimensions: 50 x 30 cm. The dashed circle corresponds
to 25º geometrical acceptance, the solid ellipse to a horizontally extended acceptance in order to provide efficient reconstruction of low momentum electrons.
Fig. 3 Standard STS geometry v11b,
first station side view, beam along
the Z axis. Units of both scales are
in cm.
Fig. 2. Standard STS geometry v11a, first
station side view, beam along the Z axis.
Units of both scales are in cm.
One cable has same thickness to radiation length ratio as 116 μm of silicon. As
we can see in Fig. 4, first line of sensors has initial thickness (300 μm), next line
has thickness increased on 232 μm (overall, 532 μm = sensor and 2 cables), following lines: 764, 996 and 1228 μm In the terms of thickness divided by radiation
length, it is 0.32%, 0.57%, 0.82%, 1.05% and 1.31%. The thickness map for the
first STS station is shown in Fig. 5.
Differences between sts_v11a.geo and sts_v11b.geo are significant from the
geometrical point of view. Nevertheless, the reconstruction results are nearly the
same, that is why we can compare RCM and SEM and do not take into account
the fact that these models have different positions of sensors.
Realistic Cable Model
In principle, the realistic model has some advantages. First of all, the drawings
have realistic look: every geometrical volume have prototype (see Fig. 6). Secondly,
78
description of model allows easily change such parameters as cable size, thickness
and media. Also, we may suppose that such description looks realistic for GEANT
and expect trusted results. On the other hand, in that model we have over 9000
GEANT volumes instead of about 1000. This slows down simulations significantly.
Another problem is Kalman Filter (KF). KF is used at the reconstruction stage to
Fig. 6. Realistic Cable Model, first station side view. Units of both scales are in cm.
decrease noise. Current algorithm of KF does not take into account any media except
sensors, which, moreover, should be similar in thickness. This fact led us to concept
of Silicon-Equivalent Model and also we realized the necessity of simulations with
models that contains sensors of similar thickness (more thick than standard). Such
models have been created and studied, results are presented below.
Main parameters to evaluate cable influence is reconstruction efficiency at
1 GeV, power of the K0s and Λ0 peaks and signal to noise ratio for these peaks.
Obviously, cable presence in the active volume of the detector leads to signal suppression and noise increase. Graphically it is shown in Fig. 7. Other reconstruction
characteristics are compared in Table 1. Power of the peak is measured in relative
units due to the fact that all reconstructions have equal number of Monte-Carlo
events and calculated as integral under the peak with eliminated background.
Peak power (K0s)
± 0.03
S/N (K0s)
± 4%
Peak power (Λ0)
± 0.03
S/N (Λ0)
± 4%
v11a
RCM
v11b
SEM
1.22
0.95
1.22
1.08
0.56
0.16
0.57
0.12
1.54
1.22
1.54
1.07
1.38
0.51
1.37
0.32
Table 1. Reconstruction parameters for models with and without cables.
79
Silicon-Equivalent Model
The main idea of the SEM is to use only silicon sensors. Such model contains
about 1000 GEANT volumes. Unfortunately, SEM is still working improperly
with Kalman Filter. But it seems much easier to adjust KF to work with sensors of
different thicknesses than with passive cable volumes. Nowadays, reconstruction
results with the SEM is nearly the same as with the RCM, that shown in Table 1.
Fig. 8 shows signal suppression and background increase in the SEM comparing
to the standard value.
Fig. 7. Suppression of the Λ0 peak by
cable presence (Realistic Cable Model).
Red filled spectrum is the result without
cables (v11a).
Fig. 8. Suppression of the Λ0 peak in the
Silicon-Equivalent Model. Red filled
spectrum corresponds to the standard
result (v11b).
Simulations with different sensor thickness
After the SEM, next logical step is to perform simulations with sensors of
equal thickness, but thicker than standard. Main feature in that case is the correct
Fig. 9. The Λ0 peak and increasing background for the different sensor thickness:
300 μm (blue line), 400 μm (magenta dashed), 500 μm (green dashed), 600 μm
(black) and 800 μm (red line).
80
work of Kalman Filter. Several thickness values were studied: standard 300, and
thicker 400, 500, 600 and 800 μm. Results for 600 μm are especially important,
because one of the possibilities to the future CBM experiment is to use back-to-back
single-sided micro-strip silicon sensors with thickness of 300 μm each. Increasing
the sensor thickness leads to increase of the background, graphically it is shown
in Fig. 9. Another reconstruction results is presented in Table 2. As we can see,
the ghost probability grows at the factor of 10 from 300 to 800 μm. The _0 peak
is not changing by absolute value but background increases and power of peak
decreases. The S/N rate has significant reduction at 600 μm. Also we should keep
in mind that this is simulations without any cable material.
Sensor thickness
300 μm
400 μm
500 μm
600 μm
800 μm
Ghost probability
0.013
0.022
0.035
0.055
0.116
Rec. efficiency at 1
GeV (±0.5%)
0.96
0.95
0.95
0.93
0.93
Ppeak(Λ0) (±0.03)
1.54
1.58
1.49
1.20
0.79
S/N (Λ0) (±4%)
1.38
0.92
0.62
0.34
0.16
Table 2. Reconstruction parameters for different sensor thickness.
Summary
Based on the data obtained are the following conclusions. The Realistic Cable
Model and the Silicon-Equivalent Model provide roughly the same results. The
S/N ratio for Λ0 is better in the RCM, but the SEM may be improved by adjusting
Kalman Filter. Nevertheless, both the RCM and the SEM provides passable results.
The usefulness of back-to-back single-sided micro-strip sensors for reaction Au+Au
at 25 GeV is being doubted by the reconstruction results with “thick” sensors.
References
1. Heuser J. M. et al. // CBM Progress Report 2010, p. 14 (2011).
81
Application of graph theory to modeling of the complex
hydraulic systems
Strizhenko Olga
bzixilu@gmail.com
Scientific supervisor: Prof. Dr. Slavyanov S.U., Department of
Computational Physics, Faculty of Physics, Saint-Petersburg
State University
Introduction
This work covers the actual problem of design and maintenance of complex
hydraulic systems (or pipeline network), which play very important role in the life
of mankind. People use given systems for oil and gas transportation, water and
heat supply in the cities, irrigation etc.
Hydraulic system is defined as a set of different facilities (pump stations, pressure regulators, shutters etc.) and connecting them pipelines, closed or open channels involved in the transportation of compressible and incompressible fluids (water,
oil, gas etc.) [1]. Hydraulic systems are a well known example of a complex and
large scale distributed parameter system. By this reason the modeling approaches,
numerical methods and optimization of operating modes of fluid transport networks
are of permanent interest for researchers and engineers who create more and more
perfect simulators (OLGA [2], PipeSim [3], Stoner etc). But all these simulators
have common concepts of graph representation of hydraulic network, which appeared several decades ago at the inception of the theory of hydraulic circuits.
In this work author described common concepts of graph representation of
pipeline network, supplement topology matrices of graph model to formulate
Kirchhoff’s laws for flow distribution problem solving, structure of the hydraulic
simulator prototype «Pipeline Network» and extension of the prototype for the
tasks of leak detection as well.
Graph Representation of Pipeline Network
For the beginning we need to understand what is pipeline network. Pipeline
network is defined as a set of diffrerent facilities (pump stations, pressure regulators, shutters and etc.) and connecting them pipelines, closed or open channels
involved in the transportation of compressible and incompressible fluids (water,
oil, gas and etc.).
It should be noted that hydraulic network contains points of branch and connection of pipes, and additional elements. For example if we discuss leak detection
problem we should consider detectors as aditional element of pipeline network
which is located in specific point in the pipeline or can move along pipeline over
time that depends on particular type of detector used in the network to gather data
(temperature, pressure, rate) about current state of system or a single pipeline as
well. Using data from detectors we can identify leakage.
82
Let’s consider the graph representation of the basic elements of pipeline network (Fig. 1 (a)).
Edges represent straight sections of pipelines, which have concrete diameter,
length, hydraulic resistance and another properties.
Nodes are elements that have a dual functionality:
1) Node describes changes of geometry and properties of the hydraulic network like turn, branch, connection of different types of pipeline and hold local
resistances, corresponding to given changes.
2) Node is a holder of facilities like pumps, pressure regulators and so on, which
affect a flow rate, pressure and resistance in the hydraulic system.
Detectors can be represented as external for graph nodes with known values
of pressure, flow rate or temperature. Data from these points can be interpreted
as boundary conditions for the flow distribution problem or as control values for
alarm generation in the leak detection system (Fig. 1 (b)).
(a) (b)
Fig. 1. (a) –Graph representation of hydraulic network; (b) – Leak detection system.
Formulation of hydraulic Kirchhoff’s laws
Modeling of flow distribution in the hydraulic system (finding of pressure in
the all nodes and rates in the all edges in the network) is used to design pipeline
network optimally or predict a behavior of pipeline network after changes in the
topology of network, after adding new facility to consideration and so on. This
modeling is based on two Kirchhoff’s hydraulic laws which can be easily described
by incidence and loop matrices of pipeline graph model:
The first Kirchhoff’s law, describing local mass conservation law of fluid flow
for each node can be easily represented by using incidence matrix:
(1)
Ax = Q where A – incidence matrix of graph, x – flow rate vector, Q – vector of additional
flow from outside the system.
The second Kirchhoff’s law, which can be interpreted as energy conservation
law in the loop of the network, is represented by using loop matrix:
By = 0 (2)
where B – loop matrix of graph, y – drop pressure vector.
83
Law of the state flow in the pipeline is:
(3)
y + H = SXx where H – push vector, S – matrix of pipeline hydraulic resistance, X – diagonal
matrix with elements ǀxiiǀβ-1, β – some coefficient.
Generally speaking parameters S, H, Q in the equations are not constants and
depend on current flow distribution in the system:
(4)
S = S ( x, p ), H = H ( x, p ), Q = Q( x, p ) Finally we get a complete system of non-linear equations:
 Ax = Q
(5)

 By = 0
 y + H = SXx

Solving this system we can find flow distribution in the pipeline network.
The development of leak detection model
The first stage in these research activities is the model development of leak
problem in the oil pipeline. The model will be developed by taking into consideration that a leak is treated as an outlet segment, in a part of pipeline network.
The outlet segment has certain diameter and very short length. In the model, the
diameter of the outlet segment will represent the size of leak, where the diameter
could be adjusted to represent the leak opening. The length of outlet segment is
taken short; short enough to make the pressure drop effect along the short segment
is negligible, compared with the pressure drop along the main line [4].
Leak detection techniques
Pipeline failure can be caused by corrosion and wear, intentional damage, unintentional damage or operation outside the design boundaries. When leak occurs at
any point in a pipeline, there is a sudden change in pressure, flow, etc. characteristics
[5]. Analyzing data from sensors about current state of pipeline, we can detect a
leakage. In this work only two leak detection techniques were used: Volume Balance
Method (VBM) and Pressure Point Analysis Method (PPA) (Fig. 1 (b)).
First method is based on the principle of conservation of mass. For a pipeline the
flow entering and leaving the pipe can be measured. The mass of the fluid can be
estimated from the dimensions of the pipe and by measuring process variables like
volumetric flow rate, pressure and temperature. When the mass of the fluid exiting
from the pipe section is less than estimated mass, a leak is determined [6].
The principle behind the operation of single point analysis is that the pressure in
a pipeline will decline as result of a leak. Further, certain statistical properties can
be computed to determine if a pressure is declining in a significant manner. PPA
requires that all events other than leaks that may cause a pressure to decline, such
as operational changes to the pipeline, must be identified so that the leak detection can be inhibited until such time as the pipeline returns to a steady operation.
The technique operates as follows. A buffer of he most recent measured values of
84
pressure is kept for analysis. The data is divided into two periods and the mean
and variance of two samples are computed.
µ old − µ new
n n (n + nnew − 2)
t0 = old new old
(6)
2
nold + nnew
(nold − 1)σ 2new + (nnew −1)σ old
where μ – is the sample mean of the data, σ2 – is the sample variance of the data,
n – number of points in the sample.
The result of equation (6) is an observed value of a random variable, which
has a Student’s t-distribution with n1 +n2 -2 degrees of freedom. Using the value
computed in (6) and number of freedom the significance level of the test can be
found from a table of Student’s distribution. This is taken as probability of a leak
being present [7]
Results and discussions
On the course of research the simulator prototype with following functionalities
has been developed (Fig. 2):
Fig. 2. Graphical User Interface.
Graphical editor of hydraulic model
• Flow distribution
• Leakage detection (Fig. 3)
85
(a)
(b)
Fig. 3. (a) –Pressure log over time, red line – time of leak occurrence; (b) –
Probability of leak over time (PPA output).
References
1. Merenkov A., Hasilev V. Theory of hydraulic circuits.- Moscow: Nauka, 1985
– 276 p (in Russian).
2. Bendiksen K.H., Malnes D., Moe R., Nuland S. The Dynamic Two-Fluid
Model OLGA: Theory and Application. SPE Production Engineering, May 1991,
pp. 171-180.
3. José M. Chaves-González, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido,
Juan M. Sánchez-Pérez. PipeSim: Pipeline-Scheduling Simulator // 8th International
Symposium on Computers in Education (SIIE'2006), pp. 109-116. León, Spain,
October 2006.
4. Pudjo Sukarno, Kuntjoro Adji Sidarto, Amoranto Trisnobudi, Delint Ira
Setyoadi, Nancy Rohani & Darmadi. // J. Eng. Sci. Vol. 39 B, No. 1, 2007.
5. Olunloyo V.O. S., Ajofoyinbo A. M. A model for real time leakage detection in pipelines: A case of an integrated GPS receiver // Proceedings of the 3rd
International Conference on Applied Mathematics, Simulation, Modeling, 2009.
6. Stuart L. Scott, Maria A. Barrufet. Worldwide Assessment of Industry Leak
Detection Capabilities for Single & Multiphase Pipelines, OTRC Library Number:
8/03A120, 2003.
7. Whaley R.S., Nicholas R.E., Van Reet J.D. Tutorial on software based leak
detection techniques, Pipeline Simulation Interest Group, 1992.
86
F. Optics and Spectroscopy
A modern implementation of Rozhdestvenski
interferometer
Agishev1 N.A., Medvedeva2 T.A., Ryabchikov1 E.L.
kolyan_a-ev@mail.ru
1
Faculty of Physics, Saint-Petersburg State University, Russia.
2
State educational institution lyceum №419 Petrodvorets, SaintPetersburg, Russia.
Scientific supervisor: Ass. Prof. Anisimov Yu.I., Department of
General Physics-1, Faculty of Physics, Saint-Petersburg State
University
Introduction
Historically, Rozhdestvenski interferometer was used to study the anomalous
dispersion in the material. Researches of anomalous dispersion, performed at the
beginning of last century, gave a significant contribution to the development of
quantum mechanics and helped to build a reliable model of the atomic nucleus.
Various data can be derived from the interference pattern with the absorption
lines such as: the concentration of atoms in the vapor substance, the oscillator
strengths of atoms. Moreover for reliable measurements it is needed high quality
of the image. In his works Rozhdestvenski was faced with several challenges: the
photographic method of recording the spectrum required bright light source, which
led to uneven heating of the mirrors of the interferometer and its misalignment.
In the present work a number of modern methods for reception of the better
image of an interferential picture are offered: use of sources of intensive radiation
in a narrow spectral range (light-emitting diodes), and application of systems of
registration of a picture on the basis of CCD-matrixes of the high resolution is
considered.
Rozhdestvenski interferometer coupled with spectrograph is used in researching discharge tubes produced by EDD technology [1]. They may be promising in
laser researches, for example for creating new active media.
The purpose of this study was to modernize the "classical" experimental assembly with a Rozhdestvenski interferometer, coupled with a spectrograph, for use
in further experiments by the definition of non-stationary concentrations of vapors
of chemical substances in the discharge tubes produced by EDD technology.
Results and Discussion
Modernization of the experimental assembly has touched sources of light and
photodetectors. Scheme of the installation is presented in Fig. 1. LED emitted (S1)
beam of light gets to an interferometer through collimator L1. Rozhdestvenski
interferometer consisted of semitransparent mirrors A1, A2 and complete reflective
mirrors B1, B2. Optical cell with vapours of investigated substance was placed
in one of interferometer arms. The parallel-sided plate k2 was situated in other
interferometer arm. Plate k was used to adjust the interferometer. At the output of
88
the interferometer the light beam passed the lens L2 and entered the spectrograph.
Interference fringes decomposed in spectrum was registered by photodetector CCD.
Fig. 1. Scheme of the installation.
By changing photographic plates to CCD-matrixes of modern cameras processing of the image of hooks had been essentially simplified, but spectral range had
been limited. Work is underway on the use of CCDs in the UV range. The resonance
transitions of most studied substances contained in this range.
Powerful (10–20 A) spectral lamps of high and average pressure and the highvoltage discharge in a porcelain capillary which was earlier used as a sources of
optical radiation in Rozhdestvenski interferometer has been replaced by modern
light-emitting diodes. Their advantages are obvious. Above all this is ease of use
in continues and pulsed modes and the almost complete absence of electrical interference. The spectral characteristics of LEDs were studied by use of specially
designed automatical monochromator coupled with a PC. The results of these
measurements are presented in Fig. 2.
The spectral range of used LEDs covers the range from near UV to near IR
range. Research into the LEDs’ response speed showed the possibility of creation
of experimental installation with a time resolution of 0.2–0.5 microseconds.
This is 10 times less then registration time of known "classic" schemes based on
Rozhdestvenski interferometer.
89
The used Rozhdestvenski interferometer has the great historical heritage. This
optical device has half a century ago received a silver medal at an exhibition in
Brussels. The mechanical and optical quality of the device is that it took only minor
adjustments to use it nowadays.
Intensity, relative units
1,0
0,8
0,6
0,4
0,2
0,0
400
500
600
λ,
700
Fig. 2. Spectral characteristics of light-emitting diodes.
To demonstrate the working capability of the experimental assembly
Rozhdestvenski hooks on the sodium doublet were obtained. This experimental
demonstration became possible by assistance of staff of the Departments of Optics
and General Physics I.
Fig. 3. Hooks in the sodium doublet. A thickness of the plate k = 2.75 mm.
Fig. 4. Hooks in the sodium doublet. A thickness of the plate k = 9.12 mm.
Fig. 5. Hooks in the sodium doublet. A thickness of the plate k = 16.40 mm.
90
Specially designed sodium cell filled with helium under the pressure of 76 Torr
was used as a vapor source. The cell was heated up by electric oven to a temperature of 450 K. Using the dependence of the saturated sodium vapor’s pressure on
temperature, one can estimate the concentration of sodium vapors. For a given
temperature the vapor concentration was 5.3±0.4·1012 cm-3. A white LED working
in continuous mode was used as light source. The LED’s voltage at 100 mA current
was 3 V. Interferogram registration was made by digital camera (Olympus Sp-350)
with the 8 Mpx CCD. The obtained images are shown in Figs. 3-5.
Conclusions
There was obtained a value of (5.5±0.7)·1012 cm-3 of atomic concentrations of sodium vapors on basis of the interference pattern. Value calculated from dependence
of the saturated sodium vapor’s pressure on temperature was (5.3±0.4)·1012 cm-3.
The both values had the same order.
The ultimate goal of our researches based on Rozhdestvenski interferometer
coupled with spectrograph is creation of experimental assembly operated in pulsed
mode with high time resolution (0.5 microseconds). This will allow to measure
non-stationary concentration of vapors of chemical substances in the discharge
tubes produced by EDD technology by Rozhdestvenski hooks.
References
1. Anisimov Yu.I., Mashek A.Ch., Metel’skii K.E., Ryabchikov E.L. // Optics and
Spectroscopy, 2009, Vol. 107, No. 3, pp. 368–370.
91
Investigation of the two-photon induced fluorescence
in Rb vapor excited by Ti:Sapphire femtosecond laser
pulses
Bondarchik Julia
bonnejuliette@gmail.com
Scientific supervisor: Prof. Dr. Pastor A.A., Department of Optics,
Faculty of Physics, Saint-Petersburg State University
Introduction
Nowadays the tendency is that the 21st century will likely be known as the
century of the photon. Optics is one of the key technologies in the 21st century
because today photonic technologies replace traditional ones. It is connected with
the advantages of optics, such as speed, lack of interactions, massive parallelism
and small energy consumption. On the other hand it has some cons, for instance
an all-optical transistor does not exist yet. The solution will appear with using
nonlinear optical materials [1].
Therefore, it is important to develop this chapter of optics devoted to nonlinear
processes. With reference to the earlier research [2], we would like to investigate
coherent two-photon interaction of femtosecond laser radiation and resonant medium in rubidium vapor. The basic idea is to excite rubidium atoms from ground
5S state to state 5D through an intermediate 5P state by means of laser pulses with
a wavelength 790 nm, generated by the Ti:Sapphire coherent femtosecond laser.
By variation of pulses time delay and changing the input wavelengths it might be
possible to observe different temporal dynamic.
In this article special attention will
be paid to the experiment’s background,
specifically theoretical overview and
preparation for experimental setup.
General concepts and definitions
As far as we are going to investigate
two-photon absorption it is necessary to
give a definition for this process, namely:
two-photon absorption (TPA) is the simultaneous absorption of two photons in order
to excite a molecule from the ground state
to a higher energy electronic state [3]. For
TPA identical as well as different frequencies can be used, in our experiment two
identical laser pulses were used. Hence,
the sum of the energies of the two photons
Fig. 1. Rubidium energy level scheme. has to be equal to the energy difference
92
between the involved ground (5S) and upper (5D) states of the molecule. According
to the theory, excited rubidium atoms emit lines with wavelength 5 µm (5D-6P)
and 420 nm (6P-5S). Rubidium energy level scheme is presented in Fig. 1.
Two-photon absorption occurs only in nonlinear optical molecules and even
then only at high light intensity. Thus, only with the invention of lasers (Light
Amplification by Stimulated Emission of Radiation) it became possible to observe
nonlinear processes. The essence of the laser is the gain – the factor by which an
input beam is amplified by a medium. A laser can be classified by different types
of gain medium (gas, liquid, solid or plasma) and each of them is used for different purposes.
Experimental setup
In our experiment for the excitation of the pulses Ti:Sapphire femtosecond laser
“Pulsar 10” designed by Amplitude Technologies company was used. The Pulsar
laser system is a compact femtosecond laser source providing more than 10 mJ
pulse energy at 10 Hz repetition rate. The pulse duration shorter than 45 fs leads
to a high peak power. The system is a Ti:Sapphire laser based on the so-called
“Chirped Pulse Amplification” (CPA) scheme. The system amplifies pulses from a
Ti: Sapphire oscillator and consists of a stretcher, a regenerative amplifier, as well
as multiphase amplifiers with respective pump lasers and a compressor.
For our research the cell containing rubidium was made from fused silica and
thereby allows the heating of the vapor in the range of 300 – 500 К. Visible fluorescence at a wavelength of 420 nm was obtained and fixed by computer-controlled
spectrograph Ocean Optics SD 2000. The schematic diagram of the experiment
is presented in Fig. 2.
Fig. 2. Schematic diagram of the experiment.
As a light source Ti:Sapphire femtosecond laser was used; then, cell containing rubidium vapor under investigation was located between focusing optics.
Consequently, output signal from substance under study was registered by computer-controlled spectrograph and indicated occurring nonlinear process, which
can be explained as two-photon transition between ground 5S and upper 5D states
in rubidium vapor.
Theoretical model
Nonlinear optics is the utilization of the fact that the when the electric field
of the light wave is sufficiently high, the induced polarization in a medium is not
linearly proportional to the electric field, but depends on its higher power as well.
The application of an electric field E to a dielectric material causes the constituent
atoms and molecules to become polarized. The medium responds to the field E by
developing a polarization P which represents the net induced dipole moment per
93
unit volume. In a linear dielectric medium the induced polarization P is proportional
to the electric field E at that point, and they are related by
P = ε 0 χE ,
where χ is the electric susceptibility. However, the linearity breakdown at high
fields and the P vs. E behavior deviates from the linear relationship.
As soon as P becomes a function of E, we can expand it in terms of increasing
powers of the electric field E. It is customary to represent the induced polarization as:
P = ε 0 χ1 E + ε 0 χ 2 E 2 + ε 0 χ3 E 3 ,
where χ1, χ2, and χ3 are the linear, second-order and third-order susceptibilities. The
coefficients decrease rapidly for higher terms and are not shown in the above equation. The importance of the second and third terms, i.e. nonlinear effects, depend
on the field strength E. Non-linear effects begin to become observable when fields
are very large, that invariably require lasers.
In our research special attention goes to the nonlinear third-order susceptibility,
since it consists of two parts: real and imaginary. Real part of third-order susceptibility is responsible for third-harmonic generation, stimulated Raman scattering
and etc, while imaginary part describes two-photon absorption [4].
Thus, in case under review it is necessary to consider third-order nonlinear
optical processes, which is a phenomenon due to an induced material polarization
that is proportional to the third power of the electric field. In this section we present
a theoretical model to fit the present experiment. Firstly, it is important to review
semi classical approximation for nonlinear effects.
According to quantum mechanics, electron movement inside the atom we can
describe with wave function which constitutes from Schrödinger’s equation:
∂ Ψ
i
= Hˆ Ψ ;
∂t
and besides using perturbation theory we can assume
Hˆ = Hˆ 0 + Hˆ '(t ) ,
where Ĥ0 is defined by interaction between the electron and the nucleus, whereas
Ĥ' – by interaction of the electron with the external field.
In case that the interaction energy of dipole with the field in the dipole
approximation is defined by multiplication of dipole moment μ by field intensity
we consider:

Hˆ '(t ) = −µˆ E (t ) .
The basic reason why we use dipole approximation is that at constant external
field reacts only dipole, while quadrupole responds to field gradient. As far as in
the visible and ultraviolet range the wavelength of the light is much bigger than
the characteristic dimensions of the atom, we can assume that at each period of
time the atom is inside an almost constant field.
For Ĥ =Ĥ0 solution of Schrödinger equation is described by hydrogen atom:
Hˆ 0 i = ε i i .
94
So any function can be presents as:
∞
Ψ = ∑ ai i where ai = i Ψ
i =1
And besides for Ψ=Ψ(t) → ai=ai(t).
Since the normalization of wave function, any change its in time constitutes as
rotation. As any operator can be expressed as
Aˆ = ∑ Aij i j , Aij = i Aˆ j ij
solution of Schrödinger’s equation will look like:
 i t
 Ψ(t ) = Ψ(t0 ) ⋅ exp  − ∫ Hˆ (t )dt  .
  t0

 i

For Hˆ = Hˆ
Ψ(t ) = Ψ(t0 ) ⋅ exp  − Hˆ 0 (t − t0 ) .
0
 

We should take into consideration that by definition the evolution operator of
the system is
Ψ(t ) = Uˆ (t , t0 ) ⋅ Ψ(t0 ) .
Foregoing result was obtained in Schrödinger’s conception. The next step is
to represent wave function in Heisenberg’s conception, where this function does
not depend on time:
Ψu = Uˆ −1 ⋅ Ψi (t ) = Uˆ −1 ⋅ Uˆ ⋅ Ψ(t0 ) = Ψ(t0 ) .
Since the mean values of the physical quantities should be equal for different
representations of wave function, it is possible to assume that
Aˆ = Ψi Aˆi Ψi = Ψu Aˆu Ψu ,
and moreover
Aˆu (t ) = Uˆ −1 ⋅ Aˆi (t ) ⋅ Uˆ .
Consequently we obtain that the time evolution operator describes the
transition from one representation to another [5].
Conclusion
Thus, observed visible fluorescence at 420 nm should be explained as a result
of two-photon excitation of rubidium atoms from the ground state 5S to the excited
5D state through the intermediate quasiresonant 5P state. The process mentioned
above can be associated with two various options of two-photon transition: through
state 5P1/2 or state 5P3/2.
Achieved results will be used in our further experiments devoted to coherent
excitation of the atomic transition in rubidium by femtosecond pulses of laser
radiation at the wavelength of 790 nm.
References
1. Vartiainen E. Class lecture, Topic: Applied Optics. Faculty of Technology,
Lappeenranta University of Technology, Lappeenranta, Finland, 2011.
95
2. Ariunbold G.O., Sautenkov V.A., Scully M.O. // Journal of the Optical Society
of America – Optical Physics, Vol. 28, No. 3, pp. 462-467, Mar. 2011.
3. Two-photon absorption. http://en.wikipedia.org/wiki/Two-photon_absorption,
Oct. 25, 2011 [Nov. 23, 2011].
4. Yariev A. Third-Order Optical Nonlinearities – Stimulated Raman and Brilloun
Scattering in Quantum Electronics, 3rd ed. New York: John Wiley & Sons, 1989,
pp. 453- 458.
5. Pastor A. Class lecture, Topic: Nonlinear optics. Faculty of Physics, Saint
Petersburg State University, Saint Petersburg, Russia, 2010.
6. Krausz F. Photonics I. The theory of light & its advanced applications. http://
www.attoworld.de/Documents/pdf/lectures/ photonicsI/Vorlesung_photonics1.pdf,
2004 [Dec. 7, 2011].
96
The research of optical spectra of oil fraction in IR-area
Chernova Ekaterina
chernova.katerina@gmail.com
Scientific supervisor: Prof. Dr. Nemetz V.M., Department of Optics,
Faculty of Physics, Saint-Petersburg State University
Introduction
Nowadays spectroscopy becomes more popular in oil production due to the
short time spending for measurements. (1) In this field of science spectra interpreting is one of the important factors in an adequacy of analysis. The main goal of
researching the information value of oil fractions’ optical spectra is a generation of
a prior method for the optimization oil-products’ compounding process and for the
creation physicochemical parameters prediction model. The application of these
results can be important for the invention methods for the quantitative analysis of
oil and oil-fractions.
Traditional spectroscopic approaches have limited possibilities due to the basing on the idea of extraction narrow frequency region with one or two characteristic components from the complicated spectrum. Whereas different component
fragments overlap in oil-spectra, attempts to find solution by such method were
unsuccessful.
On the other hand, the method of principal components (2) allows identifying
fraction samples, in spite of lack possibility to separate absorption bands. It is one
of the methods of multidimensional statistic analysis. Its characteristics and application to oil products will be discussed in this paper.
Results and Discussion
Mathematical spectra processing.
Applying a mathematical tool requires formalization of spectral information. In common case the obtained data notation is of the 2×N-dimensional
matrix form (Fig. 1).
Fig.1. The form of obtained data.
In this matrix equation reference goes here λi are spectrum points, in which
measurement were made and Ii are intensities in these points. There are two cases:
as intensity can be absolute or normalized quantity of analytical signal. The obtained 2×N-dimensional matrix can be replaced by a N-dimensional vector. This
variant is qualified if the principle of a functioning automated measuring system
is taken into account. This principle is such that measuring result is a set of analytic signal steps in certain, as a rule, equidistant points. In the case of analytical
spectroscopy optical characteristics of systems (intensity, transmission density)
97
are counted in equidistant spectrum points. Moreover, for the different samples
the same spectrum points are chosen. Thus, all sample spectra can be presented
as a set of intensity values, while wavelength values, which are the same for all
samples and do not have specific information, can be neglected. As a result the
spectrum is transformed to the vector:
( I λ1 , I λ 2 , I λ 3 ....I λN )
Here λ1, λ2, λ3….. λN are certain wavelengths, N – spatial dimension. In the
case of spectroscopic approach N has a value of several thousand.
Vector representation means that the spectrum is presented as a vector in multidimensional space. The magnitude of the vector’s projection on N axis is proportional
to the analytical signal intensities at the corresponding wavelengths. Therefore, the
spectrum as an object image (sample, fraction) in the N-dimensional space can be
represent as a vector, its terminus giving corresponding point in N-dimensional
space. If there are M objects (samples, fractions) then they are represented in this
space as the points and their quantity is M. The simplest characteristic of distinction
in a points' location is Euclidean distance between them. As a result of multiple
spectrum measurements and of transforming these spectra in vectors every objectsample can be represented as a certain distribution in the N-dimensional space.
The method of principal components
There are many methods of reducing dimensions; however, in practice the
method of principal components is the most widespread. According to this technique, the set of points-patterns occupies certain area in N-dimensional space.
Coordinate’s dependencies in this space leads to the fact that these areas have a
certain shape stretched along certain axis, the direction of which can be different
from basis vectors (Fig. 2).
Fig. 2. The illustration of principal components selection in two-dimensional
space.
In the case when XY are basis axes the correlation between the points’s coordinates is being observed well (X increase – also Y increase, X decrease – also
Y decrease). As a result, the area of all samples has evidently stretched along the
98
certain axis shape. If this axis U is considered as a new basis axis and the second
axis V is chosen as perpendicular to the first, then the fact of new axes linear
independence is obvious. In addition to the scatter between points is maximal
along new axis U in a new coordinate system. Therefore, this direction is the most
informative according to the given configuration. The meaning of finding of the
new axis is to retrieve information maximum from the system.
This fully illustrates the conception of MPC – the searching new axes in an
initial space. These directions should be characterized by well-defined ranking of
information quantity called principal axes. The fist axis has an information maximum, the second one has a maximum after the first and the third has a maximum
after the first and the second. After building the new system for further consideration
suitable amount of axes can be chosen. Consideration of the correlation of amount
of information and laboriousness further calculation is very important. There are
many mechanisms for counting information quantity, which correspond to the
principal components. Among well-known algorithms there are SIMCA, NIPALS
and searching of covariance matrix’s eigenvector.
Oil and oil’s spectra
A Crude oil (petrolium) consists of compounds boiling at the different temperatures that can be divided into variety of different generic fractions by distillation.
This term ”fraction” has been used in the meaning of composition. However, the
petroleum can only be arbitrarily defined in terms of boiling point and carbon
number (3) (Fig. 3). ”The molecular boundaries of petroleum cover a wide range
of boiling points and carbon numbers of hydrocarbon compounds and other compounds containing nitrogen, oxygen, and sulfur, as well as metallic (porphyrin)
constituents” (3).
Fig. 3. Boiling point-carbon number profile for petroleum.
Research fractions were obtained by thermal petroleum distillation of different oil’s samples by apparatus “ARN-2”. Each of oil’s samples was in amount
of 4 liters. Therefore, there were 32 oil fractions; each of them was the result of
99
10-degree rectification from the boiling temperature to 390 °C. Obtained liquids
were keeping in a cooling cabinet.
It was decided to use the optical absorptive spectral method. Absorption
spectra were obtained under normal conditions: normal pressure and temperature
Fig.4.. Absorption spectrum of the fraction from 62 to 70 °C.
by Fourier-spectrometer “Bruker” in IR-area (Measurement range: 550 – 5000
cm-1 (2-18 µm)). According to literature there were no scientific papers related to
spectroscopic researching of such products. One of the spectra is presented in Fig. 4.
The method’s of principal components application.
The goal of this work is to research convergence of real compound’s spectrum
and spectrum synthesized from its rectification spectra.
Computation of synthesized spectrum was based on the formula:
D (λ ) = ( D (λ ) × C ) /( C ) ,
s
k
∑
i
i
k
∑
i
i
i
Here Ds(λk) – synthesized optical density at the wavelength λk. Di(λk) – optical
density i-fraction at the wavelength λk.
Design formula for principal components of synthesized spectrum:
P1s = (∑ P1i × C i ) /( ∑ C i )
i
i
P 2 s = (∑ P 2 i × C i ) /( ∑ C i )
i
i
….
PN s = (∑ PN i × C i ) /(∑ C i )
i
i
100
Here P1s…PNs – principal components synthesized spectrum. Amount of components is N. P1i…PNi – principal components of spectrum i-fraction (N components). Only fractions which consist in compound’s mixture are summarized.
Results of calculation synthesized spectrum and initial spectrum imagery are
presented in Fig. 5.
Fig. 5. Plane of first and second principal components.
In Fig. 5 there is experimental point scatter corresponding to fraction with
lower boiling temperature 100ºС (100а, 100в, 100с).Indicative points of se180sint
and se180real are located at center of area formed by image-points of initial fractions' spectra. It shows that se180 compound is a mixture of corresponding fractions, which means that it is possible to obtain prior information about compound
spectrum in case of availability of composing fractions' spectra. Moreover, this
result shows that intermolecular interaction is not affecting spectral properties of
compound much.
Conclusion
On the basis of the work done it has been confirmed that the method of principal components can be used for petroleum fraction researches. This means that
it is possible to obtain a prior set of physicochemical parameters of petroleum
compounds.
References
1. Vasilyev, A. V., Grinenko, E. V. Shchukin, A. O., Fedulin T. 2007. Infrared
spectroscopy of organic and natural compounds. SPb. SPbGLTA
2. Konushenko, I. O. 2008. New methods identification of liquid mixtures.
Diploma thesis. Saint-Petersburg State University, Laboratory of spectrum analysis.
3. James G. Speight. 2002. Handbook of Petroleum Product Analysis. John Wiley
& Sons, Inc., publication
101
Luminescence spectra of YVO4 and Y2O3 nanopowders
Kolesnikov Ilya
ilya-kolesnikov@mail.ru
Scientific supervisor: Dr. Kurochkin A.V., Department of General
Physic I, Faculty of Physics, Saint-Petersburg State University
Introduction
In recent years, rare-earth-doped nanocrystalline phosphors have attracted great
interest. Unique luminescence properties, especially characteristic narrow f–f bands
typical for the Ln3+ ions make them promising materials for multicolor displays and
lightning, biological labeling, lasers and optical amplifiers and optical sensors. The
main attention is paid to YVO4 and Y2O3 as the most prospective hosts.
The aim of the present work is investigation of the luminescent properties of
different nanocrystalline powders doped with europium ions and determination
the optimal composition and synthesis parameters. Also in this paper the kinetics
of luminescence was studied and the lifetime of the Eu3+ level 5D0 of europium
ions was calculated.
Results and Discussion
Excitation spectra
Luminescence properties of nanocrystalline phosphors depend on various
factors such as composition of the host, the synthesis method and parameters, the
size and shape of particle, and the concentration of the rare-earth ions.
4800
intensity, a.u.
4600
4400
4200
4000
3800
3600
200
220
240
260
280
300
320
340
360
380
wavelength, nm
Fig. 1. Excitation spectrum of YVO4:Eu3+ 16 mol.% 1000 oC.
The measured excitation spectrum of the luminescence band at 619 nm for
YVO4:Eu3+ 16 at.% powder is shown in Fig. 1.
The observed broad band with a maximum at about 280 nm can be attributed to
the charge transfer from the oxygen ligands to the central vanadium atom inside the
VO43- ion. On the other hand, this band can be also assigned to the charge transfer
102
(CT) transition between Eu3+ and O2-, an electron transfers from O2- (2p6) orbital
to the empty orbital of 4f6 for Eu3+ [1, 2]. In such a way it can be concluded that
the broad band around 280 nm is assigned to the overlap of VO43- absorption and
charge transfer transition between Eu3+ and O2-. Thus, the pump energy is absorbed
by the lattice host and then excitation is transferred to the ions of europium.
Emission spectra
In order to study the
0-2
1800000
YVO:E u 8% 800 C
5
effect of lattice host, the
DJ – 7FJ’
1600000
Y O :E u 8% 800 C
powders of YVO4 and Y2O3
1400000
doped with Eu3+ (8 mol. %)
1200000
1000000
were synthesized at 800
o
800000
C by the Pechini method.
2-6
600000
The emission spectra of
0-1
400000
the samples are shown in
1-1 0-0
0-4
0-3
200000
Fig. 2. The luminescence
0
intensity of Eu3+ in yttrium
-200000
500
550
600
650
700
vanadate was found to be
wavelength, nm
about 6 times higher than
in yttrium oxide. Therefore, Fig.2. Emission spectra of YVO4:Eu3+ 8 mol.% and
further studies were carried Y2O3:Eu3+ 8 mol.% under 325 nm excitation.
out with yttrium vanadate
host.
The emission spectrum
of YVO 4 :Eu 3+ powder is
characterized by the 5 D 0
– 7F2,4 electric dipole transitions of Eu 3+ resulted in
luminescence bands at 619
and 699 nm. Other contributions of weaker intensity
are the 5D0 – 7F1,3 magnetic
0
dipole transitions at 594 and
590
600
610
620
630
640
wavelength, nm
652 nm. The higher intensity
of the electric dipole transi- Fig. 3. Detailed emission spectrum of YVO4: Eu3+
tions can be explained by 16 mol.% 700 oC.
low symmetry of Eu3+ local
site in the YVO4 host lattices (D2d, without inversion center).
The weak intensity bands at 538 and 609 nm are corresponded to transitions
from 5D1 and 5D2 levels to 7F1 and 7F6, respectively.
The intensity ratio of transitions 5D0 – 7F2 and 5D0 – 7F1 can be used for analysis
of the local surrounding of the luminescent center and its symmetry. This ratio is
called the asymmetry coefficient k. Fig.3 shows the detailed spectrum of YVO4:
Eu3+ 16 mol.% 700 oC in spectral region from 590 to 640 nm.
o
4
normalized intensity, a.u.
intensity, a.u.
2
103
3
o
The asymmetry coefficients were calculated for the series of samples with different sintering temperatures. The values of these coefficients are listed in Table 1
.
Table 1. The intensity ratio of 5D0 – 7F2 to 5D0 – 7F1 for YVO4:Eu3+ 16 mol.% phosphors.
Intensity
Asymmetry
Temperature, °C
5
5
coefficient, k
D0 – 7F1
D0 – 7F2
700
8,26·106
1,28·108
15,50
850
4,39·107
5,93·108
13,51
7
8
900
1,29·10
1,96·10
15,19
7
8
950
2,4·10
3,48·10
14,50
1000
2,58·107
3,57·108
13,84
intensity, a.u.
intensity, a.u.
The observed dependence
of the asymmetry coefficient
demonstrates a tendency of
the Eu3+ ions local symmetry
increase with synthesis temperature growth.
Fig. 4 shows dependence
of the luminescence intensity
at 619 nm on the concentration
of Eu3+ ions (2, 4, 8, 12, 16
0
mol.%) for YVO4:Eu3+ pow-2 0
2
4
6
8 10 12 14 16 18 ders synthesized at 900 °C.
concentration, %
The luminescence intensity
Fig. 4. Emission spectra of YVO4: Eu3+ 900oC on 619 nm grows with increasing number
under 325 nm excitation.
of Eu3+ ions up to 8 mol.%.
Further increase of Eu3+ concentration leads to a reduction
in the intensity. This decrease
is due to the concentration
quenching.
Consequently the optimum
concentration of Eu3+ ions can
be determined as 8 mol.% in
yttrium vanadate host.
Fig. 5 shows the effect of
the synthesis temperature on
0
luminescence inten-sity of
700
750
800
850
900
950 1000 the band at 619 nm (transition
5
D 0 – 7F 2) for nanopowders
temperature, C
3+
Fig. 5. Emission spectra of YVO4:Eu 16 mol.% YVO4:Eu3+ 16 mol.%. It was
on 619 nm under 325 nm excitation.
found that the sample syn104
thesized at 1000 °С demonstrates the highest luminescence intensity and can be
considered as the most efficient phosphor.
Kinetics of luminescence
In order to determine life-time of the 5D0 level the luminescence kinetics has
been investigated. The experimental data were approximated by “biexponential
fitting”:
.
The observed non-exponential decay of 5D0 luminescence is most probably
connected with two different sites of the Eu3+ ions (at the surface - surface defects
of the nanoparticles and Eu3+ ions inside the particle/grain). So, the effective
lifetime of the studied YVO4:Eu3+ nanopowders can be determined from the following equation:
The efficient life-times of nanopowders YVO4:Eu3+ synthesized at different
temperatures are listed in Table 2.
Table 2. The life-times of transition 5D0 – 7F2 for YVO4:Eu3+ 16 mol.% phosphors.
Two-exponential fitting
Temperature, oС
τav , ms
τ1, ms
τ2, ms
700
0,18
0,54
0,59
850
0,16
0,54
0,54
900
0,19
0,62
0,54
950
0,18
0,50
0,46
1000
0,22
0,51
0,43
Conclusions
a) The emission spectra of YVO4:Eu3+ and Y2O3:Eu3+ were investigated.
b) Dependence of luminescence on the synthesis temperature and the concentration of rare-earth ions were studied.
c) The asymmetry coefficients were calculated. YVO4:Eu3+ 1000 oC has the
lowest coefficient.
d) The kinetics of the 5D0 level of YVO4:Eu3+ was investigated. The fluorescence life-time presents a maximal value for the samples annealed at 700 °C.
References
1. Zhou Y.H. et al. // Optical Materials 27, pp. 1426–1432 (2005).
2. Devaraju M.K. et al. // J. Crystal Growth, Vol. 311, pp. 580–584 (2009).
3. Yuhua Wang et al. // Materials Research Bulletin 41, pp. 2147–2153 (2006).
4. Georgescu S. et al. // Romanian Reports in Physics, V. 60, No. 4, pp. 947–955 (2008).
5. Hreniak D. et al. // Journal of Luminescence 131, pp. 473–476 (2011).
105
Resonance grating based on InGaAs/GaAs quantum well
Kozhaev Mikhail, Kapitonov Yury
mikhailkozhaev@gmail.com
Scientific supervisor: PhD Petrov V.V., Department of Photonics,
Faculty of Physics, Saint-Petersburg State University
Introduction
Significant problem of modern physics is creation of an optical computer that
will make calculations in purely optical way without dissipation of energy in
the element [1]. One of the most promising candidates for the working medium
of logic elements of optical computer are InGaAs/GaAs quantum wells (QW).
Resonant reflection spectroscopy associated with the birth of 2D-excitons is used
for studying optical properties of QW [2]. Light reflection from the «excitonic
mirror» is coherent in contrast to the photoluminescence. However, the resonant
reflection from the QW, located inside the specimen, is difficult to detect against
the background of non-resonant reflection from the sample surface. The only way
to obtain interpretable QW reflection spectrum is to have zero reflection from the
sample surface by using the Brewster geometry for the incident light: the angle of
incidence equals to the Brewster angle, and polarization is in the plane of incidence
(p-polarization). To overcome this fundamental limitation we created resonant
grating by spatial modulation of the QW properties. In this case diffraction peaks
were observed. Travelling direction of this signal differs from the reflection direction. In this case influence of non-resonant reflection from the sample surface is
eliminated for all incidence angles and light polarizations. Spatial modulation of
QW was made using an ion beam irradiation of the GaAs substrate prior to the
MBE growth of the heterostructure.
Results and Discussion
Lithography
The first step of sample creation was made by focused ion beam (FIB). Array
of lines with length 400 μm was created on epi-ready GaAs substrate by focused
beam of Ga+ ions with an energy of 30 keV. The steps between the lines was 9 μm,
and the linear dose was 0.1 and 2 nA*s/cm. Thus, in addition to milling, which in
this case will not be significant on the substrate surface will occur implantation,
defect formation and deposition of hydrocarbons (Fig. 1).
Molecular beam epitaxy
Then using molecular beam epitaxy (MBE) method three In0.02Ga0.98As/GaAs
QW with different thicknesses (and hence with different exciton resonance spectral position) were grown on the substrate (Fig. 2). Fig. 3 shows different sample
places: with and without lithography.
106
Ga+
hydrocarbons
Ga+
e-
10 μm
Implantation
and defect
Deposition of
hydrocarbons
Fig. 1. Left picture - image of lithography made by scanning electron microscopy (SEM). In addition to etching (whose influence in this case because of the low
dose only slightly) in lithography place occurs implantation and defect and hydrocarbons deposition.
150 n
m
4.0 nm
150 n
m
3.0 nm
150 n
m
2.0 nm
Cap
QW 3
Interm
e
diate
QW 2
Interm
e
diate
QW 1
270 n
I
layer
I
layer
m
Buffer
I
s
GaA
As
)Ga
n(2%
s
GaA
As
)Ga
n(2%
s
GaA
As
)Ga
n(2%
s
GaA
0.1 nA*s/cm
Substr
ate
lithog with
raphy
2 nA*s/cm
s
GaA
Fig. 2. Left figure shows sample layers and the depth of QW. Right figure shows
influence of different lithography doses on grown layers and QW.
10 μm
Fig. 3. Surface maps were obtained using atomic force microscopy from two different sample places: without lithography (left), and with 2 na*s/cm doses (right).
107
Optical properties of the sample
The sample was studied at a temperature 9 K using a Ti:sapphire femtosecond
laser with a wide (about 20 meV) emission spectrum using Brewster geometry
(Fig. 4). Brewster geometry allows getting interpretable reflection spectra because
s-pol
Cryostat with
the sample
(9-300 К)
Sam
ple
Ti:Sa fs-laser
diaphragm
mirror 1
mirror 3
p-pol
Monochromator CCD
mirror 2
half mirror
lens
Fig. 4. Left figure shows the incidence light polarization effect to the reflection
from the sample surface in Brewster geometry. On the right shows the optical setup scheme.
there is a polarization in which the nonresonant reflection from the sample surface
is absent. Thus the reflection spectroscopy is limited by the angle and polarization
of the incident light.
But if QW properties are periodically laterally modulated then diffraction signal
propagation direction differs from reflection signal propagation direction. Thereby
incident light angle and polarization deviation don’t lead to loss of the information
about QW. Fig. 5 shows reflection and diffraction spectra in Brewster geometry for
Fig. 5. Reflection and diffraction spectra comparison for p-and s-polarized incident light. Sample temperature is 9 K. Brewster geometry is used.
s- and p- polarization. In contrast to s-pol diffraction spectra, the s-pol reflection
spectrum can’t be interpreted because of the useful signal interference with the
nonresonant reflection from sample surface.
108
Studied sample had three QW which are located at different heights from the
substrate. Reflection peaks intensities from these QW are approximately the same
order (full line in Fig. 6). However, for diffraction spectra picture is quite different.
For the sample place in which QW were modulated by 2 nA*s/cm before growth,
diffraction peaks intensities differ by an order (dotted line in Fig. 6) although distances from QW 1 to substrate and from QW 2 to substrate varies slightly more
Fig. 6. Reflection and diffraction spectra for places with preliminary lithography
linear dose 0.1 and 2 nA*s/cm. Sample temperature is 9 K. Brewster geometry is
used. Oscillations likely are related to the interference of the light which scattered
from the sample surfaces. Spectra are normalized to the first QW peak height.
than twice. For the sample place in which QW were modulated by 0.1 nA*s/cm
before growth, diffraction peaks from QW 2 and QW 3 are mostly lost in noise
(dashed line in Fig. 6). This observation allows to “turn on” only some of the QW
peaks in diffraction spectrum.
Conclusion
The diffraction grating has little effect on the quantum wells quality. However
the diffraction signal propagation direction differs from the nonresonant light
propagation direction reflected from the surface. Creation of multiple QW structure
on the substrate with lithography leads to difference in the reflection intensities
from QW, depending on their geometrical position. Thus, the resonant diffraction
grating is a new tool for a more detailed study of the physical processes occurring
in the quantum wells, including non-linear optical properties. In addition, such
facilities may have practical applications, such as an optical gate - one of the key
photonic logic elements, and other devices.
References
1. Gerlovin et I.Ya. al. // Nanotechnology, 11(4), 383-386 (2000).
2. Poltavtsev S.V. et al. // Physica Status Solidi (C) Current Topics in Solid State
Physics, 6(2), 483-487 ,(2009).
109
Application of 2D-correlation spectroscopy method for
interpretation of spectra and enhancing the spectral
resolution
Maximova Ekaterina, Lev Derzhavets
eama08@mail.ru, levann@mail.ru
Scientific supervisors: Dr. Levin S.B., Department of Mathematical
Physics, Faculty of Physics, Saint-Petersburg State University;
Dr. Bulanin K.M., Department of Molecular Spectroscopy, Faculty
of Physics, Saint-Petersburg State University
Introduction
The idea of 2D-correlation spectroscopy appeared not too long ago , so the
number of publications in this field is not significant yet. 2D-correlation spectra
consist of two orthogonal components - synchronal and asynchronal spectra. Both
of them carry significant and independent information about the behavior of the
studied system.
The aim of current work is to make acquaintance with the 2D-correlation method
for the interpretation of spectra and enhancing the spectral resolution. Also, we
considered the continuous approximation of spectral amplitude two-dimensional
surface defined in the frequency-time surface:
I ( ν,t ) = Ω ( ν) P (t )
where
nν
Ω ( ν) = ∑ A j e
−a j
( ν − nu )
2
j
j=1
and nt
P (t ) = ∑
j=1
Bj
(t − t )
j 2
0
+ η2j
This kind of approximation is convenient for analytical construction of 2Dcorrelation spectra. For approximation of the peaks, we used the Nelder-Mead
method (simplex algorithm). Furthermore, such a constructions will give opportunity to interpret more effectively complex processes being under study [1]. For
approximation of the peaks, we used the Nelder-Mead method (simplex algorithm).
Correlations obtained by using the discrete set of experimental data, proposed
by Noda [2, 3] are the additional criterion for the quality of approximation.
110
2D-correlation spectroscopy method
Nelder-Mead method
For the peaks approximation, we used the Nelder-Mead method (simplex algorithm). The Nelder-Mead algorithm is designed to solve the classical unconstrained
optimization problem of minimizing a given nonlinear function f : Rn →R.
The method
• uses only function values at some points in Rn, and
• does not try to form an approximate gradient at any of these points.
Hence it belongs to the general class of direct search methods .
The Nelder-Mead method is simplex-based. A simplex S in Rn is defined as the
convex hull of n+1 vertices x0,…, xn  Rn . For example, a simplex in R2 is a triangle, and a simplex in R3 is a tetrahedron.
A simplex-based direct search method begins with a set of n+1 points x0,…,
xn  Rn that are considered as the vertices of a working simplex S , and the corresponding set of function values at the vertices fj =f(xj) , for j=0,…, n . The initial
working simplex S has to be nondegenerate, i.e., the points x0,…, xn must not lie
in the same hyperplane.
The method then performs a sequence of transformations of the working
simplex S, aimed at decreasing the function values at its vertices. At each step,
the transformation is determined by computing one or more test points, together
111
with their function values, and by comparison of these function values with those
at the vertices.
This process is terminated when the working simplex S becomes sufficiently
small in some sense, or when the function values fj are close enough in some sense
(provided f is continuous).
The Nelder-Mead algorithm typically requires only one or two function evaluations at each step, while many other direct search methods use n or even more
function evaluations.
Results and Discussion
As approximating functions was taken the following basis:
(
y ( ν,t ) = e
)
2
− ν − ν1
0
b1
(t − t )
1 2
0
2
1
+η
+e
(
)
2
− ν − ν02
b
2
2 2
0
(t − t )
+ η22
The offered approximation allows us to interpret the complicated processes more
effectively from the point of view of chemical reactions occurring in the system
Quality ν01 and ν02 were taken value of positions of the maxima of the peaks
in real 2107.332 cm-1 and 1130.564 cm-1, respectively. Approximating with respect
to the frequency was carried out [4].
Search for factors b1,2,t01,2, ƞ1,2 produced by least squares method, i.e. was taken
following sum of squares [4]:
m
(() )
Sm = ∑ y t j − y j
j=1
2
Where y(tj) – values of analytical expression in the experimental points and yj
–experimental values at corresponding points.
With such an analytic function is not difficult to find correlations :
a
b
Fig. 1. a) synchronous spectrum, b) asynchronous spectrum.
112
References
1. Isao Noda, Yukihiro Ozaki. Two-dimensional Correlation Spectroscopy –
Applications in Vibrational and Optical Spectroscopy.- England, 2004.
2. Tonkov M.V., Filippov N.N. // Chem. Fizika, 10(7), 922 (1991).
3. Noda I. // Appl. Spectrosc., 47, 1329 (1993).
4. Lev Derzhavets. Application of two-dimensional correlation spectroscopy to improve the resolution of spectral bands. BSc. Thesis, St.-Petersburg University, 2010.
113
Observation of the fine structure for rovibronic spectral
lines in visible part of emission spectra of D2
Umrikhin I.S., Zhukov A.S.
umrikhin@mail.ru, zuk90@mail.ru
Scientific supervisor: Prof., Dr. Lavrov B.P., Department of Optics,
Faculty of Physics, St. Petersburg State University
For the first time the fine (triplet) structure of rovibronic lines has been observed in the visible part of the D2 spectrum (mainly within the 3Λg → c3Πu- band
systems). General trends are illustrated by the example of the R-branch of the
i3Πg-,v’=1 →c3Πu-,v”=1 band.
The studies of the D2 spectrum have been started just after the discovery of the
molecule. However, our knowledge of optical spectrum of molecular deuterium is
still insufficient in spite of tremendous efforts by spectroscopists over the previous
century [1]. Up to now most of spectral lines have not yet been assigned [2], and
wavenumbers of rovibronic transitions in visible part of the D2 spectrum were
obtained without resolving the fine structure of triplet-triplet spectral lines, in spite
of its observation by Fourier [3] and laser [4] spectroscopy in infrared. The goal of
present work was to study an opportunity of resolving the fine structure in visible
by means of spectroscopic technique developed in [1, 5].
The experimental setup was described in [5]. Pure D2 plasma of constricted
glow discharge with cold cathode and water cooled walls under j = 0.4 A/cm2,
T = 640±50 K was used as a light source. The flux of radiation through a hole in
an anode was focused on the entrance slit of the 2.65 m Ebert-Fastie spectrograph
equipped with additional camera lens and computer-controlled CMOS matrix
detector. In our conditions the resolving power was mainly limited by Doppler
broadening (linewidth ΔνD ≈ 0.15 cm-1) and overlap of adjacent lines.
Wavenumbers of rovibronic transitions of the D2 molecule were previously
obtained by photographic recording of spectra [2]. Our way of determining the
wavenumbers is based on linear response of CMOS matrix detector on the spectral irradiance and digital intensity recording. Both things provide an extremely
important advantage of our technique over traditional photographic recording
with microphotometric comparator reading. It not only makes it easier to measure
the relative spectral line intensities but also makes it possible to investigate the
shape of the individual line profiles and, in the case of overlap of the contours of
adjacent lines (so-called blending), to carry out numerically the deconvolution
operation (inverse to the convolution operation) and thus to measure the intensity
and wavelength of even blended lines. As is well known, it is this blending that
makes it very hard to analyze dense multiline spectra of the D2 molecule [2].
114
For small regions of the spectrum (≈0.5 nm wide) the observed spectral intensity distribution is approximated by superposition of a certain number of Voigt


profiles:


2
∞
exp −t


Fj ( x) = Aj ∫ 
dt ,
(1)
2
2
 x − xj
 
−∞  ∆υ
D
 2∆υ 2 +  2∆υ − t  


L
L
( )
where ΔνD and ΔνL are Doppler and Lorentzian linewidths equal for all the profiles
within a spectral region under the study. Aj is the intensity and xj – center of the
profile. One region corresponds to one third of the matrix 550 − 600 pixels wide.
Thus for each region we obtain 550-600 experimental values of the observed
intensity Jiobs from each photodetector of the CMOS matrix. To find the optimal
values for adjustable parameters (line centers, relative intensities and one common values ΔνD and ΔνL for all profiles) we need to solve the system of 550-600
nonlinear equations:
J iobs − J synt ( xi ) = 0 ,
(2)
K
where J synt ( xi ) = ∑ F j ( xi )
j =1
is “synthesized” intensity value with Fj in the form (1) and K – number of approximated Voigt profiles, xi – coordinate of the photodetector on the CMOS matrix.
We approximate the observed intensity distribution by the minimal number of
profiles K, which give us random spread of the deviations (2). For different spectral
regions we use K=10÷50 profiles for one region, thus we obtain from 20 to 100
parameters for 550-600 equations in the system (2). Therefore system of equations
(2) is overdetermined and, hence, is inconsistent because the experimental data
always involve measurement errors.
The straightforward general solution for solving such problems is the leastsquares method. In our case, it consists in minimizing the sum of mean-square
M
deviations (2):
2
r = ( J iobs − J synt ( xi ) 2 (3)
i =1
where M indicates a number of experimental intensity values Jiobs in the spectral
region under the study. To find optimal values of the profile parameters minimizing
expression (3) we used special computer program based on Levenberg-Marquardt's
algorithm. If the experimental errors are random and distributed according to a
normal (Gaussian) law, the solution obtained by least-squares method corresponds
to the maximum likelihood principle. The obtained values for K, Aj, xj, j=1…K,
ΔνD, ΔνL are optimal for the observed intensity distribution. Thus it is possible to
obtain the optimal values of the intensity, center coordinate and linewidth for each
resolved spectral line in the spectral region under the study.
In the case of long-focus spectrometers the dependence of the wavelength on the
coordinate along direction of dispersion is close to linear in the vicinity of the center
115
∑
of the focal plane. It can be represented as a power series expansion over of the
small parameter x/F (The x-coordinate actually represents small displacement from
the center of the matrix detector, F is the focal length of the spectrometer mirror),
which in our case does not exceed 2x10-3 [5]. On the other hand, the wavelength
dependence of the refractive index of air n(λ) is also close to linear inside a small
enough part of the spectrum. Thus, when recording narrow spectral intervals, the
product λvac(x) = λ(x) n(λ(x)) has the form of a power series of low degree. This
circumstance makes it possible to calibrate the spectrometer directly in vacuum
wavelengths λvac = 1/ν, thereby avoiding the technically troublesome problem of
accurate measuring the refractive index of air under the various conditions under
which measurements are made.
We used for calibration experimental vacuum wavelength values from [2] as
standard reference data. Those data show small random spread around smooth
curve representing dependence of the wavelengths on positions of corresponding
lines in the focal plane of the spectrometer. Moreover those random errors are in
good accordance with normal Gaussian distribution function. Thus it is possible
to obtain precision for new wavenumber values better than that of the reference
data due to smoothing. The calibration curve of the spectrometer was obtained by
the polynomial least-squares fitting of the data. Our measurements showed that,
using a linear hypothesis is inadequate and a third-degree polynomial is excessive, while an approximation by a second-degree polynomial provides calibration
accuracy better than 2x10-3 nm.
The results are illustrated in Fig. 1 by the example of the R-branch of the i3Πg,v’=1 → c3Πu-,v”=1 band. Observed wavenumber values (in cm−1) and Iv/Ir for all
spectral lines of the branch are presented in Table 1. One can see that separation of
60
J, counts
R4
50
40
R6
R3
30
R5
20
R7
10
0
17020
17022
17024
17026
-1
Fig. 1. Part of the observed D2 spectrum containing 5 spectral lines of the R-branch
of the i3Πg-,v’=1 →c3Πu-,v”=1 band. Fine structure components (visible doublet)
are marked by solid bold vertical lines. Experimental intensity J in relative units is
shown by open circles. Solid line represents the intensity distribution calculated as
a sum of optimal Voigt profiles. Dotted lines represent wavenumbers of observed
spectral lines, solid vertical lines – wavenumbers of lines reported in [2].
116
observed doublets is close to 0.2 cm−1. This is in coincidence with results obtained by
means of laser [4] and FTIR [3] spectroscopy in infrared part of the spectrum.
Joint analysis of splitting in such visible "doublets" (about 0.2 cm-1) and relative
intensities of two main components of visible "doublets" (about 2.0) show that they
represent partly resolved fine structure of lines determined by triplet splitting of
lower rovibronic levels of various 3Λg→ c3Πu- electronic transitions.
The observed ratio of intensities of the doublet components is close to 2.0 predicted by Burger-Dorgello-Ornstein sum rule when one assumes that the multiplet
splitting in upper rovibronic states may be neglected while in the lower rovibronic
states c3Πu,v=1,N” two fine structure sublevels (J”=N”-1 and J”=N”+1) are close to
each other and located noticeably lower than that with J”=N”. These assumptions
are in agreement with IR tunable laser observations (EJ"=2 ≈ EJ"=0<EJ"=1) for the fine
structure of the a3Σg,v=4,N=3←c3Πu,v=3,N=1 rovibronic transition reported in [4].
Thus our ability to measure both the intensities and splitting values gives us an
opportunity to get information about an order and separation of the fine structure
sublevels. It should be stressed that we are working in visible part of the spectrum,
most suitable for various applications.
In particular, the results of present work show that an observation of even
partly resolved fine structure of spectral lines provides an opportunity to expand
the existing identification of triplet rovibronic lines by detecting those doublets in
experimental spectra. Within the spectral region under the study (545-627 nm) we
already found more than 200 pairs of unassigned lines which may represent visible
doublets of partly resolved triplet structure of rovibronic transitions between 3Λg
and c3Πu- electronic states of the D2 molecule [6].
Present work
N′′
1
2
3
4
5
6
7
8
9
10
νv
16996,58(3)
17010,39(3)
17019,74(3)
17025,13(3)
17026,99(3)
17025,70(3)
17021,57(4)
17014,91(4)
17005,86(4)
16994,57(4)
νr
16996,36(4)
17010,21(3)
17019,57(4)
17024,96(3)
17026,82(4)
17025,53(4)
17021,40(4)
17014,70(4)
17005,70(5)
16994,40(4)
Δνvr
0,22(5)
0,18(4)
0,17(5)
0,17(4)
0,17(5)
0,17(5)
0,18(6)
0,21(6)
0,16(7)
0,16(6)
Iv/Ir
1,93(5)
1,88(2)
1,99(4)
1,94(3)
2,05(7)
1,84(4)
1,82(11)
1,90(60)
1,80(20)
1,89(14)
Data reported in [2]
ν
16996,58
17010,36
17019,71
17025,11
17026,97
17025,71
17021,60
17014,87
17005,79
16994,53
Present work was financially supported in part by the Russian Foundation for
Basic Research, Grant No. 10-03-00571-a.
117
References
1. Lavrov B.P., Umrikhin I.S. // J. Phys. B. 2008. v. 41. 105103. (25pp);
2. Freund R.S., Schiavone J.A., Crosswhite H.M. // J. Phys. Chem. Ref. Data.
V.14. No 1. P. 235, 1985.
3. Dabrowski I and Herzberg G. // Acta Phys. Hung. 1984. v.55, n.1-4, pp. 219228.
4. Davies P.B., Guest M.A. and Johnson S.A. // J. Chem. Phys. 1988. v. 88, n.5,
pp. 2884-2890.
5. Lavrov B.P., Mikhailov A.S., Umrikhin I.S. // J. Opt. Technol., v. 78, I. 3, p.
180, 2011.
6. Lavrov B.P., Umrikhin I.S. // e-Print arXiv:1112.2277v1 [physics.chem-ph]
10 Dec 2011.
Table 1. Wavenumber values (in cm−1) and relative intensities Iv/Ir for violet and
red components of the R-branch lines for (1 − 1) band of the i3Πg- → c3Πu- electronic transition. The νv and νr are wavenumbers of violet and red components of
the doublets; Δνvr = νv - νr .
118
G. Theoretical, Mathematical
and Computational Physics
Conservation laws and energy-momentum tensor in
Lorentz-Fock space
Angsachon Tosaporn
banktoss@yahoo.com
Scientific supervisor: Prof. Dr. Manida S.N., Department of High
Energy and Elementary Particles Physics, Faculty of Physics,
Saint-Petersburg State University
Introduction
In this paper we consider the conservation laws for classical particles in AdS4.
At first we parameterize a geodesic line and construct the conserved quantities
with analog of five dimensional Minkowski space M(2,3). Consequently we change
AdS4 space to AdS-Beltrami space and write out conserved quantities in Beltrami
coordinates. Furthermore we take a limit for small velocity (xi ≪ c) and we get
the conserved quantities in Lorentz-Fock space. And finally the energy-momentum
tensor for dust material is constructed.
Consererved quantities in embedding anti de-Sitter space(AdS4)
We define anti de-Sitter space as four dimensional hyperboloid
AdS4 = {XA ϵ M(2,3), X2 = ηABXAXB = R2}
(1)
embedded in five dimensional Minkowski space. Induced metric for this space
can be expressed
ds2 = (ηABdXAdXB)∥AdS
(2)
where ηAB = diag{1,1,-1,-1,-1} and indices A,B are running values -1,0,1,2,3.
Now we consider an action for the classical massive particle, which can be
expressed in five dimensional coordinates
S = -mc∫[(V2)1/2+a(X2-R2)]dλ,
(3)
where X(λ) is a parameterized timelike curve with a constraint X2 = R2.
VA(λ) = dXA/dλ are the velocity the particle. We know that an action (3) is invariant
under transformation:
XA → XA+ωABXB,
(4)
where ωAB is and antisymmetric infinitesimal parameter. From Noether theorem
we can find the conserved quantities along a timelike geodesic line
m
(5)
K AB =
( X AVB − X BVA )
R V2
We can write this conserved quantities in the form of two timelike vectors ξ and η
m(η Aξ B − η Bξ A ) (6)
K AB =
2 2
2
η
ξ
−
(
η
⋅
ξ
)
where ξ = (0,c,ξi) , η = (R,0,ηi).
Finally we can find a mass-shell equation
(7)
K AB K AB = 2m 2 120
Conserved quantities in Beltrami coordinates
In this part we will write our conserved quantities in Beltrami coordinates.
Beltrami coordinates are determined by the relation
xμ = RXμ/X-1 = (λc,λξi+ηi) (8)
Let H, Pi, Ki, Ji be conserved quantities in Beltrami coordinates and these
quantities are expressed in the formulas
m
H = K 0( −1) =
(9)
  2
xi ( xi − λ xi ) 2 ( x × x ) 2
+ 2 2
1− 2 −
c
R2
Rc
m( xi − λ xi )
K i = RK 0i =
(10)


xi2 ( xi − λ xi ) 2 ( x × x ) 2
1− 2 −
+ 2 2
c
R2
Rc

ε ijk x j xk
J i = Rcm
(11)


xi2 ( xi − λ xi ) 2 ( x × x ) 2
+ 2 2
1− 2 −
c
R2
Rc
mxi
Pi = cK i ( −1) = Rcm
(12)


xi2 ( xi − λ xi ) 2 ( x × x ) 2
+ 2 2
1− 2 −
c
R2
Rc
Conservation laws for nonrelativistic cosmological particles
We consider the conservation laws for particles which are moving with the small
velocity (ẋi ≪ c) in the space with constant curvature R. The conserved quantities
and mass-shell equation will be written as
m
H=
(13)
(λ xi − xi ) 2
1−
2
R
m( xi − λ xi ) Ki =
(14)
(λ xi − xi ) 2
R 1−
R2
mxi
(15)
π i ≡ lim c →∞ cPi =
(λ xi − xi ) 2
1−
R2
2
2
2
(16)
H − Ki = m The space with this mass-shell equation is called Lorentz-Fock space. If we put λ
= T+t, R/T ≡ c0 and consider the small vicinity point in this space t ≪ T, ‫׀‬xi‫≪׀‬R,
then we get
m
(17)
H=
xi2
1− 2
c0
121
πi =
mxi
x 2
1 − i2
c0
(18)
π i m( xi − txi )
(19)
=
c0
xi2
1− 2
c0
Therefore we see that this “cosmological” dynamics under the nonrelativistic limit
does not differ from the standard relativistic dynamics with speed of light c0.
Ki +
The energy-momentum tensor in Lorentz-Fock space
In this part we consider the construction of energy-momentum tensor for dust
particles in Lorentz-Fock space. We are starting from the 4-velocity in this space
which is expressed in the formula
1
vi
(20)
uµ =
(1, ) c
(vi t − xi ) 2
1−
R2
From this velocity we can down the index of 4-velocity with the help of a metric
of Lorentz-Fock space in this form
2
ds = gμνdx dx = R
2
μ
ν
R 2 − xi2 2 2 R 2 xi
R2
i
j
dt
+
dtdx
−
δ
i
ij 2 2 dx dx
c 2t 4
c 2t 3
ct
(21)
Therefore the 4-velocity in the lower index is written out in the formula
xi − vi t
R 2 R 2 − xi2 + xi vi t ,
R2
u0 = 4 4
ui = 3 3
(22)
ct
ct
(vi t − xi ) 2
(vi t − xi ) 2
1
1
−
−
R2
R2
The energy-momentum tensor for dust particles is defined in this equation
(23)
Tµν = ρ uµ uν Finally, this tensor can be expressed in each component
T00 = ρ u0u0 =
R 4 ρ ( R 2 + xi (vi t − xi )) 2
(v t − x ) 2
c 8t 8
1− i 2 i
R
R 4 ρ ( R 2 + xk (vk t − xk ))( xi − vi t )
(v t − x ) 2
c 7t 7
1− i 2 i
R
R 4 ρ ( xi − vi t )( x j − v j t )
Tij = ρ ui u j = 6 6
2
(v t − x )
ct
1− i 2 i
R
T0i = ρ u0ui =
122
(24)
(25)
(26)
References
1. Cacciatori A., Gorini V., Kamenshchik A.Yu. Special Relativity in the 21-st
century. // Ann. Phys. 17, p. 728-768 (2008), arxiv:hep-th/0807.3009.
2. Manida S.N. Fock-Lorentz transformations and time-varying speed of light,
arxiv:gr-qc/9905046.
123
Renormalization-group and ε- expansion: representation
of anomalous dimensions as nonsingular integrals
Batalov Lev
zlokor88@gmail.com
Scientific supervisor: Prof. Dr. Adzhemyan L.Ts., Department
of Statistical Physic, Faculty of Physics, Saint-Petersburg State
University
Introduction
The analytic calculation of critical exponents in critical behavior models with
renormalization-group method (RG) and ε- expansion is very difficult in high order
of perturbation theory [1], because it needs to compute singular in ε integrals
(ε= 4 - d, where d is dimension of space). In this paper we find such implementation
of RG-method with ε-expansion, in which the problem is reduced to computing of
finite integrals and we can to automate it.
It’s importantly to choose the most usable scheme of renormalization for solution
of this problem. The operation of renormalization for diagram Г is written:
ΓR = ΓR (1)
It names R-operation of Bogolubov-Parsuk often and removes divergences in
subgraphs and after remained surface divergence. First it needs to find the divergent
part of diagram.
The scheme of null momentum (NM) it most usable for calculations. It is reduced to subtraction from function F(k) n initial terms of a Taylor series:
1
n
km
1
n
n +1
(2)
F
(
k
)
|
da
(1
a
)
F
(
ak
)
−
=
−
∂
∑
k
a
=
0
n ! ∫0
m=0 m !
For R-operation for graph χ we get:
Rχ = ∏
i
1
1
dai (1 − ai ) ni ∂ ai ni +1 χ ({ai })
∫
ni ! 0
(3)
In equation (3) The product consists of all divergent subgraphs (including whole
diagram χ) with canonical dimension ni ≥ 0, a is the parameter of extension inside
i –subgraph of momentums flowing into this subgraph.
The NM-scheme is not easy, but it become such after some modification.
However, in new scheme we are able to write interesting for us values in usable
for computation form (3).
Renormalization-group functions
We shall consider the solution of this problem for example φ4 - theory. The space
dimension is d = 4-ε. The renormalization action for this theory is [2]:
1
S = − (m 2 Z1 + p 2 Z 2 + δ m 2 )ϕ 2 + ...
(4)
2
124
Surface UV-divergences for ε = 0 there are in diagrams of 1-irreducible functions Γ2 = <φφ> and Г4 = <φφφφ> having quadratic and logarithmic divergence
respectively.
Counterterms δZi = Zi-1 ; i = 1,2,3 and δm2 remove these divergences in each
order of perturbation theory at the coupling constant g. Calculation of counterterms in (4) is changed R-operation for diagrams of basic theory. The scheme of
renormalization is determined the K-operation, which selects divergent subgraphs
of 1-irreducible diagrams.
We write this operation in NM-scheme, and we will change the normalization
point after. K-operation for NM-scheme is:
(5)
K Γ 4 ≡ K 0 Γ 4 , K 0 f ({p}) ≡ f | p =0 (6)
K Γ 2 = ( K 0 + p 2 K 2 )Γ 2 , K 2 f ({p}) ≡ (∂ p2 f ) | p =0 We separated the initial partial sum of series in momentum. The quantity of
terms in it is determined the dimension of diagram.
Renormalization constants are convenient in notation of normalized
functions:
KΓ
(7)
Γ 2 = − K 2 Γ 2 , Γ 4 = − 0 ε3 gµ
In notation Γ renormalization constants are:
(8)
Z i = 1 − KRΓi |m = µ , i = 1, 2,3 We select the renormalization
scheme, which corresponds previous conditions (8), but it is defined in normalization point m=μ. This scheme is named
NP-scheme.
Schemes NM and NP differ only by the finite renormalization of parameters
g, m2 and constants Zi. It is conditioned through the relation between residues at
higher poles of renormalization constants and in first pole. This relation provides
UV-finity of renormalization functions. Values of critical exponents in NM and
NP-scheme are equal.
We consider renormalization constants Z1,Z2. these constants define β- function
and anomalous dimensions of field. Proposed method of computation approach
for both renormalization schemes NM and NP. However in NP-scheme the result
is formulated the simplest way.
Renormalization constants in NM-scheme and NP-scheme depend only from
dimensionless charge g and space dimension. This way, the form of renormalizationgroup equations (if the renormalization mass μ is arbitrary) and relation between
RG-functions and renormalization constants in these schemes are the same.
The system of RG-functions includes the β - function:
ε

β = −g  + γ g 
(9)
2

and γ- functions. These functions relate with renormalization constants Zi by the
equation:
γ i = β∂ g ln Z i
(10)
125
In particular case,
From (9)-(11) we get:
γ g = β∂ g ln Z g β =−
gε
1
2 1 + g ∂ g ln Z g
(11)
(12)
gε g ∂ g ln Z i
(13)
2 1 + g ∂ g ln Z g
The RG-functions defined singular in ε renormalization constants haven’t poles
in ε [3]. The removing of these poles for renormalized models is guaranteed by
the renormalization theory.
The removing of poles in the calculation shows us, that it is true. It is interestingly, that in our case the removing of poles has analytic character without
calculation of diagrams. It takes the problem of approximation for the high poles
in numerical calculations away.
2 f2
The result is:
γ2 =
(14)
1 + f2 2 f3 γ3
(15)
1 + f3
where
γi = −
fi = − R(m 2 ∂ m2 Γi ) |m = µ , i = 2,3
Conclusion
Equations (14-15) are the main result of this work. We realized the automatic
program for computation of critical exponents with them. We used the GRC program from the GRACE packet to construct diagrams. We added to it a program for
finding relevant subgraphs. The corresponding integrand was generated for each
diagram needed to calculate values (15) using representation for the R-operation
(3). We calculated integrals using the Monte-Carlo method. We applyed the spherical d-dimensional coordinate system and the Feynman representation to compute
these integrals.
The anomalous dimensions γ2 and γ3 were calculated in the four-loop approximation (it was needed to compute 204 diagrams). Based on them, we found β- function
(12) and defined the coordinate of the fixed point u* up to ε4 inclusively, and the
critical exponent η=γ2|u=u* was computed after.
References
1. Kleinert H., Schulte-Frohlinde V. Critical Properties of φ4-Theories. - Word
Sci. Publ., River Edge, NJ, 2001.
2. Васильев А.Н. Квантовополевая ренормгруппа в теории критического
поведения и стохастической динамике. – СПб.: изд-во ПИЯФ, 1998.
3. de Alcantara Bonfim O.F., Kirkham J.E., McKane A.J. // J. Phys A, 13:7
L247–L251 (1980).
126
Surface states in semi-infinite superlattice with rough
boundary
Bylev Alexander
alexbylev@gmail.com
Scientific supervisor: Prof. Dr. Kuchma A.E., Department of
Statistical Physics, Faculty of Physics, Saint-Petersburg State
University
Abstract
Surface states in semi-infinite superlattices with rough boundary are studied
using simple one-band approximation for nondegenerate bands of a crystal lattice.
Superlattice is modeled by δ-like potential wells, surface potential well has different depth than other potential wells and roughness is modeled by dependence
of surface potential power on coordinates in surface plane. It is shown that wave
function of surface state will decrease in spreading direction as a result of scattering
on roughness. Expression for coefficient of damping of surface state in longitudinal
direction is derived; contributions caused by scattering along the surface and by
transformation of surface wave to volume wave were distinguished; numerical
calculations of this coefficient are done.
Introduction
Boundary between solid body and vacuum or other media could be a source of
special states of electrons, called surface states. Such states are localized near the
boundary of the body. Possibility of existence of such states was firstly studied in
[1]. In our work we studied surface states in semi-infinite superlattices.
Superlattices are referred to call solid body structures with additional periodic
potential with bigger period than period of a crystal lattice. Parameters of superlattice potential could be widely changed, due to it is possible to controllable
change band structure of a spectrum. Superlattices were discovered early in the
20th century and were theoretically studied in [2]. Now superlattices are widely
used in electronics and optics.
In general case surface is not a sharp change from nonpertubed periodic potential to the outer space. It’s also necessary to take into account, that surface might
be covered by adsorbent layer. This kind of roughness will cause scattering of a
surface wave, which is an electron surface state, on the surface roughness. This
scattering will also cause transformation of a surface wave to volume waves. It’s
interesting to estimate damping of a surface state caused by this scattering.
Model of considered system
In one band approximation [3] for nondegenerate electron bands in crystal
Schrodinger equation for envelope wave function is:
2




∆Ψ(r ) + V (r )Ψ(r ) = E Ψ(r )
−
2m
127
In order to simplify the problem we consider the following potential consisting
of δ-wells (Fig. 1). Roughness of surface is described by dependence of power of
surface potential on coordinates in surface plane [4].
Fig. 1.
→
Here ρ = (x,y) z- axis in direction perpendicular to the surface of superlattice,
→
a- period of superlattice, ν0 - superlattice potential, u0+ u1( ρ ) - surface potential
→
and u1(ρ ) describes roughness of surface layer, u̅ - outer space potential
So we are solving the following equation:
∞
2 

  
∂2 
−∆ ρ − 2 + u θ(− z ) − (u0 + u1 (ρ))δ ( z ) − v0 ∑ δ ( z − na )  Ψ(ρ, z ) == E Ψ(ρ, z )
2m 
∂z
n =1


We suppose that roughness is random function with zero average value. We also
accept that roughness is statistically homogeneous with correlation function:


 
u1 ( ρ )u1 ( ρ ′) = ∆U 2 W ( ρ − ρ ′) Damping of surface state
According to our model we suggest that wave function of surface state is
 
 

Ψ ( ρ , z ) = e ik0 ρ Ψ (q0 , z ) + ∑ a (k )e ikρ Ψ (q, z ) ,
k ≠ k0
 

where e ik0 ρ Ψ (q0 , z ) - surface wave, e ikρ Ψ (q, z ) - surface waves or Bloch waves
(these functions are solutions of Schredinger equation for z>0) when
2mE  
q02 + k02 = q 2 + k 2 = ε  ε = 2 

 
and
ε <u .
On the surface z=0 the derivative of wave function must satisfy the following
conditions (functions Ψ(q,z) is chosen so that Ψ(q,0)=1)

 

− Ψ ′(q0 ,+0) + Ψ ′(q0 ,−0) − u0 − ∑u1 (k 0 − k )a (k ) = 0


 k ≠ k
 


′
′
′
′) = 0
(
)
−
Ψ
+
+
Ψ
−
−
−
−
−
−
(
q
,
0
)
(
q
,
0
)
u
a
(
k
)
u
(
k
k
)
u
(
k
k
)
a
(
k
∑
0
1
0
1

 
′
≠
k
k

128
0
0
We solve these equations by perturbation theory in the 2nd order. Then we get
 
 
q0
u1 (k0 − k )u1 (k − k0 )
λ (q0 ) − cos q0 a )− η(q0 ) + u0 = ∑
(

q
sin q0 a
k
(λ (q) − cos qa) − η(q) + u0
sin qa
where
v0
sin qa , η(q ) = u − q 2 2q
This equation determines wave vector of surface state of electrons. Rhs of this
equation is complex quantity because of complexity of λ(q) for Bloch waves and
imaginary contributions from zeros of denominator. That’s why wave number of
surface state of electron k0 becomes a complex quantity too. Imaginary part of
wave number k0 describes the damping of surface wave propagating along rough
boundary. Phase velosity of surface wave is changing too.
In our analysis we use the following correlation function
λ (q ) = L ± L2 − 1, L = cos qa −
2
−
 
 
2π ∆U
u1 (k0 − k )u1 (k − k0 ) =
e
(2l ) 2 pc 2
 
( k − k0 ) 2
2 pc 2
where pc is inverse correlation length of roughness. →
Assuming that damping is small, we put q̅0 and k̅0 of plane surface case into
rhs of previous equation. In this case rhs of this equation will simply give us correction to surface potential u0 and we get the following expression for imaginary
part of this correction
ε − q 2 + k0 2
−
 ε − q2 k 
∆U 2 ε
2
dq 2
q
2p
0
2
Im ∆u0 = −
∫ 2 F (q) 2 sin qa 1 − L (q) e c I 0  pc 2  −
pc 2 −∞
− π∑
where
i
∆U
−
2
pc 2
⋅
F (q ) ≡
ki 2 + k02
2
kk 
qi e 2 pc
I 0  i 20 
F ′ (qi )
 pc 
q
(λ (q) − cos qa) − η(q) + u0
sin qa
In this equation integral gives us contribution caused by transformation of
surface wave to volume Bloch waves, and the sum gives us contribution caused
by scattering to the states propagating along the surface of superlattice. As a result
we get the correction to k0̅
δ k0 = −
q 0 ∆u 0 (q 0 , k 0 ) k 0 F ′(q 0 )
129
In case of u̅ =0 and u0 >ν0 estimations of contributions of scattering along the
surface and of transformation to volume wave are
k 2 + k02
−
 
∆U 2
 kk
2
(Im δk0 )s = − p 2 π (u0 a) e 2 pc I 0  p 20 


c
(Im δk0 )Bloch = −
∆U 2
2
c
p
c
(v0 a )3 e− v a
(u0 a − v0 a )2 + (v0 a )2 e− v a
0
0
e
−
k 2 + k02
2 pc2
 
 kk
I 0  20 
 pc 
As we can see, contribution of transformation of surface waves to volume waves
is strongly suppressed by e−ν a factor. And both contributions become comparable
when ν0 goes to u0 . Contributions to parameter of damping caused by scattering as
functions of inverse correlation length of roughness are illustrated by Figs. 2, 3.
0
0.20
0.15
0.10
0.05
2
4
6
8
10
6
8
10
Fig. 2.
0.00020
0.00015
0.00010
0.00005
2
4
Fig. 3.
130
Fig. 2 gives contribution of scattering to the states propagating along the surface and Fig. 3 gives contribution of transformation to volume waves. In case of
small and large inverse correlation length (in comparison with k0 ) damping goes
to zero that is physically proved true. Small inverse correlation length practically
corresponds to the plane surface and in case of large inverse correlation length
averaging of roughness occurs.
References
1. Tamm I.E. // Sow. Phys, 1932, v.1, p. 733.
2. Keldysh L.V. // FTT, 1962, v.4, p. 2265.
3. Peter Y.Yu, Manuel Cordona. Fundamentals of semiconductors. - Springer,
2010.
4. Kuchma A.E., Kovalevsky D.V., Voronin N.V. // Vestnik S.-Peterburgskogo
Universiteta, Ser. 4. 2008. Number 4. pp. 3-15.
131
Modeling of thermal-hydraulic processes in complex
domains by conservative immersed boundary method
Chepilko Stepan
Ar.noion@gmail.com
Scientific supervisor: Yudov Y.V., Alexandrov Research Institute
of Technology
Introduction
Today main approach within Computational Fluid Dynamics (CFD) for numerical flow simulations in domains with complex boundaries is so-called boundaryfitted methods on unstructured grids. Due to its universality they are implemented
in most widely used CFD-codes (CFX, Star-CD). In these methods grid is generated
basing on boundary representation inside computational domain. But in recent
decade there is a growing interest in developing of alternative immersed boundary methods, where computational domain is embedded in Cartesian grid, what
significantly simplifies spatial discretization of equations in most cells and reduces
volumes of data for storage.
The aim of present work is implementation of conservative Cartesian cut-cell
immersed boundary method based on developing of work [1] for modeling of
two-dimensional viscous incompressible laminar flows of Newtonian fluids in
Boussinesq approximation. Stated goal was realized in F77 program with following
key features: spatial discretization by finite volume second-order accurate method
on collocated Cartesian grid, temporal discretization by two-step fractional step
procedure, direct numerical solver for pressure Poisson equation, piecewise linear
modelling of complex boundaries, formation of computational control volumes
of rectangular fully-fluid cells and trapezoidal partly-fluid ones near the boundary
with cell-merging technique for small cut cells and, finally, accurate discretization
of governing equations at cut cells by means of second-order accurate interpolation
stencils, slightly modified as offered in [1].
Accuracy and fidelity of the programmed solver have been validated by
simulating a number of benchmark laminar flows including natural convection in
closed cavities. Ability to simulate flows with arbitrary complex boundaries has
been demonstrated.
Numerical scheme
The governing equations are Navier-Stokes system for incompressible
Newtonian viscous fluid with scalar transport equation written in integral form
for each control volume CV with control surface CS

∫ vndS = 0
∂t
∫
CV

v = − ∫ ∇pdV −
CV
∫
CS
CV
 
 
v (vn ) dS + η ∫ (n∇)v dS +
CS
132

∫ s (ϕ) dV
CV
∂t


∫ φ dV = − ∫ φ(vn ) dS + Γ ∫ (n∇)φ dS
CV
CS
,
CS
where ν is velocity vector, n – external normal, p – pressure, φ - scalar quantity,
s(φ) – source term, η - kinematical viscosity, Γ – diffusive coefficient.
Two-step fractional step method [1] is used for advancing solution in time. In
this time-stepping scheme, the solution is advanced from time level “n” to “n+1”
through an intermediate convection-diffusion step, where the momentum equations without the pressure gradients terms are first advanced in time using fully
explicit scheme. This step is followed by pressure-correction step, when pressure
Poisson equation has to be solved and divergence-free velocity field at time level
“n+1” is obtained by pressure correction. Finally, scalar field at time level “n+1”
is computed explicitly. The spatial discretization is performed on a 2-D Cartesian
mesh using a cell-centered collocated arrangement of primitive variables p, u, v,
φ with introduction of face-center velocities U, V, which are used for surface flux
computation, for strict implementation of mass conservation at pressure-correction
stage and for elimination of unphysical pressure field problem.
Inclusion of immersed boundaries is realized as follows. The immersed boundary is first represented by a series of piece-wise linear segments. Based on this
representation one determines Cartesian cells that are cut by the boundary. Cut-cells
which center lies in the fluid are reshaped by discarding the part of these cells that
lies in the solid. Pieces of cut cells whose center lies in the solid are absorbed by
neighboring fluid cells for removing numerical instability of the solver. This results
in the formation of trapezoidal control volumes for cells near the boundary and of
rectangular ones for internal fluid cells (Fig. 1).
The key aspect of immersed boundary
solver is an accurate discretization of governing
equations at cut cells, which includes evaluation
of mass, convective, diffusive fluxes and pressure gradients on the cell-faces of trapezoidal
cells from neighboring cell-center values with
second-order accuracy. For this aim surface
integrals for fluxes are computed by mid-point
rule, which requires accurate evaluation of integrand at the face-center. For regular internal
cells the integrand is calculated with second- Fig. 1. Variables arrangement and
order accuracy using a linear profile between control volumes formation.
nodes. But for cut faces spatial interpolation stencils are used (slightly modified
as offered in [1]), which generally represent value at node P as weighted average
of values at neighboring nodes nb.
Results and discussions
Accuracy and fidelity of the programmed solver was validated by simulating
a number of benchmark problems with laminar recirculating flows in closed en133
closures (cavities) with complex geometry. Here we collect results of computation
both of hydrodynamic flows and of natural convection ones. Each flow problem is
fully characterized by a set of specific dimensionless numbers - Reynold’s (Re),
Prandtl’s (Pr), Rayleigh’s (Ra), which are combinations of characteristic dimension parameters.
The first problem is laminar flow in cavity with inclined side walls and moving lid.
Streamlines and velocity profiles, compared with [2], are shown in Figs. 2, 3.
Fig. 2. Llid-driven cavity with
inclined side walls, streamlines
for 30° (top) and 45° (bottom)
at Re=1000.
Fig. 3. Lid-driven cavity with inclined side
walls, velocity centerline profiles u(y) (left) and
v(x) (right) for 30°at Re=1000.
Fig. 4. Local heat transfer coefficients for Ra=48000, Pr=0.706, ε=0.652 (top) and
Ra=49300, Pr=0.706, ε=-0.623 (bottom). Top curve is for internal wall circle, bottom – for external.
The second problem is natural convection between eсcentric circles with eccentricity ε and hot internal and cold external boundaries. Angle distribution of
134
Fig. 5. Streamlines for positive (a) and negative (b) eccentricity. Parameters are
the same as in Fig. 4.
local heat transfer coefficients along hot and cold walls and its agreement with
experimental data [2] and other numerical studies [3] (DINUS-code) are shown in
Fig. 4. Small difference for ε>0 for top part of internal circle might be caused by
local flow instability. Streamlines for both cases are shown in Fig. 5.
The last investigated problem is flow in square cavity with hot and cold side
walls and adiabatic top and bottom ones. For demonstration of abilities of the
solver there were studied two possibilities: computational domain embedding
into ordinary non-rotated grid and into 45°-rotated grid, which is equivalent to
simultaneous rotation of cavity and gravitational force. Streamlines and centerline
velocity profiles, compared with [4], are depicted in Figs. 6, 7.
Fig. 6. 45 -rotated square cavity,
streamlines for Ra=106, Pr=0.71.
°
Fig. 7. 45°-rotated square cavity, velocity centerline profiles profiles u(y) (left)
and v(x) (right) for Ra=106, Pr=0.71.
Detailed numerical validation includes comparison of maximum centerline
velocities (umax, vmax), their locations (y,x) and Nusselt (Nu) number (full heat flux
over heated wall) with benchmark simulations on fine grids [4] and is summarized
in Table 1 for various Rayleigh numbers 104-106. Full comparison is performed
135
for non-rotated cavity (α=0°) and partial – for 45°-rotated one (α=45°). One could
observe small difference <0,8% for non-rotated case and <1,1% for rotated one.
Table 1. Comparative study with benchmark solutions for Ra=104-106.
Simulation
128x128, α=0
Benchmark solution [12]
128x128, α=0
128x128, α=45○
Benchmark solution [12]
128x128, α=0
128x128, α=45○
Benchmark solution [12]
u max
y
4
Ra=10
16.182
0.824
16.176
0.826
Ra=105
34.88
0.856
34.53
34.74
0.855
Ra=106
64.77
0.848
64.13
64.83
0.850
v max
x
Nu
19.626
19.624
0.121
0.12
2.244
2.245
68.53
68.43
68.64
0.0663
0.066
4.519
4.522
219.22
217.48
220.5
0.0351
0.039
8.83
8.827
Conclusions
There was realized conservative 2-dimensional cut-cell immersed boundary
method for laminar incompressible flow modelling. There were investigated main
features of proposed approach in spite of many simplifications, such as explicit
convective-diffusion scheme, direct pressure solver, Dirihlet flow boundary conditions, etc. Universality of approach was checked for various complex geometries.
The program was tested for numerous benchmark flow problems including natural
convection in complex cavities.
References
1. Ye T., Mittal R., Udaykumar H.S., Shyy W. // Journal of Computational Physics,
V. 156, pp. 209-240 (1999).
2. Ray S., Date A. // Numerical Heat Transfer. Part B, Vol. 38, pp. 93-131
(2000).
3. Yudov Yu.V. // Mathematical models and computer simulations, Vol. 3, 2, pp.
185-195 (2011).
4. Darbandi M., Schneider G.E. // Numerical Heat Transfer. Part A, Vol. 35, pp.
695-715 (1999).
136
Development of functional integration techniques for
drift-diffusion processes on Riemannian manifolds
Chepilko Stepan
Ar.Noion@gmail.com
Scientific supervisor: Dr. Dmitrieva L.A., Faculty of Physics, SaintPetersburg State University
Introduction
Today complete self-consistent rigorous mathematical foundation of functional
integration (FI) is absent. Moreover, there exists a deep gap between theoretical
physicists, who successfully use so-called path integrals as a formal tool, and
mathematicians, who obtain different exact FI results in various fields. The main
problem in FI foundation, to our knowledge, is a strong interplay between stochastic
processes, evolutionary equations and Riemannian geometry.
So the aims of present work are follows: to consider most general problem of
drift-diffusion Ito process and to describe its relation with variable coefficients
parabolic type PDE and Riemannian geometry; to present rigorous functional
integral representation for Cauchy problem of associated PDE based on measure,
generated by PDE fundamental solution [1]; to present path integral construction
based on multiplicative semigroup representation in which local asymptotic either
of exact fundamental solution or equivalent approximating kernel operator with
the same generator could be used [2, 3]; to connect both pictures together and to
point out an ambiguities in formal limiting expressions and local path integral
discretizations [4].
Functional integration
Let us consider time-homogeneous Ito stochastic differential equation (SDE)
in space X = Rm
dq i (t ) = bi (q )dt + eai (q )dW a (t ) ,
where Wa(t) is m-dimensional Wiener process with property dWadWb~δabdt as dt→0.
Solution of this equation is stochastic Ito process qx (t), which for 0≤τ < t ≤T satisfies following integral equation
t
t
τ
τ
qxi (t ) = x i + ∫ bi (q ( s ))ds + ∫ eai (q( s ))dW a ( s ), qx (τ) = x ∈ X (1)
Here the last integral is stochastic integral over Wiener process (in Ito sense). Since
dimensions of Wiener and Ito processes are the same Ito process qi(t) is diffusion
process with drift coefficient bi(q) and diffusion matrix eia(q). Moreover, every
Ito process is markovian process, so its transition probability function generates
evolutionary family in Banach space B(X) of functions on X
U (t − τ) f ( x) = E[ f (qx (t ))] = ∫ f ( y ) P (t − τ, x, dy ) X
137
(2)
where P(t-τ,x,dy) is transition probability function of process qi(t). In time- homogeneous case evolutionary operator family forms simply operator semigroup
in terms of variable s=t-τ
U ( s1 )  U ( s2 ) = U ( s1 + s2 )
with generator D defined as
U (s) − I
Df ( x) = lim
f ( x)
s →0
s
From Ito formulae one could find that generator is
D = 12 eai (q )eaj (q)∂ ij + b i (q )∂ i
It could be shown, that function u(s,x)=U(s)f(x), s=t-τ is solution of following
initial Cauchy problem for parabolic type PDE
∂ s u ( s, x) = Du ( s, x), u (0, x) = f ( x)
called backward Kolmogorov equation. Important fact is that transition probability
density p(s,x,y) of Ito process is a fundamental solution of Kolmogorov (backward
and forward) equations
∂ s p ( s, x, y ) = Dx p ( s, x, y ) ∂ s p ( s, x, y ) = D y * p ( s, x, y )


p (0, x, y ) = δ ( x − y )

 p (0, x, y ) = δ ( x − y )
Stochastic process q(t’), τ ≤ t’ ≤ t is defined by a system of finite-dimensional
distributions
µt1 ...tn ( A1 × ... × An ) = P(q(t k ) ∈ Ak , k = 1,..., n)
where Ak is subset of σ-algebra Σ of subsets of space X. This system introduces
finite-additive measure μ on algebra of cylindrical sets Q
Qt1 ...tn ( M n ) = {q (t ) : (q (t1 ),..., q (tn )) ∈ M n }, M n ∈Σ n
by relation
µ[Qt1 ...tn ( M n )] = µ t1 ...tn ( M n )
According to Kolmogorov’s measure extension theorem measure μ admits
σ-additive extension on minimal σ-algebra of all cylindrical sets Q in space of
functions [τ,t]→X. Let us consider as simple example a diffusion equation in Rm
∂ t u = 12 σ 2 ∆ R n u = 12 σ 2 ∂ i ∂ i u
which has following well-known fundamental solution
2
p(t , x, y ) = (2πtσ )
−
m
2
e
−
( x− y )2
2 tσ 2
So probability that Brownian particle, started at point x(t0)=x0=x will occupy
cylindrical set {a1≤ x(t1)≤b1,..., an≤ x(tn)≤bn} is
n
µ = ∏ (2π(ti − ti −1 )σ )
2
i =1
−
m b1
2
bn
a1
an
∫ ...∫ e
−
n
(x −x
)2
∑ 2 (tii − ti−i−11) σ2
i =1
dx1 ...dxn
Extension of this measure to space of continuous but nowhere differentiable
functions Sx(τ,t)(X) is called Wiener measure μW. Since one has a measure on functional space Sx(τ,t)(X) of trajectories x(t) one could define an ordinary Lebesque
integral for functionals F[x(t)]
I=
∫ F[ x]d µ
W
S x (τ ,t ) ( X )
138
But this expression is unconstructive, since explicit expression for μW is still
undefined. So one could consider “path” integral
n
I * = lim ∏ (2π(ti − ti −1 )σ )
n →∞
2
−
i =1
m
2
∫ F ( x ,...x )e
n
−
n
1
n
(x −x
)2
∑ 2 (tii − ti−i−11) σ2
i =1
dx1 ...dxn
Xn
where Brownian trajectory x(t) is approximated by piecewise constant curve
xn(t)=xk=x(tk), tk ≤ t ≤ tk+1 1 and Fn (x1,...,xn)=F(xn(t)). It turns out, that for Wiener
measure I=I* for some set of functionals F.
Measure on functional space of trajectories of Ito process, generated by fundamental solution of corresponding backward Kolmogorov equation
∂ s u ( s, x) = ( 12 eai (q)eaj (q)∂ ij + b i (q )∂ i )u ( s, x)
we will call Ito measure. The following statement is very important. If q1, q2 are
two Ito processes with drift coefficients b1,b2 and the same diffusion matrix eai
dqk (t ) = bk (qk )dt + (e(qk ), dW (t )), k = 1, 2
then corresponding Ito measures μ1, μ2 are absolutely continuous with density
t
t

d µ2
[q1 (⋅)] = exp  ∫ (α (q1 ( s )), dW ( s )) − 12 ∫ || α (q1 ( s )) ||2 ds
d µ1
τ

τ
where α=e-1(q)(b2(q)- b1(q)). Let us now introduce geometric objects.
Ito process qi(t) induces structure of Riemannian manifold (M,g) in Rm. Let us
equipe Rm with inverse metric gij(x)=eai(x)eaj(x) and Levi-Chevita connection Γijk
Then one could rewrite generator in covariant form
D = 1 e i ( q )e j ( q ) ∂ + b i ( q ) ∂ = 1 ∆ + f i ∂ a
2
a
ij
i
2
M
i
where f i (x) is covariant drift vector (bi (x) is not a vector due to Ito formulae)
f i = bi + 12 g lj Γ lji
and Δ – invariant Laplacian on M
1
1
M
−
∆ M = g ij ∂ ij − g ij Γijk ∂ k = g 2 ∂ i ( g 2 g ij ∂ j )
at interval [τ,t]
Solution of initial Cauchy problem for drift-diffusion on M
∂ s u ( s, x) = ( 12 ∆ M + f i ∂ i )u ( s, x) (3)
(3)

(
0
,
)
(
)
u
x
=
f
x

could be expressed by Lebesque functional integral over condition Ito measure μIy
u ( s, x) = ∫ dy f ( y )
∫
d µ Iy (q )
( τ ,t )
Sx , y ( X )
where space S (τ,t)(X) consists ofX trajectories
of Ito process
(1), started at point x
x,y
and finished at y. Using measure equivalence theorem one could express solution
u(s,x) as FI over condition Riemannian Wiener measure μWy
u ( s, x) = ∫ dy f ( y )
X
∫
S x( τ, y,t ) ( X )
t
t

d µWy (q ) exp  ∫ gij f i eaj dW a − 12 ∫ gij f i f j dt '
τ

τ
where μ is generated by fundamental solution of diffusion equa0tion on M
139
y
W
∂ t u (t , x) = 12 ∆ M u (t , x)
and space Sx,y (τ,t)(X) consists of solutions of following Ito SDE (Brownian motion on M)
t
t
q i (t ) = x − ∫ 12 g lj (q (t '))Γ lji (q (t '))dt '+ ∫ eai (q (t '))dW a (t ')
τ
τ
Path integration
Using Chapman-Kolmogorov equation for transition probability density p(s,x,y)
(or, equivalently semigroup property of fundamental solution) and taking limit of
n→∞, one obtains “path” integral multiplicative representation of fundamental
solution
 s

 s

p ( s, x, y ) = lim ∫ p 
, x, y1  ... p 
, yn , y  dy1 ...dyn
n →∞




1
1
n
n
+
+
Xn
Existence of the limit is a consequence of following general Chernoff theorem
for operator semigroups. Let S(t) be an operator semigroup with generator D and
property ||S(t)||=1+O(t), t→0. Then for every f from B(X) for n→∞, ∑ni=1ti→t,
S (t1 ) ⋅ ... ⋅ S (tn ) f → etD f For S(t) one could take Ito evolutionary family U(s) (2) with exact transition probability density p(s,x,y). So from the above theorem one obtains coincidence of
Lebesque FI representation of fundamental solution and its “path” integral limit.
Exactly this relation (I=I*) we have already cited for Wiener integral with explicitly
known fundamental solution.
The key issue of path integral approach is an ability to replace exact p(s,x,y)
for finite time s by its local asymptotic for s→0 and “close” in some sense points
x,y. Chernoff's theorem allow one not only to use local asymptotic of exact transition probability density in multiplicative semigroup representation, but to use in
some sense equivalent operator family S(s) (usually of kernel operators), called
approximating, instead of exact one U(s). Condition of equivalency of multiplicative representations by approximating operator family are follows: ||S(s)||=1+O(s),
s→0 and
S ( s) − I
lim
= D = 12 ∆ M + f i ∂ i
s →0
s
One way to find1 exact local asymptotic is to try following WKB-ansatz for
fundamental solution [3] for ρ2(x,y)<s and s→0
m
2
ρ2 ( x , y )
+ a1 ( x , y ) + sa2 ( x , y ) + O ( s 2 )
2s
(5)
p ( s, x, y ) ≅ (2 sπ)
g ( y )e
Substitution of this ansatz to parabolic equation (3) and requirement for coincidence
of asymptotic series in both sides give rise to recurrent equations for determination of functions ak(x,y) and ρ(x,y). Leading term of this expansion is Riemannian
geodesic distance ρ(x,y), which is defined as integral
ρ( x, y ) = ∫ gij ( x(t )) x i (t ) x j (t )dt
−
−
C
along geodesic curve C connecting x and y
xi (t ) + Γkji ( x(t ) x k (t ) x j (t ) = 0
140
Each approximating operator family S(s) gives rise to corresponding formal
continuous “action” S and “Lagrangian” L for limiting multiplicative representation (4)
p ( s , x, y ) =
∫
S x( τ ,t ) ( X )
t
Dxe − S [ x ] , S [ x] = ∫ L( x(t '), x (t '))dt '
τ
(6)
However, continuous formal expression (6) must be interpreted only as a limit
of multiplicative representation. For our kernel example (5) one has to interpret
(6), where L is given in work [3], only as limit (4) with p replaced by p. There are
several variations of formal “Lagrangians” (6) that are suggested in literature. Very
often formal “Lagrangians” are written in discretized from (see [4]), but correctness
and fixation of ambiguities in these expressions in general case on non-constant
metric g is still an open question.
Conclusions
Problems in rigorous FI foundations were reviewed. The following existed, but
not clearly formulated in literature items and results were put together: general
problem of relation between drift-diffusion process, Riemannian geometry and
evolutionary equations of parabolic type was investigated. Basing of this interplay
solution of backward Kolmogorov equation was represented as Lebesque functional integral over Ito measure for corresponding stochastic process. Difference
in functional and path integral definitions was mentioned including problem of its
correspondence. Path integral multiplicative representation was build using local
asymptotic of fundamental solution; problem of variety and ambiguity in formal
continuous Lagrangians for path integrals was discussed.
1.
2.
3.
4.
References
Daleckiy Ju. // Mathematical Analysis (Russian), pp. 83–124 (1966).
Smolyanov O., Weizsacker H., Wittich O. // arXiv:math/0409155v3 (2008).
Alimov A.L., Buslaev V.S. // Vestnik LGU (Russian), 1, pp. 1-14 (1972).
Kleinert H. Path integrals // World Scientific (2009).
141
Multifractal generalization of the detrending moving
average approach to time series analysis
Ganin Denis
vitamin.diego@gmail.com
Scientific supervisor: Dr. Kuperin Y.A., Department of Nuclear
Physics, Faculty of Physics, Saint-Petersburg State University
Introduction
The analysis of financial time series has been the focus of intense research
by the physics community in the last years [1, 2]. An important aspect concerns
concepts as scaling and the scale invariance of return fluctuations [3]. The aim
was to characterize the statistical properties of the series with the hope that a better
understanding of the underlying stochastic dynamics could provide useful information to create new models which are able to reproduce experimental facts. It has
been recently noticed that time series of returns in stock markets are of multifractal
(multiscaling) character.
Among other approaches the detrending moving average (DMA) algorithm [4]
is a widely used technique to quantify the long-term correlations of non-stationary
time series. In the present paper we develop a multifractal generalization of DMA
that is multifractal detrending moving average (MFDMA) algorithms for the
analysis of one-dimensional multifractal measures and multifractal time series.
The performance ability of the elaborated MFDMA methods has been investigated
using synthetic multifractal measures, namely binomial measure with the parameter
p=0.25, and fractal Brownian motion with different values of the parameter H. We
have found that the estimated multifractal scaling exponent τ(q) and the singularity spectrum f(α) are in good agreement with the theoretical values. It has been
shown that the MFDMA algorithm also out-performs the multifractal detrended
fluctuation analysis (MFDFA) proposed in [5]. The proposed MFDMA method was
applied to analyzing the time series of the DJIA Index and its multifractal nature
has been fairly detected.
Results and Discussion
For the beginning we have tested the method by applying it to the binomial
measure, for which there is a analytical representation for the form of the singularity
spectrum f(α) and for the set of Renyi dimensions [6]. Compare our results with
the theoretical ones. The graph of the singularity spectrum f(α) (Fig. 1 (c)) has the
form, which was predicted theoretically (see, for instance [6]). Also, we can calculate the spectrum using the program FracLab. For clarity, depict both depending
in one graph (Fig. 1 (c)). It is obvious that the graphs are practically identical; it
indicates good accuracy of the proposed here method. In addition, if we construct
a spectrum of the generalized Renyi dimensions (D(q)) (Fig. 1 (d)), we find that
the data obtained are completely consistent with theoretical predictions.
142
b
a
c
d
Fig. 1. a) Binomial measure (parameter p=0.25); b) Multifractal scaling exponent
τ(q) for the binomial measure (p=0.25); c) Graphs of the singularity spectrum f(α) for binomial measure (p=0.25), constructed by the method MFDMA (green line)
and using FracLab (blue line); d) Spectrum of Renyi dimensions for the binomial
measure (p=0.25) [6].
We also apply a new method to a monofractal Brownian motion [5]. For the
calculations we take initial time series with Hurst exponent H=0.5, containing 4096
points. From the graph of the spectrum Renyi dimensions (Fig. 2 (d)), one can see
that D(q)=1, as expected, since the Renyi dimension must be equal to dimension
of measure, that is unity. The graph of the multifractal scaling exponent τ(q) is a
linear function (Fig. 2 (b)), which is consistent with the theory, because the time
series in question is monofractal.
Applying the method MFDMA, we find that the singularity spectrum f(α) is
almost completely degenerated to a point (Fig. 2 (c)), as it should be for the monofractal Brownian motion. It is known that FracLab program builds the same form of
the singularity spectrum f(α) for a multifractal series and for a monofractal series,
that is obviously wrong. For clarity, we represent both results in the same graph
(Fig. 2 (c)). It is obvious that method MFDMA has the advantage in comparison
the methods used in FracLab program, because the method MFDMA gives a true
form of the singularity spectrum f(α).
143
a
b
c
d
Fig. 2. a) Series of Brownian motion (Hurst exponent H=0.5); b) Multifractal
scaling exponent τ(q) for the Brownian motion (H=0.5); c) Graphs of the singularity spectrum f(α) for the Brownian motion (H=0.5), constructed by the method
MFDMA (blue line) and using FracLab (green line); d) Spectrum of Renyi dimensions for the Brownian motion (H=0.5).
After that we have applied the our method to real data, such as the values of the
Dow Jones Index [3]. In this case, we use the logarithmic increments of time series
rather than the Dow Jones Index itself. We can determine the spectrum of Renyi
dimensions D(q) which is practically constant and equal to unity (Fig. 3 (d)). We
see in the graph (Fig. 3 (c)) that the singularity spectrum f(α) isn’t degenerated to
a point, as it was for a monofractal Brownian motion. It indicates that economic
time series of values of the Dow Jones Index have multifractal nature [3].
Now, compare the form of the singularity spectrum f(α) calculated for a fractal
Brownian motion and for the Dow Jones Index. The graph (Fig.4) clearly shows
that the peak of spectrum of the Dow Jones Index is shifted relative to peak of the
spectrum of fractal Brownian motion, and the scope of the spectrum Dow Jones
index is much broader. This result suggests that the time series of economic indexes
have multifractal nature, as expected.
Conclusion
The implemented method MFDMA gives better results compared with calculations in Fraclab program. Algorithm MFDMA detects multifractality of time
series much better. This method may be considered as one of the most accurate in
identifying the main multifractal characteristics of nonstationary time series.
144
a
c
b
d
Fig. 3. a) Series of Dow Jones indexes (it contains 3000 points); b) Multifractal
scaling exponent τ(q) for Dow Jones; c) Graphs of the singularity spectrum f(α)
for Dow Jones, constructed by the method MFDMA (blue line) and using FracLab
(green line); d) Spectrum of Renyi dimensions for Dow Jones.
Fig. 4. The singularity spectrum f(α),
obtained for a fractal Brownian motion (blue line) and for a values of the
Dow Jones Index (green line), (enlarged scale).
References
1. Bouchaud J.P., Potters M. Theory of Financial Risk. - Cambridge University
Press, Cambridge, 2000.
2. Mantegna R.N., Stanley H.E. An introduction to Econophysics. - Cambridge
University Press, Cambridge, 1999.
3. Mantegna R.N., Stanley H.E. // Nature, 376, 46–49 (1995).
4. Alssio E., Carbone A., Castelli G., Frappietro V. // Eur. Phys. J. B, 27, 197
(2002).
5. Kantelhardt J.W. at al // Physica A, 316, 87-114 (2002).
6. Feder A. Fractals. - Moscow: Mir, 1991, -254 p.
145
Propagation of photons and massive vector mesons
between a parity breaking medium and vacuum
Kolevatov Sergey
kss2005@list.ru
Scientific supervisor: Prof. Dr. Andrianov A.A., Department of
High Energy and Elementary Particles Physics, Faculty of Physics,
Saint-Petersburg State University
Introduction
The problem of crossing the boundary between the vacuum and a parity breaking
medium, which may occur in the presence of so-called axion fields, is important
for modern physics because axions are realistic candidates for the role of dark
matter. This assumption is attractive by the fact that these particles were result
of the hypothesis which explained the strong CP problem in QCD, and only after
some time it was shown that they, in principle, could be dark matter. However,
the experiments showed that on scales comparable to the size of the Universe
axion fields are not observed. Yet, not applicable to the specific effects at large
distances, does not preclude the existence of these fields on the scale of stars and
even galaxies. One may think of an axion background accumulated by very dense
stars like neutron ones or even of bosonic axion stars [1]. Another interesting area
for observation of parity breaking is the heavy ion physics [2]. In the occurrences
of axion-like background in astrophysics or heavy ion physics the existence of
a boundary between the parity-odd medium and the vacuum is quite essential.
For star condensed axions there is evidently a boundary where axion background
disappears and photons distorted by it escape to vacuum.
Results and Discussion
We start from the Lagrange density which describes the propagation of a vector
field in the presence of a pseudoscalar axion-like background,
(1)
where Aµ and acl stand for the vector and background pseudoscalar fields respectively,
is the dual field strength, while B is the auxiliary Stuckelberg scalar field with
real
. The positive dimensionless coupling g > 0 and the (large) mass parameter M»m do specify the intensity and the scale of the pseudoscalar-vector
interaction. Notice that we have included the Proca mass term for the vector field
because, as it is discussed in [2], the latter is required to account for the strong
146
interaction effects in heavy ion collisions supported by massive vector mesons
(ρ, ω, ...) in addition to photons. Moreover, as thoroughly debated in [3], the
mass term for the vector field appears to be generally necessary to render the
dynamics self-consistent in the presence of a Chern-Simons lagrangian and is
generally induced by radiative corrections from the fermionic matter lagrangian.
The auxiliary Stuckelberg lagrangian, which further violates gauge invariance
beyond the mass term for the vector field, has been introduced to provide – just
owing to the renowned Stuckelberg trick – the simultaneous occurrences of
power counting renormalizability and perturbative unitarity for a general interacting theory. Moreover, its presence allows for a smooth massless limit of the
quantized vector field.
We shall consider a slowly varying classical pseudoscalar background of
the kind,
(2)
where θ(...) is the Heaviside step distribution, in which a fixed constant four
vector ζ with dimension of a mass has been introduced, in a way to violate
Lorentz and CPT invariances in the Minkowski half space ζ ⋅x < 0. In what
follows we shall suppose that ζ2 ≠0. If we now insert the specific form (2) of
the the pseudoscalar background in the pseudoscalar-vector coupling lagrangian we can write the equivalent Lagrange density,
(3)
in which the gauge invariance is badly broken by all the terms but the one, i.e. the
Maxwell’s radiation lagrangian. Then the field equations read,
(4)
This system gives us different solutions in different half-spaces. The first is wellknown Proca-Stuckelberg vector field, the second - Maxwell-Chern-Simons vector
field, which have been extensively discussed and applied in [3] for the massive
case and in [4] for the massless case.
This solutions face one another at the hyperplane ζ ⋅x = 0. Hence locality of the
quantized wave fields does require equality on the surface separating the classical
pseudoscalar background from the vacuum: namely,
(5)
We discuss the case of a spatial Chern-Simons vector ζµ = (0, -ζx, 0, 0) so that
δ(ζ⋅x) = -δ(x)/ζ x .
If we now set such objects:
147
and look at the boundary conditions, we have the result that there are two different
Fock vacua: namely,
The operator equalities can be written,
Using this relations one can find,
(6)
(7)
Moreover we get,
The latter quantity,
can thereof be interpreted as the relative probability amplitude that a birefringent particle of mass m, frequency ω and wave vector (k1A, k2, k3) and chiral
polarization vector
is transmitted from the left face to the right face through the hyperplane x =
0 to become a Proca-Stuckelberg particle with equal mass m, frequency ω and
wave vector (k1A, k2, k3) but polarization vector
1
As an effect of this transmission, the first component of wave vector of a
birefringent massive particle changes, while the longitudinal massive quanta
do not change it’s wave vector.
The similar result was obtained in the classic solutions of the Euler-Lagrange
equations. Moreover, classical solutions give us the coefficient of reflection. It
means that we know which part of coming into the vacuum particles is reflected
and which is passed through:
k 1A− k 10 K =
(8) (8)
refl
k 1A+ k 10
where k is the first component of wave vector, corresponding for the polarization A
1A
148
(9)
Easy to see that the longitudinal polarization does not feel the boundary and
completely passes into the next area without reflection.
Finally, using our classical solutions and the conventional Strum-Liouville
theory we can determine the Green function in the momentum space,
All calculations can be found in [5].
Conclusion
The main results were obtained for the space-like Chern-Simons vector, such
symmetry breaking is possible in the fireball, around neutron stars or in the axion
stars. Relations, describing the passage through and reflection of incoming and
outgoing particles from the area of any polarization were obtained. In particular,
it was shown that the longitudinal polarization does not feel the boundary and
completely passes into the next area without reflection. However, we need methods
for testing the described phenomena. Since we cannot detect outgoing particles
near the bound, we have to know what is going on after they are released outside.
The influence of a boundary between parity-odd medium and vacuum on the decay
width of photons and vector mesons represents a very interesting problem which
deserves to be a subject of further investigation.
149
Acknowledgements. The work was partially supported by the non-profit foundation “Dynasty”. I am grateful to A.A. Andrianov and R. Soldati for productive and
valuable colaboration.
References
1. Mielke E.W. and Perez J.A. // Phys. Lett. B 671 (2009) 174.
2. Andrianov A., Andrianov V.A., Espriu D. and Planells X. // Abnormal dilepton
yield from local parity breaking in heavy-ion collisions, arXiv:1010.4688.
3. Alfaro J., Andrianov A., Cambiaso M., Giacconi P. and Soldati R. // Int. J. Mod.
Phys. A 25 (2010) 3271 [arXiv:0904.3557].
4. Andrianov A.A., Giacconi P. and Soldati R. // Journal of High Energy Physics
02 (2002) 030 [hep-th/0110279].
5. Andrianov A.A., Kolevatov S.S. and Soldati R. // Journal of High Energy
Physics, 11 (2011) 007 [arXiv:1109.3440].
150
Analytical solution of two-dimensional Scarf II model by
means of SUSY
Krupitskaya Ekaterina
e.v.krup@yandex.ru
Scientific supervisor: Prof. Dr. Ioffe M.V., Department of High
Energy and Elementary Particles Physics, Faculty of Physics,
Saint-Petersburg State University
Introduction
The importance of each new exactly solvable model in one-dimensional quantum mechanics is well known. The approach of supersymmetric quantum mechanics
(SUSY QM) and, in particular, shape invariance [1] has been fully exploited for
construction and investigation of such models by generating a partnership between
pairs of dynamical systems which allows us to establish the solvability of one in
terms of other by means of intertwining relations with supercharges of first order
in derivatives. Within the search for larger class of problems which can be solved
by supersymmetrical methods, extensions of SUSY QM have been elaborated with
different realizations if the intertwining operators (supercharges). The next step was
the suggestion that for two-dimensional models one can use ordinary intertwining
relations but with supercharge of second order in derivatives [2, 3]. The two main
methods of SUSY-separation of variables were formulated. One of the methods
leads to partial (quasi-exact) solvability of the model [3, 4], and another one - to
complete(exact) solvability of two-dimensional generalizations of Morse [5] and
Pӧschl-Teller [6] models.
In our paper one more two-dimensional model - with potential, which is naturally associated with solvable one-dimensional hyperbolical version of Scarf model
(Scarf II) [7] - can be solved analytically as well by means of supersymmetrical
separation. This potential was obtained recently among new two-dimensional models with shape invariance property [8, 10]. Just this property will allow to solve the
problems with the whole hierarchy of generalized Scarf II potentials. At first, the
two-dimensional generalized Scarf II model was completely solved for the specific
parameter value a = -1: both energy values and corresponding wave functions of
all bound states were built analytically. And then the procedure was generalized
to the models with arbitrary negative integer values of parameter a.
Investigation
We start from the supersymmetrical intertwining relation
H (1) Q + = Q + H ( 2 ) ; Q − H (1) = H ( 2 ) Q −
for two partners two-dimensional Hamiltonians of Schrödinger form


H (i ) = −∆ (2) + V (i ) ( x ); i = 1, 2; x = ( x1, x2 ); ∆ (2) ≡ ∂12 + ∂ 22
151
(1)
(2)
with mutually conjugated supercharges Q± of second order in derivatives.
We have considered two-dimensional generalization of Scarf II potential:
1
1

V (1),(2) ( x ) = −2λ 2 a(a  1)(
)−
−
cosh 2 (λx+ ) cosh 2 (λx− )
(2k sinh (2λx )+ k )+ (2k sinh (2λx )+ k ),
−
(4cosh (2λx ))
(4cosh (2λx ))
1
1
2
2
1
2
2
1
2
And the second order supercharges are:
Q + = (Q − )† = 4∂ + ∂ − + 2atanh(λx+ )∂ − + 2acoth(λx− )∂ + +
(2k1sinh(2λx1 ) + k2 )
a 2 tanh(λx+ )coth(λx− ) +
−
(4cosh 2 (2λx1 ))
−
(4)
(2k sinh (2λx )+ k )
(4cosh (2λx ))
1
2
2
(3)
2
2
2
It is evident that potentials are not amenable to standard separation of variables.
The first step of the our approach was to choose such values of parameters, that
one of the Hamiltonians H(2) does allow standard separation of variables. Then, we
have a chance to find the spectrum and wave functions of the partner Hamiltonian
H(1) which does not allow standard separation. Indeed, one can choose the parameter
a = −1 to cancel the terms prohibited from separation. For simplicity, we have also
fixed the parameter λ = 1/2. Thus
1
1
(1) 
H ( x ) = −∆ (2) − (
) + U ( x1 ) + U ( x2 )
−
2
2
cosh ( x+ / 2) sinh ( x+ / 2)
(5)

H (2) (x ) = −∆ (2) + U (x1 )+ U (x2 )
where one-dimensional potential U is defined as:
−2k1 sinh ( x ) + k2
.
U ( x) =
4cosh 2 ( x )
(6)
The next step was to solve the model which does allow separation of variables.
That means, we need to find the solution of one- dimensional model with potential
U. For the general case of U, this is impossible to perform analytically. But for
specific form (6) for U(x), the solution is known explicitly [9]. After some transformations we have obtained the eigenfunctions and eigenvalues of H(2):
(2)
2
2
En(2)
, m = Em , n =∈n + ∈m = − ( A − n) − ( A − m) ;
(7)
(2) ±
(2) ±
ψ n , m = ±ψ m , n = ηn (x1 )ηm (x2 )± ηm (x1 )ηn (x2 ).
where
ηn ( x) = (cosh( x)) − A exp (− Barctan( sinh( x))) ×
× Pn( − iB − A −1/ 2, + iB − A −1/ 2) (isinh( x))
152
In the formulas above, A and B are positive parameters (A,B > 0), and Pn(α,β) are
the n−th power Jacobi polynomials of their argument. The positive parameters A,
B can be expressed in terms of coupling constants k1 < 0, k2 :
A = −1 / 2 − 1 / 2k1 ( (k2 + 1) 2 + 4k12 − (k2 + 1))1/ 2
1
(8)
B=
( (k2 + 1) 2 + 4k12 − (k2 + 1))1/ 2
2 2
The condition n, m <A is necessary for normalizability of bound state wave functions (9).
Further we have to find the discrete spectrum and normalizable wave functions
for the quantum problem with the Hamiltonian H(1)(x) with parameter a = −1. To
do this one have to use the supersymmetrical intertwining relations (1), which
provide the links between spectra and wave functions of partner Hamiltonians. In
general, these Hamiltonians are isospectral. But some properties of intertwining
operators Q± are crucial at this stage, such as their singularities and zero modes.
In general case, three kinds of bound states of H(1) may exist :
(i). The levels obtained by means of intertwining relations from the levels of the :


H (2) ψ (n1,)m (x ) = Q + ψ (n2,)m± (x )
(9)
2
2
En(1), m = En(2)
, m = − (A − n ) − (A − m ) ,
(ii) The levels that are absent in the spectrum of H(2), if there are zero-modes of Q̶
among the wave functions of H(1);
(iii) The levels that are absent in the spectrum of H(2), if some wave function of H
become non-normalizable after Q̶ act on it.
After the investigation these possibilities we obtain, that there are no levels of
the second and the third type. And the first type levels could correspond only to
antisymmetric function Ψ(2) .Thus, the discrete spectrum of H(1) is nondegenerate
and consists of bound state levels with such energies with corresponding wave
functions:
En(1), m = En(1), m = −( A − n) 2 − ( A − m) 2 , 

(10)
ψ (1) (x ) = Q + ψ (2)− (x )
n,m
n,m
To solve the problem with arbitrary parameter we had to extend our results.
To do this it is necessary to construct the hierarchy of Hamiltonians H(1)(ak) with
ak = −k, k = 1,2,..., with the previous one H(1)≡ H(1)(a1). The hierarchy is based on
the alternate application shape-invariance and intertwining relations. One can see
that Hamiltonians are actually shape-invariant:
H (1) ( ak ) = H (1) ( ak +1 ) (11)
Therefore, the infinite chain (hierarchy) of Hamiltonians can be built:
H ( 2 ) (a1 ) ÷ H (1) (a1 ) = H ( 2 ) (a2 ) ÷ H (1) (a2 ) = ...
÷ H (1) (aN −1 ) = H ( 2 ) (aN ) ÷ H (1) (aN ) = ...,
153
where the sign ÷ means that the corresponding Hamiltonians H(1),(2)(ak) are intertwined by supercharges Q±(ak). After analysis, which we have carried out, one can
conclude that eigenfunctions and eigenvalues of Hamiltonian H(1) are:
2
2
En(1,)m = En(2, m) = − ( A − n) − ( A − m) ,
(12)
( 2)
+
+
+
( 2)
(1)
ψ (an ) = ψ (an −1 ) = Q (an −1 )ψ (an −1 ) = Q (an −1 )Q (an − 2 )...
...Q + (a1 )ψ ( 2 ) (a1 ),
where |n-m|≥k.
The latter condition follows from the existence of zero modes of Q+(ak).
Conclusions
It was demonstrated that the two-dimensional quantum model, which can be
called as two-dimensional Scarf II model is exactly solvable for arbitrary value
ak = −k. All hamiltonians of this hierarchy of Hamiltonians has nondegenerate
spectrum. The values of energies are given by (7) for |n-m|≥k, and the corresponding wave functions are given by (12). Together with generalized Morse model
[11] and generalized Pӧschl-Teller model [12] the complete analytical solution
of this two-dimensional model, demonstrates that supersymmetrical approach is
a powerful method to solve the problems which are not amenable to conventional
separation of variables.
The extended version of this work - M.V. Ioffe, E.V. Krupitskaya, D.N. Nishnianidze
“Analytical solution of two-dimensional Scarf II model by means of SUSY methods” - is accepted for publication in “Annals of Physics”.
Acknowledgements. The work was partially supported by the grant RFFI 09-0100145-a. I am also indebted to the non-profit foundation “Dynasty” for financial
support. I am grateful to M.V. Ioffe and D.N. Nishnianidze for productive and
valuable colaboration.
References
1. Junker G. Supersymmetric Methods in Quantum and Statistical Physics,
Springer, Berlin, 1996.
2. Andrianov A.A., Ioffe M.V., Nishnianidze D.N. // Phys.Lett., A201 (1995)
103; Andrianov A.A., Ioffe M.V., Nishnianidze D.N. // Theor. and Math.Phys.,
104 (1995) 1129.
3. Ioffe M.V. // J.Phys.A37 (2004) 10363.
4. Cannata F., Ioffe M.V., Nishnianidze D.N. // J.Phys.:Math.Gen., A35 (2002)
1389.
5. Andrianov A.A., Cannata F. // J. Phys. A 37 (2004) 10297.
6. Andrianov A.A., Borisov N.V., Ioffe M.V., Eides M.I. // Phys. Lett. A 109
(1985) 143; Andrianov A.A., Borisov N.V., Eides M.I., Ioffe M.V. // Theor. Math.
Phys. 61 (1984) 965.
7. AndrianovA.A., Borisov N.V., Ioffe M.V. // Phys. Lett. B 181 (1986)141.
8. Andrianov A.A.,Ioffe M.V. // Phys. Lett. B 205 (1988) 507.
154
9. Aoyama H., Sato M., Tanaka T. // Phys. Lett. B 503 (2001) 423; Aoyama H.,
Sato M., Tanaka T. // Nucl. Phys. B 619 (2001) 105.
10.Ioffe M.V., Nishnianidze D.N. // Phys. Rev. A 76 (2007) 052114.
11.Ioffe M.V., Nishnianidze D.N., Valinevich P.A. // J. Phys. A 43 (2010)
485303.
155
Effects of turbulent mixing on critical behaviour:
Renormalization group analysis of the ATP model
Malyshev Aleksei
alvlamal@gmail.com
Scientific supervisor: Prof. Dr. Antonov N.V., High Energy and
Elementary Particles Physics, Faculty of Physics, Saint-Petersburg
State University
Critical behaviour of a system, subjected to strongly anisotropic turbulent
mixing, is studied by means of the field theoretic renormalization group (RG).
Specifically, relaxational stochastic dynamics of a non-conserved multicomponent
order parameter of the Ashkin-Teller-Potts (ATP) model, coupled to a random velocity field with prescribed statistics, is considered. The velocity is taken Gaussian,
white in time, with correlation function of the form ∝ δ(t-t′ )/|k⊥|d−1+ξ, where k⊥ is the
component of the wave vector, perpendicular to the distinguished direction ("direction of the flow"). It is shown that, depending on the values of parameters that define
self-interaction of the order parameter and the relation between the exponent ξ and
the space dimension d, the system exhibits various types of large-scale behaviour.
In addition to known asymptotic regimes, existence of a new, non-equilibrium and
strongly anisotropic, type of critical behaviour (universality class) is established,
and the corresponding critical dimensions are calculated to the leading order of
the double expansion in ξ and ε = 6 − d (one-loop approximation).
Numerous systems of very different physical nature reveal interesting singular
behaviour in the vicinity of their critical points (second order phase transitions).
Their correlation functions exhibit self-similar (scaling) behaviour with universal critical dimensions. Most typical equilibrium phase transitions belong to the
universality class of the On-symmetric φ4 model of an n-component scalar order
parameter φ. Universal characteristics of the critical behaviour in this case depend
only on n and the space dimension d and can be calculated within systematic
perturbation schemes.
Another important example is provided by the Ashkin-Teller-Potts class of
models. Such models have numerous physical applications: magnetic materials and
solids with nontrivial symmetry, Edwards-Anderson spin-glass models within the
replica formalism, and so on. In general, the ATP models describe systems which
locally have n states, but the energy of any given configuration depends on whether
pairs of neighboring sites are in the same state or not. The case n = 2 corresponds
to nematic-to-isotropic transitions in the liquid crystals, while the formal limit
n = 0 corresponds to the percolation problem.
The behaviour of a real system near its critical point is extremely sensitive to
external disturbances, gravity, geometry of the experimental setup, presence of
impurities and so on. Some disturbances can change the type of the phase transition
(second-order to first-order one, and vice versa) and even produce completely new
types of critical behaviour with rich and rather exotic properties. In the presence
156
of a distinguished direction, scaling behaviour can become strongly anisotropic. In
this paper we study effects of turbulent mixing on the dynamical critical behaviour
of the systems, described by the generalized ATP model, paying special attention
to anisotropy of the flow.
Relaxational dynamics of a non-conserved n-component order parameter φa(x)
with x≡{t, x} is described by a stochastic differential equation
δ H (ϕ )
(1)
∂ tϕa (x ) = −λ0
+ ηa (x ),
δϕ
x
(
)
a
where ∂t=∂/∂t, λ0 is the (constant) kinetic coefficient, and ηa(x) is a Gaussian random
noise with zero mean and the pair correlation function
(2)
η (x )ηb (x′ ) = δ ab Dη (x − x′ ), Dη (x − x′ ) = 2λ0δ (t − t ′)δ ( d ) (x − x′ ),
d beinga the dimension
H(φ)
of the x space. Near the critical point, the Hamiltonian
of the ATP model is taken in the form
τ
g
 1

H (ϕ ) = ∫ dx − ϕa (x )∂ 2ϕa (x ) + 0 ϕa (x )ϕa (x ) + 0 Rabcϕa (x )ϕb (x )ϕc (x ) , (3)
2
3!
 2

where ∂i = ∂/∂xi is the spatial derivative, ∂2= ∂i∂i is the Laplacian, τ0∝(T−Tc) measures deviation of the temperature (or its analog) from the critical value and g0 is
the coupling constant. Summations over repeated indices are always implied
(a, b, c =1, ..., n and i =1, ..., d); after the functional differentiation in (2) one has
to replace φ(x)→ φ(x).
In the original ATP model Rabc is the irreducible invariant third-rank symmetric
tensor of the symmetry group of the hypertetrahedron in n dimensions. The Rabc
components are conveniently expressed in terms of the set of n+1vectors eα which
define vertices of the hypertetrahedron [1]
(4)
Rabc = ∑ α eaα ebα ecα , α
where e satisfy
n +1
n +1
n
∑ e = 0, ∑ e e = (n + 1)δ , ∑ e e = (n + 1)δ − 1.
(5)
α =1
α
a
α =1
α α
a b
ab
a =1
α β
a a
αβ
Using
equations (5) all the contractions with the tensor Rabc can be calculated.
For
example, the contractions of two and three tensors have the forms
Rabc Rabe = R1δ ce , Raec Rchb Rbfa = R2 Rehf ,
(6)
2
2
R1 = (n + 1) (n − 1), R2 = (n + 1) (n − 2 ). Coupling with the velocity field v={vi(t,x)}, which describes the turbulent mixing, is
introduced by the replacement ∂t →∇t = ∂t+vi∂i in (1), where ∇t is the Lagrangian (Galilean
covariant) derivative. The velocity ensemble is defined as follows. Let n be a unit constant
vector that determines some distinguished direction ("direction of the flow"). Then any vector
can be decomposed into the components perpendicular and parallel to the flow, for example,
x = x⊥+nx|| with x⊥⋅n = 0. The velocity field will be taken in the form v = nv(t,x⊥). For v(t,x⊥)
we assume a Gaussian distribution with zero mean and the pair correlation function:
v (t , x ⊥ )v (t ′, x′⊥ ) = δ (t − t ′ ) D0 ∫
dk ⊥
(2π )
d −1
157
exp (ik ⊥ ⋅ (x ⊥ − x′⊥ ))k⊥ − d +1−ξ .
(7)
Here D0 is a constant amplitude factor, ξ is an arbitrary exponent (with the most
realistic Kolmogorov value ξ = 4/3) and k⊥=|k⊥|.
In order to ensure multiplicative renormalizability of the model, it is necessary
to split the Laplacian in (3) into the parallel and perpendicular parts ∂2→∂||2+f0∂⊥2
by introducing a new parameter f0 > 0. In the anisotropic case, these two terms will
be renormalized in a different way. According to the general theorem [2], the stochastic problem (1)−(7) is equivalent to the field theoretic model of the extended
set of fields Φ={ϕa′,ϕa,v} with action functional
ϕ a′ Dηϕ a′
λ g f 1/ 4
S (Φ ) =
+ ϕ a′ −∇t + λ 0 (∂ ⊥2 + f 0 ∂||2 − τ 0 ) ϕ a − 0 0 0 Rabc ϕ a′ ϕ b ϕ c + Sv ( v ), (8)
2
{
}
2
where we segregated the factor f from the charge g0. The first few terms represent
the De Dominicis-Janssen action functional for the stochastic problem (1)−(3) at
fixed v; it involves the auxiliary scalar response field ϕa′. All the required integrations over x≡{t, x} and summations over the vector indices are implied. The last
term in (8) corresponds to the Gaussian averaging over v with the correlator (7)
and has the form
1
Sv (v ) = − ∫ dt ∫ dx ⊥ dx′⊥ v (t , x ⊥ )Dv−1 (x ⊥ − x′⊥ )v (t , x′⊥ ), Dv−1 (r⊥ ) ∝ D0−1r⊥2(1− d ) −ξ .
2
This formulation means that statistical averages of random quantities in the original
stochastic problem coincide with the Green functions of the field theoretic model
with the action (8), given by functional averages with the weight exp S(Φ). This allows one to apply the standard Feynman diagrammatic technique, the field theoretic
renormalization theory and renormalization group to our stochastic problem.
From the dimensional analysis it follows that the model is logarithmic (both
coupling constants g0 and w0=D0/λ0f0 are simultaneously dimensionless) at d = 6 and
ξ = 0, so that the UV divergences in the correlation functions manifest themselves as
poles in ε = 6−d, ξ and their linear combinations. The careful analysis, augmented
by symmetry considerations, shows that all the counterterms needed to cancel the
UV divergences in our model are present in the action (8). Here important role is
played by the Galilean symmetry and the invariance with respect to the symmetry
group of the hypertetrahedron.
Thus our model appears multiplicatively renormalizable and we conclude that
the renormalized action can be written in the form
ϕ a′ Dηϕ a′
S R (Φ ) = Z1
+ ϕ a′  − Z 2 ∇t + λ Z 3 ∂ ⊥2 + Z 4 f ∂||2 − Z 5 τ  ϕ a −
2
(9)
λg µ ε / 2 f 1/ 4
Rabc ϕ a′ ϕ b ϕ c + Sv ( v )
− Z6
2
Here λ, τ, f, g and w are renormalized analogs of the bare parameters (with the
subscripts "0'') and μ is the reference mass scale. The one-loop calculation of the
renormalization constants Zi is easily performed. In the minimal subtraction scheme
they contain only simple poles in ε and ξ and have the forms:
uR
uR
uR w
2uR1
2uR2
Z1 = Z 2 = 1 − 1 , Z 3 = 1 − 1 , Z 4 = 1 − 1 − , Z 5 = 1 −
, Z6 = 1 −
. (10)
2ε
3ε
3ε ξ
ε
ε
158
1/4
0
(
)
Here we have passed to more convenient coupling constants u→g/128π3 and
w→w/24 π3. The parameters R1 and R2 are related to the dimension n of the order
parameter by the expression (6). Although we are especially interested in the
cases n=2 and 0, for completeness the coefficients R1 and R2 in what follows are
assumed to be arbitrary.
Expression (9) can be obtained by the multiplicative renormalization of the
fields φa→ φaZφ, φa'→ φa'Zφ' and the parameters:
(11)
λ0 = λ Z λ , τ 0 = τ Zτ , f 0 = fZ f , g 0 = g µ ε /2 Z g , w0 = wµ ξ Z w
(no renormalization of the velocity field is needed: Zv=1). The renormalization
constants in Eqs. (9) and (11) are related as follows:
Z1 = Z λ Zϕ2′ , Z 2 = Zϕ ′ Zϕ , Z 3 = Zϕ ′ Z λ Zϕ , Z 4 = Zϕ ′ Z λ Z f Zϕ , (12)
2
Z 5 = Zϕ ′ Z λ Zτ Zϕ , Z 6 = Z λ Z u1/2 Z 1/4
f Zϕ ′ Zϕ , Z w Z f Z λ = 1.
The last equality is a corollary of the fact that Sv(v) is not renormalized.
The RG equation for the renormalized Green functions GR=⟨Φ…Φ⟩R in our
model has the following form
{DRG + Nϕ γ ϕ + Nϕ ′γ ϕ ′ }GR (e, µ ,) = 0,
where Nϕ and Nϕ′ are the numbers of corresponding fields entering into GR, e denotes the set of the renormalized counterparts for the bare parameters e0, DRG is the
operator dµ=µ∂µ for fixed e0 expressed in the renormalized variables
(13)
DRG ≡ Dµ + βu ∂ u + β w∂ w − γ f D f − γ λ Dλ − γ τ Dτ . Here we have written Dx=x∂x for any variable x, the anomalous dimensions are
defined as γF ≡dµ lnZF and the β-functions for the two dimensionless couplings u
and w from (13) are
(14)
βu ≡ d µ u = u [−ε − γ u ], β w ≡ d µ w = w [−ξ − γ w ]. Using (10) and (12) one can find, that (Zu= Zg2, and hence γu= 2γg)
uR
uR
5uR
uR
γ ϕ′ =
3
1
, γϕ =
, γ f = w, γ τ =
, γλ = −
1
,
6
uR
w γ w = 1 − w, γ u = (4 R2 − R1 )u − ,
6
2
6
1
3
1
(15)
with corrections of order u2, w2, uw and higher.
It is well known that possible large-scale regimes of a renormalizable model
are associated with IR attractive fixed points of the corresponding RG equations.
The coordinates g* of the fixed points are found from the requirement that the
β-functions, corresponding to all renormalized couplings gi, vanish. The type of
a fixed point is determined by the matrix Ωij=∂βi /∂gj, where βi is the full set of
β-functions and gj is the full set of couplings. For IR attractive fixed points the
matrix is positive, i.e., the real parts of all its eigenvalues are positive. In our case,
159
gi={u,w}. The functions βi, calculated in the one-loop approximation from (14)
and the explicit relations (15), have the forms:
(16)
β u = u [−ε + Ru + w / 2], β w = w [−ξ − uR1 / 6 + w],
where we have introduced a new convenient parameter R=R −4R .
1
2
The analysis of the functions (16) reveals four possible IR attractive points
(coordinates of the fixed points and corresponding domains of stability):
1. u∗ = 0, w∗ = 0; IR attractive for ε < 0, ξ < 0;
2. u∗ = 0 (exact result to all orders), w∗ = ξ; IR attractive for ξ >2ε, ξ > 0.
3. w∗ = 0 (exact result to all orders), u∗ = ε /R;
this scaling regime exists if R>0; IR attractive for ε >0, ξ <−εR1/6R.
4. u∗ = (12ε−6ξ)/(12R+R1), w∗ = (12Rξ+2R1ε)/(12R+R1);
this scaling regime exists if R+R1/12>0. IR attractive for ξ<2ε, ξ >−εR1/6R if R>0
and ξ<2ε, ξ <−ε(6R+R1)/3R if R<0.
The first fixed point is a Gaussian (free) one. In the scaling regime corresponding to the fixed point 2, the nonlinearity φ2 in the stochastic equation (1) becomes
irrelevant due to the exact relation u∗ = 0. Thus we arrive at the linear advectiondiffusion equation for a passive scalar field φ. In turn, the effects of the velocity
field become irrelevant in the third regime (fixed point 3). The isotropy violated
by the velocity ensemble is restored and the leading terms of the IR behaviour
coincide with those of the equilibrium dynamic model ATP. Finally, the last point
represent a new nontrivial IR universality class, in which the both nonlinearities
of the model are simultaneously important.
The critical dimensions ∆F of the IR relevant quantities F are given by the relations ∆F = dF⊥ +∆|| dF|| + ∆ω dFω +γF∗ with the normalization condition ∆⊥=1, here
dF⊥,||,ω are the canonical dimensions of F and γF∗=γF (u∗,w∗). Results for ∆||, ∆ω, ∆τ
for all the universality classes are given in the following table:
№
FP1
FP2
FP3
FP4 (∆R=R+R1/12):
∆||
1
1+ξ/2
1
1+(6Rξ+R1ε)/12∆R
∆ω
2
2
2+R1ε/6R
2+ R1(2ε−ξ)/12∆R
∆τ
2
2
2+R1ε/3R
2+ 5R1(2ε−ξ)/6∆R
One can easily see that the most realistic values of the model parameters (for
example, d = 3 and the Kolmogorov exponent ξ = 4/3) belong completely to the
region of stability of the most nontrivial fixed point 4 (for both physically interesting cases n = 2 and n = 0).
More detailed presentation of this work can be found in [3]. The author thanks
the Dynasty Foundation for the financial support.
References
1. de Alcantara Bonfim O.F. et al. // J. Phys. A: Math. Gen. 13 L247 (1980).
2. Vasil'ev A.N. The field theoretic renormalization group in critical behavior
theory and stochastic dynamics.- Boca Raton: Chapman & Hall/CRC, 2004.
3. Antonov N.V., Malyshev A.V. // arXiv:1111.6238v1.
160
Inertial-range behaviour of a passive scalar field in a
random shear flow: Renormalization group analysis of a
simple model
Malyshev Aleksei
alvlamal@gmail.com
Scientific supervisor: Prof. Dr. Antonov N.V., High Energy and
Elementary Particles Physics, Faculty of Physics, Saint-Petersburg
State University
Infrared asymptotic behavior of a scalar field, passively advected by a random
shear flow, is studied by means of the field theoretic renormalization group (RG)
and the operator product expansion (OPE). The advecting velocity is Gaussian,
white in time, with correlation function of the form ∝δ(t-t′ )/k⊥d −1+ξ, where k⊥=|k⊥|
and k⊥ is the component of the wave vector, perpendicular to the distinguished
direction ("direction of the flow"). The structure functions of the scalar field in the
infrared range exhibit scaling behavior with exactly known critical dimensions.
It is strongly anisotropic in the sense that the dimensions related to the directions parallel and perpendicular to the flow are essentially different. In contrast
to the isotropic Kraichnan’s rapid-change model, the structure functions show no
anomalous (multi)scaling and have finite limits when the integral turbulence scale
tends to infinity. On the contrary, the dependence of the internal scale persists in
the infrared range.
The problem of turbulent advection, being of practical importance in itself,
has become a cornerstone in studying fully developed hydrodynamical turbulence
on the whole. On one hand, deviations from the classical Kolmogorov theory –
intermittency and anomalous scaling – are much stronger pronounced for a passively advected scalar field (temperature of the fluid or concentration of impurity)
than for the advecting turbulent field itself. On the other, the problem of passive
advection appears easier tractable theoretically. Most remarkable progress was
achieved for Kraichnan’s rapid-change model: for the first time, the anomalous
exponents were derived on the basis of a dynamical model and within controlled
approximations.
In Kraichnan’s model, the turbulent velocity field is modeled by the Gaussian
distribution with the pair correlation function of the form
(1)
vi v j ∝ D0δ (t − t ′ ) Pij k − d −ξ ,
where P =δ −k k /k2 is the transverse
the wave number, D >0 is
projector,
k
≡
|k|
is
ij
ij
i j
0
an amplitude factor, d is the dimension of the x space and ξ is an arbitrary exponent
with the most realistic (Kolmogorov) value ξ = 4/3.
The issue of interest is the behavior of the equal-time structure functions
n
S n (r ) = θ (t , x ) − θ (t , x′ ) , r = x − x′
(2)
161
of the scalar field θ(x) with x ≡{t, x} in the inertial range l<<r<<L, where l is
the dissipation length and L is the integral turbulence scale. Within the so-called
zero-mode approach, it was shown that in the inertial range the functions (2) are
independent of the diffusivity coefficient and have the forms:
S 2 n (r ) ∝ D0− n r n (2−ξ ) (r / L ) n , ∆ n = −2n(n − 1)ξ / (d + 2) + O (ξ 2 ). ∆
(3)
Thus the functions (2) depend on the integral scale and diverge for L→∞ (the anomalous exponents Δn are negative), in contradiction with the classical Kolmogorov
theory.
In [1] the field theoretic renormalization group and operator product expansion were applied to Kraichnan’s model. The anomalous scaling for the structure
functions emerges as a consequence of the existence in the corresponding OPE of
"dangerous" composite fields (composite operators in the field theoretic terminology) of the form (∂θ)2n, whose negative critical dimensions are identified with the
anomalous exponents Δn. In this work we apply RG+OPE to the model of a passive
scalar field in a random shear flow. We show that the inertial-range behavior of this
model appears essentially different from the isotropic Kraichnan’s model.
The advection-diffusion equation for the scalar field θ(x) has the form
(4)
∇tθ = ν 0 ∂ 2θ + ζ , ∇t = ∂ t + vi ∂ i ,
here ∇t is the Lagrangian Galilean covariant derivative, ∂t=∂/∂t, ∂i = ∂/∂xi, ∂2 is
the Laplacian, ν0 is the diffusion coefficient and ζ(t,x) is a Gaussian random noise
with zero mean and the pair correlation function
Dζ = ζ (t , x )ζ (t ′, x′ ) = δ (t − t ′ )C (r ), r = x − x′. (5)
The function C(r) is finite at r = 0 (we assume the normalization C(0) =1) and
rapidly decays for r→∞; its precise form is inessential.
The velocity ensemble is defined as follows. Let n be a unit constant vector
that determines some distinguished direction ("direction of the flow"). Then any
vector can be decomposed into the components perpendicular and parallel to the
flow, for example, x = x⊥+nx|| with x⊥⋅n = 0. The velocity field will be taken in the
form v = nv(t,x⊥). For v(t,x⊥) we assume a Gaussian distribution with zero mean
and the pair correlation function of the form:
v (t , x ⊥ )v (t ′, x′⊥ ) = δ (t − t ′ ) D0 ∫
dk ⊥
(2π )
d −1
exp (ik ⊥ ⋅ (x ⊥ − x′⊥ ))k⊥ − d +1−ξ , k⊥ = k ⊥ . (6)
Here and below d is the dimension of the x space, D0 is a constant amplitude factor,
ξ is an arbitrary exponent (the most realistic Kolmogorov value ξ = 4/3).
In order to ensure multiplicative renormalizability of the model, it is necessary
to split the Laplacian in (4) into the parallel and perpendicular parts ∂2→∂||2+f0∂⊥2
by introducing a new parameter f0 > 0. In the anisotropic case, these two terms
will be renormalized in a different way. According to the general theorem [2], the
stochastic problem (4)−(6) is equivalent to the field theoretic model of the extended
set of fields Φ={θ', θ, v} with action functional
1
S (Φ ) = θ ′Dζ θ ′ + θ ′ −∇t + ν0 ∂ 2⊥ + f 0 ∂||2 θ + Sv (v ).
(7)
2
162
{
(
)}
The first few terms represent the De Dominicis–Janssen action functional for
the stochastic problem (4), (5) at fixed v; it involves auxiliary scalar response field
θ'(x). All the required integrations over x≡{t, x} are implied. The last term in (7)
corresponds to the Gaussian averaging over v with the correlator (6)
1
Sv (v ) = − ∫ dt ∫ dx ⊥ dx′⊥ v (t , x ⊥ )Dv−1 (x ⊥ − x′⊥ )v (t , x′⊥ ), Dv−1 (r⊥ ) ∝ D0−1r⊥2(1− d ) −ξ .
2
This formulation means that statistical averages of random quantities in the original
stochastic problem coincide with the Green functions of the field theoretic model
with the action (7), given by functional averages with the weight exp S(Φ). This
allows one to apply the field theoretic renormalization theory and renormalization
group to our stochastic problem.
The action (7) corresponds to the standard Feynman diagrammatic technique.
The role of the bare coupling constant (expansion parameter in the ordinary perturbation theory) is played by the parameter w0=D0/ν0 f0. Dimensional analysis
shows that for our model superficial UV divergences, whose removal requires
counterterms, can be present only in 1-irreducible Green function ⟨θ'θ⟩1-ir. The corresponding counterterm must contain two symbols ∂|| and therefore reduces to θ'∂||2θ.
Inclusion of this counterterm is reproduced by the multiplicative renormalization
of the action (7) with the only independent renormalization constant Zf:
(8)
ν = ν , f = fZ f , w0 = wµ ξ Z w , Z w = Z −f 1.
Here the reference0 scale μ 0is an additional
parameter of the renormalized
theory,
ν, f and w are renormalized analogs of the bare parameters (with the subscript
"0") and Zi are the renormalization constants. Their relation in (8) results from the
absence of renormalization of the contribution with D0 in (7), D0=w0ν0f0=wμξνf.
No renormalization of the fields is required.
It turns out that in our model all the multiloop diagrams needed for calculating
of the renormalization constant Zf vanish [3]. This means that Zf is given exactly
by the one-loop approximation. This calculation is easily performed. In the MS
scheme we obtain
(9)
Z f = 1− w / ξ ,
where we have absorbed the factor S /2(2π)(d−1)
into the coupling constant.
d−1
The RG equation for the renormalized Green functions GR(e,μ...) is
{DRG + γ G }G R (e, µ ,) = 0, e denotes the set of the renormalized
counterparts for the bare parameters e0, DRG
is the operator dµ=µ∂µ for fixed e0 expressed in the renormalized variables
DRG ≡ Dµ + β∂ w − γ f D f . Here we have written Dx=x∂x for any variable x, the anomalous dimensions are
defined as γF ≡dµ lnZF and the β-function for the dimensionless couplings w is
β ≡ d µ w = w [−ξ − γ w ]. Using (9) one obtains exact expressions
γ f = −γ w = w, β = w [−ξ + w]. 163
(10)
It is well known that IR asymptotic behavior of the Green functions is governed
by IR attractive fixed points of the RG equations, defined by the relations β(w*)=0
and β'(w*)>0. From (10) it follows that for our model
w * = ξ , β ′ (w * ) = ξ .
This fixed point is positive and IR attractive for ξ>0. The critical dimensions ∆
F
of the IR relevant quantities F are given by the relations
∆ F = d F⊥ + ∆ || d F|| + ∆ ω d Fω + γ F* ;
∆ ω = 2, ∆ θ = 1 + ξ / 2, with the normalization condition Δ ⊥=1, here dF⊥,||,ω are the canonical dimensions
of F and γF∗=γF (w∗). For example, Δθ = −1.
The key role in the following will be played by the critical dimensions of certain
composite fields. We begin with the simplest operators θn. The analysis shows that
such operators are UV finite and requires no counterterms: θn =Z[θn]R with Z = 1.
It then follows that their critical dimensions are simply given by the sum of the
critical dimensions of the constituents: Δn=nΔθ=−n, because γF∗=0.
In what follows we will be interested also in the critical dimensions of the operators with minimal canonical dimension (namely, dF = 0) that are invariant with
respect to the shift θ→θ+const. In Kraichnan’s rapid-change model, the anomalous
exponents (3) are identified with the negative critical dimensions of such operators.
For that isotropic case, the scalar operator of the needed form is unique for any
given n: Fn=(∂iθ∂iθ)n. In the case at hand one can construct n+1 different operators
of the form (∂θ)2n, invariant under the residual symmetry:
k
s
(11)
Fk , s = ∂i⊥ θ ∂i⊥ θ ∂||i θ ∂||i θ , k + s = n. It was shown that in spite of renormalization mixing of the operators (11) it is
possible to find exact expressions for their critical dimensions (for details see [3]).
They have the forms:
(12)
∆ k , s = 2k + 2 s∆ θ − (k + s )∆ ω = sξ.
In contrast to the results (3)
for the isotropic Kraichnan’s model,
the expressions (12)
have no corrections of order O(ξ2) and higher and are positive for all k, s and ξ>0.
The last results allow one to find the IR scaling behavior of the correlation
functions. For generality, consider the different-time structure functions
(
(
)(
)
)
S 2 n τ, r⊥ , r|| = θ (t , x) − θ (t ′, x ′ ) ; τ = t ′ − t , r⊥ = x ⊥′ − x ⊥ , r|| = x||′ − x|| . (13)
The function (13) is a linear combination of the two-point correlators ⟨θ(t,x)kθ(t',x')s⟩ with
the fixed k+s=2n. Due to simple exact relations Δn= −n for the operators θn, the
critical dimensions of these correlators are all equal, Δk+Δs=−(k+s) =−2n =2nΔθ,
and hence the function S2n in the IR range behaves as a single object. From the
dimensional considerations
(
)
S 2 n = ν − n r⊥2 nQ µ r⊥ , w, r⊥ / f 1/2 r⊥ , ντ r⊥2 , r⊥ / L , (14)
where Q(...) are some functions of completely dimensionless arguments. Using (14)
and RG equation the following asymptotic expression for the structure functions
in the IR range (l<<r⊥) can be obtained:
164
(
)
(
)
S 2 n = ν 0− n r⊥2 nQ 1, w* , r⊥ / ( f 0 w0 ) r⊥1+ξ /2 , ν 0τ / r⊥2 , r⊥ / L = r⊥−2 n∆θ R r⊥ / r⊥∆ ⊥ , τ / r⊥∆ω , r⊥ / L . (15)
1/2
with certain scaling functions R. In the isotropic Kraichnan’s model, the structure
functions in the IR range depend only on the IR (integral) scale L and the amplitude
D0=w0ν0 entering the velocity correlator (1), but not on the diffusivity coefficient ν0
and the coupling constant w0 separately; see (3). This fact is in agreement with the
second Kolmogorov hypothesis about the independence of the correlation functions
in the IR range of the parameters, related to the UV (dissipation) scale. In the case
at hand, the UV parameters ν0, f0 and w0 survive in the IR asymptotic expression
(15) for the structure functions (they do not form the combination D0=w0ν0 f0 even
if we set f0=1). Thus we may conclude that, in contrast to the isotropic case, the
second Kolmogorov hypothesis is invalid for the shear flow.
The asymptotic representations (15) hold in the IR asymptotic range, specified
by the inequality l<<r⊥, while the other arguments of the scaling functions R are
kept finite. Inertial range corresponds to the additional condition r⊥<<L. The form
of the scaling functions Q or R is not determined by the RG equations alone. In
order to study the limit r⊥/L→0 in the structure functions, one should combine the
plain RG with the OPE techniques. For simplicity, let us return to the equal-time
functions. From (15) and OPE one can get
∆
S (r ) = r 2 n
A r / r ∆θ , r / L (r / L ) F , (16)
2n
⊥
∑
F
F
(
||
⊥
⊥
)
⊥
were F are all possible renormalized local composite operators allowed by the
symmetry and coefficients AF are regular in (r⊥/L)2. Due to the invariance of the
action functional (7) and the structure functions with respect to the shift of the
field, θ→θ+const, the operators entering the OPE must also obey this symmetry.
The leading contributions are determined by the operators with minimal critical
dimensions ΔF=dF+O(ξ). It then follows that the leading terms of the small- r⊥/L
behavior of the expression (16) are given by the scalar operators (11) with k+s<2n.
Their dimensions (12) are all nonnegative; the leading term is given by the operators with s=0 (including the simplest F=1), the operators with s≥1 determine the
corrections vanishing for r⊥/L→0. We conclude that the function S2n remains finite
at L=∞. Thus the inertial-range behavior in the rapid-change model of a shear flow
differs essentially from that in the isotropic model. There is no anomalous (multi)
scaling in the case at hand.
More detailed presentation of this work can be found in [3]. The author thanks
the Dynasty Foundation for the financial support.
References
1. Adzhemyan L.Ts., Antonov N.V., Vasil’ev A.N. // Phys. Rev. E 58, 1823–1835
(1998).
2. Vasil'ev A.N. The field theoretic renormalization group in critical behavior
theory and stochastic dynamics.- Boca Raton: Chapman & Hall/CRC, 2004.
3. http://www.springerlink.com/content/k7328k18m3745733/
165
Effects of Stefan’s flow and concentration-dependent
diffusivity in binary condensation
Martyukova Darya
darya.martyukova@gmail.com
Scientific supervisor: Prof. Dr. Kuchma A.E., Department of
Statistical Physics, Faculty of Physics, Saint-Petersburg State
University
Introduction
The subject of this paper is studying the problem of droplet growth in the atmosphere of two condensable vapors and noncondensable carrier gas. The results
of such study are essential for fundamental and applied problems of the theory of
decomposition of solid and liquid solutions, and the theory of phase transitions in
the Earth atmosphere. A typical example of binary condensation under the conditions of the Earth atmosphere is condensation of water and sulfuric acid vapors
in rain drops.
Discussion
We consider a spherical supercritical droplet of binary solution, isothermally
growing in the diffusion regime. Let n1, n2 – the numbers of molecules of the first
and second components of the vapor per unit volume, n3 – the number of molecules
of carrier gas per unit volume, and ñ = n1+n2+n3 - total number of molecules per
unit volume in the vapor-gas mixture.
We introduce a spherical coordinate so that r is distance from the center of the
droplet. Denote by R the radius of the droplet.
The boundary conditions in this problem have the form:
ni (r ) r →∞ = ni 0
(1)

ni (r ) r = R = ni∞ ( xi )
ni0 – is the concentration of the i-th (i = 1,2) component of the vapor far from the
droplet, ni∞( xi ) – equilibrium concentration of the i-th (i = 1,2) component of the
vapor at the droplet of flat boundary, and xi – current value of the molar concentration of the i-th (i = 1,2) component in the droplet.
Denote by ji – flow density of the i-th (i = 1,2) vapor component. For the case
of stationary diffusion, we have:
R2
(2)
ji (r ) = ji ( R) 2 r
Total flow density j(r) of molecules of both components of the vapor is:
j (r ) = j1 (r ) + j2 (r ) (3)
The density of a stationary flow of molecules of i-th component ji(r) taking into
account the hydrodynamic flow of vapor-gas mixture with the radial velocity v(r)
can be written as:
166
− Di (r )
∂ni (r )
+ ni (r )υ(r ) = ji (r )
∂r
(i = 1, 2, 3) (4)
Flow density j3 corresponds to carrier gas. The flow of noncondensable carrier
gas is equal zero. So using (4) we have:
∂n ( r )
− D3 (r ) 3
+ n3 (r )υ(r ) = 0
(5)
∂r
Because the sum of diffusion flows in the reference system moving with velocity v(r), is equal to zero, we obtain:
∂n ( r )
∂n ( r )
∂n ( r ) − D3 3
= D1 1 + D2 2
(6)
∂r
∂r
∂r
Expressing the density of the total flow through the flow densities of both
components of the vapor through (3) and using (4) - (6), we have:
(7)
j (r ) = nυ(r ) For accounting concentration-dependent diffusivity, we approximate Di(r) as
follows:
n (r )
n (r ) 

Di (r ) = Di 0 1 + ε i1 1 + ε i 2 2  (8)


n
n 
where Di0 – diffusion coefficient of molecules of i-th (i = 1,2) component vapor
in a pure carrier gas with a concentration ñ and εi1, εi2 – numerical constants of
order unity.
Using (2), (3), (7), (8) we obtain from (4):
n (r )
n ( r )  ∂n ( r )
j (r ) + j2 (r ) R2

ji ( R) 2 = − Di 0 1 + ε i1 1 + ε i 2 2  i
+ ni 1
(9)

r
n
n  ∂r
n
In the lowest approximation for the concentration (linear approximation), we have:
(10)
∂n (1) (r ) R2
ji (1) ( R) 2 = − Di 0 i
∂r
r
Integrating this equation and using boundary conditions (1), we obtain the following expression for the flows in the lowest approximation:
D
(11)
ji (1) ( R) = − i 0 [ ni 0 − ni∞ ( xi ) ]
R
Substituting (11) in (10), separating variables and integrating, we obtain the
expression for the concentrations in the lowest approximation:
R
(12)
ni (1) (r ) = ni 0 − [ ni 0 − ni∞ ( xi ) ]
r
We return to equation (9), which contains terms both linear and quadratic for the concentration of vapor contributions and substitute flows
in linear approximation for the concentration (11) and concentrations
in linear approximation (12) into the quadratic terms. Then we have:
167
R
n10 − [ n10 − n1∞ ( x1 ) ]
∂n1 (r )
R2
r
j1 ( R) 2 = − D10
− D10 (1 + ε11 )
⋅
∂r
r
n
R
n20 − [ n20 − n2 ∞ ( x2 ) ]
R
r ⋅
⋅ 2 [ n10 − n1∞ ( x1 ) ] − D10 ε12
r
n
R
n10 − [ n10 − n1∞ ( x1 ) ]
R
r R n − n (x )
⋅ 2 [ n10 − n1∞ ( x1 ) ] − D20
[ 20 2∞ 2 ]
n
r
r2
R n20 − [ n20 − n2 ∞ ( x2 ) ]
∂n2 (r )
R2
r ⋅
j2 ( R) 2 = − D20
− D20 (1 + ε 22 )
n
∂r
r
R
n10 − [ n10 − n1∞ ( x1 ) ]
R
r ⋅
⋅ 2 [ n20 − n2 ∞ ( x2 ) ] − D20 ε 21
r
n
R
n20 − [ n20 − n2 ∞ ( x2 ) ]
R
r R n − n (x )
⋅ 2 [ n20 − n2 ∞ ( x2 ) ] − D10
[ 10 1∞ 1 ]

r
r2
n
Integrating the last two expressions with the boundary conditions (1), we obtain
the required expressions for the flow densities:
D
n + n (x )
n + n2 ∞ ( x2 ) 
j1 ( R) = − 10 ( n10 − n1∞ ( x1 ) ) 1 + ε11 10 1∞ 1 + ε12 20
+

R
2n
2n

(13)
n10 + n1∞ ( x1 ) n10 + n1∞ ( x1 ) D20 [ n20 − n2 ∞ ( x2 ) ] 
+
+

D10 [ n10 − n1∞ ( x1 ) ] 
2n
2n
D20
n20 + n2 ∞ ( x2 )
n10 + n1∞ ( x1 )

j2 ( R ) = −
+ ε 21
+
(n20 − n2∞ ( x2 )) 1 + ε 22
R
(14)
2n
2n

+
n20 + n2 ∞ ( x2 ) n20 + n2 ∞ ( x2 ) D10 [ n10 − n1∞ ( x1 ) ] 
+

D20 [ n20 − n2 ∞ ( x2 ) ] 
2n
2n
As can be seen from the expression for the flow density of the first vapor
component ji(r) (13), corrections to the flows obtained in the lowest approximation are caused by different mechanisms. Contributions with ε11 and ε12 come
from concentration-dependent diffusivity. Fourth contribution describes the effect
of Stefan’s flow, and the last - the influence on the Stefan’s flow of the second
component vapor.
Conclusions
The equations (13) and (14) express the dependence of the flow densities caused
by Stefan’s flow and concentration dependence of diffusion coefficients.
The obtained results can be used to clarify the description of binary condensation which developed in [1-3].
168
References
1. Kulmala M., Vesala T., Wagner P.E. // Proc. Royal Soc., Vol. 441, P. 589
(1993).
2. Kuchma A.E., Shchekin A.K., Kuni F.M. // Colloid Journal, Vol. 73, P. 215
(2011).
3. Kuchma A.E., Shchekin A.K., Kuni F.M. // Physica A, Vol. 390, Issue 20, P.
3308 (2011).
169
A matrix approach for dyadic Green's function in
multilayered elastic media
Nikitina Margarita
margaritnikitina@yandex.ru
Scientific supervisor: Dr. Val’kov A.Y., Department of Statistics
Physics, Department of Physics, Saint-Petersburg State
University
Introduction
The Green’s function method is widely used for studies of layered media [1, 2],
in particular for analysis of the synthetic seismograms. Problems of this type have
a long history going back to the articles of Kelvin and Stokes. They are extremely
important for geophysical applications and a lot of papers in mathematical physics
were devoted to them.
Generals
An elastic medium is described with the field u(r,t) which is the displacement
of matter in point r at the time t. The motion equation (Navier-Cauchy) in an elastic
medium can be written as
∂σ αβ
∂ 2u
(1)
ρ(r ) 2α =
+ Fβ
∂rβ
∂t
where ρ is the mass density, σ is the stress tensor, and F is the body force per unit
volume. The stress tensor is related to the strain tensor epsilon, defined for small
displacements with the constitutive relations.
σ αβ = Cαβγς ε γζ , (2)
1  ∂uγ ∂uζ 
ε γζ = 
+
(3)
2  ∂rζ ∂rγ 
In the linear approximation the constitutive relations know as Hook law (1).
And C is the stiffness tensor of 4-th-order (4)
2


(4)
Cαβγζ (r ) = K (r )δ αβ δ γζ + µ (r )  δ αγ δ βζ + δ αζ δ γβ − δ αβ δ γζ 


3
Differential equation (1) must be added with proper boundary condition.
We use two types of boundary conditions:
• on the boundary S of two homogeneous medium the matter deformations
u(r) are to satisfy two conditions of continuity, where n is the normal vector to
the boundary. The first equation means the continuity of the displacement vector,
and the second one corresponds to the equality of pressures on the opposite sides
of the boundary.
(5)
u(r ) S = u(r ) S + ,
−
σn S = σn S + ,. (6)
−
170
• in case of elastic medium boundary with the vacuum one requires only the
pressure components on the boundary should be zero.
(7)
σn S = 0 Presently we consider displacement field harmonic in time. In this case the
solution for deformations can be presented in form of the plane waves (8) where
k is the wave vector and ω is the circular frequency.
u (r, t ) = Aei (kr −ωt ) (8)
There are three wave modes, longitudinal one, with polarization vector A being parallel to wave vector k and two transverse ones with A perpendicular k. The
wave numbers of these modes are determined by dispersion relationship, which
you can see in Fig. 1.
(a) (b)
Fig. 1. a) General information about longitudinal wave and direction of polarization vector and wave vector; b) General information about transverse wave and
direction of polarization vector and wave vector.
The Green function (fundamental solution) G of equation (1) is defined by
The physical sense of Gγη(r,r’;t-t’) nis the γ-component of the displacement u(r,t)
initiated by a source that is impulse of impact point force directed along η-axis,
localized in point r’ and running at the moment t’. In addition G obeys to • Causality principle:
• Infinity- the Sommerfeld radiation condition
• Medium-medium
Gˆ
Gˆ (r, r ' , t − t ' ) = 0, t < t '.
r ∈S−
= Gˆ
r ∈S +
, nβ Cαβγζ
• Medium-vacuum
nβ Cαβγζ
∂Gγη
= nβ Cαβγζ
∂rζ
r ∈S−
∂Gγη
∂rζ
r ∈S
171
=0
∂Gγη
∂rζ
r ∈S +
Method for arbitrary layered medium
In this article we consider layered medium wherein the elastic modules depends
on the z-axis only. Let the system occupies the region l1 ≤z ≤ l2
The Green’s function obeys to equation
 ˆ

∂2
∂
+ Kˆ 1 ( z ) + Kˆ 0 ( z ) Gˆ (q ⊥ , z , z ', ω ) = Iˆδ ( z − z ')
K
(
z
)
2
2
(1)

∂z
∂z


where matrices K2(z),K1(z),K0(z) depend on the mediums constants.
Kˆ 2 ( z ) = ct2 ( z ) Iˆ + (cl2 ( z ) − ct2 ( z ))n ⊗ n,
Kˆ ( z ) = (c 2 ( z ) − c 2 ( z ))(Q ⊗ n + n ⊗ Q),
l
1
t
(2)
Kˆ 2 ( z ) = (ω 2 − ct2 ( z )q⊥2 ) Iˆ − (cl2 ( z ) − ct2 ( z ))Q ⊗ Q,
The most generalized form of the third-type (Robin) boundary condition
∂ ˆ
ˆ
 ˆ
= 0; j = 1, 2
 B1 (l j ) + B0 (l j ) G (q ⊥ , z , z ', ω )
∂z
z =l
(3)
j
2


Bˆ1 ( z ) = µ ( z )Q ⊗ n +  K ( z ) − µ ( z ) n ⊗ Q,


3
(4)
1


ˆ
ˆ
B0 ( z ) = µ ( z ) I +  K ( z ) + µ ( z ) n ⊗ n.


3
At the same way we can write the equations for deformations and boundary
condition. And there are six linear independent solutions of equation. Three of
them satisfy one of boundary condition (with l2)
u>(1) , u>(2) , u>(3) , (5)
and three another condition (with l1)
u<(1) , u<(2) , u<(3) (6)
For convenience we present these solutions as two 3×3 matrices
Vˆ ( z ) = u (1) , u (2) , u (3) ,Vˆ ( z ) = u (1) , u (2) , u (3) (
>
>
>
>
)
<
(
<
<
<
)
(7)
In these terms the Green’s function can be represented in this form
Vˆ> ( z )Vˆ>−1 ( z )Wˆ −1 ( z '), z ≥ z ',
ˆ
G ( z , z ') = 
−1
−1
(8)
Vˆ< ( z )Vˆ< ( z )Wˆ ( z '), z < z ',
where W can be understood as the generalized matrix Wronskian of the set of
functions u.
Wˆ ( z ) = K ( z )(Vˆ<′( z )Vˆ< ( z ) − Vˆ>′( z )Vˆ> ( z )) (9)
To obtain matrices V>(z) and V<(z) we express them in terms of six independent solutions of the wave equation (2), u(1)(z),…, u(6)(z). We arranged conveniently
the 3×6 matrix
Vˆ ( z ) = u (1) ( z ), u (2) ( z ), u (3) ( z ), u (4) ( z ), u (5) ( z ), u (6) ( z ) (10)
(
)
172
Since solutions u<(1),(2),(3) and u>(1),(2),(3) are the linear combinations of vectors u(1)
(z),…, u(6)(z), we can present the columns of the matrices V>(z) and V<(z) as linear
combinations the columns of matrix V(z). That’s why we can get this formula
(11)
Vˆ< ( z ) = Vˆ ( z ) Jˆ< , Vˆ> ( z ) = Vˆ ( z ) Jˆ> where J1,2 are the 6×3 matrices of coefficients of the linear combinations. For
obtaining J1,2 we use formulae (3,11)
 Bˆ1 l j Vˆ ' l j + iBˆ0 l j Vˆ l j  J j = 0, j = 1, 2 

() ()
()()


 Iˆ Iˆ
Jˆ2 =  -1
, Aˆ1 = (−iBˆ1 (l1 )Vˆ '(l1 ) + Bˆ0 (l1 )Vˆ (l1 ))   ,

 0ˆ 
 − Aˆ 2 (l2 ) Aˆ1 (l2 )
 − Aˆ -1 (l ) Aˆ (l )
 0ˆ 
Jˆ 1 =  1 1 2 1  , Aˆ 2 = (−iBˆ1 (l2 )Vˆ '(l2 ) + Bˆ 0 (l2 )Vˆ (l2 ))  
Iˆ


 Iˆ
Method for multi-layered medium
But the explicit analytical results for solutions of the homogeneous wave
equation (i.e. for matrix V) can be obtained only for some specials cases. In what
follows we consider one such case, which is important for applications. In case of
piecewise – homogeneous medium we can write normal modes in the layer in the
explicit form. In (q perpendicular, z)-presentation in particular we have
± iλ ( q ) z
u ±jp (q ⊥ ; z ) = e ±jp (q ⊥ )e jp ⊥ , λ jp = k 2jp − q ⊥2 ,
e1±p =
 λ jp q ⊥ + q⊥2 n ±
q ⊥ ± λ jp n
q⊥ ⊗ n ±
, e2 p =
, e3 p =
,
q⊥
q⊥ ktp
klp
where j=1,2- are transverse waves, and j=3- is longitudinal wave. Here we can see
the result for fields deformation matrices V< and V>.


 − Aˆ1-1 (l1 ) Aˆ 2 (l1 ) Iˆ
<
ˆ
ˆ
ˆ
,
(
)
(
)
Vˆ> ( z ) = Uˆ p ( z ) Mˆ p>  -1
V
z
=
U
z
M
 <
.
p
p 
Iˆ


 − Aˆ 2 (l2 ) Aˆ1 (l2 )
And after that I’ll show using formulae.
Uˆ ( z ) = (Uˆ + ( z ) Uˆ − ( z )),
p
(
p
Uˆ p± ( z ) = e1±p (q ⊥ )e
p
± iλ1 p ( q ⊥ ) z
; e ±2 p (q ⊥ )e
± iλ 2 p ( q ⊥ ) z
ˆ ± + Bˆ (l )Uˆ ± (l )) Eˆ
Aˆ1,2 (l1 ) = ( Bˆ1 (l1 )Uˆ1± (l1 ) Λ
1
0 1
1
1
1,2
; e3± p (q ⊥ )e
± iλ 3 p ( q ⊥ ) z
)
 Iˆ
 0̂
Eˆ1 =   , Eˆ 2 =   .
 0ˆ 
 Iˆ


Uˆ p ( z )
Sˆ p ( z ) = 
,
ˆ + Bˆ Uˆ ( z )
 Bˆ1 pUˆ p ( z ) Λ
p
op p
where Λp is diagonal matrix 6×6 with wave numbers ±λip , i=1,2,3 on the diagonal.
173
Mˆ p = Sˆ p−1+1 ( z p ) Sˆ p ( z p )  ( M N −1 ...M p ) −1 , p < N ,
Mˆ p> = 
p = N,
 Iˆ,
p = 1,
 Iˆ,
Mˆ p< = 
 M p −1 ...M 1 , p > 1
Discussion
Let’s discuss finally GF in the r-representation. Performing the inverse Fouriertransform we obtain
dq⊥
Gˆ (R ⊥ ; z , z ') = ∫ Gˆ (q ⊥ ; z , z ')eiq⊥ R ⊥
(2
π) 2 There exists inputs to GF several types, such as
• Far field: the stationary phase points vicinity
• Poles (Rayleigh , Love, Stoneley waves)
• Head wave (λ=0)
• Near-field (wide area q^ input).
The final formula for the Green’s function is contributed by the body waves,
transverse and longitudinal, and by surface waves generated by the poles of the
matrix Wronskian. This formula has been used for numerical computation of the
harmonic Green function in the layered media.
References
1. Aki K., Richards P.G. Quantitative Seismology: Theory and Methods, v.1,
1980; Freeman W.H., Co. Shikin A.M. et al. // Phys. Rev. B 62, pp. 13202–13208,
2000.
2. Wapenaar K., Fokkema J. // Geophysics, v. 71, Iss. 4, pp. SI33-I46 Suppl. (2006);
Liu E., Zhang Z., Yue J., Dobson A. // Commun. Comput. Phys., v. 3, pp. 52-62,
2008.
174
Detectable effects in classical supergravity
Niyazov Ramil
ramilniyazov@gmail.com
Scientific supervisor: Dr. Shadchin S.V., Department of High
Energy and Elementary Particles Physics, Faculty of Physics,
Saint Petersburg State University
Introduction
Supersymmetric theories and, in particular, supergravity manifest a symmetry
between fermions and bosons. Despite the popularity of such theories, they miss
experimental evidences. That’s why it’s important to search for possible experiments to confirm these theories.
Supersymmetric corrections
In N=1 supergravity exist two particles: quantum of gravitational field - graviton - particle with spin 2 and its superpartner - gravitino - particle with spin 3/2,
described by Rarita-Schwinger field. Supergravity Lagrangian is sum of Einstein
Lagrangian (formulated in vierbein (tetrad) formalism)
g µν ( x) = eµa ( x)eνb ( x) ηab
(for possibility of introduction spinors) so Rarita-Schwinger Lagrangian read [1]
1
1
(1)
| g |R − ieeaµ ebν ecρ ψ µ γ abcD ν ψ ρ
L = L E + L RS = −
4
2
where
1


Dµ ψ =  ∂ µ + ω µab γ ab  ψ


4
and ωμab – gauge fields called spin-connection introduced for invariance of RaritaSchwinger Lagrangian under Lorentz transformations. Supergravity Lagrangian is
invariant under three kinds of local symmetries [1].
•
general coordinate transformations with parameters ξμ(x):
δ G (ξ)eµa = −ξ ν ∂ ν eµa − ∂ µ ξ ν eνa , (2)
(3)
δ G (ξ)ψ µ = −ξ ν ∂ ν ψ µ − ∂ µ ξ ν ψ ν .
•
•
local Lorentz transformation with parameters λab(x) (λab= - λba):
δ L (λ )eµa = −λ a b eµb ,
1 ab
δ L (λ )ψ µ = − λ γ ab ψ µ .
4
local supertransformations with parameter εα(x) (εc=ε):
175
(4)
(5)
δ Q (ε )eµa = −i εγ a ψ µ ,
δ Q (ε )ψ µ = Dµ ε.
(6)
(7)
Follows from (1) that General Relativity can be seen as supergravity with
vanishing gravitino so with the supersymmetry transformations applied to the
Schwarzschild solution one can obtain solution in classical supergravity.
ψ µ = 0 +∆ψ µ ⇒ (8)
a
a (0)
a
eµ = eµ +∆eµ ⇒
(9)
(0)
g µν = g µν
+ ∆g µν 
GR
(10)



SUSY
Supersymmetric metric calculated in paper [3]. Consider its in equatorial plane then
(11)
1 ,
g rr = −
rg
1−
r
1  rg 
gθθ = − r 2 − k 2 1 −  ,
(12)
2 
r
1  rg 
gϕϕ = − r 2 − k 2 1 −  ,
2 
r
2 rg 1 rg
gtt = 1 − + k 2 4 ,
r 8 r
 3 rg 
g ϕt = g t ϕ = A  1 −
 2 r 
where
A = εγ 5 γ 2 ε
k ≡ εε
(13)
(14)
(15)
Motion of a probe particle
To calculate equation of the motion of a probe particle, we use approach developed in [2]. Consider Hamilton-Jacobi equation
∂S ∂S
g µν µ ν − m 2 c 2 = 0
(16)
∂x ∂x
where m is particle mass. Solution of this equation is sought as
S = − Et + Lϕ + S r (r ) (17)
Hence for radial part of action we have
1
 2 2 g tt E 2
2
g ϕt LE
ϕϕ 2  1
(18)
S r = ∫  m c − 2 + 2
− g L  rr  dr
c
c
g 

We introduce a new variable r' in such way that
r (r − rg ) = r ′ 2 , ⇒, r ′ ≈ r − rg / 2 .
176
Denote ratio rg/r‫ ׳‬as Greek letter γ. We need the contravariant components of
the metric, we obtain from inverting (11). Introduce the non-relativistic energy
E‫≡׳‬E-mc2, expand the metric tensor up to terms of order γ2 and k2 and obtain
rg  E ′ 2
 E ′2

S r = ∫  2 + 2 E ′m +  2 2 + 4 E ′m + m 2 c 2  −
r′  c
 c

2 2
1  2 3 2 2 2 2 E ′ rg
− 2 L − m c rg −
− 4 E ′mrg2 +
2
r ′  
c2

GR corrections
(19)
1
2


2 LEA  2 k 2   E ′ 2
2 2 
+
+  A −   2 + 2 E ′m + m c    dr ′.
c
2  c


  
SUSY corrections

The substitution L‫=׳‬L+EA/c will remove the first-order supersymmetric in the
constant redefinition of the angular momentum L. This is legitimate since actually
the angular momentum which we measure is the L‫׳‬. We introduce constant B so
that the coefficient of 1/r‫׳‬2 has the form L‫׳‬2 – B . In B leave in the order of с2
3
E2 (20)
B = m 2 c 2 rg2 + k 2 2
2
2c
Next we use the fact that the angle change during one rotation of a probe
particle in its orbit is
∂
∆ϕ = −
∆S r ∂L
where ΔSr corresponded change in Sr. It expanded in powers of a small correction
B then we obtain
B ∂∆S r(0) (21)
∆S r = ∆S r(0) −
,
2 L ∂L ′
where ∆S r(0) corresponds to classical motion that is
∂
−
∆S r(0) = ∆ϕ (0) = 2π
∂L
As a result
Bπ
∆ϕ = 2π + 2 . (22)
L′
We distinguish the corrections relating to supergravity
1
BSUSY = m 2 c 2 k 2
2
and to General Relativity
3
BGR = m 2 c 2 rg2 2
and their ratio
BSUSY
k2
(23)
= 2 .
BGR
3rg
Thus we can assert the smaller the gravitational radius of the black hole, the
greater the supersymmetric correction. Also this phenomena can be observed in
177
the formula of the correction to angle changing at 2π by only supersymmetry with
expressing the momentum through the focal parameter of the orbit
L ′2 = m 2 pGM k 2π
(24)
∆ϕ =
.
2 prg
Suppose that a probe particle rotates in the field of a black hole with mass
M = 1 kg with a period T = 1 day, the eccentricity is e = 0.5 and the anomalous
displacement of perihelion Δφ=1°. Focal parameter can be expressed through a
rotation period of the particle. Note that the semimajor axis of the orbit about
0.25 meter. Then we can estimate the order of k:
2rg 
3rg π 
−15
k=
p∆ϕ −
~ 10 m.
(25)
π 
2 
Motion of a light ray
For light ray we will modify our considerations assuming m = 0 and passing
from particle energy E to the light frequency ω, from L to ρ=cL/ω. Having a similar
argument as above (16) - (22) one can obtain:
2rg R
∂
∂
∆ψ r = −
∆ψ (0)
.
(26)
r +
∂L
∂L
(ρ + A) R 2 − (ρ + A) 2 We limit R to infinity and note that the straight ray Δψr(0) changing equal π so
2rg ∆ϕ = π +
.
(27)
ρ+ A
∆ϕ = −
It is seen that the first-order corrections can be removed by redefinition of
constant ρ. Besides such supplement, we can not measure.
Conclusions
First order corrections (on a suitable parameter) are not observable and can be
eliminated by redefinition of momentum and impact parameter for the particles,
and for the light beam. Second order corrections exist only for the motion of probe
particle. Such corrections appear in the anomalous displacement of perihelion of
the probe particle. Thus effects of second-order supergravity can be detected while
observation of probe particle perihelion displacement for the small black holes.
To make effect measurable, some conditions should be imposed to the laboratory
system. The size (according to calculations about 0.25 meter) and the weight (about
1 kilogram) of this system are quite human-natural, but complete isolation of the
laboratory from external fields is required.
178
References
1. Tanii Y. Introduction to Supergravities in Diverse Dimensions, 1998.
2. Landau L.D., Lifshits E.M. Course of Theoretical Physics, V. 2. The Classical
Theory of Fields. Butterworth-Heinemann.- М., 4th edition, 1975.
3. Baaklini N., Ferrara S. van Nieuwenhuizen P. // Lettere Al Nuovo Cimento
(1971 – 1985), 20:113-116, 1977.
179
Calculation of characteristics of critical behavior in
logarithmic dimensions
Artem Pismenskiy
artem5085@mail.ru
Scientific supervisor: Prof. Dr. Pis’mak Yu.M., Physical Faculty,
Saint Petersburg State University
We study the asymptotic behavior of Green functions for the theories φ3 and φ4
in a critical point by means of renormalization group equation and also by means
of self-consistent equation.
In a critical point all the dimensional coupling constants become equal to zero,
only dimensionless coupling constants remain. The action for the φ3 theory is of
the form
λ
1

S [ϕ ] = ∫d d x  (∂ϕ ( x )) 2 − ϕ 3 ( x ) . 2

3!
Here φ(x) is a scalar field, d is the space dimension, λ is a coupling constant,
d
(∂ϕ ( x)) 2 = ∑∂ µ ϕ ( x ) ∂ µ ϕ ( x).
µ =1
For the φ4 theory,
g
1

S [ϕ ] = ∫ d d x  (∂ϕ ( x)) 2 − (ϕ 2 ( x)) 2  . 2

4!
where φ(x) is n-component vector field, d is the space dimension, g is a coupling
constant,
n
d n
(∂ϕ ( x)) 2 = ∑∑∂ µ ϕ i ( x ) ∂ µ ϕ i ( x), ϕ 2 = ∑ϕ i ϕ i
µ =1 i =1
i =1
First, let’s consider the renormalization group equation [1]:
 ∂

∂
+
µ
+
β
g
m
γ
g
W
p
,
µ
,
g
=
0
(
)
(
)
(
)
 ∂µ
 m
∂g
Here μ is a scale parameter, g is a coupling constant, Wm is m-point connected
Green function, p denotes a set of momentums, β(g) is a beta-function , γ(g) is an
anomalous dimension of the field φ.
δm
Wm = m lnG ( J ) |J = 0 .
δJ
Here G(J) is generating functional of Green function:
G ( J ) = c ∫Dϕe − S [ϕ ]+ ϕJ
c −1 = ∫Dϕe − S [ϕ ]
We are interested in two-point correlation function (propagator): D = W2
(m=2).
(μ ∂/∂μ + β(g) ∂/∂g + 2γ(g) ) D(p,μ,g) = 0.
There is other equation, which contains ∂/∂μ and ∂/∂p [1]:
180
(μ ∂/∂μ + p ∂/∂p – 2) D (p,μ,g) = 0.
Combining these 2 equations we exclude ∂/∂μ.
 ∂

∂
−
p
+
β
g
2
³
g
−
2
D
p
,
¼
,
g
=
0
�
(
)
(
)
(
)
 ∂p

∂g
(1)
We want to find the dependence of the propagator D on the momentum p. The
solution of (1) reads
−2
 g ( p ) 2 γ ( x ) 
 p
D ( p) = D0   exp  − ∫
dx  ,
 p0 
 g0 β ( x ) 
where D0, p0, g0 are constants and the function g(p) fulfills the equation:
− ln
p
=
p0
g
dx
∫ β ( x)
g0
We study infrared (p→0) or ultraviolet (p→∞) asymptotic of the propagator
when g→0. From the renormalization group equation we can obtain only one of
two asymptotics. To understand which of them we can obtain we should consider
the beta-function. For both theories φ3 and φ4 in logarithmic dimension (d=6 for
φ3 and d=4 for φ4) the main approximation of the beta-function has the form
β(g) = b2g2,
and we receive the equation:
1  1 1
p
− ln
=  − 
p0 b2  g0 g 
For the φ3 theory it holds b2<0 and g~λ2. If λ is real then g>0 and we obtain the
ultraviolet asymptotic. Usually one considers λ to be imaginary then g<0 and we
receive the infrared asymptotic. For the φ4-theory we have b2>0, g>0 and we obtain
the infrared asymptotic.
For the φ3 theory in the logarithmic dimension d=6 it holds [1]:
β(g) = – 3g2/2 – 125g3/72 + …
γ(g) = g/12 + 13g2/432 + …
and we receive
p


ln ln( )
−2
−1/ 9

 p  p 
p0
125
D ( p) = D0    ln  1 +
+ …
(2)
 p0   p0   1458 ln( p )



p0
For the φ4 theory in logarithmic dimension d=4 we have [1]:
β(g) = g2(n+8)/3 – g3(3n+14)/3 + …
γ(g) = g2(n+2)/36 – g3(n+2)(n+8)/432 +…
and we obtain
p


lnn ln( )
−2

 p 
p0
n+2
1
2(n + 2)(3n + 14)
D ( p ) = D0   1 −
−
+ … (3)
2
4
p
 p0   2 ( n + 8) ln p
2 ( n + 8)

ln 2 ( )


p0
p0
181
Now, let us consider the self-consistent equation.
The full propagator D fulfills the Dyson equation
D–1 = Δ–1 – Σ (4)
Here Δ is the bar propagator, Σ is the mass operator. Σ is presented by infinite
number of 1-irreducible diagrams.
We study infrared asymptotic of the full propagator D. The problem is nonperturbative, because every next term in Σ is more significant than previous one
and we need to take into consideration infinite number of terms. If D and Δ have
different asymptotics then we drop out from the right-hand side of (4) the bar
propagator Δ and we receive the self-consistent equation: D–1 = – Σ
The mass operator Σ in 1-loop approximation represents a product of 2 propagators in the coordinate representation.
First of all we consider the φ3 theory with ∑ =
λ2
2
The line is the full propagator D.
It is technically simpler to make calculations in the coordinate representation then
the equation (2) in our approximation is written as
λ2
2 D −1 ( x ) = − D ( x ) .
2
For solution we use the ansatz
A
D ( x) = 2a
.
x (ln x 2 )α
In the logarithmic dimension d=6 we obtain a=2, α=1/3 and
A
D ( x) = 4
2 1/ 3
(5)
x (ln x )
12
with λ 2 A3 = 6
π
and in the momentum representation it looks like
A1
D ( p) = 2
p (ln p / p0 )1/ 3
where A1 is a function of A.
This result does not coincide with the main approximation of (2).
Now we consider the φ4 theory. Introducing auxiliary scalar field ψ
4
 ψ2

+ ψϕ 2 
e Λϕ = c ∫Dψ exp  −
 4Λ

we obtain the system of 2 self-consistent equations [1]
182
The solid line is the full propagator of the field φ and the dashed line is the full the
propagator of the field ψ. That is
 Dϕ−1 ( x) = − Dϕ ( x) Dψ ( x)

 −1
n
2
 Dψ ( x) = − Dϕ ( x )

2
In the logarithmic dimension d=4 we receive the following result
A
n/4

Dϕ ( x ) = 2 1 +

x  ln x 2 ⋅ ln ln x 2 
B
x 4 (ln x 2 ) 2 ⋅ ln ln x 2
where
n
A2 B = 4
2π
In the momentum representation the propagator Dφ of the field φ looks like

A 
A'
Dϕ ( p ) = 12 1 +
p  ln( p / p0 ) ⋅ ln ln( p / p0 ) 
Dψ ( x ) =
(6)
A1 is a function of A and A' is a function of n.
We see that results (3) and (6) coincide in the main approximation, but it is not the
case for next corrections.
Logarithmic corrections at d=4 were also considered by Y. Okabe [2]. He investigated characteristics of the system behavior in the neighborhood of the critical
point, which we did not calculate.
Conclusions
We have investigated the infrared asymptotic of propagators for the theories
φ3 and φ4 by two different methods: renormalization group equations and selfconsistent equations. For the φ4 theory the results coincide only in the main approximation, and for the φ3 theory the results disagree. The possible reason is that in
the renormalization group equation we wrote the beta-function taking into account
the renormalization of vertex. In the self-consistent equation we did not make this.
To correct result one should insert the full vertexes into the mass operator
.
References
1. Vasil'ev A.N. The field theoretic renormalization group in critical behavior
theory and stochastic dynamics. SPb, 1998.
2. Okabe Y. // Progress of Theoretical Physics, vol. 59, № 2, 1978.
183
Deal.ii library as a tool to study three-body quantum
systems
Shmeleva Yulia
julia.shmeleva@gmail.com
Scientific supervisor: Dr. Yarevsky E.A., Department of
Computational Physics, Faculty of Physics, Saint-Petersburg
State University
Introduction
Few-body quantum mechanical problem is the well-known challenging problem
in quantum mechanics. Provided that interparticle interactions are known, it poses
the eigenvalue or scattering problem. Their solutions should be calculated with
high accuracy and can be compared with experiment so that physical model and
numerical approach are tested [1]. Several successful approximation techniques
have been developed for few-body problem, including the Hartree Fock method,
finite difference methods and various variational approximations.
The finite element method (FEM) has originally been used in technical applications like elastic and fluid mechanics [2]. The implementation of the FEM in
quantum mechanics was rather rare in spite of its advantages. However, starting
since 1985 some works applying the FEM to three-body problems have appeared
and demonstrated good results in this area [1, 3].
One of the available implementations of the FEM is the Deal.ii library. Deal.
ii is the powerful general purpose object oriented finite element differential equations analysis library. It provides a wide collection of tool classes such as adaptive
meshes, a variety of finite elements, parallelization, support of various formats
and many other routines [4]. Use of such a frameworks will significantly simplify
the process of writing the program code and let the researcher concentrate on the
physical or technical problem itself.
The aim of the present work is to explore deal.II and to evaluate its efficiency,
stability and accuracy for quantum mechanical problems.
Results and Discussion
The helium atom with the zero total angular momentum has been chosen as a
benchmark of the three-body Coulomb problem. The system of the helium atom
is described by the six-dimensional Schrodinger equation. It is unpractical to be
solved directly by FEM. After expanding wave function in terms of Wigner D functions and using their orthogonality relation one can obtain exact three-dimensional
Schrödinger equation [1]

 2 1  ∂2
∂ 1 − c2 ∂ 
 − ∑ 2  ri 2 ri +
 +V r1, r2, c  Ψ = E Ψ ,
∂c
∂c 
 i=1 2ri  ∂ri



(
)
184
(
)
(
)
V r1, r2, c = −
2 2
1
− +
.
2
r1 r2
r1 − 2r1r2 c + r22 In this work the atomic units are used, i.e.
Bohr radius for the length, Hartree for the
energy and the standard Jacobi coordinate
system (Fig. 1), where r1 and r2 are distances
between electrons and nucleus and
c = cos (rˆ1 , r2 )
By integrating the Schrödinger equation
over the inter-electronic angle (rˆ1 , r2 ) one can
obtain the two-dimensional equation for the
S-wave model
Fig. 1. Jacobi coordinate system.
 2 1 ∂2

r +Vˆ r1, r2  Ψ = E Ψ , −∑
2 i
 i=1 2ri ∂ri

2 2
1
Vˆ r1, r2 = − − −
|
r
−
r
|
−
|
r
+
r
|
.
[1 2 1 2]
r1 r2 2r1r2
( )
( )
Both two- and three-dimensional equations have been investigated using FEM
techniques implemented in the deal.ii library. The three-dimensional space formed
by r1, r2 and chas been divided into some number of rectangular boxes numbered
by i. The problem has
been solved on static
(Figs. 2, 4) and automatically refined grids
(Fig. 3).
The wave function
has been expanded in
terms of finite-element basis functions
such that
Fig. 2. 2D problem,
Fig. 3. 2D problem, adapinput grid.
tive grid.
Fig .4. 3D problem, input grid.
(
)
(
)
Ψ r1, r2, c = ∑ aim f im r1, r2, c .
im
The Lagrange polynomials have been
chosen as the basis functions, the polynomial degree varies from 1 to 5. The
coefficients aim and the energy E are
obtained by minimization of the functional
185
Ψ|H |Ψ
That leads to the eigenvalue problem

 = ESa,
Ha
H im, jk = f im | H | f jk ,
Sim, jk = f im | f jk .
All matrix elements are computed by the standard Gaussian integration formula. Deal.ii library provides an interface to SLEPc library, which implements
few eigenvalue solvers including the Krylov-Schur solver. This solver supports all
types of eigenvalue problems. It can be used for searching any part of spectrum. In
this work it has been applied to solve the generalized eigenvalue problem.
The ground state energy has been calculated and compared to the high precision reference results. The Table 1 and Figs. 5, 6 show the relative error of the
approximate eigenvalue for the ground state.
Table 1. FEM results for the ground state.
2D problem,
static grid
Reference results
-2,879194803
Obtained approximate
-2,87901
results
Number of DOF
1681
Number of GPs
64 x 64
CPU time T (sec)
30
2D problem,
adaptive grid
-2,879194803
3D problem,
static grid
-2,903724377...
-2,87903
-2,90358
5645
64 x 64
77
10086
16 x 16 x 40
~14000
Here DOF – the number of degrees of freedom, GPs – the number of Gaussian
points, T – the computational time.
Fig. 5. Error of the solution in 2D
problem.
Fig. 6. Error of the solution in 3D
problem.
186
With the number of degrees of freedom increasing (achieved by increasing
the number of elements or polynomial degree), the error of the solution becomes
smaller. In both two-dimensional and three-dimensional problems the value of
obtained error is 10-4 a.u. This non-zero limit might be a consequence of KrylovSchur solver application. Further investigation of different eigenvalue solvers is in
progress. The graph for the error in the three-dimensional problem (Fig. 6) does not
reflect the limit of the error. The three-dimensional problem demands substantional
computational resources such as CPU time and memory, and further increasing of
degrees of freedom is rather problematic.
References
1. Elander N., Yarevsky E. // Phys. Rev. A, 57(4) (1998) 3119–3122.
2. Zienkiewicz O.C., Taylor R.L. The finite element method. The basis. Vol. 1.Butterworth-Heinemann, Fifth Edition, 2000.
3. Levin F.S., Shertzer J. // Phys. Rev. A, 32(6) (1985) 3285–3290.
4. http://dealii.org/ (A Finite Element Differential Equations Analysis Library).
187
Second order effects in the hyperfine and Zeeman
splittings in highly charged ions
Mikhail M. Sokolov
pbo-ii@yandex.ru
Scientific supervisor: Dr. Glazov D.A., Department of Quantum
Mechanics, Faculty of Physics, Saint-Petersburg State
University
Introduction
Development of experimental techniques for cooling and trapping of individual
charged particles has lead to the measurements of the g factor of light hydrogenand lithium-like ions with a relative accuracy of 10-9 [1-3]. In turn, it motivates for
high-precision theoretical calculations [4]. In the boron-like systems, which are of
particular interest presently [5], the non-linear effects in magnetic field become
considerable. In the present work the second- and third-order effects for the ground
state Zeeman splitting of the boron-like argon ion 40Ar13+ are investigated.
Possible influence of the second-order effects in the hyperfine interaction on the
observed transition energies between the fine-structure components is estimated
as well. The case of boron-like argon is considered, where high-accuracy experimental values are available [6]. The results for various argon isotopes 33Ar, 35Ar,
37
Ar and 39Ar are presented.
Relativistic units (ћ = c = m = 1) and the Heaviside charge unit (e2 = 4πα) are
used in the work.
Zeeman splitting
High-precision experimental and theoretical investigations of the g factor of
highly charged ions aim at stringent tests of QED in strong field domain. These
investigations allow as well for accurate determination of the values of some fundamental constants such as the electron mass and the fine structure constant. The
proposed experiment with boron-like argon is based on spectroscopic determination of the frequency of forbidden transitions between the Zeeman sublevels of an
isolated ion placed in a trap.
In the present case of boron-like argon ion 40Ar13+ the spin of the nucleus is zero
and the total angular momentum of an ion is determined only by the electrons. The
hyperfine structure is absent; the scheme of the energy levels is shown in Fig. 1.
Energy shift of the Zeeman sublevels of the fine structure due to the interaction
with external magnetic field is determined by the expression:
∆Eγ , M ( B) = g ( B)µ 0 BM
Here, μ0 – Bohr magneton, B – magnetic field strength,
M – projection of the total
angular momentum of the ion, γ – quantum numbers defining the level of the fine
structure. The function g(B) tends to the g factor at B→0. However, in any ex188
Fig. 1. Energy levels scheme for the 2p state in magnetic field. The notation:
(1)
(2)
(0)
(0)
∆EFS (2p) = E2p
− E2p
, a1 = ∆E2 p1 2 , M =±1 2 , a2 = ∆E2 p1 2 , M = ±1 2
32
12
a3 = ∆E2(3p)1 2 , M =±1 2 , b1 = ∆E2(1p)3 2 , M =±1 2 , b2 = ∆E2(2p)3 2 , M = ±1 2 , b3 = ∆E2(3p)3 2 , M = ±1 2
perimental setup the value of B is small but finite. Thus, to accurately determine
the g factor from the measured value of ∆Eγ,M the nonlinear terms in magnetic field
might be relevant. We consider the perturbation theory expansion up to the third
order,
∆E2 p1 2 , M = ∆E2(1)p1 2 , M + ∆E2(2)p1 2 , M + ∆E2(3)p1 2 , M There is one electron in the 2p state over the closed 1s and 2s shells in the ground
state of the boron-like ion. Therefore we employ the one-electron approximation.
The corrections of the first, second and third orders are given by standard formulas
of the perturbation theory,
(1)
∆E2p
= 2p1 2 , M Vˆhom 2p1 2 , M
1 2 ,M
(2)
∆E2p
≈ 2p1 2 , M Vˆhom 2p3 2 , M
1 2 ,M
∆E
(3)
2p1 2 , M
(
2
(E
(0)
2p1 2
(0)
− E2p
32
)
≈ 2p3 2 , M Vˆhom 2p3 2 , M − 2p1 2 , M Vˆhom 2p1 2 , M
× 2p1 2 , M Vˆhom 2p3 2 , M
2
(E
(0)
2p1 2
(0)
− E2p
32
)
)×
2
We restrict the sums over the spectrum to the fine structure component 2p3/2, since
it provides the dominant contribution. The omitted terms are smaller by three orders
189
of magnitude. The formulas involve the matrix elements of the relativistic operator, describing the electron interaction with the external magnetic field,
Vˆ = 1 eB ⋅ (r n × α ), n ≡ r r hom
2
where α – the Dirac matrices. The results of numerical calculations can be presented
in the following form:
(*)
∆E [eV] = z B[T]M 1 + z B[T]M + z ( B[T]) 2 2p1 2
1
(
2
3
)
The values of the coefficients calculated for hydrogen-like wave functions are
presented in the second column of Table 1. In order to take into account in a
simple way the interelectronic interaction with the 1s and 2s electrons we use the
approximation of effective nuclear charge. The value of Zeff is taken as to reproduce
the experimental value of the fine structure ΔEFS(2p) =2.810 eV within the Dirac
equation for the nuclear charge Zeff. The results obtained with Zeff =15.7504 are
presented in the third column of Table 1.
Hyperfine structure
The hyperfine splitting in highly charged ions can serve as a tool to test QED in
strong electromagnetic fields [7] and to probe the nuclear properties. Investigations
of the hyperfine splitting of low- and middle-Z ions are also motivated by astronomical search in hot astrophysical plasma [8]. In particular, the most accurate
up-to-date calculations of the hyperfine splittings in boron-like ions in the range
Z=7–28, including QED, correlation and nuclear effects, were presented in [9].
In this paper, we consider the effects of higher orders in the hyperfine interaction
that can affect both fine and hyperfine splittings.
The relativistic operator of magnetic dipole hyperfine interaction is
Vˆhfi = 41π eµ ⋅ (r −2 n × α ) where μ – nuclear magnetic moment operator. This interaction leads to the shift of
the energy levels depending on the total angular momentum F. Due to the conservation of F and the large values of the nuclear excitation energies on the atomic
scale, only the following matrix elements are considered as non-zero,
γ 1 F M F I j1 Vˆhfi γ 2 F M F I j2 The state vector
γFM I j =
C FM F I M γ jm F
∑
M ,m
IMjm
describes the ion with the angular momentums of the electron shell (j), of the
nucleus (I), and of the ion (F). Here
C FM F IMjm
190
Fig. 2. The hyperfine structure of the 2p level for a positive nuclear magnetic moment (the nuclear spin I≥3/2). The notation:
AF = γ F M F I , j = 1 2 Vˆhfi γ F M F I , j = 1 2
BF = γ F M F I , j = 3 2 Vˆhfi γ F M F I , j = 3 2
DF = E3(1)2, F − E1(1)2, F
are the Clebsch-Gordan coefficients, M and m – the projections of the angular
momenta. To the first order of the perturbation theory the energy shift due to the
hyperfine interaction is given by the diagonal in γ and j matrix elements of Vhfi.
Evaluation of this matrix elements with hydrogen-like functions leads to the known
expressions for the hyperfine splitting. In order to take into account the higher-order
effects for the 2p state we diagonalize the corresponding Hamiltonian matrix. Since
the dominant contribution is provided by the 2p state with different j, we restrict
the matrix to fine-structure components. It consists of the blocks of the form
 E 1(1),I ± 1 CI ± 1  2
2
2


 CI ± 1 E 3(1),I ± 1 
2
2
2
on the diagonal. Here we introduce the following notation for the non-diagonal
matrix elements and the first-order energies of the hyperfine components:
C = γ F M I , j = 1 2 Vˆ γ F M I , j = 3 2
F
E
(1)
j,F
F
=E
(0)
j
hfi
F
+ γ F M F I j Vˆhfi γ F M F I j
191
j = 12 , 32
where Ej(0) – the fine-structure components Dirac energy. Only the levels with equal
total angular momenta of the ion are mixed (see Fig. 2). The diagonalization of
the Hamiltonian matrix leads to the following explicit expressions for the mixingcorrection to the ground-state energy in case of the nuclear spin I≥3/2,
∆E (mix ) = −∆E (mix ) = 1 D 2 + 4C 2 − D
for F = I ± 1 3
,F
2
∆E
(mix )
3
,F
2
2
1 ,F
2
(
F
F
F
)
2
for F = I ± 32
=0
Here
DF = E3(1)2, F − E1(1)2, F In the case of the nuclear spin I=1/2 and I=1 the correction has a similar form.
The centers of gravity of the hyperfine multiplets coincide with the corresponding fine-structure levels, when the hyperfine interaction Vhfi is taken to the first order.
The higher-order (mixing) corrections lead to the shift of the center of gravity,

 

)
)
∆ε (mix
=  ∑ ∆E (mix
∑ (2 F + 1)
j
j , F (2 F + 1)

 
F
F
In principle, this effect can be observed as a shift of the transition energy between
the fine-structure components. The Table 2 shows the calculated values for the
mixing corrections and the shifts of the centers of gravity.
Acknowledgments. Valuable conversations with V.M. Shabaev are acknowledged.
The work was supported by the grant of the President of the Russian Federation
(Grant No. MK-3215.2011.2) and by RFBR (Grant No. 10-02-00450).
References
1. Hermanspahn N. et al. // Phys. Rev. Lett. 84, 427 (2000); Häffner H. et al. //
Phys. Rev. Lett. 85, 5308 (2000).
2. Verdú J.L. et al. // Phys. Rev. Lett. 92, 093002 (2004).
3. S. Sturm et al., Phys. Rev. Lett. 107, 023002 (2011).
4. Mohr P.J., Taylor B.N., Newell D.B. // Rev. Mod. Phys. 80, 633 (2008), and
refs. therein.
5. Vogel M., Quint W. // Phys. Rep. 490, 1 (2010).
6. Draganic I. et al. // Phys. Rev. Lett. 91, 183001 (2003).
7. Shabaev V.M. et al. // Phys. Rev. Lett. 86, 3959 (2001).
8. Sunyaev R.A., Churazov E.M. // Sov. Astron. Lett. 10, 201 (1984).
9. Volotka A.V. et al. // Phys. Rev. A 78, 062507 (2008).
192
Hamiltonian Mechanics in Spaces of Constant Negative
Curvature
Stepanov Vasiliy
vps88@yandex.ru
Scientific supervisor: Prof. Dr. Manida S.N., Department of High
Energy and Elementary Particles Physics, Faculty of Physics,
Saint-Petersburg State University.
Introduction
Investigations of non-flat spaces are motivated by existence of non-nil term in
Einstein equation, corresponding to cosmological constant. Non-nil cosmological
constant means curved space, and at first spaces of constant curvature must be
investigated. Lagrangian and integrals of motion have been introduced for spaces
of constant curvature.
The aim of the present work is to introduce Hamiltonian mechanics in the
anti-de Sitter space (AdS), i.e. in space of constant negative curvature, described
by radius R and velocity of light c. Impulse and Hamiltonian were deduced and 10
integrals of motion were also calculated due to Noether theorem and Lagrangian
symmetries in terms of dynamical variables.
At first all calculations have been carried out in Beltrami coordinates. In those
coordinates free particle equations of motion are linear, but free particle Hamiltonian
does not coincide with free particle energy. In order to make them match new coordinates are required. Such coordinates are derived in the present work and in those
coordinates operation of time translation is trivial, and Lagrangian, Hamiltonian
and energy are time-idependent.
Results and Discussion
Lets consider anti-de Sitter space – it is a section by hyperplane of a cone in
six-dimensional pseudo-euclidic flat space zμ with metric (+,+,-,-,-,-). Anti-de Sitter
space is a four-dimensional space of constant negative curvature.
Symmetry group in this space are produced by a rotations in hyperplanes (zμ,
ν
z ) where μ ≠ ν and consists of 10 movement and 5 conformal transformations.
Usually Beltrami coordinates used for calculations:
Time translation is a transformation of particular interest:
193
Lagrange function is known for anti-de Sitter space, it writes as follows:
10 integrals of motion can be derived due to Noether theorem: energy E, im→
pulse P (this impulse is an integral of motion and does not coincide with dynamical
→
→
→
variable p ), dual impulse K and angular momentum J. Dual impulse appears due
to symmetry of Lagrangean with respect to boost transformations.
Hamiltonian can be deduced by Legendre transformation of Lagrangian.
→
where p – impulse equal to
Obviously Hamiltonian is time-dependent and consiquently not coincide with
energy because energy is an integral of motion deduced by Noether theorem from
Lagrangian.
To make Hamiltonian time-independent new coordinates were introduced:
In these coordinates time-translation is trivial
and Lagrangian, Hamiltonian, energy and other integrals of motions writes as
follows:
194
→
→
It is interesting, that P and - K are rotated by time translation.
Also it is known that anti-de Sitters space is described by two constants: light
velocity (c), and radius of curvature (R). Minkowsky space is a flat limit of anti-de
Sitter space (with R→∞).
Galilei space is a non-relativistic limit of Minkowsky space (with c →∞).
Lorenz-Fock space is a non-relativistic limit of anti-de Sitter space (with
c →∞).
ParaGalilei space is a flat limit of Lorenz-Fock space (with R→∞).
Let c tend to infinity in order to expand Hamilton function in a Taylor series
and consider the two highest terms:
Let R tend to infinity and consider two highest terms in the new expansion of
Hamiltonian:
The most interesting term is
Although both R and c are infinite we will consider theirs ratio as finite. Lets
introduce new constant T=R/c and denote mc2 by self-energy Eself. Now the highest
term of Schrodinger equation writes as follows:
The same equation can be obtained by the same expension but with other order
of tension to infinity to R and c. This equation describes a harmonic oscillator:
with energy of
and oscillation frequency
This solution does not coincide with well-known plane-wave solution of free
particle Schrodinger equation in Galilei space. But plane-wave solution is a limit
of T→∞ of the obtained one. Because T is a very big constant (about age of the
Universe), its conribution is small, and neglectable in most cases.
Therefore new solition of free particle Schrodinger equation is obtained due
to non flat structure of the space.
Conclusions
In this article hamiltonian function is obtained for anti-de Sitter space and
integrals of motion are calculated. Due to time dependence of hamiltonian new
coordinates are introduced.
In these new coordinates some new facts appear:
195
1) New integrals of motion inherit the rotation symmetry of AdS.
2)Hamiltonian Taylor series in nearly flat and non-relativistic case show new
Schrodinger equation for free paticles. Consequently new solution appears.
3) New solution desribes a harmonic oscillator with period about the age of the
Universe instead of a plane wave.
4) New results does not contradict with old ones, because plane wave is a limit
of obtained solution.
References
1. W. de Sitter. Monthly Notices of the Royal Astronomical Society, Vol. 71, p.
388-415, 1911.
2. Manida S.N. Additional Chapters of Physics. Mechanics, p. 57, Saint-Petersburg,
Faculty of Physics printing establishment, 2007.
196
3D isotropic random walks with exponential distribution
on free paths. Application to evaporation of a droplet at
transient Knudsen numbers
Telyatnik Rodion
statphys@narod.ru
Scientific supervisor: Prof. Dr. Adzhemyan L.Ts., Department of
Statistical Physics, Faculty of Physics, Saint-Petersburg State
University
1. History and basic theory of random walks
The classical problem of random walks (RW), first formulated in the letter
to Nature journal in 1905 by Pearson [1, 2], who investigated the spread of mosquito populations, is the question about the probability to find a random walker
at certain distance from the start point after n steps, which can occur in general at
random direction and random distance with given probability distributions (the
simplest Pearson’s case is the isotropic walks with constant step length λ). Earlier
in 1880 for similar problem of composition of n iso-periodic vibrations with unit
amplitude and random phase in the limits n→∞ and λ→0 Rayleigh found Gaussian
asymptotic solution [2] that reflects modern Central Limit Theorem (CLT) about
the sum of random independent identically distributed values. In 1905 Einstein
obtained [3] this asymptotic solution as Green function for diffusion equation,
which he derived in the same limits from the recurrent equation for Markov chain
of RW that expresses the full probability of a position by the sum of all conditional
probabilities, also known as Kolmogorov-Chapman equation [4], first proposed by
Bachelier in his Ph.D. thesis (1900) for modeling random prices in a stock market
[1]. We generalize this equation for RW chain with k-steps links:


 
  


 

ρn (rn ) = ∫ drn −1 P1 (rn − rn −1 )∫ ...∫ dr0 P1 (r1 − r0 )ρ0 (r0 ) = ∫ drn − k Pk (rn − rn − k )ρn − k (rn − k ) (1.1)
Here ρn(rn) is the probability density function (PDF) for a particle to occur at point
rn after n steps of RW, and Pk(rn-rn-k) is the k-steps transition probability density
function (TPDF) for a particle to occur at point rn after k steps of RW from the
place at point rn-k (in general TPDF can depend on step number n). In essence PDF
represents the local concentration of particles normalized to their total amount in
the system. If we make change r’= rn-rn-1 in (1.1) for k=1 and make Taylor expansion of ρn-1(rn-r’) near rn with Einstein’s condition |r’∇ln ρn-1(rn)| << 1 of relatively
small change of concentration on step distance r’, we will get equation, which we
write for generality for d-dimensional space:

  ∂ 2ρn −1 (rn )
1 d d

 

 

+ ... dr ′ (1.2)
ρn (rn ) = ∫ P1 (r ′ ) ρn −1 (rn ) − r ′∇ρn −1 (rn ) + ∑ ∑ rα′rβ′
2 α =1 β =1
∂rα ∂rβ


Let P1(|r’|) be an isotropic one-step TPDF and let angle brackets <…> denote
averaging with the TPDF. As it is normalized to unity (the probability to occur
anywhere), the equation (1.2) in notation r≡rn turns to:
197
1



ρn (r ) = ρn −1 (r ) +
r ′ 2 ∇ 2ρn −1 (r ) + ...
2d
(1.3)
The expansion (1.3)by m-th moments <r’m> of TPDF is known as Kramers-Moyal
expansion [1,4]. Setting ρ(r,t=nτ)≡ρn(r) for n→∞, where τ is the mean time of RW
step, and dividing by τ→0 one can obtain diffusional equation:
∂ρ(r , t )
1
1

= D∆ρ(r , t ) , D ≡
lim r ′ 2
(1.4)
τ
→
0
∂t
2d
τ
Rigorously we should expand (1.3) by times as well as by space [1, 5]. For d = 3
and <r’2> = 2λ2 (e.g. for exponential TPDF, see (2.1)), where λ is mean free path
of a particle, one can see usual expression for D = <v> λ/3, where <v> is mean
velocity. So called anomalous diffusion happens if TPDF hasn’t finite moments.
For one particle flying out the origin of coordinates with ρ0(r0)=δ(r0) according
to (1.1) PDF after n steps is essentially TPDF for n steps, i.e. ρn(r)=Pn(r), and it is
described for n→∞ by Green function of (1.4) that represents CLT:
r2
−
1



 4 Dt
Pn (r ) = ρ(r , t = nτ) →
e
, ρ0 (r ) = δ (r )
(1.5)
d
2
n →∞
( 4πDt )
Corrections to the asymptotic (1.5), known as Gram-Charlier expansion [1, 6] (by
Chebyshev-Hermite polynomials defined by the derivatives of Gaussian exponent),
can be constructed by Laplace asymptotic calculation for Fourier transform (FT) of
PDFs, i.e. by characteristic functions with their moments and cumulants in terms
of probability theory. Indeed, chain equation (1.1) is the convolution, which is
turned by FT to the algebraic expression:
ρn (r ) = ( P1 ∗ ... ∗ P1 ∗ρ0 )(r ) , ρˆ n (q ) = PPˆ1n (q )ρˆ 0 (q ) , Pˆn (q ) = Pˆ1n (q ) (1.6)


n
FT of an isotropic function in spherical coordinates can be expressed with Bessel
functions of first kind Jν(z) [2, 7], e.g. for one-step TPDF:
d 2 −1
∞


 

d 2  1 
d −1
− iqr
ˆ
P1 (| q |) = ∫ e P1 (r )dq = (2π ) ∫  
r J d 2 −1 (qr ) P1 (| r |)dr (1.7)
 qr 
0
Formal solutions obtained by reverse FT of (1.6) can’t be evaluated in elementary
functions for most cases of n, d and P1(r), e.g. for n-steps TPDF:
d 2 −1
∞

 1
1
1


iqr ˆ n 
d −1
ˆ n (| q |)dq
=
(
)
(
)
Pn (| r |) =
e
P
q
dq
q
J
qr
P
(1.8)
1
d 2 −1
1
d ∫
d 2 ∫

(2π)
(2π)
0
 qr 
For isotropic RWs it is simpler to work with radial PDF (RPDF) σn(r) and radial
TPDF (RTPDF) Sn(r) expressed by volume density probabilities multiplied by the
surface area of d-dimensional sphere with radius r (Γ(z) is Gamma function):


(1.9)
σ n (r ) ≡ ρn (| r |)r d −1 2π d 2 Γ (d 2) , S n (r ) ≡ Pn (| r |)r d −1 2π d 2 Γ (d 2) In essence σn(r) is the normalized surface concentration in centrally symmetric
system, and Sn(r)dr is the probability for a particle to occur after n steps from the
origin of coordinates at the distances in the interval [r,r+dr]. Asymptotic (1.5) for
198
Sn(r) takes form of d-dimensional Rayleigh distribution. In spherical coordinates
chain equation (1.1) hasn’t convolution form, it can be rewritten as:
∞




σ n (r ) = ∫ S k (r , rn − k )ρn − k (rn − k )drn − k = ∫ S k (r ,| rn − k |)σ n − k (rn − k )drn − k (1.10)
0
Here Sk(r,rn-k) is the probability, which we call spherical TPDF (STPDF), to fly
out the point rn-k and to occur after k steps at the distance r from the origin of coordinates, at that Sk(r,0)=Sk(r) defined in (1.9). Although STPDF doesn’t depend
on direction of the vector rn-k (from the symmetry of the problem) we keep its
designation to remember its point-to-sphere transition nature.
2. Random walks with exponential distribution on free paths
If a lot of particles experiencing RWs have constant mean free path λ (in effective sense for long-ranged forces), which is defined by surrounding system and
independent on concentration of particles themselves (e.g. Brownian particles or
vapor in passive gas), then their random free paths at mean free time τ have exponential distribution [8] that becomes RTPDF (1.9) for one step:
∞
∞
1 −r
(2.1)
S1 (r ) = e λ , ∫ S1 (r )dr = 1 , r m = ∫ r m S1 (r )dr = λ m m !
λ
0
0
Random free times have the same exponential distribution for fixed free path equal
to λ if we substitute r=(λ/τ)t in the integral in (2.1), that’s how RW problem can
be formulated in terms of continuous time RW
(CTRW model of Weiss [1, 2]), which is widely
used in RWs on lattices. Another than (2.1)distribution on free paths Tait derived in 1886 by
another definition of averaging of free paths (by
amount of collisions not in unit time, but from
the chosen moment to the next collision, see
differences in [9]). STPDF can be derived from
RTPDF by means of the probability dPr to occur
in spherical layer segment with center at point
r0 and volume dV’=dφ’sinθ’dθ’r’2dr’ (where
r’=r-r0, see Fig.1) written in corresponding Fig. 1. RW in spherical coorspherical coordinates that can be expressed dinates.
from original ones:
r ′ (r , θ | r ) = r 2 − 2rr cos θ + r 2 , ϕ ′ = ϕ 0
0
0
(2.2)

sin
θ
=
sin
θ
⋅
r
r
co
s
=
cos
−
,
θ
r
θ
r
r
′
′
′
′
(
)
0

dV ′
1 − r ′ (λr ,θ ) r sin θ ∂θ ′ ∂θ ∂θ ′ ∂r
d Pr = S1 (r ′ )
=
e
d ϕd θdr (2.3)
r ′ ( r , θ ) ∂r ′ ∂θ ∂r ′ ∂r
4πr ′ 2 4πλ
Integral of dPr over φ and θ will give us S1(r,r0)dr. After evaluating Jacobian determinant in (2.3) with the help of (2.2) and integrating upon changing variable θ
to ξ=-r’(r, θ)/λ ∈ [-(r+r0)/λ , -|r-r0|/λ] we will finally obtain STPDF:
199
− (r + r
0
1 r

S1 (r , r0 ) =
v.p. ∫
2λ r0
− r − r0
)
λ
λ
 r − r0
1 ξ
1 r   r + r0 
e dξ =
− Ei  −
 Ei  −

ξ
2λ r0  
λ 
λ



 
(2.4)
Principle-value integral in (2.4) (as r=r0 is logarithmic singularity, but we neglect
the probability to remain on 2D surface after 3D RW step) can’t be taken in elementary functions and it is represented by exponential integrals Ei(x). Modulus
in range of definition of ξ expresses the ambiguity of particle position, which is
observed under angle θ (see Fig.1). From the chain equation with links of single
steps and σ0(r0)= δ(r0) one can evaluate Sn(r)=Sn(r,0):
∞
∞
∞



S n (r ) = v.p.∫ dr0 S n (r , r0 )δ (r0 ) = v.p.∫ drn −1 S1 (r , rn −1 )...∫ dr0 S1 (r1 , r0 )δ (r0 )
(2.5)
0
0
0
So, RTPDF (2.5) plays role of Green
function whose asymptotic for big

enough n (factually for n≥30) is described

by (1.5) with (1.9). For any n RTPDF
Sn(r) can be approximated within simulation data (Fig. 2) by the general function
ar2exp(-rb/c) with varying constants a,b,c
connected by normalization (power of r2
ahead of exponent is from space dimension, so it is fixed). RTPDF (2.5) can be

roughly evaluated analytically near appreciable singularities rk=rk-1 (k=1,…,n)
Fig. 2. Sn(r) for n=10 compared to its from (2.4) except the special cases rk close
asymptotic.
to zero with the help of asymptotes of
Ei(x) in zero and infinity [10]. In this way for 2<n≤20 two bounded by Heaviside
functions approximations with varying parameters a,b,c,d connected by conditions
of normalization and continuous differentiability are:
S (r ) ≈ ar 2 Θ(b − r ) + ce − r λ ln ( 2r λ ) − e 2 r λ + d Ei ( − 2r λ )  n −1 Θ(r − b) (2.6)
n


0.25
RW simulation data
(r)=0.09 r2exp(-r 1.4/3.4)
10
Asymptotic (1.5) for t=n
S
0.2
Formula (2.6) with a=0.07,
d b=0.77, c=0.0055
S (r)
n
0.15
0.1
0.05
2
4
6
8
10
12
r ( =1)
RW simulation is based on generation of the exponentially distributed free path
r’=-λln(u), where u ∈ [0,1] is uniformly distributed [11] as well as φ’ and cosθ’ in
notations (2.2) for isotropy (i.e. for uniform measure dφ’dcosθ’).
3. Evaporation of a droplet at transient Knudsen numbers
Developed above theory allows us to generalize molecular flows, which can’t be
described in the limits of entirely free flight or indiscrete diffusive mixing, e.g. for
mass transfer between a droplet (with radius R) and surrounding vapor in passive
gas in case of transient Knudsen number Kn≡λ/R≈1. Particles entering the droplet
(with the uniform concentration ρ of its particles) without reflection during one
step of RW cause the growth of droplet’s radius on the value dR+:
200
R
∞
0
R

ρ ⋅ 4πR 2 dR+ = v.p.∫ dr ∫ dr0 S1 (r , r0 )σ 0 (r0 ) (3.1)
While growing up on dR+ the droplet catches neighboring vapor particles resulting
in the new increment accounting from iterative procedure similar to (3.1). Particles
flying out the surface of the droplet with constant probability α (in the unit time;
quasi-stationary case) with the same λ (temperature) and initial surface concentration σ0(r0)=α𝜏ρ4𝜋R2𝛿(r0-R) have new surface concentration σ1+(r) like but with
condition θ’ ∈ [0,π/2] in for the limiting tangent line (see Fig. 1 meaning r0=R)
that results in new STPDF SR(r) in place of (2.4):
−1
S R (r ) ≡ ( 2λ ) r R ⋅  Ei − r 2 − R 2 λ − Ei ( − ( r − R ) λ )  , r > R (3.2)



+
+
2
2
(3.3)
ρ1 (| r |)4πr = σ1 (r ) = ατρ4πR S R (r )
(
)
But such evaporation causes radius decrease R(t)=R0-αtl, where l3 is the mean
volume of liquid phase attributed to one particle. TPDFs, i.e. RW steps, are defined
only on time interval τ, so we can’t use (3.2) in principle if decrement dR-=-ατl is
significant, nevertheless by substituting R(t) into (3.3) and replacing τ by t (linear
interpolation) and integrating (3.3) by t ∈ [0,𝜏] we can obtain continuous interpolation between RW steps that smooths numerical noise and cancels logarithmic
singularity r=R in (3.2). Stationary vapor concentration profile can be derived by
differentiating by R the equation with change dR+ on dR=dR+-dR-=Const that
leads to Volterra integral equation of second kind:
R
∞




ρ(| R |)4πR 2 v.p.∫ S1 (r , R)dr = v.p.∫ S1 ( R, r0 )ρ(| r0 |)4πr02 dr0 − 8πConstρR
(3.4)
R
0
First iteration of (3.4) with constant ρ(r)=ρ∞ for Kn≪1 gives Maxwell’s diffusive
stationary solution ρ(r)=ρ∞+(ρR(0)-ρ∞)R/r [12] with boundary values ρR(0) and ρ∞,
and for arbitrary Kn it gives required correction to this asymptotic:
2r
− 
1 − L ( R )
1 + L(r )
1 + L( R )  R
λ 

λ
(3.5)
1− e
ρ(| r |) ~
ρ∞ + 
ρR −
ρ∞ 
, L(r ) ≡
1 − L(r )
 1 − L(r )
1 − L(r )
r
2r 

Concentration (3.5) with its boundary value ρR=(ρR(0)+ρ∞L(R))/(1-L(R)) exceeds
the Maxwell’s one, that is known as Langmuir “jump” of concentration near drop’s
surface. Other treatment based on kinetic Boltzmann equation is in [13].
This work was supported by the RFFI grant № 10-03-00903 and by SPbSU
research effort № 0.37.138.2011.
1.
2.
3.
4.
References
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Einstein A. // Annalen der Physik (ser. 4), v. 17, p. 549–560 (1905).
Risken H. // The Fokker-Planck Equation, ch. 2, 4 (1989).
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5. Aranovich G., Donohue M. // Molecular Physics, v. 105, №8, p. 1085
(2007).
6. Gnedenko B., Kolmogorov A. // Limit distributions for sums of independent
random variables, ch. 8 (1968, Russian edition in 1949).
7. Kingman J.// Acta Mathematica, v. 109, p.11-53 (1963).
8. Feynman R.P. et al. // The Feynman lectures on physics, ch. 43 (1963).
9. Whitman J.// Ph.D. Thesis, Johns Hopkins University, Baltimore (2010).
10.Gradshteyn, Ryzhik // Table of integrals, series and products, ch.8.2 (1980).
11.Gentle J.// Random number generation and Monte Carlo methods (2005).
12.Fuks N.// Evaporation and droplet growth in gaseous media, ch. 1, 5 (1959).
13.Lushnikov A. et al // Fizika aerodispersnyh sistem, v. 37, p.7 (2001).
202
Investigation of the dependence of the number of binary
interactions and the number of participants on the class
of centrality in ultrarelativistic heavy ion collisions
Vorobyev Ivan
vorobyov_ivan@mail.ru
Scientific supervisor: Prof. Dr. Vechernin V.V., Department of High
Energy and Elementary Particles Physics, Faculty of Physics,
Saint-Petersburg State University
Introduction
The investigation is connected with the activity of the SPSU research team,
including into the ALICE at LHC and NA61/SHINE at SPS collaborations at
CERN. The SPSU team searches for the long-range correlations, which can be
considered as one of the signs of quark-gluon strings fusion in high energy heavy
ions collisions. In the usual version of quark-gluon string model [1] there is no
interaction between the strings. In hadron interactions the quark-gluon strings are
formed and then in the process of their fragmentation the observable hadrons are
produced. In the case of high energy heavy ion collisions one needs to take into
account interaction between strings [2], which in turn changes their fragmentation.
As quark-gluon string is an extended object in the rapidity space, i.e. contributes by
fragmentation to wide rapidity range, one can expect appearance of the correlations
between observable quantities in distant rapidity intervals (see, for instance [3]).
The number of strings Nstr formed in the high energy nucleus-nucleus collision
depends on the number of binary nucleon-nucleon (NN) collisions Ncoll and the
number of nucleon-participants Npart. This dependence can be parameterized as
follows [4]:
NN
N str = xN str
N part + (1 − x) N coll (1)
where the parameter x and the number of strings per NN-interaction NNNstr depends
only on the energy of interaction.
Usually for simplicity one supposes that the event by event fluctuations of the
number of strings at fixed impact parameter value obey the Poisson distribution
[4], which means that
DN str
= 1
(2)
N str
In present work it was shown numerically that the fluctuations of the number of
binary interactions and the number of participants don’t obey the Poisson distribution, which affects the fluctuations of the number of strings. Using the C++ MC
simulation code of AA collision, developed in the present work, the normalized
dispersions of the numbers of wounded nucleons and binary interactions were
calculated both at fixed impact parameter and at one corresponding to certain
centrality classes.
203
Model of AA scattering
In this work one uses Glauber model with the standard Woods-Saxon approximation for profile function of colliding nuclei [5]:
r − RA −1 (3)
TA (a ) = ∫ dzρ(r ),
r 2 = a2 + z 2 ,
ρ(r ) = ρ0 (1 + exp
)
k
with RA=R0A1/3, R0=1.07 fm, k=0.545 fm, ∫daTA(a)=1. For NN-interaction crosssection we use empirical formula [4]:
σ inNN = (32.08 − 1.574 ln E + 0.6622 ln 2 E ) ⋅10−1 fm (4)
where E is the energy in GeV per NN-interaction in the center-of-mass system.
We also consider two alternative versions for the probability of NN-interaction,
the so-called “black disc”:
σ (r ) = Θ(rN − r ),
σ inNN = πrN2 (5)
2
r
and the Gaussian one:
− 2
σ (r ) = e rN ,
σ inNN = πrN2 (6)
where r is the distance between the centers of colliding nucleons in the impact
parameter plane, and rN is the radius of NN-interaction, which depends on the
energy of collision (see (4)-(6)).
For numerical modeling of heavy ion high-energy collisions the following algorithm was realized as C++ code. First, according to the nucleus profile function
(5) we distribute certain number of nucleons (207 for Pb) for each of colliding
nuclei in the impact parameter plane. Then taking into account the impact parameter
we determined the distance r between all pairs of nucleons. If σ(r) has occurred
more than some parameter t (t is the random parameter, evenly distributed in the
interval [0, 1]), then we consider these nucleons to interact with each other. Next
we calculate number of wounded nucleons and number of NN-interactions in each
collision, repeating (up to 50000 times) this cycle for fixed impact parameter (or
centrality class) and storing required average values. So it is possible to calculate
various values for any AA collisions (it is not even necessary for colliding nuclei
to be the same) at given profile functions of colliding nuclei and probability of
NN-interaction.
Verification of the code
For this purpose let’s consider the simulation of Pb-Pb collisions for SPS energy
(Super Proton Synchrotron, 17 Gev per NN-interaction in c. m. system) at fixed
impact parameter, because similar calculations have been carried out earlier in [5].
For E=17 GeV according to (4)-(6) we have σNNin = 3.14 fm2 and rN =1 fm.
We see in Fig. 1 that all our calculation results are in a good accordance with
the results obtained earlier (Fig. 1).
Subdividing into centrality classes
In real experiment, of course, we can’t fix impact parameter, so one have to
deal with impact parameter fluctuations of order about 2-3 fm. So in a real experiment the impact parameter range is divided into several parts, so-called centrality
classes. One classifies events by centrality classes and then in each class we calculate necessary values and average them on the events. There are different methods
using in a real experiment for dividing events into centrality classes, but usually
204
it can’t be done with accuracy better than 3 fm, at best 1.5 fm. We will consider
both these cases: 3 fm and 1.5 fm.
Fig. 1. 17 GeV Pb-Pb collisions at fixed impact parameter. Left – doubled correlator between the numbers of wounded nucleons in colliding nuclei vs impact
parameter; right – relative dispersion of NN-collisions number vs impact parameter. Results of our code marked as line with triangles
For a generation of random impact parameter one uses a following formula:
b = bmax y (7)
where bmax is a constant, at which the probability of nucleus interaction is insignificant and y is a random parameter, evenly distributed in range [0, 1]. It is easily
2
to see from (7) that we have:
bmax
2
(8)
=
d
b
dy 2π
which corresponds to uniform distribution in impact parameter plane.
Calculations results
Let’s consider results for Pb-Pb collisions simulation at SPS energy (1 GeV per
NN-interaction). As we can see in Fig. 2, relative dispersions both for number of
participants and number of collisions don’t equal to 1 as it would be for Poisson
distribution, so it is obvious that in real experiment the distributions of these quantities considerably differ from Poisson one. Moreover, with the size of centrality
class grows up, dispersions also increase.
In Fig. 3 similar results are represented for the case of LHC energy (E=5.5 TeV
per NN σNNin = 6.75 fm2, rN = 1.45 fm), and again we can see huge difference between Ncoll and Npart relative dispersions and Poisson case.
Comparing results at different energies
With the energy increases from 17 GeV to 5.5 TeV per NN interaction, the
radius of this NN interaction according to (4)-(6) enlarge from 1 fm to 1.45 fm,
that is, nucleons become enlarged (expanded). One can explain this fact something
like this way: with the energy increases, additional virtual quark-antiquark pairs
are formed a bit further from nucleon’s center of mass in impact parameter plane,
and nucleons effectively become larger.
Number of wounded nucleons in one nucleus increases insignificantly, which
is quite natural, as well as relative dispersion of this value (Fig. 4).
205
<DN_W/N_W>
10
5
black_disc_1.5fm
black_disc_3fm
gauss_1.5fm
gauss_3fm
black_fixb
gauss_fixb
poisson
30
<DN_C/N_C>
black_disc_1.5fm
black_disc_3fm
gauss_1.5fm
gauss_3fm
black_fixb
gauss_fixb
poisson
20
10
0
0
8
0
16
0
impact parameter b, fm
8
16
impact parameter b, fm
Fig. 2. 17 GeV Pb-Pb collisions for different centrality classes. Left – relative
dispersion of number of wounded nucleons vs impact parameter; right – relative
dispersion of NN-collisions number vs impact parameter. Poisson case is represented as straight line equals to 1 at the bottom of graph.
black_disc_1.5fm
black_disc_3fm
gauss_1.5fm
gauss_3fm
black_fixb
gauss_fixb
poisson
60
<DN_C/N_C>
<DN_W/N_W>
10
5
0
0
black_disc_1.5fm
black_disc_3fm
gauss_1.5fm
gauss_3fm
black_fixb
gauss_fixb
poisson
8
30
0
16
0
impact parameter b, fm
8
16
impact parameter b, fm
Fig. 3. 5.5 TeV Pb-Pb collisions for different centrality classes. Left – relative
dispersion of number of wounded nucleons vs impact parameter; right – relative
dispersion of NN-collisions number vs impact parameter.
black_LE
gauss_LE
black_HE
gauss_HE
6
black_LE
gauss_LE
black_HE
gauss_HE
<DN_W/N_W>
<N_A>
200
100
0
0
8
impact parameter b, fm
16
3
0
0
8
16
impact parameter b, fm
Fig. 4. Pb-Pb collisions at different energies - 17 GeV and 5.5 TeV. Left – Number
of wounded nucleons in incoming nucleus vs impact parameter; right – relative
dispersion of number of wounded nucleons in both nuclei vs impact parameter.
Results for higher energy are marked as black squares and circles, for lower energy
as white ones. All results are represented for case of 1.5 fm centrality class.
But for number of collisions picture is completely different (Fig. 5). For all of
centrality classes number of interactions increases two times, which is a consequence of nucleons “swell” – now each nucleon interact with greater amount of
nucleons in other nucleus, and general number of collisions increases significantly.
206
30
black_LE
gauss_LE
black_HE
gauss_HE
black_LE
gauss_LE
black_HE
gauss_HE
20
<DN_C/N_C>
<N_C>
2000
1000
0
0
8
impact parameter b, fm
16
10
0
0
8
impact parameter b, fm
16
Fig. 5. Pb-Pb collisions by different energies - 17 GeV and 5.5 TeV. Left – Number
of NN-collisions vs impact parameter; right – relative dispersion of number of
wounded nucleons in both nuclei vs impact parameter. Results for higher energy
are marked as black squares and circles, for lower energy as white ones. All results are represented for case of 1.5 fm centrality class.
As consequence, relative dispersions of this value also differ of one another quite
considerably.
One can explain this result as follows: number of wounded nucleons increases
insignificantly only thanks to nucleons on the periphery of interacting area of
nucleus. But as nucleons become thicker, each nucleon inside the interacting area
now interacts with greater amount of nucleons in other nucleus, and general number of
NN-interactions increases significantly (about two times in each centrality class).
Conclusions
a) Relative dispersions of number of wounded nucleons and number of NN collisions differ from Poisson one even for case of fixed impact parameter. Moreover,
as we allow the impact parameter to fluctuate, relative dispersions increase even
more, so in real experiment distributions of Ncoll and Npart (therefore of Nstr as well)
are far away from the Poisson one.
b) There is week dependence on energy for number of participants Npart. By
contrast, number of collisions (as well as its relative dispersion) enlarges quite
significantly.
c) There is almost no difference between results obtained with using of different
versions for the probability of NN-interaction - the Gaussian and the “black disc”
at condition that the total cross-sections are the same.
References
1. Kaidalov A.B. // Phys. Lett. B 116, 459, (1982); Kaidalov A.B., Ter-Martirosyan K.A.
// Phys. Lett. B 117, 247, (1982); Capella A., Sukhatme U.P., Tan C.-I., J. Tran Thanh
Van // Phys. Lett. B 81, 68 (1979); Phys. Rep. 236, 225 (1994).
2. Braun M.A., Pajares C. // Phys. Lett. B 287, 154 (1992); Nucl. Phys. B 390,
542; 549 (1993).
3. Braun M.A., Kolevatov R.S., Pajares C., Vechernin V.V. // Eur. Phys. J. C 32,
535 (2004).
4. Vechernin V.V., Kolevatov R.S. // Phys. of Atom. Nucl. 70, 1797; 1809 (2007).
5. Vechernin V.V., Nguyen H.S. // Phys. Rev. C 84, 054909 (2011); Vechernin V.V.
// Relativistic Nuclear Physics and Quantum Chromodynamics, vol.2, JINR, Dubna,
2008, pp.88-94; hep-ph/0702141.
207
H. Biophysics
Application of Surface Plasmon Resonance for Detection
of DNA Immobilization on Gold Surface
Fironov Alexander
alexfiron@gmail.com
Scientific supervisor: Prof. Dr. Kasyanenko N.A., Department of
Molecular Biophysics, Faculty of Physics, Saint-Petersburg State
University
Introduction
Nanoplasmonics is a rapidly developing branch of science. Phenomena studied
by nanoplasmonics have found their application for wide variety of tasks, such as
creation of biosensors, therapy and visualization of tumors and much more.
Creation of a thin metal film modified by DNA is the first step to in-depth
investigation of DNA interaction with various compounds using surface plasmon
resonance (SPR).
Functionalized self-assembled monolayers (SAMs) have been employed to
immobilize DNA on a gold surface, based on covalent bonding attachment. For
example, 11-mercaptoundecanoic acid (MUDA) SAMs were employed to immobilize DNA on a gold surface for AFM imaging, based on carbodimide covalent
coupling. Divalent cations, such as Mg2+, Ni2+, Zn2+, and so on, have been widely
employed to immobilize DNA on a mica surface for AFM imaging This method
is based on a ‘‘salt bridge’’ effect mediated by divalent cations between the negatively charged mica surface and the negatively charged DNA. By choosing the
appropriate cation as a bridge ion, a weak electrostatic attachment to mica could
be obtained. In this paper an attempt to immobilize DNA on gold surface using
thioglycolic acid (HSCH2COOH) and Mg2+ ions as connecting agents was made.
The thioglycolic acid (TGA) was first self-assembled onto a gold surface to produce a negatively charged surface. Then DNA was attached onto this surface via
the divalent cation bridge.
The technique based on a surface plasmon resonance, which is very sensitive
to modification of the metal surface was used for the detection of DNA biding.
SPR phenomenon connects with the conduction electrons oscillation in the metallic
lattice. This collective oscillation when excited by light with specific wavelength
at the angle of total inner reflection, TIR (the light frequency must be the same as
for inner oscillation of nonvalent electrons in metallic lattice) produces the surface
plasmons. Known as a surface plasmon resonance (SPR), this phenomenon results
in unusually strong scattering and absorption properties. In the reflected light the
SPR frequency disappears. One can see the darkness in spectra. The device fixes
the intensity dependence on the angle for the determination of TIR angle which
depends on reflecting constant and, therefore, on mass of adsorbed substrate.
In addition to data obtained from SPR method, gold surface was studied with
atomic force microscope (AFM).
210
Results and Discussion
Gold chip was immersed in TGA for 24 hours in order to obtain TGA-modified
sample. After sample was taken out of TGA and rinsed with distilled water. Then
during another 24 hours sample was reacted by (covered by) NaCl solution of
DNA with ionic strength of 0,005M and DNA concentration of 0,0135%. The
measurement of SPR curves was carried out with Nanospr8 model 481 device.
The dynamic of interaction can be observed in the experiment.
Eight different combinations of reacting compounds were taken into consideration. In the Figs.1-3 a 24 hours dynamics of relative shift of SPR angle is
presented.
Fig. 1. Dynamics of gold surface modification with TGA.
211
Fig. 2. 24 hours SPR measurement for non-modified gold surface.
Fig. 3. 24 hours SPR measurement for modified gold surface.
212
According to the information presented on these figures we suppose that TGA
interacts with gold and forms a film on its surface. Also we can observe large
shift of SPR angle for DNA with Mg2+ ions interaction with TGA-modified gold
surface. But AFM image, obtained with NanoScope 4a (Veeco), of this sample
doesn’t confirm our suggestion.
Fig. 4. AFM image of DNA with Mg2+ ions on TGA-modified gold surface.
Fig. 5. Dependence of SPR angle shift on different ionic strength of the solution.
213
Conclusions
According to experimental data the modification of gold surface with thioglycolic acid and DNA is observed, but further investigation of mechanics of interaction
is suggested. Subsequent consideration of the possibilities of SPR technique for
DNA examination and the improvement of measurements should be carried out.
References
1. Yonghai Song, Zhuang Li, Zhiguo Liu,Gang Wei, Li Wang, Lanlan Sun //
Microscopy Research And Technique 68:59–64 (2005).
2. Klimov V.V. Nanoplasmonics.- Moscow: Fizmatlit, 2009.
214
DNA Interaction with Palladium Compound
K2[PdHGluCl2] in vitro
Kozhenkov Pavel
pavel.kozhenkov@gmail.com
Scientific supervisor: Prof. Dr. Kasyanenko N.A., Department of
Molecular Biophysics, Faculty of Physics, Saint-Petersburg State
University
Introduction
The coordination compounds of metals from Platinum group play the important
role in antitumor therapy. Amazing results were reached with the using of Platinum
drugs, but only for certain kinds of tumor. Most widely used and successfully
applied drug cisplatin prevents DNA replication in tumor cells. Unfortunately,
cisplatin generates the serious side-effects (toxicity and non-specific influence).
This circumstance stimulates the synthesis of new compounds for the selection of
non-toxic medicaments with high antitumor activity. The coordination compounds
of other metals – palladium, titan, ruthenium, etc. are tested for antitumor activity
via the binding with DNA in a solution.
The influence of Palladium complex K2[PdHGluCl2] on DNA conformation in
vitro was investigated in current report. The structure of Palladium complex was
calculated.
Methods and materials
The change in the Palladium coordination sphere due to the aquation is analyzed. Quantum-mechanical calculations of a molecular structure of [PdHGluCl2]
with two stages of aquation (the replacement of one and second chlorine atoms
by water molecules) have been carried out with software packages HyperChem
8.0 and GAMESS (FireFly 7.1g). The unlimited Hartree-Fock method and bases
SBKJC VDZ ECP for Palladium, DH for Hydrogen atoms and 6,31+G* for all
other atoms have been used.
DNA circular dichroism (CD) spectra were registered with Mark 4 (Jobin Ivon,
France) autodichrograph. The absorption spectra of components in a solution
were obtained with spectrophotometer SF-56 (Russia). Atomic Force Microscopic
(AFM) images of DNA and its complexes with Palladium compound (DNA- Pd
complexes) have been received by means of microscope NanoScope 4a (Veeco)
in a taping mode on air. DNA fixation on a mica surface was carried out by the
spontaneous adsorption of DNA molecules from a solution containing magnesium
ions. DNA interaction with Palladium compound was studied in 0,005 M NaCl or
0,15 M NaCl with the variation of DNA and Pd concentration.
Results and Discussion
Fig. 1 shows the absorption spectrum of K2[PdHGluCl2] solutions in distilled
water, 0,005 M NaCl and 0,15 M NaCl at room temperature. One can see that the
215
addition of salt into K2[PdHGluCl2] solution leads to the long-wave shift of the
maximum. The observed changes in the absorption spectrum associated with the
state of the coordination sphere of the complex ion.
Fig. 1. The absorption spectra of K2 [PdHGluCl2] solutions at different NaCl
concentrations.
Fig. 2 shows the experimental results obtained by circular dichroism method.
CD spectra of DNA in complex with palladium compound in 0,005 M NaCl solution for the different concentrations of the tested palladium compound C(Pd) are
presented. It is clear from the Fig. 2 that DNA interacts with compound. At low
C(Pd)=3×10-5M we can not see any changes in DNA CD spectrum. At C(Pd)>
9×10-5M the similar CD spectra are registered. The state of DNA double helix is
identical and differs from free DNA. It can indicate the filling of vacant binding
Fig. 2. CD spectra of
DNA complexes with
K 2[ P d H G l u C l 2] i n 0 , 0 0 5
M NaCl at constant concentration of DNA and
varying concentrations of
K2[PdHGluCl2]: C(Pd)×105,
М= 0 (0),25(1), 19 (2),
13(3), 9(4), 3(5).
216
after 15 min
0,3
0,05
0,3
K2[PdHGluCl2]
complex DNA-Pd
after week
K2[PdHGluCl2]
complex DNA-Pd
0,2
D
D
0,2
0,03
positions for Pd in that kind of complex. The exceeding of Pd concentration up to
25×10-5M may be leads to the perturbation in DNA-Pd complexes due to alternative binding mode.
The absorption spectrum of Pd compound has a band out of DNA spectrum.
Fig. 3 demonstrates that this band modifies during interaction with DNA (blue shift
of the maximum and drop in intensity). We can not see the significant change in
band for free Pd and Pd in complex with DNA during the time. The comparative
examination of the spectra showed that the maximum of K2[PdHGluCl2] in DNA
solution (371 nm) does not shift within a week as well as a maximum of absorption
band for free K2[PdHGluCl2] (378 nm). It should be noted that similar experiments
were carried out in solution with distilled water and 0,15 M NaCl and the identical
results were obtained (the shift of the maximum was observed). These data testify
that Pd compound in complex with DNA is more stable than in solution without
DNA. Fig. 4 indicates that in contrast to 0.005 M NaCl DNA does not bind with Pd
compound in 0.15 M – the spectral properties of Pd does not change with increase
in DNA concentration out of DNA band. The corresponding spectra were recorded
in one day after preparation of the complexes.
0,1
0,1
7 nm
7 nm
0,0
330
360
390
420
λ, nm
450
480
0,0
330
360
390
420
λ, nm
450
480
Fig. 3. Absorption spectra of Pd-compound and its complex with DNA in time.
0,6
0,8
C(PdGlu)=5*10-4M; NaCl-0,005 M
C(PdGlu)=5*10-4M; NaCl-0,15 M
0,7
C(DNA) = 0 %
C(DNA) = 0.0001 %
C(DNA) = 0.0003 %
C(DNA) = 0.0004 %
C(DNA) = 0.0005 %
C(DNA) = 0.0008 %
C(DNA) = 0.001 %
D
0,4
0,3
0,2
C(DNA) = 0 %
C(DNA) = 0.0001 %
C(DNA) = 0.0003 %
C(DNA) = 0.0004 %
C(DNA) = 0.0005 %
C(DNA) = 0.0008 %
C(DNA) = 0.001 %
0,6
0,5
D
0,5
0,4
0,3
0,2
0,1
0,0
240
0,1
270
300
330
360
λ, nm
390
420
450
480
0,0
240
270
300
330
360
λ, nm
390
420
450
480
Fig. 4. Difference in interaction of K2[PdHGluCl2] with DNA in low and high
ionic strength.
217
It can be suggested that the aquation process of Palladium atom in 0,15 M NaCl
is absent. The interaction of K2[PdHGluCl2] with DNA in 0,005 M NaCl may be
realized due to the coordination of Pd-atom to DNA or due to ligand binding with
macromolecule.
Fig. 5 shows the images obtained by atomic force microscopy in tapping mode.
The study was performed for a circular plasmid DNA (Fig. 5a). The image in
Fig. 5a shows the fixation of free individual molecules of DNA on a substrate.
b
c
Free DNA
(С(DNA) = 0,75*10-4%)
a
(b), (c) C(K2[PdHGluCl2]) = 10-5 М
(d), (e) C(K2[PdHGluCl2]) = 2·10-6 М
d
e
Fig. 5. AFM images of
DNA in complex with
K2[PdHGluCl2].
Fig. 6. Molecular structure of [PdHGluCl2] and two stages of its aquation.
218
In Fig. 6 the calculated structure of compound under the study in two stages of
its aquation is represented ([PdHGluCl2], [PdHGluH2OCl] and [PdHGlu2(H2O)]).
As the result of calculations the absolute values of energy for each molecule were
obtained. The received values of energy can indicate that in aqueous solution the
most probable process is the substitution of the chlorine atoms to a water molecule
(aquation), as in this case the energy of the system is minimal.
Atomic distances and angles between these atoms were individually obtained
for each molecule.
Conclusion
It is shown that Palladium complex interacts with DNA. The binding causes
change in DNA conformation. Complex formation is realized in solutions of low
NaCl concentration (0,005 M), whereas under physiological conditions (in 0,15
M NaCl) DNA interaction with Palladium compound is not observed.
References
1. Kasyanenko N.A., Levykina E.V., Erofeev D.C. etc. // Journal of Structural
Chemistry, 2009, V. 50, № 5.
2. Kozhenkov P.V. Quantum-mechanical calculation of the structures of potential
anticancer coordination compounds using package HyperChem and GAMESS //
Term work, Saint Petersburg State University, 2009.
3. Firefly (PC GAMESS) version 7.1.G, build number 5618. Copyright (c) 1994,
2009 by Alex A. Granovsky, Firefly Project, Moscow, Russia.
4. HyperChemTM release 8.0.9 for Windows. Copyright (c) 1995-2011 Hypercube,
Inc.
5. Stevens W.J., Krauss M., Basch H., Jasien P.G. SBKJC VDZ ECP EMSL Basis
Set Exchange Library K – RN, 1992.
6. Dill J.D., Pople J.A. // J. Chem. 6-31+G* EMSL Basis Set Exchange Library
Li – Ne, 1975.
219
Studing of the UV radiation influence on the DNA in a
solution in the presence of caffeine
Platonov Denis
KotShredengera@mail.ru
Scientific supervisor: Dr. Paston S.V., Department of Molecular
Biophysics, Faculty of Physics, Saint-Petersburg State
University
The absorption of UV-light by the nitrogenous bases of DNA (λmax=260 nm)
can lead to changes in the structure of the macromolecule, such as hydration (occurs only in the single-stranded DNA) and the formation of pyrimidine adducts
(stable products of addition of pyrimidine bases to other neighbour bases in the
same DNA strand). Maximum quantum yield is observed for the reaction of thymines dimerization [1]. These damages in the DNA structure are the major cause
of mutagenic and bactericidal action of ultraviolet radiation on a cell [2].
In the present investigation caffeine was chosen as a possible DNA protector
from UV radiation (UVC range). This choice was dictated by the fact that the
maximum of caffeine absorption spectrum is at λ=272 nm (near the maximum
absorption of the DNA nitrogenous bases). Besides caffeine weakly interacts with
DNA [3, 4], i.e. it is not a mutagen. Caffeine is a biologically active substance of
plant origin, widely consumed in the world. It has a strong stimulating effect on
the central nervous system [5], affects the cell cycle and the processes of repair
of DNA damage [6].
In this study we used the sodium salt of calf thymus DNA of molecular
weight M = 13,6 ± 0,6 MDa provided by D.Y. Lando (Institute of Bioorganic
Chemistry NAS of Belarus), caffeine purchased from «Sigma». Ionic strength of all
the investigated solutions were 0.003M NaCl. As a source of UV radiation a quartz
mercury lamp of low pressure DRB-8 was used. The output of the lamp is 8 W,
λmax = 254 nm. DNA solutions were irradiated in quartz cuvettes a 0.2 cm thick. To
prevent possible heating of the solutions in the way of UV light a quartz cell was
set a 6 cm thick, filled with distilled water, absorbing thermal infrared radiation. At
the same time the UV radiation transmission of the cell was 85% at λ = 254 nm.
The concentration of DNA in all irradiated solutions was C = 0.011%, the distance to the source was 7 cm, which corresponds to the absorption intensity Iabs =
0.88*103 J/(kg*s). The exposure time varied between 12 min–2 h 15 min.
After the UV-light exposure the alterations in DNA absorption (Fig. 1) and circular dichroism (CD) spectra (Fig. 2) are observed. The DNA absorption intensity
increases at all wave lengths (Fig. 3) and the intensities of the positive and negative
bands in the DNA CD spectrum decreases monotonously with the radiation dose
growth (Fig. 4(a,b)). Also the shift of CD spectra to the long wavelength region
is observed (Fig. 5). The alterations observed can be caused by the partial DNA
denaturation and, possibly by the modification of nitrogenous bases.
220
Fig. 1. DNA absorption spectra after UV irradiation with the different doses.
Fig. 2. The DNA CD spectra after UV irradiation with different doses.
221
D
D260
0,9
D310
D 0,06
0,8
0,05
0,7
0,04
0,6
0,03
D230
0,5
0,02
0,4
0,01
0
1
2
3
4
5
Dr, 106 J/kg
6
7
8
0
2
4
6
8
6
Dr, 10 J/kg
10
Fig. 3. Dependences of the intensity of the absorption spectrum of DNA at wavelengths of 260nm, 230nm and 310nm on the dose of UV irradiation
1,9
∆ε275, Μ−1cm−1
1,8
-1,8
1,7
1,6
-2,0
1,5
-2,2
1,4
-2,4
1,3
-2,6
1,2
-2,8
1,1
1,0
∆ε240, Μ−1cm−1
-1,6
0
2
4
Dr, 106 J/kg
6
-3,0
8
0
2
4
Dr, 106 J/kg
6
8
Fig. 4. The intensities of the positive (a) and negative (b) bands in the DNA CD
spectrum.
λ, nm
262
261
260
259
258
257
256
255
254
0
2
4
Dr, 106 J/kg
6
Fig. 5. The position of ∆ε = 0 in the DNA CD spectrum.
222
8
DNA
DNA+caffeine (1.3*10-3mol/l)
1,0
ηr/ηr0
0,8
0,6
0,4
0
3
Dr, 106 J/kg
6
Fig. 6. Dose dependency of the DNA relative viscosity at C(DNA)=0.011%.
Table 1. Characteristic viscosity of DNA.
Ccaf, mol/l
Dr, 106 J/kg
[η] DNA, dl/g
0
0
135
0
0,64
82
0,0013
0,64
118
UV-irradiation also leads to the diminution of the volume of macromolecule in a
solution. In our work it is obtained by the measurement of DNA intrinsic viscosity.
The experiment shows that in the presence of caffeine in the DNA solution under
the UV-light exposure the diminution of the DNA volume is smaller (Table 1).
Investigation of the dose dependences of the relative viscosity of DNA solutions
(Fig. 6) also reveals the photoprotective action of caffeine. We estimated the dose
reduction factor (DRF80) at Ccaf=1.3*10-3 mol/l according to the expression [7]:
D (DNA+caf ) DRF = r 80
80
D (DNA)
r 80
(values Dr80(DNA+caf) and Dr80(DNA) are obtained from the Fig. 6).
It was found that for Ccaf=1.3*10-3 mol/l DRF80=13±3. The photoprotective action of caffeine may be explained by the fact that this substance and DNA absorb
in the same spectral region (λmax(caffeine) = 272 nm) that is why the intensity of
UV light in solution containing caffeine is lower then in a pure water. One can say
that caffeine shields DNA from UV light.
223
References
1. Smith K.C., Hanawalt P.C. Molecular Photobiology. – Acad. Pr., New York
and London, 1969.
2. Rubin A.B. Biophysics, v.2. – Moscow: High School, 1987.
3. Tarasov A.E. Study of the radioprotective properties of caffeine, catechines
and aliphatic alcoholes in DNA solutions. Master's thesis, Saint Petersburg State
University, Faculty of Physics, 2011 (in Russian).
4. Osipov N.D., Kondrat'eva O.P., Frisman E.V. // Vestnik LGU, № 4, 98 – 101,
1979 (in Russian).
5. Caffeine. PubChem public chemical database (http://pubchem.ncbi.nlm.nih.
gov/summary/summary.cgi?cid=2519#Synonyms)
6. Conney A.H., Zhou S., Lee M.-J. et.al. // Toxicol. and Appl. Pharm., v. 224, p.
209 (2007).
7. Kudryashov Yu.B. Radiation Biophysics (Ionizing Radiations). – New York:
Nova Science Publishers, Inc.; 2008.
224
Entropic sampling of thermodynamic and structural
properties of polymer chains and stars within WangLandau algorithm
Silantyeva Irina
sila3@yandex.ru
Scientific supervisor: Prof. Dr. Vorontsov-Velyaminov P.N.,
Department of Molecular Biophysics, Faculty of Physics, SaintPetersburg State University
Introduction
Different types of numerical methods are used now for studying of polymers,
such as molecular dynamics, entropic sampling [1, 2], numerical treatment in
self-consistent field method [3]. Entropic sampling method allows us to obtain the
density of states functions. Using these functions we can calculate canonical averages in a wide temperature range. In our work the entropic sampling (ES) combined
with Wang-Landau (WL) algorithm is used for studying of polymer chains and
stars. Polymer star is not an abstract model. Such molecules are synthesized (for
example in University of Helsinki [4]) and used for transport of DNA and drugs
into living cells.
Results and Discussion
The aim of our work is calculation of the density of states and the thermodynamic properties of lattice models of polymer chains and stars on 3D simple cubic
lattice using WL algorithm. The semi-phantom random walk is used for generating
models. The 6-arm polymer star with the length of arms Narm=5, 12, 20 segments
is considered. So the overall number of segments is N = 30, 72, 120. In athermal
case, when interaction between monomers is reduced to exclusion of intersections,
the ratio of self-avoiding walk (SAW) conformations is calculated for stars with
total number of segments up to N=720 (Table 1).
N
6
12
18
24
30
36
42
48
54
60
72
Narm
1
2
3
4
5
6
7
8
9
10
12
Ω0
1
5.67E-01
1.64E-01
1.06E-01
5.19E-02
3.24E-02
1.81E-02
1.12E-02
6.71E-03
4.21E-03
1.49E-03
N
120
180
240
300
360
420
480
540
600
660
720
Narm
20
30
40
50
60
70
80
90
100
110
120
Ω0
4.03E-05
4.96E-07
7.47E-09
1.26E-10
1.93E-12
3.23E-14
6.12E-16
1.32E-17
1.96E-19
3.46E-21
7.75E-23
Table 1. The ratio of SAW conformations among semi-phantom for stars.
225
a
0,2
b
30
0,15
72
C/N
0,1
120
0,05
цепи
0
0
1
2
3
4
5
T
0,5
с
0,45
d
120
72
30
0,4
0,35
120
C/N
0,3
0,25
72
30
0,2
0,15
0,1
0,05
0
0
2
T
3
4
e
-0,2
-0,3
30
-0,15
-0,4
∆S/N
72, 120
-0,2
72
-0,5
-0,6
0
2
4
β
6
8
120
72
72
-0,8
30
30
120
-0,7
-0,25
-0,3
5
f
-0,1
-0,1
ΔS /N
1
0
-0,05
-0,35
0
30
-0,9
-1
10
0
1
2
β
3
4
5
Fig. 1. Specific energy (a, c), specific heat capacity (b, d) dependences on temperature and specific excess entropy (e, f) dependences on inverse temperature β for
stars with overall number of segments N = 30, 72, 120 (thick lines) and for chains
with the same number of segments (thin lines); ε>0(a, c, e), ε<0(b, d, f). Dashes
on energy axis (a, c) denote limiting values of energy at T→∞.
226
Fig. 2. Mean square radius of inertia dependence on temperature; ε>0 (black
line), ε<0 (grey line). Chain length N =30, 72, 120 segments. Mean square radius of inertia determines the size of the molecule. Horizontal dashed lines denote
the limiting value of mean square radius of inertia at T→∞.
600
500
<h ^2>
400
120
300
200
72
100
30
0
0
2
4
6
8
10
T
Fig. 3. Mean square end-to-end distance dependence on temperature; ε>0 (black
line), ε<0 (grey line). Chain length N =30, 72, 120 segments. Horizontal dashed
lines denote the limiting value of mean square end-to-end distance at T→∞.
227
Fig. 4. Square root of mean square end-to-end distance dependence on inverse
temperature for chain with length N=30; ε>0 (diamonds), ε<0 (squares). These
dependences are in agreement with data from work [1].
Fig. 5. Distribution of the mean square radius of inertia over number of contacts
for stars. Total number of segments in stars N=30, 72, 120 (Narm =5, 12, 20).
In the thermal case we attribute an energy ε to each pair of non-neighboring
monomers occurring at closest contact, ε>0 or ε<0. In this case the SAW conformations are selected and the distributions Ω0m over contacts between monomers are calculated. Using these distributions the equilibrium properties, such
as conformational energy (Fig. 1a, c), heat capacity (Figs. 1b, d), entropy
(Figs. 1e, f) are obtained as dependences on temperature. Results for chains
and stars are presented in common figures for comparison. The temperature in
all figures is in energy units. Also the mean square radius of inertia <R2> and
mean square end-to-end distance <h2> dependences on temperature for chains
(Figs. 2, 3) are obtained using equation:
228
mmax
R 2 (T ) =
R2
m
=
∑
m=0
R2
mmax
m
exp − εm / kT Ω 0 m
∑ exp
m=0
− εm / kT
Ω0 m
where Ω0m – density distribution of SAW over contacts and <R2>m – distribution of
mean square radius of inertia over number of contacts are obtained in our numerical experiment. For chains in the attraction case (ε<0) the radius of inertia monotonically increases with temperature. In the repulsion case (ε>0) it monotonically
decreases with temperature. The limiting value the radius of inertia at T→∞ can
be calculated using equation
mmax
lim R 2 = ∑ R 2 m Ω0 m
T →∞
m=0
At T→∞ the radius of inertia both for ε<0 and for ε>0 tend to the same limit
(dashed line in Figs. 2, 3). The comparison of our results (Fig. 4) with that from
work [1] shows good agreement.
The distributions of mean square radius of inertia <R2>m over contacts (Fig. 5)
allow to obtain the corresponding temperature dependence for stars.
Conclusion
As a result of our work we can make the following conclusions:
1) For repulsion case (ε>0) the specific heat capacity dependences on temperature have one maximum both for chains and stars. But for attraction (ε<0) the
dependences are more complicated. They have several maxima. Maxima for stars
are shifted to lower temperatures.
2) Heat capacity maxima denote the significant changes in polymer conformations.
3) Further studying of structure properties is required.
Acknowledgment. The work is supported by the grant RFBR 11-02-00084 a.
References
1. Vorontsov-Velyaminov P.N., Volkov N.A., Yurchenko A.A. // J. Phys. A, 2004.
37, pp. 1573-1588.
2. Volkov N.A., Yurchenko A.A., Lyubartsev A.P., Vorontsov-Velyaminov P.N. //
Macromol. Theory and Simul. 2005, 14, pp. 491-504.
3. Birshtein T.M., Mercuryeva A.A., Leermakers F.A.M., Rud O.V. //
Macromolecular compounds A, 2008, 50, № 8, pp. 1-18 (In Russia).
4. Anu Alhoranta, Julia Lehtinen, Arto Urtti, Sarah Butcher, Vladimir Aseyev,
Heikki Tenhu. Book of abstract of the 14th IUPAC International Symposium of
Macromolecular Complexes (MMC-14) August 14-17, 2011, Helsinki, Finland,
p. 195.
229
Silver nanoparticles and their interaction with polymers
in solution and on a surfaces
Varshavskii Mikhail
varshavskiimiha@mail.ru
Scientific supervisor: Prof. Dr. Kasyanenko N.A., Department of
Molecular Biophysics, Faculty of Physics, Saint-Petersburg State
University
Introduction
Nanoparticles (NPs) are small objects with a narrow size distribution and
diameter in the range of 10 – 200 nm. Optical and other physical properties of
nanoparticles depend greatly on their size and shape, in contrast to a bulk material
with the physical properties regardless of its size. Wide research activity takes place
currently in the field of nanosensors based on NPs. For example biosensors based
on surface plasmon resonance effect are used for biochemical tests for glucose
and urea, for immunoassays of proteins, hormones, drugs, steroids, viruses, DNA
testing and finally to study the kinetics of drug action in real time.
In this paper the spectral properties of metal nanoparticles and their interaction
with charged macromolecules in aqueous and aqueous-salt solutions were studied.
CD-spectrophotometer Mark 4 (Jobin Ivon, France), UV-spectophotometers SF-56
and SF-2000 (Russia) were used for circular dichroism and UV-adsorption spectra
studies. Visualization of the obtained structures was done using atomic force microscopy (NanoScope 4a, Veeco) in tapping mode in air. Silver NPs was synthesized
in National Technical University of Ukraine, Kyiv Polytechnic Institute by means
of a new electric-spark dispersion technique. Calf thymus DNA (Sigma) with the
molecular mass M=8×106 Da, determined from the value of the DNA intrinsic
viscosity in 0.15 M NaCl solution.
Results and Discussion
One of the most interesting phenomenon among the unique properties of metallic
NPs is a local surface plasmon resonance (LSPR). The strong interaction of silver
nanoparticles with light in this case is determined by the collective oscillation of
conduction electrons within the metal. The “electron gas” motion due to quantum
effects can be presented as the movement of quasiparticles plasmons. LSPR can be
observed if the size of metallic particles is smaller than light wavelength as a result
of light interaction with NPs. LSPR is realized when light frequency coincidents
with the intrinsic plasmon frequency. The investigation of silver nanoparticles
and their interaction with charged macromolecules is interesting for the different
applications. It was shown that Ag NPs have a plasmon absorption band (peak is
about 400 nm). This band does not overlap with the absorption spectrum of DNA.
The absorptions spectra of different metal nanoparticles in water for two series of
preparations are shown in Fig. 1.
230
Fig. 1. The absorptions spectra of metal nanoparticles in water for two series of
preparations.
Our experiments demonstrate that silver nanoparticles can interact with negatively charged DNA and positively charged polyallylamine (PAA). This follows
from the change in the absorption spectrum of nanoparticles within the wavelength
region where PAA and DNA do not absorb light. A solution of silver nanoparticles
with polyallylamine is not so stable (the precipitation of NPs can be observed after
14 days). The solution of NPs with DNA is stable even after 2 weeks storage.
The influence of salt concentration on the absorption spectrum of silver NPs
was studied (Fig. 2a). The intensity of plasmon resonance peak (400 ± 1 nm in
aqueous solution) increases and the maximum shifts to shorter wavelengths for 10
± 1 nm with the increasing of NaCl concentration up to 0.1 M
Fig. 2. The absorption spectra of silver nanoparticles in aqueous salt solutions
of different ionic strength (a) and the dependence of the intensity of the plasmon
resonance peak on the salt concentration (b). 1 – 0 M; 2 – 0.001 M; 3 – 0.002 M;
4 – 0.003 M; 5 – 0.005 M; 6 – 0.01 M; 7 – 0.03 M; 8 – 0.05 M; 9 – 0.1 M; 10 –
0.15 M; 11 – 0.3 M; 12 – 0.5 M; 13 – 1 M NaCl.
With further increasing of salt concentration the intensity of plasmon peak
decreases. The band changes the shape and shifts to the red region. Such changes
may be associated with the precipitation of some NPs due to the aggregation at
231
high salt concentrations. We emphasize the essential difference in the effect of
large (> 0.1 M) and low (<0.1 M) NaCl concentration on the spectral properties of
nanoparticles. The shift of the maximum of the plasmon band is usually associated
with the change in the size of the nanoparticles. For the systems under investigation
this may be observed as a result of NPs interaction with ions. It was also found that
silver nanoparticles in aqueous salt solution eventually precipitate over time.
Fig. 3 shows the absorption spectra
of silver nanoparticles in different solutions: 1) in water, 2) in the aqueous
salt solution (0,005 M NaCl), 3) in the
DNA solution (C(DNA)=0.005%).
Maximum of plasmon peak shifts to
shorter wavelengths in the presence of
NaCl (0,005M) and DNA (0,005%).
This result indicates the decrease in the
size of silver clusters.
Circular dichroism method showed
Fig. 3. The absorption spectra of silver that silver nanoparticles are not optinanoparticles in water, in aqueous salt cally active in the range 220-480 nm
solution (0.005 M NaCl), in 0.005% solu- (Fig. 4). In Fig. 4 one can see also
tion of DNA.
CD spectra of DNA in a solution with
different amount of silver NPs. The
increase in NPs concentration causes
the great change in DNA circular dichroism spectra. These data show us
the interaction of silver nanoparticles
with DNA. The influence of silver NPs
on DNA conformation is observed.
Atomic force microscopy was also
used for the study of silver nanoparticles (Fig. 5). NPs were deposited
on a mica substrate using a magnetic
Fig. 4. Circular dichroism spectra of DNA stirrer. The sample was rotated about
in the solutions with silver nanoparticles. 3 minutes. The size of NPs is about
30nm (Fig. 5a,b). The images contain
the track of NPs under centrifugal force. Silver nanoparticles and DNA are not fixed
on mica in the absence of MgCl2. In the presence of MgCl2 DNA and nanoparticles
can be fixed on mica. For the DNA fixation from the solution with NPs one can
observe knobs at the ends of DNA strands and crossing parts of DNA (Fig. 6a, b)).
Diameter of the knobs is about 30 nanometers (Fig. 6c). So we can propose that
NPs are localized at the ends of double stranded helix.
232
Fig. 5. Silver nanoparticles on a mica substrate.
(a)
(b)
(c)
Fig. 6. DNA-nanoparticles complex on a mica (a), (b). NPs section profile (c).
233
Conclusions
• It was shown that the aqueous solutions of NPs are stable in time.
• The increase of NaCl concentration in a solution influences on the state of silver
nanoparticles. The low (<0.1 M) and high (> 0.1 M) NaCl concentrations cause
the different changes in plasmon resonance peak.
• The addition of NaCl into water solution provokes a partial precipitation of
NPs in time.
• The addition of DNA into NPs water-salt solution stabilizes the system regardless on the concentration of NaCl.
• The presence of anionic polymer (DNA) and cationic polyallylamine in NPs
solution influences on the plasmon resonance band.
• AFM images of the systems were obtained. It was shown that the size of nanoparticles is about 30 nm.
234
I. Resonance Phenomena
in Condensed Matter
Phase transitions in magnesium: ab initio study
Klyukin Konstantin
konstantin.klyukin@gmail.com
Scientific supervisor: Dr. Shelyapina M.G., Department of
Quantum Magnetic Phenomena, Faculty of Physics, SaintPetersburg State University
Introduction
Magnesium-based hydrides stand as one of the most promising candidates for
hydrogen storage due to its high hydrogen uptake (up to 7.6 w% in MgH2), large
reserves and low cost. The main disadvantages of MgH2 are high hydrogen release
temperature, slow sorption kinetics and a high reactivity toward air and oxygen.
However, transition metals additives (TM), such as Ti, V or Nb, greatly enhance
hydrogen sorption kinetics [1].
Numerous theoretical and experimental investigations have been made in
order to understand the mechanism of hydrogen uptake by magnesium. But there
is no clear conclusion. Moreover, in situ X-ray diffraction experiments have
shown that penetration of hydrogen into Mg occurs through TM “gates” [2]: at
first, a TM hydride appears and after that hydrogen penetrates into magnesium.
Therefore, investigation of the interface border of magnesium with TM could be
helpful to understand the role of TM additives on the hydrogen sorption process
in magnesium.
In our previous theoretical investigations of Mg/Nb thin films we have found
that near the interface Mg keeps the bcc structure of Nb [3], that means that Nb
layers (or particles) favor the bcc structure of Mg. Since Mg layers deposited on a
Nb-(011) surface may keep the bcc stacking mode, we assume that there are two
pathways for Mg → MgH2 transformation. The first one, corresponds to the direct
Mg (hcp) → MgH2 (rutile) transition. But even small additions of Nb or V with
bcc structure can lead to the the second two-step hydrogenation scheme: the first
step is the Mg(hcp) → Mg(bcc) transformation, which occurs near the Mg/TM
interface border, the second step is that upon hydrogenation of the bcc magnesium
a MgHx → MgH2 transformation occurs.
In this contribution we report on the results of our theoretical study of phase
transitions in metallic magnesium.
Method of calculation
The electronic structures of all systems were calculated within a DFT fullpotential linearized augmented plane-waves (FLAPW) method using the Perdew–
Burke–Ernzerhof GGA exchange and correlation potential. In all calculations,
self consistency was achieved with a tolerance in the total energy of 0.1 mRy. The
number of k-points in the irreducible Brillouin zone was equal to 1000. For Mg
and Nb the radius of nonoverlapping muffin-tin spheres was chosen equal to 2.0
236
a.u, for hydrogen – 1.1 a.u. The calculations were carried out using the package
WIEN2k [4].
Results
Hcp-bcc phase transition. To describe the hcp-bcc transition we have considered the distortion model proposed by Burgers [5]. The simplest path of transition
from bcc lattice into hcp structure includes two independent processes (see Fig. 1):
• a shear deformation from the bcc (110) plane to the hexagonal basal plane;
• a slide along the (110) planes.
Fig. 1. Scheme of hcp-bcc phase transformation of magnesium.
To describe the transformation we used a two-dimensional parameter space
(λ1, λ2), where λ1 represents the shear deformation and λ2 represents the slide displacement. The hcp and bcc structures correspond to (0,0) and (1,1), respectively.
The lattice parameters of the primitive cell can be written as
a = (1 − λ1 ) ⋅ ahcp + λ1 ⋅ abcc , b =  3 ⋅ λ1 + 2 ⋅ (1 − λ1 )  ⋅ a


 c
c = (1 − λ1 ) ⋅   + 2 ⋅ λ1  ⋅ a
 a  opt


with Mg atoms at positions
1
1
 1 1
 5 1
0, 3 + 6 ⋅ λ 2 , 2  , and 0, 6 + 6 ⋅ λ 2 , 2 
In Fig. 2 we represent the counter plot of potential energy surface for magnesium
hcp-bcc transition. The bcc structure is located at an unstable point. However, the
energy difference between hcp and bcc phases is only 1.41 kJ/mol per atom.
As it is seen in Fig. 2, during the hcp-bcc phase transition, at first, the shear
deformation dominated, but at the end of transformation the slide displacement
becomes stronger.
237
Fig. 2. Counter plot of potential energy surface (hcp-bcc transformation).
Bcc-fcc phase transition. However, bcc Mg structure is unstable and is held
only by presence of Nb. So, tetragonal distortion of free bcc Mg cell leads to fcc Mg
structure. As it is seen in Fig. 3, the energy curve has two minima. The first one corresponds to metastable bcc Mg structure, the second one to stable fcc Mg structure.
Fig. 3. Bcc-fcc phase transition of magnesium.
As the normal state of magnesium hydride is the rutile phase, we also considered
rutile structure of magnesium with hydrogen vacancies. The relative stability ∆E
of all studied structures (relative to the hcp structure) was estimated by using the
following expression:
∆E (Mg structure ) = Etot (Mg structure ) – Etot Mg hcp (1)
(
)
The results, together with the equilibrium lattice parameters and total energy per
atom are listed in Table 1.
238
Structure
a (Å)
c (Å)
Etot (Ry/atom)
Mg (hcp)
Mg (bcc)
Mg (fcc)
Mg (rutile)
3.218
3.5763
4.5190
4.2526
5.108
3.5763
4.5190
2.8382
-400.6672
-400.6650
-400.6663
-400.6531
∆Ehf
(kJ/mol*atom)
0
1.41
0.56
9.26
Table 1. Equilibrium lattice parameters, total energy and heat of formation.
(a)
(b)
(c)
(d)
Fig. 4. The total, s- and p-states resolved DOS’s calculated for different magnesium structures: (a) – hcp, (b) – fcc, (c) – bcc, (d) – rutile. The Fermi level corresponds to E = 0 and is marked by a vertical solid line.
For better understanding of the nature of stability (or instability) of different
magnesium structures we have studied their electronic structure. The calculated
electronic density of states (DOS) are shown in Fig .4.
As it is seen in all considered magnesium structures the DOS near the Fermi
level (EF) is formed by the delocalized s- and p-states. For the hcp structure (Fig. 4a)
EF falls into a local minimum of both s- and p-DOS that characterizes a stable state,
whereas for the rutile structure (Fig. 4d) EF corresponds to the local maximum of
the p-states and a rather high population of the s-states. That, by-turn, character239
izes an unstable structure. The fcc and bcc structures are in the intermediate position: they have the Fermi level in the minimum of one state and in the maximum
of another (Figs. 4b and 4c). This result is in agreement with total energy values
listed in Table 1 and confirms that the rutile structure with hydrogen vacancies is
the most unstable.
Conclusions
The FLAPW calculations of different phases of magnesium, namely, hcp, bcc,
fcc and rutile structure of MgH2 with hydrogen vacancies, have shown that the
hcp phase is the most stable, whereas the rutile structure with hydrogen vacancies
is the most unstable. However, as soon as one creates a bcc phase, for example
by deposing Mg on a Nb-(011) surface, the bcc structure may relax into the hcp
structure. At the beginning of the bcc-hcp phase transition the the slide displacement dominates, but at the end of transformation the shear deformation becomes
stronger. Nevertheless, the bcc phase may be transformed into a fcc phase as well.
The energy difference between the bcc and fcc phases is only 0.85 kJ/mol per atom.
Hence, the fcc-Mg can be an intermediate phase during the bcc-hcp transition.
Calculations of the fcc-hcp phase transition in magnesium are under evaluation.
References
1. Charbonnier J., P. de Rango, Fruchart D. et. al. // Alloys Compd. Vol. 383,
p. 205, 2004.
2. Pelletier J.F., Huot J. // Phys. Rev. B, Vol. 63, Is. 5, 052103, 2001.
3. Klyukin K., Shelyapina M.G., Fruchart D. // Solid State Phenomena 170,
p. 298-301, 2011.
4. Blaha P., Schwarz K. and Luitz J. Computer code WIEN2k,Vienna University
of Technology, 2000.
5. Burgers W.G. // Physica, Amsterdam 1, 561, 1934.
240
NMR study of spin relaxation
Nefedov Denis
iverson89@yandex.ru
Scientific supervisor: Prof. Dr. Charnaya E.V., Department of
Solid State Physics, Faculty of Physics, Saint-Petersburg State
University
Introduction
The influence of size effects on various physical properties of materials, including atomic mobility, relaxation phenomena, and liquid fluidity, in confining systems
has been recently attracting considerable interest. NMR is known to yield valuable
information on dynamics in condensed media and has found wide application in
studies of the mobility of liquids inserted into nanosized pores. A decrease in atomic
mobility, first observed to occur in liquid gallium incorporated in nanosized pores
[1], manifested itself in an increase in the nuclear spin relaxation rate by a few
times due to enhancement of the role of the quadrupole contribution. In [1] only
nanosized gallium particles were studied, which did not permit one to reveal the
characteristic dimensions at which size effects in atomic mobility become noticeable in a gallium melt.
The aim of this work was to process the data obtained in the study of the relaxation times of the nuclei of isotopes of gallium (69Ga and 71Ga) in confined geometry
with the use of NMR. The samples were isolated particles of gallium the size of
about 50 microns, thin film of gallium on an opal the thickness of about 10 microns,
gallium injected into the pores of the artificial opal and porous glass.
Results and discussion
The gallium relaxation process can be described by the relation [2]:
4


 t  C τ cñt  1
Cτ t  
M (t )
= 1 − b  exp  −
+ exp  −
exp  −
2 2
2 2
M0
 T1m 
 1 + 4ω 0 τ c  5
 1 + ω0 τc  
5
Here M(t) and M0 are the time-dependent (t) and equilibrium magnetizations,
respectively; 1 – b is the relative magnetization immediately following the inverting pulse; ω0 is the Larmor frequency; C is a constant proportional to Q2 and τc is
the correlation time.
The magnetization restoration process is described by a single exponent, and
C·τc is equal to the inverse quadrupole spin-lattice relaxation time,
C ⋅ τ c = T1−q1 = Rq1 . Since ω0τc<<1, the relation of magnetization restoration process takes the form

 t 


 t 
 t 
 t 
M (t )
= 1 − b exp  −   exp  −
= 1 − b exp  −   = 1 − b exp  −   
M0
 T1  
 T1  
 T1m 
 T1q  



241
According to this formula the exponential approximation of experimental points
is performed.
Designations :
T171- the longitudinal relaxation time 71Ga
T169 - the longitudinal relaxation time 69Ga
R171 - the longitudinal relaxation rate 71Ga; R171=1/T171
R169 - the longitudinal relaxation rate 69Ga; R169=1/T169
R1q71 - the quadrupole contribution in longitudinal relaxation rate 71Ga
R1m71 - the magnetic contribution in longitudinal relaxation rate 71Ga
R1q69 - the quadrupole contribution in longitudinal relaxation rate 69Ga
R1m69 - the magnetic contribution in longitudinal relaxation rate 69Ga
It is necessary to find out contributions of magnetic and quadrupole relaxation
rates to the overall rate.
For the sample 1.
Using the conditions
R169q
Θ 69 2
=
(
) ≈ 2,51
R171q
Θ 71
Plan of calculations
R169m
γ 69 2
=
(
) ≈ 0, 62
R171m
γ 71
(*),
(**)
And knowing, that:
R171 = R1m71 +R1q71; R169 = R1m69 +R1q69,
form the system of this four equations. R171=1/T171 and R169=1/T169 and T169 are
obtained from the processing of experimental curves. Having solved this system
of equation we obtain R1q71, R1m71, R1q69, R1m69.
For the other three samples:
Let’s consider Korringa ratio [3]:
T1mTK s2 = const / ( γK )
Here T is the temperature; KS is the Knight shift; K is a factor which takes into
account the correlation effects. The experiment revealed, that the Knight shift did
not vary from sample to sample. The temperature in this experiment was constant.
Thus we can make the assumption, that T1m remained constant during the transition
from sample to sample. Then, after obtaining overall relaxation rates R1, we find
contributions R1q71, R1m71, R1q69, R1m69:
R1q71 = R171 +R1m71; R1q69 = R169 +R1m69.
Results R1q71, R1m71, R1q69, R1m69 must satisfy the conditions (*) and (**).
242
Results
1) Sample 1 (isolated particles of gallium, the size of about 50 microns). The
dependence of the longitudinal magnetization on the time.
R169 = 1480 (c-1)
T169= 680·10-6 (c)
T171 = 530·10-6 (c) R171 = 1870 (c-1)
By solving the system of equations:
R169q
R 69
Θ 69
γ 69
= ( 71 ) 2 ≈ 2,51; 171m = ( 71 ) 2 ≈ 0, 62
71
R1q
Θ
R1m
γ
R171 = R171m + R171q ; R169 = R169m + R169q
We obtain:
R1q69 = 420 (c-1) ; R1m69 = 1060 (c-1)
R1q71 = 170 (c-1) ; R1m71 = 1710 (c-1)
Results of all samples:
1) 50 microns
2) Thin film
3)Opal
4) Porous glass
R1q71 (c-1)
170
170
2940
3730
R1q69 (c-1)
420
420
4790
9700
R1m71 (c-1)
1710
1710
1710
1710
R1m69 (c-1)
1060
1060
1060
1060
R171 (c-1)
1870
1870
4650
5440
R169 (c-1)
1480
1480
5850
10750
243
Conclusions
Magnetic relaxation dominates in first two samples, and quadrupole relaxation
in the two last ones.
Magnetic contribution to the relaxation rate is constant from sample to
sample.
R169m
γ 69
= ( 71 ) 2 ≈ 0, 62 71
R1m
γ
It is confirmed experimentally within error.
As a result of processing the experimental data and calculations magnetic
and quadrupole contributions to the spin-lattice relaxation rate in liquid gallium
were obtained.
Reduction in the size of liquid gallium sample is followed by:
1) Increase in the quadrupole relaxation rate;
2) Decrease in the atomic mobility.
3) Increase in the correlation time.
References
1. Charnaya E.V., Loeser T., Michel D. et al. // Phys. Rev. Lett. 88, 097602
(2002).
2. Tien C., Charnaya E.V., Sedykh P. and Kumzerov Yu.A. //Physics of the Solid
State, V. 45, N. 12, 2352-2356.
3. Слиткер Ч. Основы теории магнитного резонанса.- М: Мир, 1981.-173 с.
244
Magnetic properties of cubic magnetite Fe3O4: a density
functional theory study
Irina Shikhman
shikhman.irina@gmail.com
Scientific supervisor: Dr. Shelyapina M.G., Department of
Quantum Magnetic Phenomena, Faculty of Physics, SaintPetersburg State University
Introduction
Magnetite is the earliest discovered magnet. Extensive studies of Fe3O4 have
been carried out over the past 60 years. Because of interesting electronic and magnetic properties as well as potential industrial applications in magnetic multilayer
devices and spintronics, magnetite has still attracted much attention.
Theoretical studies can be very helpful to understand the nature of physical
phenomena observed in magnetite, both bulk and multilayered. However, even
for the bulk Fe3O4, the calculated properties, such as density of states, magnetic
moments, magnetic anisotropy energy etc., are rather sensible to the method of
calculation [1, 2].
The aim of the present work is to study structural and magnetic properties of
magnetite by different density functional theory methods (GGA, LSDA, LDA+U).
It will be applied further to study Fe/Fe3O4 multilayes.
Method of calculations
The calculations have been done within the framework of the full-potential
linearized augmented plane waves (FLAPW) method that is one of the most accurate realizations of the DFT methods. We have tried several types of exchangecorrelation potentials, such as GGA, LSDA and LDA+U. Non-overlapping
“muffin-tin” spheres (RMT) were chosen equal to 1.75 a.u. and 1.55 a.u. for the
Fe and O atoms, respectively. For LDA+U method the parameter Ueff were chosen
equal to 4.5 eV and 4.0 eV for atoms FeA and FeB, respectively. In all calculations,
self-consistency was achieved with a tolerance in the total energy of 0.1 mRy. All
calculations have been carried out using the Wien2k package [3].
Crystal structure
Magnetite crystallizes in the inverse cubic spinel structure (Fd3m) above the
so-called Verwey transition temperature (tV≈120 K). But at temperature lower
than tV it takes the orthorhombic structure (Imma) [4 – 6]. Unit cells of the high
temperature (HT) and low temperature (LT) phases are presented in Figs. 1a and
1b, respectively.
245
(a)
Fig. 1. Unit cells of the HT (a) and LT (b) Fe3O4.
(b)
In both structures the iron atoms occupy octahedral and tetrahedral sites (the
nearest neighboring of oxygen atoms forms an octahedron or a tetrahedron). The
iron atoms of A-type (FeA) with valence 3+ held tetrahedral sites, whereas the iron
atoms of B-type (FeB) with mixed 2.5+ valence held octahedral sites. Besides, in
low temperature phase the B-site is separated into two positions FeB1 and FeB2 with
different valences, 3+ and 2+, respectively. Positions of atoms in both structures
are described in Table 1. Structural differences cause the difference in properties
of magnetite, in particularly, conductivity of HT structure is higher than conductivity of LT one.
The experimental value of the lattice parameters [2, 6] and positions of atoms
for the high and the low temperature phases of Fe3O4 are presented in Table 1.
High temperature phase
Low temperature phase
Space group
Fd3m (227)
Imma (74)
aexp (Å)
8.394
5.912
bexp (Å)
8.394
5.945
cexp (Å)
8.394
8.388
FeA positions
8a (0 0 0)
4e (0 1/4 1/8)
FeB positions
16d (5/8 5/8 5/8)
O positions
u
32e (u u u)
0.3798
4b (0 0 ½)
4d (¼ ¼ ¾)
8h (0 2u u)
8i (2u-¼ ¼ ¼-u)
0.2548
Table 1. Experimental lattice parameters and positions of atoms for HT and LT
phases of Fe3O4.
246
Results
The properties of HT Fe3O4 were calculated within the framework of LSDA,
GGA and LDA+U methods. The results of the geometry optimization and calculated
atomic magnetic moments are listed in Table 2.
As is it seen from Table 2 all methods indicate that the magnetic sublattice of FeA
and FeB atoms are ordered antiferromagnetically. As the number of FeB sites is two
times greater than the number of FeA sites, the Fe3O4 is a ferrimagnetic. However
the LSDA method leads to the extremely low values of the atomic magnetic moments (the value of the magnetic moment on the FeA site is about 1 µB lower the
experimental value). The gradient correction of the exchange-correlation potential
improves the values of atomic magnetic moments, but the better agreement with
experiment is achieved using the LDA+U method.
Method
LSDA
GGA
LDA+U
aopt (Å)
8.1247
8.4018
8.1247
m(FeA) (µB)
-2.86
-3.39
-3.71
m(FeB) (µB)
3.08
3.48
3.63
m(O) (µB)
0.07
0.07
0.06
Table 2. Optimized lattice parameter and magnetic moments of atoms in Fe3O4
calculated within different methods.
In Fig. 2 we plot the total density of states (DOS) calculated within three different methods. As it is seen, the different methods lead to rather different shape
of DOS. For example, for the LSDA method the energy gap for the spin-up states
is above the Fermi level (EF), whereas for the GGA method EF falls in the middle
of a narrow gap of about 1 eV. And this gap becomes larger (of about 2 eV) for the
LDA+U method. If we look at the site resolved DOS one can see that the downspin DOS at the Fermi level is formed mainly by the FeB d-states.
The LSDA method leads to the well localized d-states for the both FeA and
FeB atoms. The up and down spin states are shifted relative to each other by about
3.5 eV in such a manner that the for the FeA atom the down-spin states are occupied and the up-spin states are unoccupied. For the FeB atom, on the contrary,
the up-spin states are almost completely occupied and the down-spin states are
partly filled. The gradient correction does not change the DOS shape noticeably.
However, consideration of LDA+U method results to dramatic changes in DOS.
Namely, the shift between up- and down-spin states becomes much more important
(~ 10 eV); the up-spin states of FeB get delocalized, whereas the down-spin part
remains almost unchanged.
We do not plot the oxygen contribution to DOS, as it is of low intensity and
is not visible in the picture. The analysis shows that the O p-states are strongly
hybridized with d-states of iron atoms.
247
Fig. 3. Total spin-polarized DOS in the high temperature phase of Fe3O4 calculated within different methods. The Fermi level corresponds to E = 0 and is marked
by a vertical solid line.
Conclusions
As a conclusion, according to our calculations the most accurate and suitable
method to study Fe3O4 is LDA+U, and exactly this method will be used for studying of properties thin films Fe/Fe3O4.
1.
2.
3.
4.
5.
6.
References
Wenzel M.J. et al. // Phys. Rev. B, vol. 75, p. 214430 (2007).
Novak P. et al. // Physica B, 312-313, p. 785 (2002).
Blaha P. et al. // Comput. Phys. Commun. 59, p. 399 (1990).
Jeng H.-T. et al. // Phys. Rev. B, v. 65, p. 094429 (2002).
Verwey E.J.W. et al. // J. Chem. Phys., vol. 15, p. 174 (1947).
Verwey E. J.W. et al. // J. Chem. Phys., vol. 15, p. 181 (1947).
Hamilton W.C. // Phys. Rev., vol. 110, number 5, p. 1050 (1958).
248
Table of Content
A. Chemistry....................................................................................................... 5
Solid-contact ion-selective electrodes with ion-to-electron transducer layer
composed of nanostructured materials
Ivanova Nataliya.................................................................................................. 6
Using of semiconductor oxide films for detection of volatile organic compounds
in gases
Lopatnikov Artem............................................................................................... 11
Digital spectrographic analysis of human biological fluids for determination of
microelements
Savinov Sergey................................................................................................... 15
Synthesis of condensed imidazole derivatives with a Nodal nitrogen atom pyrido[1,2-a]benzimidazoles
Sokolov Alexandr............................................................................................... 19
С. Mathematics and Mechanics...................................................................... 23
Algebraic approximation of global attractors of discrete dynamical systems
Malykh Artem..................................................................................................... 24
Taken’s time delay embedding theorem for dynamical systems on infinitedimensional manifolds
Popov Sergey...................................................................................................... 28
A two-phase problem arising from a microwave heating process in nonhomogeneous
material
Serkova Nadezhda.............................................................................................. 32
Lyapunov functions in upper Hausdorff dimension estimates of cocycle
attractors
Slepukhin Alexander.......................................................................................... 37
D. Solid State Physics....................................................................................... 43
Intercalation of Al as a method of formation of quasifreestanding graphene
Anna Popova, Alexander M. Shikin................................................................... 44
Calculation of Sound Speed in Artificial Opal
Andrey Uskov..................................................................................................... 49
250
Modification of spin and electronic structure of graphene by intercalation of Bi
Evgeny Zhizhin................................................................................................... 54
E. Applied Physics............................................................................................ 59
Usage Pocket Comsol for the Numerical Nonstationary Nonlocal Plasma
Modeling
Burkova Zoya..................................................................................................... 60
Factorization of charge formfactors for clusterized light nuclei in reactions e+16O
and e+12C
Danilenko Valeria.............................................................................................. 64
Study of interaction forces between constant magnet and high-temperature
superconductor
Marek Veronika.................................................................................................. 68
Usage of stereoscopic 3D-visualization technologies
Marek Veronika.................................................................................................. 73
Evaluation of the influence of readout cables in the CBM Silicon Tracking
System
Prokofyev Nikita................................................................................................. 77
Application of graph theory to modeling of the complex hydraulic systems
Strizhenko Olga.................................................................................................. 82
F. Optics and Spectroscopy............................................................................. 87
A modern implementation of Rozhdestvenski interferometer
Agishev1 N.A., Medvedeva2 T.A., Ryabchikov1 E.L............................................ 88
Investigation of the two-photon induced fluorescence in Rb vapor excited by
Ti:Sapphire femtosecond laser pulses
Bondarchik Julia................................................................................................ 92
The research of optical spectra of oil fraction in IR-area
Chernova Ekaterina........................................................................................... 97
Luminescence spectra of YVO4 and Y2O3 nanopowders
Kolesnikov Ilya................................................................................................. 102
251
Resonance grating based on InGaAs/GaAs quantum well
Kozhaev Mikhail, Kapitonov Yury................................................................... 106
Application of 2D-correlation spectroscopy method for interpretation of spectra
and enhancing the spectral resolution
Maximova Ekaterina, Lev Derzhavets............................................................. 110
Observation of the fine structure for rovibronic spectral lines in visible part of
emission spectra of D2
Umrikhin I.S., Zhukov A.S................................................................................ 114
G. Theoretical, Mathematical and Computational Physics....................... 119
Conservation laws and energy-momentum tensor in Lorentz-Fock space
Angsachon Tosaporn........................................................................................ 120
Renormalization-group and ε- expansion: representation of anomalous dimensions
as nonsingular integrals
Batalov Lev...................................................................................................... 124
Surface states in semi-infinite superlattice with rough boundary
Bylev Alexander............................................................................................... 127
Modeling of thermal-hydraulic processes in complex domains by conservative
immersed boundary method
Chepilko Stepan............................................................................................... 132
Development of functional integration techniques for drift-diffusion processes on
Riemannian manifolds
Chepilko Stepan............................................................................................... 137
Multifractal generalization of the detrending moving average approach to time
series analysis
Ganin Denis..................................................................................................... 142
Propagation of photons and massive vector mesons between a parity breaking
medium and vacuum
Kolevatov Sergey.............................................................................................. 146
Analytical solution of two-dimensional Scarf II model by means of SUSY
Krupitskaya Ekaterina..................................................................................... 151
252
Effects of turbulent mixing on critical behaviour: Renormalization group analysis
of the ATP model
Malyshev Aleksei.............................................................................................. 156
Inertial-range behaviour of a passive scalar field in a random shear flow:
Renormalization group analysis of a simple model
Malyshev Aleksei.............................................................................................. 161
Effects of Stefan’s flow and concentration-dependent diffusivity in binary
condensation
Martyukova Darya........................................................................................... 166
A matrix approach for dyadic Green's function in multilayered elastic media
Nikitina Margarita........................................................................................... 170
Detectable effects in classical supergravity
Niyazov Ramil.................................................................................................. 175
Calculation of characteristics of critical behavior in logarithmic dimensions
Artem Pismenskiy............................................................................................. 180
Deal.ii library as a tool to study three-body quantum systems
Shmeleva Yulia................................................................................................. 184
Second order effects in the hyperfine and Zeeman splittings in highly charged
ions
Mikhail M. Sokolov.......................................................................................... 188
Hamiltonian Mechanics in Spaces of Constant Negative Curvature
Stepanov Vasiliy............................................................................................... 193
3D isotropic random walks with exponential distribution on free paths. Application
to evaporation of a droplet at transient Knudsen numbers
Telyatnik Rodion.............................................................................................. 197
Investigation of the dependence of the number of binary interactions and the
number of participants on the class of centrality in ultrarelativistic heavy ion
collisions
Vorobyev Ivan................................................................................................... 203
253
H. Biophysics.................................................................................................. 209
Application of Surface Plasmon Resonance for Detection of DNA Immobilization
on Gold Surface
Fironov Alexander........................................................................................... 210
DNA Interaction with Palladium Compound K2[PdHGluCl2] in vitro
Kozhenkov Pavel.............................................................................................. 215
Studing of the UV radiation influence on the DNA in a solution in the presence
of caffeine
Platonov Denis................................................................................................. 220
Entropic sampling of thermodynamic and structural properties of polymer chains
and stars within Wang-Landau algorithm
Silantyeva Irina................................................................................................ 225
Silver nanoparticles and their interaction with polymers in solution and on a
surfaces
Varshavskii Mikhail.......................................................................................... 230
I. Resonance Phenomena in Condensed Matter.......................................... 235
Phase transitions in magnesium: ab initio study
Klyukin Konstantin.......................................................................................... 236
NMR study of spin relaxation
Nefedov Denis.................................................................................................. 241
Magnetic properties of cubic magnetite Fe3O4: a density functional theory study
Irina Shikhma................................................................................................... 245
254