CONFERENCE PROCEEDINGS
International Student Conference
“Science and Progress”
German-Russian
Interdisciplinary
Science Center
St. Petersburg – Peterhof
November, 12-16
2012
Organizing committee
Prof. Dr. S.F. Bureiko, Prof. Dr. A.M. Shikin, E.I. Spirin,
E.V. Serova, T.A. Zalyalyutdinov, A.A. Rybkina, Dean of Faculty of Physics, SPbSU
Coordinator of G-RISC, SPbSU
Dean-assistant of Faculty of Physics, SPbSU
Head of Academic Mobility Department, SPbSU
G-RISC office, SPbSU
G-RISC office, SPbSU
Program Committee
Prof. Dr. E. Rühl, Prof. Dr. C. Laubschat, Prof. Dr. A.M. Shikin, Prof. Dr. V.N. Troyan, Coordinator of G-RISC, FU Berlin
Faculty of Physics, TU Dresden
Coordinator of G-RISC, SPbSU
Faculty of Physics, SPbSU
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.g-risc.org
3
Heads of sections
A. Chemistry – Dr. A.A. Manschina, Faculty of Chemistry, SPbSU
B. Geo- and Astrophysics – Prof. Dr. V.N. Troyan, Faculty of Physics, SPbSU,
C. Mathematics and Mechanics
– Prof. Dr. V. Reitmann,
Faculty of Mathematics and Mechanics, SPbSU
D. Solid State Physics – Prof. Dr. A.P. Baraban,
Faculty of Physics, SPbSU,
E. Applied Physics
– Prof. Dr. A.S. Chirtsov,
Faculty of Physics, SPbSU,
F. Optics and Spectroscopy
– Prof. Dr. Yu.V. Chizhov,
Prof. Dr. N.A. Timofeev,
Faculty of Physics, SPbSU,
G. Theoretical, Mathematical and Computational Physics
– Prof. Dr. Yu.M. Pis’mak,
Faculty of Physics, SPbSU,
H. Biophysics
– Prof. Dr. N.V. Tsvetkov,
Faculty of Physics, SPbSU,
I. Resonance Phenomena in Condenced Matter
– Prof. Dr. V.I. Chizhik,
Faculty of Physics, SPbSU,
4
A. Chemistry
Quantum-chemical investigation of 2,1-benzisoxasoles
formation by aromatic nucleophilic hydrogen
substitution process
Adreeva K.V., Tsivov A.V., Orlov V.Yu.
Ksuka2009@rambler.ru
Scientific supervisor: Prof. Orlov V.Yu., Department of Organic
and Biologic Chemistry, Faculty of Biology and Ecology, Yaroslavl
Demidov State University
Reactions of aromatic nucleophilic hydrogen substitution (SNArH) is one of
perspective methods for nitroaromatic compounds functionalization. The arylacetonitriles with nitroarenes interaction processes is one of the examples of
nucleophilic hydrogen substitution reaction.
The two reaction ways are possible:
1) interaction between arylacetonitriles and para-substituted nitroarenes
(Scheme 1, Х≠Н) leads to 5-X-3-phenyl-2,1-benzisoxazole formations [1];
2) interaction between arylacetonitriles and nitroarenes without substitute in
para-position (Scheme 1, Х=Н) leads to formation of phenylacetonitriles as a
final product [2].
Investigated reaction is carrying out with solvent (alcohol) and alkali
environment.
H-OR
CN
CH2
Ar
NO2
X
1
-
O O
N+
-
X=H
H2O OHX
(A)
CN
+ CHAr
-
CN
CH Ar
H
2
-
CN
C Ar
H
X
(B)
H-OR
O ON+
H CH
Ar CN
3
O OH
N
RO-
X=H
RO-
H-OR
-
-
O
N
OR
Ar
CN
-H2O
+H2O
X
ROH
(C)
H-OR
C H
CN RO-
-
-H2O
+H2O
4
N O
A
r
-CN-
RO-
O OH
N
Ar
-
O
-
OR
X
(D)
HO
N
N
H+
C
Ar CN
OH-
C
Ar CN
ROH
Scheme 1.
1 X=Cl; 2 X=Br; 3 X=Ph; 4 X=OPh; 5 X=СН(ОСН2СН2О);
6 X=С(ОСН2СН2О)СН3;7 X=COOH; 8 X=C6H4CH(OCH2OCH2).
In this work the quantum-chemical modeling was carried out for cyclization
stage (2,1-benzisoxazolic cycle formation process) of arylacetonitriles with nitroarenes interaction.
6
The modeling was performed for various para-substituted nitrobenzenes (1-8),
and for various solvents [3] with para-substituted nitroclorinebenzene (1). Such
solvents as methanol, ethanol, propanol, iso-propanol, butanol were used during
calculation.
The quantum-chemical modeling was carried out with using a PC
GAMESS /Firefly software [4]. The
non-empirical (ab initio) method
of calculation (UHF, Unrestricted
Hartree-Fock approximation), and
6-31G/6-31G(d,p) basis functions
set were used. ChemCraft 1.6 [5]
was used for result visualization.
It was investigated, that cyclizaa
tion process performs during O- and
C- atoms (molecular reaction centers) oncoming, from 3.0 Å distance,
corresponds to (C) intermediate, to
1.3-1.5 Å, corresponds to final 2,1benzisoxazole (D) formation. This
process is accompanied by -CN
group separation (Fig. 1).
The PES (Potential energy surface) profiles of cyclization process
for «intermediate - final product»
modeling system was obtained, for
various substitutes in substratum
structure with methanol solvent
(Fig. 2), and for various solvent
b
consideration (Fig. 3) for para-nitroclhorinebenzene (X=1 substitute). Fig. 1. -CN group separation and 2,1On the X axis the C7-O1 inter- benzisoxazolic cycle formation (by C7atomic distance was shown. On the O1 distance 1,3 -1,5 Å) a) by non-solvent
Y axis the relative energy (∆Er) scale modeling; b) with solvent presence (two
was shown - value of minimal energy molecules of methanol).
for each substitute is a zero value.
Next, the correlation analysis between experimental and obtained modeling
data was performed. As experimental reaction properties, kef - effective constants
of process rate [6] were chosen, and Eact - calculated activation energies of cyclization stage were taken.
7
Fig. 2. PES profiles for various substitutes in substratum structure with methanol solvent.
Fig. 3. PES profiles for various solvents consideration with X=1 chlorine
substitute.
The linear correlation dependence between calculated quantum-chemical
parameters and experimental values of effective process rate constants kef was
obtained. These results are presented in Fig. 4 and Fig. 5.
8
Fig. 4. In kef - ∆Eact approximation and correlation parameters for various substitutes in substratum structure.
Fig. 5. In kef - ∆Eact approximation and correlation parameters for various solvents
with X=1 chlorine substitute.
Values of obtained correlation coefficient r: 0,974 for various para-substituted
nitrobenzenes modeling, and 0,857 for various solvents calculation.
Analysis of obtained results allows to make a definition, that 2,1-benzisoxasolic
cycle formation stage for arylacetonitriles with nitroarenes interaction process is
9
limiting stage, and this stage defines the specific characteristics of investigated
process.
Also, it is established, that solvents as iso-propanol and butanol are more effective, than the solvents with smaller molecular mass (ethanol or methanol).
References
1. Davis R.B., Pizzini L.C., Benigni J.D // J. Am. Chem. Soc. 1960. V. 82. N 11.
P. 2913-2914.
2. Davis R.B., Pizzini L.C. // J. Org. Chem. – 1960. – Vol. 25, N 11. – P. 18841888.
3. Reichardt Ch. Solvent and solvent effects in organic chemistry. Marburg,
2004.
4. Granovsky A.A. http://classic.chem.msu.su/gran/games/index.html
5. ChemCraft: http://www.chemcraftprog.com/
6. Sokovikov Ya.V. Nucleophilic hydrogen substitution with nitroarenes and
phenylacetonitriles carb-anion interaction // PhD diss., Yaroslavl, YarSU, 1998.
10
А Synthesis diaryl esters in the presence of oxides of iron
(III) as the promoting agent
Lyutkin A.S., Orlov V.Yu., Volkov E.M.
andrewstudent@rambler.ru
Scientific supervisor: Dr. Orlov V.Yu., Department of Organic and
Biological Chemistry, Demidov Yaroslavl State University
Synthesis of esters is usually carried out diaryl nitrohalogenbenzenes reaction
(or other activated substrates) with phenoxides anions. This reaction proceeds by
nucleophilic aromatic substitution mechanism (SNAr).
Fig. 1.
The process is carried out in two ways. In the first case, two-stage scheme is
implemented. Phenoxides previously prepared from the corresponding phenols
interaction with favoring protons agents, and then introduced into the reaction
system. In the second case, the synthesis of diaryl ethers performed, getting nucleophilic attacking particles in situ [1-5]. Usually in this case, carbonates of alkali
metals are favoring protons agents [4-8]. Among them one of the most effective
and at the same time available and relatively cheap is a potassium carbonate. It
should be noted that potassium carbonate is practically insoluble in aprotic dipolar
solvents, which is usually carried out the synthesis of diaryl ethers. The resulting
heterophase system has a significant effect on the rate and completeness of the
implementation of process of nucleophilic substitution [9-10]. This is due to the
fact that that the key stage of the process occurs at the interface. On this basis, to
intensify the fusion reaction diaryl ethers used a variety of approaches. In the first,
it are the physical methods - increase the intensity of mixing, varying the particle
size of the solid phase. There are examples of various catalytic systems. The authors
studied the effect of iron complexes in the solid-phase nucleophilic substitution
reactions involving N-nucleophiles. At the same time very satisfactory results were
obtained on the yields of target compounds. It shows the influence of organometallic structures on the regularities of the process, nature of the activating reagent,
which consists in the interaction of iron with π-electron system of the aromatic
substrate. It can be assumed that the promoting effect may have iron and inorganic
compounds, including those in the solid phase. At the same time centers have an
activating effect. They allegedly associated with features (including defects) of the
surface structure. As a model for the study of nucleophilic aromatic substitution of
traditional nucleofuge (halogen) by O-nucleophiles was chosen as the reaction of
11
4-clorinenitrobenzene with phenolate, formed in situ by the interaction of phenol
with potassium carbonate.
Fig. 2.
As was mentioned reaction of p-nitrochlorinebenzene with phenol occurs in
the presence of the solid phase (potassium carbonate). Iron oxide (III) is used
as the promoting of solid additives. Comparison of the percolation model of the
process in the presence and absence of additives is shown in Fig. 3. Thus, effect
is observed of accelerating the reaction of p-nitrochlorinebenzene with phenol in
the presence of iron oxide (III).
Due to the fact, that the course of the process is largely determined by the localization of the reaction zone at the interface of reagents and products, becomes
essential genesis of solid samples, most important value is the genesis of solid
samples. Genesis defines the nature of the surface and, therefore, the localization
of active centers. Table 1 presents the characteristics and genesis of the samples
Fe2O3.
Table 1. Characteristics and genesis of the samples Fe2O3.
№ of the samples
Genesis
Characteristics
1
Fe2O3obtained from FeSO4
S = 4,5 m2/g
2
Fe2O3 obtained from Fe(OH)2
S = 20,3 m2/g
3
Fe2O3 obtained from salt Mohr
S = 10,5 m2/g
4
Fe2O3 obtained from FeСO3
S = 7,8 m2/g
The experiments using different samples of hematite in the reaction system of
p-nitrochlorinebenzene / phenol / potassium carbonate / DMF showed that the addition of this solid-phase component of the accelerating flow process. The greatest
influence on the process under study has iron oxide (III), prepared from ferrous
sulfate and passed additional mechanical treatment (Fig. 3).
12
Fig. 3. Dependence of the yield 4-nitrodiaryl ethers from time to time the reaction
by adding samples of hematite with different genesis (molar ratio of reagents 4-chlo
rinenitrobenzene: phenol: K2CO3 : Fe2O3 = 1 : 1 : 1,2 : 0,015, Т = 125-130 oC).
Similar results were obtained for the interaction of 4- nitrochlorinebenzene
with 4-nitrochlorinebenzene, 4-nitrophenolate, and p-cresol (Fig. 4). Reaction
with O-nucleophiles data is much more intense than when using phenol as the
use of additives Fe2O3 and without them. The presence of substituents slightly
reduces the effect of adding samples of hematite in the reaction system, but the
trend remains the promoting action.
Fig. 4. The dependence of the reaction product from the use of O-nucleophile, and
from reaction time.
13
A preliminary interpretation of this effect could be next. Iron oxide performs
supporting role. It comes as a promoter action of potassium carbonate. This effect
may be to ionizing effects on the crystal lattice of potassium carbonate in places
of contact phases or by antidiffusion potassium and iron ions in the surface layers
of the lattice of hematite and potassium carbonate, respectively. Presumably, that
this effect leads to a weakening of ties of K-O and Fe-O in the crystalline phases,
which does not reducible to date in the literature data on similar systems.
References
1. Whito D.M., Takekoshi T., Williams F.J. // J. Polym. Sci. 1981. V.19. N.7.
P.1635.
2. Radlmann E., Schmidt W., Nischk G.E. // J. Makromol. Chem. 1969. V.130,
P.45.
3. Heath D.R., Takekoshi T. Patent № 3879428 USA. 1976.
4. Heath D.R., Wirth J.G. Patent № 3869499 USA. 1975.
5. Heath D.R., Wirth J.G. Patent № 3763210 USA. 1974.
6. Williams F.J. Patent № 4017511 USA. 1978.
7. Relles H.M., Johnson D.S. Patent № 4054577 USA. 1978.
8. Johnson D.S., Relles H.M. Patent № 4020069 USA. 1978.
9. Kaninskii P.S., Abramov I.G., Yasinskii O.A. // J. Org. Chem. 1992. 28.
P.1232.
10.Milto V.I., Orlov V.Yu., Sokolov A.V., Mironov G.S. Izv. Vyssh. Uchebn. Zaved.,
Khim. Khim. Tekhnol. 2005. V. 48. №. 1. P. 95.
14
A study into thermodynamics and structure of smeared
charges fluids: the hypernetted-chain closure of the fluid
state theory
Nikolaeva Alexandra
alexandra.l.nikolaeva@gmail.com
Scientific supervisor: Assoc. Prof. Dr. Vlasov A.Yu., Department
of Physical Chemistry, Faculty of Chemistry, Saint-Petersburg
State University
Introduction
Some conceivable applications of systems composed of big charged particles
immersed in a neutralizing “sea” of smaller particles of a solvent are related to
possible development of phases having charge waves. Examples include i.a., colloidal suspensions, water solutions of polymers and poly-electrolytes, mixtures of
ionized isotopes in the stars and so-called dusty plasmas [1-4]. Theory and computer
simulation predict, that un-damped periodic oscillations of density may evolve in
a macro-phase of such systems, the phenomenon heralding an appearance of the
so-called “cluster” or “striped” phases (Fig. 1) [5];
a
b
Fig. 1. “Striped”(a) and “cluster” (b) structures at ρ=0,4 T=0,5 and ρ=0,1 T=0,32,
correspondingly [5].
one may speak about the micro-phase separation in the system. Theoretical analysis
of fluid structure is based on investigation into the behavior of the total pair correlation functions (TPCF) [6]. As soon as periodic variations of density develop
the asymptotic behavior of the TPCF crosses over from monotonous decay to the
exponentially damped oscillations. A change of the behavior signals a presence of
a crossover point on a phase diagram. Varying the conditions for a given system
gives rise to a crossover line. This line limits instability with respect to a transition to a modulated phase. There are two types of such instability. The first one
represents an appearance of clusters. Then, a value of a TPCF oscillation period is
noticeably greater than the size of particles and clusters are composed of some tens
or even hundreds of particles. In this case, the crossover line is termed a Kirkwood
15
line. If the period of oscillations equals a particle diameter, then instability with
respect to a crystalline lattice arises. Its boundary in the state phase is designated
as a Fisher-Widom line.
In the present work one-, two- and three-component fluids where particles
interact through modified Coulomb potential with addition of soft repulsion are
considered. To calculate thermodynamic properties in the aforesaid systems and
determine the asymptotic behavior of the TPCF in a wide range of thermodynamic
conditions we have used the hypernetted-chain closure of the Ornstein-Zernike
equation.
Results and discussions
Modified Coulomb potential with addition of soft repulsion describes the
spherically-symmetric interaction between big particles in the fluid:
vij (r )
l
r
 r  Aij
= (qi q j ) B erf   +
(1 − ) 2 H (rC − r )
(1)
kT
r
rC
 2σ  2
x
where
 t2 
2
 r 
erf
=
exp
 
∫ − 2  dt π0
 2σ 
is an error function; r is the inter-particle separation; qi are particle charges; lB is
Bjerrum length; σ is diameter of a molecule (molecular core); Aij stands for an
amplitude of repulsion; rC is width of repulsion zone; H(x) - Heaviside function.
The first term helps to exclude a “conglutination” of charges by their smearing.
It is seen from Fig. 2 that this modified part provides non-singular behavior of a
given potential at the point of zero separation giving an opportunity to use it in the
meso-scopic simulation which cannot be carried out for point charges. A plausibility
of particles interpenetration is taken into account by the second term. The “sea” of
smaller particles is considered to be a structureless medium and its individuality
is contained in the Bjerrum parameter. Thus using this potential one can model
the solution of a polyelectrolyte, of two oppositely charged poly-electrolytes or of
poly-electrolytes and a polymer in a solvent.
A comparison of the aforesaid potential with the simple Coulomb potential and
modified one with parameters
lB / σ = 10, rC = σ = 1, Aij σ = 25
is given in Fig.2
Having solved the Ornstein-Zernike equation
 
 
hν µ (r ) = cν µ (r ) + ρ∑ xλ ∫ cν λ ( r − r ′ )hλ µ (r ′ )dr ′ (2)
λ
(cij is direct correlation function, ρ is the density of the fluid, xλ is molar fraction
of component λ) within the hypernetted-chain closure:
g νµ (r ) = exp −vνµ (r ) / kT ⋅ exp hνµ (r ) − cνµ (r ) ; (3)
g ν µ = hν µ + 1
we obtained a full set of functions hij. Knowledge of these functions allows to
find the thermodynamics of a fluid by integrating of radial distribution function
gνμ [7].
16
{
} {
}
Fig. 2. A comparison of simple Coulomb potential (diamonds) with modified one
and modified Coulomb potential with addition of soft repulsion (solid lines).
Investigation of the TPCFs behavior helps to predict the structure of the fluid
under definite thermodynamic conditions. Fig. 3 illustrates the dependence of hij
upon inter-particle separation at various Bjerrum lengths (0.1; 1.0; 10.0) which
are proportional to inverse temperature. Data on two- and three-component systems are presented. It is obvious that at high temperatures (small values of lB) an
exponential decay of the TPCFs has monotonous mode, the latter being changed
to damped oscillatory one as soon as temperature drops. This transition is termed
a crossover. The period of oscillations is also not constant. As seen from this figure
at middle temperatures its value equals to a few particle diameters, while at low
temperatures it becomes smaller and attains a particle diameter. The first case corresponds to formation of cluster phases and is described by Kirkwood line at phase
diagram. The second one relates to an appearance of somewhat like crystalline
lattice where big particles are located in its sites. Then, Fisher-Widom line takes
place. Having looked at Fig. 3 one should expect an appearance of Kirkwood line
since the period of oscillations is greater than the diameter of one particle but a
location of crossover point (exact value lB) at a chosen section of phase diagram
should be determined.
A nature of the crossover and a locus of crossover points can be determined
by both analysis of complex poles of Fourier component of the TPCFs [3] and
empirical relations between period of oscillations and Bjerrum length suggested
by R. Evans and P.B. Warren [2-4] for Fisher-Widom line and Kirkwood line cor1/ 2
respondingly
 const 
λ
=
(5.a)
 (l − l FW )  B
B


const
λ=
Kirkwood 
l
−
l
(
)
 B B

17
1/3
(5.b)
a
In our work we used only the second
method, i.e. we have carried out scanning of
the mode of oscillations along the section of
the state space at fixed density and composition of the system for various temperatures
(scanning the Bjerrum lengths). Then,
we have fitted obtained data by aforesaid
relations. Dependence of period of oscillations of hij vs. Bjerrum length described by
equation (5.b) for three-component fluid is
shown in Fig. 4. This equation proved to be
the best one under given conditions.
b
Value of lB Kirkwood in Fig. 4 is Bjerrum
length corresponding to a crossover point
in a definite section of the state space. Thus,
Kirkwood line can be obtained by varying
density and composition and calculating lB
at each point of state space. This line signals
the development of micro-inhomogenieties
represented by clusters which consist of
particles having identical charge.
It is worth mentioning that the system
c is thermodynamically stable with respect to
spinodal de-composition what is confirmed
by positive-defined main cofactors of isodynamic matrix of Helmholtz energy
Fig. 3. The behavior of the total
pair correlation functions for twoand three-component fluids with
following parameters l B /σ=10,
r C=σ=1, A ij/σ=25,|qi|=1, total molar fraction of charged particles is
0.03, density of big particles is 0.3
(in three-component fluid including
neutral particles).
procedures described.
Conclusion
Thermodynamic properties of one-, twoand three-component fluids composed of
inter-penetrable particles bearing smeared
identical or opposite charges are investigated. The crossover in the behavior of the total
pair correlation functions is established.
Its nature determined by phenomenological relations is found to be of a Kirkwood
type. One could expect that in real systems
with potentials close to a considered above,
evolvement of self-organizing pre-cursor
matrices may be procured for certain thermodynamic conditions, definable through
18
Fig. 4. Correlation between the period of oscillations of the total pair correlation
functions and Bjerrum length in three-component fluid. rC=σ=1, Aij/σ=25,|qi|=1,
total molar fraction of charged particles is 0.03, density of big particles is 0.3
(in three-component fluid including neutral particles).
References
1. Hopkins P., Archer A.J., Evans R. // J. Chem. Phys. 124, p. 054503 (2006).
2. Archer A.J., Pini D., Evans R., Reatto L. // J. Chem. Phys. 126, p. 014104
(2007).
3. Hopkins P., Archer A.J., Evans R.// J. Chem. Phys. 124, p. 054503 (2006).
4. Archer A.J., Evans R. // Phys. Rev. E 64, p. 041501 (2001).
5. Imperio A., Reatto L. // J. Phys.: Condens. Matter 16, pp. 3769–3789 (2004).
6. Fisher M.E., Widom B. // J. Chem. Phys. 50, 3756 (1969).
7. Hansen J.-P., McDonald I.R. Theory of Simple Liquids. - Academic Press,
Amsterdam, 2006.
19
B.Geo- and Astrophysics
Ionosphere profile reconstruction with oblique sounding
ionogram
Muldashev Iskander
iskander.muldashev@gmail.com
Scientific supervisor: Losseva T., Institute of Geosphere Dynamics
Russian Academy of Science
Introduction
Investigation and analysis of the ionosphere state are very complicated problems
that cannot have any absolute solution because of various ionospheric perturbations. Nowadays the most popular tool for receiving ionosphere profile is IRI
(International Reference Ionosphere). IRI is the system that had been developed in
respect to Radio propagation via the ionosphere. This system is based on a plenty
of stations of vertical sounding all over the world.
Results and Discussion
An Appleton-Hartree equation is a mathematical expression that describes
the complex refractive index for electromagnetic wave propagation in a cold
magnetized plasma. (1)
This equation is used perfectly for frequency from 5 to 20 MHz. Equation
includes ratios of plasma frequency, gyro frequency, electron collision frequency
and wave frequency.
(2)
(3)
(4)
Where ω0 is plasma frequency, ωH is gyro frequency, ν is electron collision
frequency and ω is plasma frequency.
Using refractive index the Hamiltonian for ray can be written as
(5)
Having Hamiltonian (5) one can consider differential equation system
(6)
(7)
22
(8)
(10)
(9)
(11)
(12)
(13)
This system can be solved with Runge Kutta method.
Now the solution of this system depends on the values of magnetic field, collision frequency and electron density.
The program uses Earth’s centered dipole for imitation of Earth’s magnetic
field.
The electron density profile may be calculated from
(14)
where
(15)
and
(16)
Where NE (15) is a relative concentration of E-layer, NF (16) is a relative concentration of F-layer, zm is height of maximal concentration for given layer and z0
is half height of given layer. Having this model we can reconstruct any condition
of anisotropic ionosphere.
Fig. 1. Modulated propagation of 13 MHz in one-layer ionosphere with perturbations.
23
Fig. 2. Propagation of 10 MHz rays in two-layers ionosphere.
As we can from Fig. 2 there are not only one radio trace for given frequency
between injector and receiver. And it is clear to see that having ray tracing program
it is easy to make algorithm for searching trajectories between stations.
Using this algorithm one can write another program for reconstructing ionosphere profile. Reconstructing of ionosphere profile is very complex problem and
it can have some solutions because of perturbations. That’s why sometimes it is
better to reconstruct in manual mode.
Fig. 3. Reconstruction of ionosphere profile by means of Sodanklya – Mikhnevo recordings.
Black line is ionogram of reconstructed ionosphere.
24
Conclusion
There has developed a three-dimensional high frequency ray tracing program
for numerical modeling in approximation of geometrical optics. This code has
several ionospheric models for electron density, perturbations of electronic density,
the Earth’s magnetic field and electron collision frequency. Refractive index is
represented by Appleton-Hartree formula.
The program implements the solution of the direct problem of oblique sounding
and using algorithm of finding ray trace between two stations (for given ionospheric
profile and frequency of transmitter) can solve the problem of reconstructing the
ionosphere profile for either one or many hops ray trace. In contrast to IRI this
approach helps to determine ionosphere perturbations in radio traces.
References
1. Maeda K., Kimura I. A Theoretical Investigation on the Propagation Path of
the Whistling Atmospherics, Rept. Ionosphere Res., 1956. p. 105-123.
2. Jones R.M., Stephenson J.J. A versatile three dimensional ray tracing computer program for radio waves in the ionosphere. OT Report 75-76. PB2488567,
1975.
3. Davidson D.G. Plasma Waves. 2nd ed. 2003. IOP. - 458 p.
25
С. Mathematics and Mechanics
Consensus in Stochastic Systems with Uncertainties in
Measurements with Simulation in JADE
Amelina Natalia
natalia_amelina@mail.ru
Faculty of Mathematics and Mechanics, Saint-Petersburg State
University
Department of Telematics, Norwegian University of Science and
Technology
Distributed coordination in networks of dynamic agents has attracted an interest numerous researchers in recent years. It is mostly due to broad applications
of multi-agent systems in many areas including formation control [1], flocking,
distributed sensor networks, congestion control in communication networks,
cooperative control of unmanned air vehicles (UAVs) [2], attitude alignment of
clusters of satellites, and others [3]. Many of these problems can be reformulated
in terms of achieving consensus in multi-agent systems [4-6].
In [7] the stochastic approximation algorithm for solving consensus problem
was proposed and justified for the group of cooperating agents that communicate
with imperfect information in discrete time, under condition of switching topology and delay. Stochastic gradient algorithms were used for such problems before
[8-10]. Stochastic approximation with decreasing step-sizes allows each agent both
to extract state information from its neighbors and to reduce the noise influence.
Under dynamic changes of the external conditions (getting new task, etc.),
stochastic approximation algorithms with decreasing step-size are not efficient.
In [11, 12] the efficiency of stochastic approximation algorithms with constant
step-size was studied. Their applicability to the problem of load balancing in
centralized network system where noisy information about load and productivity
of nodes was analyzed in [13, 14].
Analyzing of discrete stochastic systems may be complicated in practical applications. On the one hand, it is because of imperfect information exchange, which
is, moreover, usually measured with noise. On the other hand, it is due to the effects
of quantization effect common to all digital systems. Additional complication may
be due to switching topology of networks.
To analyze the dynamics of the stochastic discrete systems the method of continuous models (ODE approach or Derevitskii-Fradkov-Ljung (DFL)-scheme) was
developed and used in the control theory, dynamical systems theory and nonlinear
mechanics. This method was described in [15-18]. In [19-22] the method of continuous models was used for consensus problem in stochastic networks.
These problems show the relevance of study in the properties of stochastic approximation type algorithms with small constant or not decreasing to zero step-size
in the nonlinear formulation of the problem with switched topology and noise.
28
In this work the new simulation results for stochastic network of large number
of agents with switched topology and noise are presented. The local voting protocol
with not decreasing to zero step size is used. The software framework JADE (Java
Agent DEvelopment Framework) is used for simulation.
Problem Statement
We consider a system that separates the same type of jobs among different
agents, for parallel computing or production with feedback. Denote N={1,...,n}
as a set of intelligent agents, each of which serves the incoming requests using a
first-in-first-out queue.
At any time t, the state of agent i, i = 1,...,n is described by two characteristics:
• qti is a queue length of the atomic elementary jobs of the agent i at time t;
• pti is a productivity of the agent i at time t.
The dynamics of each agent are described by
qti+1 = qti − pti + zti + uti ; i ∈ N , t
= 0,1,  , T , (1)
where zti is the new job received by agent i at time t, uti is the result of information
redistribution between agents, which is obtained by using the selected protocol of
information redistribution. For redistribution we’ll use the local voting protocol.
For the considered case, the dynamics of closed loop system with local voting
protocol is as follows:
xti+1 = xti − 1 + zti /pti + α t
bti , j ( yti , j /ptj − yti ,i /pti ), (2)
∑
j∈ N ti
where α t are step sizes of control protocol, yti , j are noisy observation about j -th
agents queue length.
Model Implementation
The software framework JADE will be used for simulation. The development
of JADE agents is similar to the development of JAVA-applications. To implement
the algorithm the traditional integrated development environment JAVA Eclipse
will be used.
The system consists of three modules:
1. Loading and configuration module.
2. Simulation module.
3. Visualization module.
We set the initial parameters of each agents and the number of agents. Since
we consider the problem statement with switched topology, the adjacency matrix
consists of probabilities of edges.
The graphical tools of JADE (Remote Agent Management GUI, see Fig. 1) can
be used to monitor the activity of agents.
Simulation Results
The system provides two subclasses jade.core.Agent:
29
Fig. 1. JADE Remote Agent Management GUI.
1. JadeAgent.java – the agent that represents the node in the algorithm. Agent
parameters are:
• product – the productivity of the node
• data – initial state of the node (in practice, for example, the size of the job)
• delay – possible integer delay
• neighAgents – list of neighbors of the node
• probabilityNeigh – list the probability of having an edge with neighboring nodes
• alpha – constant step-size
• sigma – noise variance
• time – maximum implementation time of the agent
2. PlotterAgent.java – agent that collects information about nodes (JadeAgent.
java). Also it is used to interact with the user.
The implementation of the algorithm on a large number of nodes is shown in
Fig. 2. The input data was generated randomly.
30
Fig. 2. The convergence of 50 agents
to a balanced value (step-size=0,03;
variance of the
noise=10).
This work was carried out during the tenure of an ERCIM “Alain Bensoussan”
Fellowship Programme. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013)
under grant agreement No. 246016. Also author would like to thank the SPRINT
laboratory of SPbSU and Intel Corp. for supporting this work.
References
1. Fax A., Murray R.M. // IEEE Trans. Automat. Contr. Sept. 2004. Vol. 49. P.
1465-1476.
2. Antal C., Granichin O., Levi S. // Proc. of the 49th IEEE Conf. on Decision and
Control (CDC-49). Dec. 15-17. 2010. Atlanta. USA. P. 3656-3661.
3. Amelina N.O. // Stochastic Optimization in Informatics, vol. 7, 2011, PP. 149185.
4. Jadbabaie A., Lin J., Morse A.S. // IEEE Trans. Automat. Contr. June 2003.
Vol. 48. P. 988-1000.
5. Olfati-Saber R., Murray R.M. // IEEE Trans. Automatic Control. Sept. 2004.
Vol. 49. P. 1520-1533.
6. Ren W., Beard R.W. // IEEE Trans. Automat. Control. 2005. Vol. 50. No. 5. P.
655-661.
7. Huang M. // Proc. of IEEE Conf. on Decision and Control (CDC-49). 2010.
Atlanta. USA. P. 7449-7454.
8. Tsitsiklis J.N., Bertsekas D.P., Athans M. // IEEE Trans. Autom. Contr. 1986.
Vol. 31. No. 9. P. 803-812.
9. Kar S., Moura J.M.F. // IEEE Trans. Sig. Process. 2009. Vol. 57. No. 1. P. 355369.
10.Li T., Zhang J.-F. // Automatica. 2009. Vol. 45. No. 8. P. 1929-1936.
11.Vakhitov A.T., Granichin O.N., Gurevich L.S. // Automation and Remote
Control. v. 70. No. 11. P. 1827-1835. 2009.
12.Borkar V.S. Stochastic Approximation: a Dynamical Systems Viewpoint - New
York: Cambridge University Press. 2008. -164 p.
13.Granichin O.N. // Stochastic optimization in informatics. No. 6. P. 3-44.
2010.
14.Vakhitov A.T., Granichin O.N., Panshenskov M.A. // Neurocomputers: design,
application. No. 11. P. 45 - 52. 2009.
15.Derevitskii D.P., Fradkov A.L. // Automation and Remote Control. No 1. P.
59-67. 1974.
16.Ljung L. // IEEE Trans. Aut. Control. 1977. No. 4. P. 551-575.
17.Kushner H.J. // IEEE Trans. Aut. Control. 1977. No. 6. P. 921-930.
18.Derevitskii D.P., Fradkov A.L. Applied theory of discrete adaptive control
systems. -Moscow: Nauka (in Russian). 1981.
19.Amelina N.O. // Neurocomputers: design, application, No. 6, 2011, PP. 5663.
20.Amelina N.O., Fradkov A.L. // Stochastic Optimization in Informatics, Vol. 8,
2012, PP. 3-39.
31
21.Amelina N.O., Fradkov A.L. // Automation and Remote Control, 2012, Vol.
73, No. 11, PP. 1765–1783.
22.Amelina N.O. // Vestnik Sankt-Petersburgskogo University. Mathematics, Vol.
45, No. 2, 2012, PP. 56--60.
32
Existence of a global B-pullback attractor for a
periodically forced mechanical system
Maltseva Anastasia
maltseva.anastacia@gmail.com
Scientific supervisor: Prof. Dr. Reitmann V., Department of
Applied Cybernetics, Faculty of Mathematics and Mechanics,
Saint-Petersburg State University
1 Introduction
In this paper a class of periodically forced mechanical systems with angular
coordinates is considered [1, 2]. This class can be treated as non-autonomous
differential equations on the cylinder. The theory of cocycles [3] is used for the
mathematical description of such non-autonomous differential equations. The existence of the cocycle and its global B-attractor in the form of an invariant curve
is shown. The method of Poincare maps [4] is also used to investigate periodically
forced mechanical systems. Time-discrete dynamical systems are obtained as the
result of the application of the Poincare map's method. Similar to the cocycle approach in this case the existence of a global B-attractor in the form of an invariant
curve is shown.
2 Basic tools for cocycle theory
Let (Q, ρQ ) be a compact complete metric space. A base flow on the metric
t
t
space (Q, ρQ ) is a pair ({τ }t∈ , (Q, ρQ )) , where τ : Q → Q is a family of
mappings, satisfying:
0
1) τ (·) = id Q ;
t+s
t
s
2) τ (·) = τ (·)  τ (·) for all t , s ∈  ;
(·)
3) τ (·) :  × Q → Q is continuous.
Let ( M , ρ M ) be another compact complete metric space which we call the
t
phaset space. A cocycle on M over the base flow ({τ }t∈ , (Q, ρQ )) is a pair
({ϕ (q,·)}t∈ , , ( M , ρ M )) , satisfying :
q∈Q
0
1) ϕ ( q,·) = id M , for all q ∈ Q ;
t+s
t
s
s
2) ϕ ( q,·) = ϕ (τ ( q ), ϕ ( q,·)) , for all q ∈ Q, t , s ∈  ;
t
3) ϕ ( q,·) is continuous for all q ∈ Q and t ∈  .
t
In the sequel we shortly denote a cocycle ({ϕ ( q,·)}t∈ , , ( M , ρ M )) over
q∈Q
t
the base flow ({τ }t∈ , (Q, ρQ )) by (ϕ ,τ ) . The basics of the cocycle theory
one can find in [3].
For any Z1 , Z 2 ⊂ M define dist ( Z1 , Z 2 ) := sup inf ρ M ( p, q ) .
p∈Z1 q∈Z 2
A g l o b a l B - p u l l b a c k a t t r a c t i n g s e t f o r t h e c o c y c l e (ϕ ,τ )
i s a b o u n d e d n o n a u t o n o m o u s s e t Zˆ = {Z ( q )}q∈Q s a t i s f y i n g :
33
t →∞
dist (ϕ t (τ − t (q ), B), Z (q )) 
→ 0 for all q ∈ Q and any bounded set
B⊂M .
A global B - pullback attractor for the cocycle (ϕ ,τ ) is a compact invariant nonautonomous set Zˆ = {Z ( q )}q∈Q , where Ẑ is globally B - pullback
attracting.
3 Periodically forced mechanical systems
Consider the differential equation
σ + β (σ − 1) = g (σ )uT (t ), (1)
4
where β > 0 is a parameter, g :  →  is a C -smooth nonlinear 1-periodic
1
function (i.e. σ can be treated as σ ∈ S ), uT (·) is an external periodically
pulsed perturbation.
Equation (1) is equivalent to
σ = ϑ + 1,
(2)

1

ϑ = − βϑ + g (σ )uT (t ), (σ , ϑ ) ∈ S × , t ∈ .
Rewrite the system (2) in the following form:
u = Au + φ (t , u ), where
σ 
0 1 
 1 
u =  , A = 
 , φ (t , u ) =  
 , g (t , σ ) = g (σ )uT (t ).
ϑ 
 0 −β 
 g (t , σ ) 
4 Existence of the cocycle
({τ t }t∈ , (Q, ρQ )) be the base flow on the metric space (Q, ρQ ) ,
( M , ρ M ) be the phase space with M := S 1 ×  and the metric ρc defined by
2
ρc ( p, q ) := inf 2 x − y , where · is the  norm,
Let
x , y ∈ ,
[ x ]= p ,[ y ]= q
p = [ x] = { y ∈  2 y = me1 + x, m ∈ , e1 = (1, 0)}
Theorem 1. ([6]) System (2) generates a cocycle
on the metric space
is
({ϕ t (q,·)}t∈ , , ( M , ρc ))
q∈Q
M := S ×  over the base flow ({τ }τ ∈ , (Q, ρQ )). Here
t
1
Q :=  and the base flow is defined as τ t (q ) := q + t for all q ∈ Q .
The proof of the Theorem 1 and next theorems is presented in [6].
5 Existence of the global B-pullback attractor
Let us introduce the additional assumption:
(A1) There exists C0 > 0 such that ∥F (t , u )∥a C0 for all u ∈ M and for
all t ∈  + ;
Theorem 2. ([6]) Suppose that the assumption (A1) is satisfied, then the
t
1
cocycle ({ϕ ( q,·)}t∈ , , ( M , ρ c )) on the metric space M = S ×  over the
q∈Q
34
t
base flow ({τ }t∈ , Q ) has a global B-pullback attractor
which is given by
A(q ) =   ϕ s (τ − s (q ), Z (τ − s (q))), where
Aˆ = { A(q )}q∈Q
t ∈ + s ≥ t ,
s ∈ +
1
Z := { p = (σ, ϑ ) ∈ S 1 ×  | ϑ |≤ R}, R = C0 . β
Let us make following assumptions:
t
(A2) g ( q, σ ) is measurable in q ∈ Q and 1-periodic in σ; g (τ ( q ), σ )
is continuous in t and smooth in σ ; there exists C1 > 0 such that
| g (q, σ ) |,| gσ (q, σ ) |≤ C1 for any (q, σ ) ∈ Q ×  ;
(A3) β > 2 C1 .
*
1
Theorem 3. ([6]) Assume (A1)-(A3). Then there exists a curve h : Q × S → 
*
*
1
such that h ( q, σ ) is measurable in q and A( q ) = {(σ , h ( q, σ ) σ ∈ S } is
t
the global B- pullback attractor for the cocycle ({ϕ ( q,·)}t∈ , , ( M , ρ c )) over
t
q∈Q
the base flow ({τ }t∈ , (Q, ρQ )) .
6 Discrete dynamical systems
Consider again the differential equation
σ + β (σ − 1) = g (σ )uT (t ), σ ∈ S , t ∈ , (1)
which equivalent to the system (2).
Here T is the period of the external forcing: uT (t ) = uT (t + T ) for any t ∈ 
′ ′′
. Assume that there exist t0 , t0 , t0 ∈ (0, T ) such that
1
1, t ∈ [0, t0 ],
0, t ∈ (t , t ′ ),

0 0
uT (t ) := 
′ ′′
δ , t ∈ [t0 , t0 ],
0, t ∈ (t0′′ , T ),
where δ < 1 is a parameter.
Let us consider the Poincare’ map for the system (2)
ϑ (t0′′)

(1 − e− β (T −t0′′ ) ) 
 σ 0   σ (T )   σ (t0′′) + (T − t0′′) +
β
PT :    
,
=

 ϑ0   ϑ (T )   ϑ (t ′′)e − β (T −t0′′ )

 0

where σ 0 := σ (0), ϑ0 := ϑ (0) .
Recall the definition of the discrete dynamical system.
A discrete dynamical system on the metric space ( M , ρ M ) is a pair
({ϕ t }t∈ , ( M , ρ M )) , where  ∈ {,  + } is a set of time moments,
{ϕ t }t∈ : M → M is a family of mappings, satisfying the conditions:
35
0
1) ϕ (·) = id M ;
t+s
t
s
2) ϕ (·) = ϕ (·)  ϕ (·) for any t , s ∈  ;
t
3) ϕ (·) : M → M is continuous for any t ∈  .
According to the Poincare’ map consider the dynamical system
({PTn }n∈+ , ( M , ρc )), (3)
where the phase space M described in the section 4.
7 Existence of the global attractor for discrete dynamical systems
For ε > 0 and a set Z ⊂ M let us define the ε - neighborhood of the set
Z as following U ε ( Z ) := {q ∈ M dist (q, Z ) < ε}.
′
We say that a set Z ⊂ M attracts the set Z ⊂ M if for any ε > 0 there
′
exists a time n0 = n0 (ε , Z ) such that for any n ∈ [ n0 , ∞) ∩  + we have
PTn ( Z ′ ) ⊂ U ε ( Z ).
A global attractor for the dynamical system (3) is a compact invariant set
A ⊂ M , which attracts all points of M.
Let us make additional assumptions according to the equation (1):
4
(H1) There is a function g 0 :  →  which in C -smooth 1 –periodical and
g (σ ) =
such that
(H2)
g0
t0 <
1
g 0 (σ ) ;
t − t0′
′′
0
1
1 1
min{ , 2 } , where K 0 = max{ g 0
10
β K 0
C4
,1},
4
C
4
:= ∑ max1 | g 0 (i ) (σ ) | ;
i:= 0
σ ∈S
β ≥ 4K 0 ;
3
′′
(H4) T ≥ t0 + .
2
(H3)
Theorem 4. ([6]) Suppose that assumptions (H1) - (H4) are satisfied. Then there
4
*
1
*
1
exists a C -smooth curve h : S →  such that A = {(σ , h (σ ) σ ∈ S } is
n
the global attractor of the dynamical system ({PT }n∈+ , ( M , ρ c )) .
References
1. Wang Q. and Young L.-S. // Commun. Math. Phys., vol. 225, issue 2, pp. 275304 (2002).
2. Zaslavsky G.M. Physics of Chaos in Hamiltonian Systems – London: ICP,
1998 – 260p.
3. Kloeden P.E., Schmalfuss B. // Numerical Algorithms, vol. 14, № 1-3, pp.
141-152 (1997).Wakeman D.R. // J. Diff. Eqs., vol. 17, pp. 259-295 (1975).
4. Guckenheimer J., Holmes Ph. Nonlinear Oscillations, Dynamical Systems,
and Bifurcations of Vector Fields – Springer. Applied mathematical science, 1983
– vol.42, 484 p.
36
5. Shen W.D.// Discrete Contin. Dynamic. Systems, vol. 18, № 2-3, pp. 597611 (2007).
6. Maltseva A.A. Rotation Number for a Periodically Pulsed Forced Mechanical
System – Diploma thesis, Saint-Petersburg State University, 2012 - 32 p.
37
D. Solid State Physics
Factor analysis of Raman spectra of whisker GaAs with
sphalerite-wurtzite structure
Chirkov Evgenii
evgen.chirkoff@gmai.com
Scientific supervisor: Prof. Dr. Karpov S.V., Department of
Solid State Physics, Faculty of Physics, Saint-Petersburg State
University
Modern experimental technique allows the experimenter to get great amount of
data. However, most of data are redundant and interdependent. In such cases necessary perform reduction through the
provision of independent components
that carry information about studied
process.
One problem is the problem of
separating curves, which obtained from
superposition of single linearly independent contributions. In our case, with
this separation was a task - to research
a complex spectrum polytype crystal,
which would consist of lines that show
Fig. 1. NWs GaAs, grown on the sub- up and disappear in different modifications. For this have been selected
strateFig.1.
Si [1].NWs GaAs, grown
nanowires (NWs) GaAs [1].
on the substrate Si [1]
Bulk crystal semiconductor gallium
arsenide GaAs at normal pressure is
growing in a stable phase with a sphalerite structure (Zinc Blende - ZB). In
microscopic crystals of GaAs, which
are characteristic for quantum dots and
nanowires, structure of wurtzite (WZ)
can be stable. This in the spectrum of
Raman scattering would change the
number and arrangement of lines.
Fig. 2. The experimental Raman spectra
It has been suggested that the reof GaAs whiskers.
searched structure of the spectrum of
whiskers formed a certain set of linearly independent contributions of the spectra
of individual crystal structures, but the contribution to the total spectral function
can vary depending on the local place of excitation (Fig. 2).
For solving this problem is traditionally used factor analysis, which is based
on principal component analysis (PCA). In this analysis, the array investigating
spectra is collected to the matrix X, whose rows - the values of
​​ the spectra. Analysis
40
provides a number of linearly independent contributions (rank of matrix). From a
mathematical point of view, the method PCA - this decomposition X and presenting
it as the product of two matrix T and P
N
X = TP t + E = ∑ tn pnt + E
n =1
In this equation, T is the matrix of scores, P – matrix of loadings, and E - matrix
residues upper icon t denotes the transposed matrix, tn the number of columns in
the matrix T and pn in the matrix P is
the rank of X. This value is called the
number of principal components, and
of course it is less than the number of
rows in the matrix X.
To test the efficiency of the method
has been investigated a set of theoretical
spectra, consisting of the sum of two
Lorentzian curves with different ratios
of intensity. The relations between the
peaks in each of the Lorentzian spectrum varies linearly from 0 to 1, and
each curves were added random noise,
resulting in signal/noise ratio to the
value of 10/1 in the first case and 2/1
in the second. As a result, each of the Fig. 3. Spectra obtained from the funcspectra in the array of initial data has a tions of Lorentz.
Fig.4. Factors of PCA with a signal to Fig.5. Factors of PCA with a signal to
noise ratio of 10 to 1.
noise ratio of 2 to 1.
complex, not discernible structure to the eye (Fig. 3).
Figuring out the number of linearly independent contributions (rank of the matrix)
was done using the procedure of factor analysis in the package Statistica 10.0.
As a method of finding the factors we used the method of principal components,
the desired number of factors extracted more than two rotation method - varimax
that maximizes the variance. Processing results shown in Fig. 4 and 5, which
show that the analysis identified the presence of two components forming the fine
41
structure of a set of spectra. Their position is exactly the initially set the center of
the Lorentzians.
With the method of factor analysis was analyzed 55 Raman spectra of the sample
NWs in frequency range of 130-400 cm-1.
The problem is analyze the matrix 55x300.
Analysis of the experimental data by a factor
analysis has identified the existence of two
linearly independent contributions (factors)
that describe each of the 55 spectra. One of
the selected factors - F1 is very similar to
the Raman spectrum of the bulk crystal of
gallium arsenide in which there are only
Fig. 6. Raman spectra and calculated two vibrational modes LO - 291 cm-1 and
factors.
TO - 268 cm-1(Fig. 7).
Emergence of additional band in the spectrum at a frequency 286 cm-1, which
Fig. 7. The theoretical spectrum of the
Fig. 8. The theoretical spectrum of the
ZB and factor F1.
WZ and factor F2.
is a result of the factor analysis (F1) was isolated from the complex spectrum can
be interpreted as a surface mode that occurs in whiskers.
Another contribution F2 is similar to the theoretically calculated vibrational
spectrum of the wurtzite structure GaAs based on data of [2] (Fig. 8).
With respect to line of 256 cm-1 there is a lot of debate. In particular it is known
that the interaction GaAs and its oxide can occur excess As, leading to the formation of crystalline or amorphous arsenic, which lies close to the line 256 cm-1, and
its intensity can be released.
Factor analysis showed that using only mathematical statistics can "filter" the
large amount of data from unnecessary information. Just FA revealed the fine structure of spectral lines are interpreted as a manifestation of the wurtzite structure of
42
NWs. The very same coexistence of crystalline sphalerite and wurtzite structures in
samples wiskers GaAs has been experimentally confirmed by electron microscopic
studies, obtained in the literature.
References
1. Dubrovskii V.G. et al. // Phys.Rev. B 77, 035414 (2008).
2. Karpov S.V., Novikov B.V., Smirnov M.B., Davydov V.Yu., Smirnov A.N.,
Shtrom I.V., Cirlin G.E., Bouravleuv A.D., Samsonenko Yu.B. // Physics of the
Solid State, (2011), 53, 1431.
43
Study of electron spin-polarization relaxation in LiF
Nikita Kan
nikitakan@mail.ru
Scientific supervisors: Dr. Ustinov A.B., Department of Physical
Electronics, Radiophysics Faculty, Saint-Petersburg State
Polytechnic University; Prof. Petrov V.N., Department of
Experimental Physics, Physics and Mechanics Faculty, SaintPetersburg State Polytechnic University
Abstract
At the end of the last century the new direction appeared in microelectronics spintronics, which uses, as opposed to traditional electronic, magnetic properties of
particles. Nowadays, scientists are creating and researching the properties of simple
prototype devices. One important and open question lies in analysis of the properties
of the flow of spin-polarized electrons after they pass within solids. To determine
the rate of reduction of electron flow polarization after passing through a film of
lithium fluoride in the present study we used the following idea. Ferromagnetic
crystal was used as a source of polarized electrons. On its surface the depositions
of lithium fluoride films of different thicknesses were made. The rate of reduction
of polarization of the secondary electrons transmitted through the films was used
for determining mean-free path for spin polarization relaxation.
Equipment
This research was made inside ultra-high vacuum chamber, which was equipped
with electron and ion guns, LiF evaporation source, cylindrical mirror energy analyzer [1] and detector of spin polarization [2] . The substrate was a monocrystal
of FeNi3 (110). Atomically-clean surface for the experiment was provided by ion
bombardment and hightemperature annealing
of the substrate. Control
of substrate's surface
purity was produced
by Auger spectroscopy.
The magnetization of
the sample was performed by passing current through the adjacent coil.
Fig. 1. A Scheme of Mott’s detector.
44
LiF Source Calibration
Calibration was made using Auger spectroscopy. Series of evaporations at different value of evaporation source heater current (2 to 3.2 A) were made. Auger
peak’s intensities were fixed in the range from 5 to 700 eV. In figure under we
can see the intensity of Ni Auger peak (61eV) as a function of evaporation time.
Time when Auger peak intensity is reduced by e gives moment when the film's
thickness is about inelastic mean free path for auger peak energy. It is about t1Ni
≈ 25 min for heater current of 2.1 A basing on Ni auger peak. Data fo Li auger
peak gives close t1Li≈ 28. Hence speed of LiF deposition is 5 Angstroms per 10
minutes for I=2.1 A.
Fig. 2. Auger spectrums for calibration.
Fig. 3. Reduction of Auger peak of Ni relatively time of evaporating.
45
Measurements
In the next stage we again evaporated the LiF and fixed the hysteresis loops
by Spin-Polarized Secondary-Electron Emission (SPSEE) method for different
fixed energies of secondary electrons. FeNi3 sample was magnetized along the
easy magnetization axis [111]. During measurements the sample was irradiated with
unpolarized electron beam. Experimental
equipment allowed to get energy N(E) and
polarization P(E) spectra of the secondary
electrons, also polarization as a function
of the sample’s magnetization current P(I)
could be measured. This latter mode of
operation was used to get the results of
this work.
The analyzer's energy of the transmission had been fixed at a certain value, and
then secondary electron polarization has
been recorded as a function of sample
magnetization current P(I) that had the
form of a ferromagnetic hysteresis loops.
On figure we can see the example of such
loop for energy E=35 . Deposition of LiF
films on the ferromagnetic surface led to
reduction of the amplitude of P(I). And
its rate determined the value of mean-free
path for spin polarization relaxation of
electrons in film.
Results
In Fig. 4 it is seen the polarization of
secondary electrons (sq) for energy of 35
eV as a function of evaporation time. We
used data from hysteresis loops with very
good statistics. There are two exponents.
Left one corresponds mainly with the
Fig. 4. Reduction of Auger peak of Ni rela- decreasing of intensity of electrons from
tively time of evaporating.
46
magnetic substrate. Right one gives us the real mean free path for spin relaxation.
For right exponent we have the t1 ≈ 130 min. In this case - the mean free path for
the spin relaxation is more than inelastic mean free paths (for energy ~ 35 eV) in
~5 times (130/25=5.2).
Total
Recording of hysteresis curves was made for the five energies in the range of
35 - 550 eV. The length of inelastic mean-free path λne is 1=1,2 nm in this energy
range, and that determines the decrease of intensity of the secondary electrons.
Such figures for reduction of polarization of the secondary electrons in the films of
lithium fluoride can be explained by a simple electronic structure and the absence
of d-electrons. This decreases quantity of spin-relaxation processes in output of
electrons through a thin film. As a total, the value of mean-free path relatively spin
relaxation of electrons λspin-relaxation for films of lithium fluoride exceeds 3-9 times
λnon-elastic and is respectively 3,10 nm.
Acknowledgements
We gratefully acknowledge RFBR for supporting this work with grant number
11-02-01092-a.
References
1. Petrov V.N., Kamochkin A.S. // Rev. Sci. Instrum., v. 75, № 5, p. 1274
(2004).
2. Petrov V.N., Grebenshikov V.V., Andronov A.N., Gabdullin P.G., Maslevtcov A.V.
// Rev. Sci. Instrum., v. 78, №2, p. 025102 (2007).
3. Ullah S., Dogar A. H., Ashraf M. // Chin. Phys. B, 2010, No. 8, Vol. 19.
47
Spin polarization of interface states in thin films of Bi on
Ag/W(110)
I.I. Klimovskikh, M.V. Rusinova, E. Zhizhin, A.A. Rybkina,
A.G. Rybkin and A.M. Shikin
Klimovskih_ilya@mail.ru
Scientific supervisor: Prof. Dr. Shikin A.M., Solid State
Electronics Department, Faculty of Physics, Saint-Petersburg
State University
Introduction
Effects of spin polarization of electronic states attract increasing attention in
last years, as they have important significance in rapidly developing spintronics.
Spin-orbit splitting, firstly founded in semiconductor heterostructure, had already
been investigated in many systems, see, for instance [1], and the simplest way to
describe this effect can be the simplest Rashba-Bychkov model developed for two
dimensional electron gas [2]. Consideration of this problem from the position of the
tight-bonding model with including the corresponding inner atomic and surface
lateral potential gradient into RB Hamiltonian allows to calculate the spin splitting
of electronic states for more complicated systems. Another way for formation of
spin-dependent electronic structure is a polarization of initial states due to the reflection of bulk Bloch states from the surface barrier. This idea was firstly developed in
[3] where it was successfully confirmed for the surface states at Bi(111). According
to this model the surface states and resonances of Bi should be characterized by
strong antisymmetric polarization with the spin inversion relative to normal emission. Later this model was applied by Rybkin et. al. [4] to analysis of spin structure
at the surface W(110). As a result a significant antisymmetric spin polarization of
surface states localized in spin-orbit pseudogap was found. Moreover, it was shown
that these states are characterized by the Dirac-cone-like spin structure similar to
that taking place for topological insulator. Modification of this spin structure by
deposited ultrathin metallic films with different structure of the valence band is very
interesting and perspective problem. The investigations in [5] showed that under
deposition of monolayers of Ag and Au on W(110) in the surface-projected gap of
W(110) the states are formed which are characterized by pronounced antisymmetric
spin-polarization. The value of the spin splitting between the formed states doesn’t
depend on the deposited metal and is the substrate-induced determined by the high
atomic number of the substrate material. Similar substrate-induced spin structure
is observed in 1 ML Al on W(110) [4]. But in this case the dispersion relations of
the formed states differ from that observed for monolayers of the noble metals.
In the Al/W system the formed states have parabolic-like dispersion, and cross
Fermi level at significantly lower values of kII. In monolayers of Ag and Au on W
these features are entirely localized in middle of surface protected gap and cross
the Fermi level at the border of the Brillouin zone.
48
The main idea of the present work is a combination of the substrate-induced
spin-orbit effects characteristic for simple of sp-metal (Bi) and noble metal (Ag)
on W(110) and analysis of the formed electronic and spin structure of the system
consisting of layer of Bi deposited on 1 ML Ag on W(110). We have investigated
the modification of the spin-dependent dispersion relations of the surface states
and d-resonances beginning from the pure surface of W(110) to that characteristic
for the Ag/W(110) and to that for the Bi/Ag/W(110) systems and figured out the
role of Ag and Bi in the formed states. Also we tried to appreciate the main factors
influencing on the spin structure of such systems and to describe the formed spin
polarized states in accordance with Krassovskii [3, 4] and Rashba [2] models.
Experiment
The experiments were carried out at the Russian-German U125/2-SGM beamline
at BESSY II with a “Phoibos” hemispherical analyzer and a Mott spin detector operating at 26 keV. W(110) surface was cleaned by annealing in oxygen at a pressure of
1x10-7 mbar and temperature of 1100°C followed by flash heating to temperatures
of ~2000°C in ultrahigh vacuum. Bi was deposited by thermal evaporation from a
tungsten basket with piece of Bi inside heated by electron bombardment. Ag was
deposited from a tungsten basket heated by direct current. The Ag and Bi layer
thicknesses ware calibrated using a quartz microbalance method with following
testing by the analysis of corresponding quantum well states. The spin-integrated
and spin-resolved spectra were measured with a p polarized light of hν = 62 eV. The
light incidence angle was fixed at 55° relative to the analyzer axis and the off-normal
emission in the direction ГS was measured by rotating the sample.
Result and Discussion
Dispersion relation for clean surface W(110) in the ГS direction of the Brillouin
zone taken from [4] is shown in Fig. 1. As it is described in [4] the electronic
a
α
γ
β
b
α
γ
β
δ
Fig. 1. Dispersion
relations
for clean
W(110)(a)
andBi/Ag/W(110).
Bi/Ag/W(110).
Fig. 1. Dispersion
relations
for clean
W(110)(a) and
All All measurement datameasurement
are obtained
62obtained
eV energy
DottedDotted
whitewhite
line represent
dataatare
at 62 of
eV photons.
energy of photons.
surface protected
band
gapprotected
in W. Red
and
correspond
line represent
surface
band
gapblue
in W.lines
Red and
blue lines to spin up and
spin downcorrespond
electronic
respectively.
to states,
spin up and
spin down electronic states, respectively.
49
structure of the W(110) is characterized by surface resonance located at 1.3 eV at
normal emission which gives rise to two oppositely spin polarized surface states
β and γ at nonzero kII. Also we can see surface state α located at 0.8eV at normal
emission which disperses to lower binding energy and cross the Fermi edge at kII
≈0.4 Å-1 This state has spin-up polarization at positive kII , and spin-down at negative kII . Surface protected band gap marked by white dotted line is arranged in the
region between the binding energy of 2 eV and the Fermi level. States β and γ are
localized at the bottom edge of surface protected gap at kII > 0.3 Å-1. In order to
analyze a formation of the system we have measured the changes of the valence
band spectra at normal emission
during deposition of Ag and Bi
on W(110).
Fig. 2 shows the changes of
the spectrum at different stages
of the system formation. After
deposition of 1 monolayer of
Ag the spectra are characterized
by the peak of the quantum well
states corresponding to 1 ML
Ag, QWS which are located at
4.6 and at 4.2 eV.
Thereat, the intensity of the
substrate-derived features is
Fig. 2. Spectra of formation of system. Black spec- decreased by 2 times that cortra represent clean W and final stages of deposi- responds to formation of one
tion of Ag and Bi.
monolayer of Ag. Next spectra
correspond to deposition of Bi. Quantum well state related to 1 ML Bi located at
3 eV is replaced by surface state at 3.4 eV corresponding to formation of 2 ML
Bi. The last spectrum represents the system after deposition of Bi and Ag. The
estimation of the depletion of the intensity of the W-derived features allows us to
conclude that the formed system consists of 2 ML of Bi arranged above the 1 ML
of Ag on W(110).
In Fig. 1b the experimental dispersion relations for the Bi/Ag/W(110) system is
shown. We can see that the state β is localized inside the surface-projected gap. It
crosses the edge of gap at kII ≈0.4 Å-1 and then does not practically disperse with increasing kII. State α disperses from 0.8 eV at kII=0 till the Fermi level at kII ≈ 0.5 Å-1.
Then it is localized near Fermi level and goes to the region of unoccupied states at
the value of kII ≈0.75Å. The states α and β are oppositely spin polarized. Additionally
in the spectra we can see the surface state of Bi with the binding energy of 3.5 eV
at normal emission which is marked as δ. This state hybridizes with d- states of
W and become spin split. Fig. 3 shows the spin-resolved photoemission spectra
measured at various polar angle relative to normal emission.
Surface state α has spin up, and has β spin down at positive polar angle and,
respectively, at positive kII. The branch γ is spin polarized with the polarization
50
opposite to that for branch
β as its spin counterpart.
The states change its spin
at kII=0, so at kII<0 state α
has spin down and state
β-spin up. Note that the
energy difference between
the oppositely spin polarized states at the center of
gap is round 0.7 eV that is
high in comparison with
classic Rashba systems.
The dispersion dependences of states α, β for
the 1 ML Al/W(110) are
parabolic-like and cross Fig. 3. Spin-resolved spectra of Bi/Ag/W(110). Polar
the Fermi level at 0.6 and angles relate to normal emission are shown at the right
0.7 eV, respectively [4]. In edge of figure.
our system the behavior of formed states is more complicated. Thin films of Bi
has its own spin polarized surface states between 0.2eV and Fermi level, they mix
with surface states of W and form new interface spin polarized states localized in
surface-protected gap. One monolayer of Ag between Bi and W plays the role of
carrier of an interaction due to strong coupling with substrate and wide sp-band.
The surface state δ of Bi becomes spin split due to hybridization with the W-derived
states with strong spin-orbit coupling.
Conclusions
Investigation of electronic and spin structure of the Bi/Ag/W(110) system was
carried out. We gave well-defined analysis of formation of the system showed that
it consist of 1 ML Ag and 2 ML Bi. Comparative analysis of the dispersion relations
of clean W(110) and Bi/Ag/W(110) showed that new formed interface states formed
in electronic structure are localized in the region of the surface-protected band gap
and are clearly spin polarized. Behavior of this states differ from that observed for
such systems as Al/W(110) and Ag/W(110) due to peculiar spin structure of Bi.
Influence of W, beside formed interface states, appears in spin splitting of QWS
of 2 ML Bi because of strong spin-orbit coupling of W.
Acknowledgment. The experiment was carried out at Helmholtz-Zentrum
Berlin (BESSY II) at the Russian-German beamline. The work was supported in
framework of G-RISC program.
References
1. Ast C.R., Henk J., Ernst A., Moreschini L., Falub M.C., Pacilé D., Bruno P.,
Kern K., Grioni M. // Phys. Rev. Lett. 98, 186807 (2007).
2. Bychkov Y.A., Rashba E.I. // JETP Lett. 39, 78 (1984).
51
3. Kimura A., Krasovskii E.E., Nishimura R., Miyamoto K., Kadono T.,
Kanomaru K., Chulkov E.V., Bihlmayer G., Shimada K., Namatame H., Taniguchi
M. // Phys. Rev. Lett. 105, 076804 (2010).
4. Rybkin A., Krassovskii E.E., Marchenko D., Chulkov E.V., Varykhalov A.,
Rader O., Shikin A.M. // Phys. Rev. B 86, 035117 (2012).
5. Varykhalov A., S´anchez-Barriga J., Shikin A.M., Gudat W., Eberhardt W.,
Rader O. // Phys. Rev. Lett. 101, 256 601 (2008).
52
DFT Calculations of the X-Ray Spectra of Wide Band
Materials Including Dynamical Screening and Auger
Effects
R.E. Ovcharenko1, I.I. Tupitsyn1, E. Voloshina2, B. Paulus2,
A.S. Shulakov1
r.e.ovcharenko@gmail.com
Scientific supervisor: Prof. Dr. Shulakov A.S., Solid State
Electronics Department, Faculty of Physics, Saint-Petersburg
State University
1
Saint-Petersburg State University, Russia
2
Freie Universitat Berlin, Germany
Introduction
Spectral photon intensity distribution in characteristic X-ray emission (XES)
and absorption (XAS) spectra contains information on the local characteristics of
valence band (VB) and conduction band (CB) electron density of states (DOS)
distribution in crystals. Despite a considerable successes of one-electron methods
in the description of electronic structure of metals, for reliable interpretation of
features of the experimental X-ray spectra it is necessary to use the theoretical methods allowing, along with one-electron description of the XAS and XES shapes, to
consider many-electron effects which are a consequence of the dynamical processes
accompanying the X-ray transition. In particular, L2,3-XEBs of simple Na, Mg and
Al metals and K- XAS of graphite contains features which are poorly described by
the one-electron model and are attributed to the many-electron processes.
The first theoretical model for dynamical screening effects was proposed by
Mahan [1] and generalized by Nozieres and De Dominicis (Mahan-Nozieres-De
Dominicis (MND) theory [2]). Their theory describes correctly only narrow part
of X-Ray spectra just before and just above Fermi level. In this work we propose
significantly improved real multiband calculation scheme described the spectral distribution in whole energy scale including MND effect where as an input the crystal
electronic structure obtained from DFT calculations were used. This scheme may
be consider as generalized and improved formalism proposed in [3, 4] papers.
There is another feature, governed by many-electron processes, which is in
charge for appearance of a prolonged “tail” in the low-energy part of the emission
spectrum [5, 6]. It is accepted that this feature is a consequence of the non-radiative
Auger process in the valence band of metal. Indeed, a vacancy, appeared in VB after
the X-ray transition, can decay due to Auger process characterizing by excitation of
one more VB electron into conduction band above the Fermi level. Moving from
the Fermi level to the bottom of the VB a lifetime of the hole in the final state of the
emission transition decreases and the energy width increases. To date non-empirical
ab initio treatment of this effect is too complicated and time consuming. Therefore
in this work we used a semi-empirical method [7] which allows reproducing accurately the shape of metal XES in low-energy range of spectrum.
53
Described calculation schemes were applied to K and L2,3 XES of Mg and Al
crystal metals. The calculated X-ray spectra were compared with the experimental
results.
MND theory
Let’s consider X-ray emission process. In the one-electron approximation,
ignoring the dependence of transition probabilities on wave vector k, XES may
be expressed as
I c ( E ) = ∑ Wµ ( E ) N µ ( E ),
µ =lc ±1
where the atom-like probability of transition Wμ(E) is
Wµ ( E ) ~ E 3 ∑ µ rα c .
2
α ,mc
Here |μ⟩, |c⟩ are atom-like wave functions of the valence and core-hole electrons,
respectively. The partial density of states (PDOS) can be expressed via the imaginary
part of the one-particle Green’s function Gμν(E) projected onto atomic sites μ, ν
Nµ (E ) =
1
Im Gµµ ( E ),
π
which may be constructed using DFT calculations in two extreme cases. In the
first case, so-called final state approximation, the influence of core-hole on valence
electrons is completely ignored. Then we have
Gµν ( E ) = ∑
n,k
µ nk nk ν
,
E − En (k ) − iδ
where |nk⟩, En(k) are unperturbed crystal wave function and band energy, δ is
small parameter.
In the opposite case, which called initial state approximation, we assume the
band wave functions are completely adjusted under the influence of the Coulomb potential of a core-hole and the expression for the Green’s function takes the form
µ nk ' nk ' ν
G µν ( E ) = ∑
,
E
− En' (k ) − iδ
n,k
where |nk'⟩, E'n(k) are perturbed crystal wave function and band energy in the
core-hole field. Green function matrixes G ( E ) , G ( E ) should satisfy SlaterKoster equation
−1
−1
V = G ( E )  + [G ( E ) ] .
Using expressions for G ( E ) , G ( E )
we get
(
)
Vµν = ∑ µ nk nk ν En (k ) − µ nk ' nk ' ν En' (k ) .
n,k
54
In contrast to previous one-electrons extreme cases, MND theory describes the
more realistic dynamical process of core-hole filling. Technically, MND emission
spectrum may be calculated as follow
∑
I cMND ( E ) ~ Re
φ( E , t ) = 1 − V
µ = lc ±1
EF
∫ dε⋅
−∞
I* ( E , t ) = 1 − V
+∞
EF
0
−∞
I* (ε, t ) − eit ( ε − E ) I* ( E , t ) 
N (ε)φ(ε, t ),
ε−E
∫ dε⋅
−∞
+∞
Wµ ( E ) ∫ dt ∫ d ε ⋅ e − it ( E − ε )  N (ε)φ(ε, t )  µµ ,
eit ( E −ε ) − 1
N (ε),
E−ε
~
where
are DOSes obtained from unperturbed G(E) and perturbed G ( E )
Green functions, respectively.
Auger effect
Auger process in the VB can affects the shape of K- and L- XES of the metals.
It leads to the suppression of the intensity at the bottom of the VB and the appearance of a tightened low-energy tail in low energy part of XES. Indeed, the final
state of X-ray transition is characterized by the presence of vacancy in the VB of
metal which can be filled by transition of electron from the high-lying occupied
VB states. Then the released energy is transferred to an Auger electron, which is
excited into the empty state of the CB. Additional decay channel leads to an increase
of the natural width of the states in the VB. Such mechanism of the VB hole-states
decay especially probable in metallic systems where there is no energy gap.
Technically, it may be taken into account by convolution of the theoretical
(MND or one-electron) spectrum I(E) with the Lorentzian L(ε,E) which Full
Width at Half Maximum (FWHM γ(E)) depends on energy according the following expression
γ ( E ) = β( EF − E ) 2 ,
where β is fitting parameter. Then emission spectrum with Auger correction is
given by
1
γ (ε) 2
J ( E ) = ∫ d ε ⋅ I (ε)
.
π (ε − E ) 2 + γ 2 (ε) 4
Results
Figs. 1, 2 show the experimental L3 XES of magnesium and aluminum metals
and theoretical spectra calculated in the one-electron and many-electron approximations. One-electron spectra were calculated in the final state approximation and
in fact reflect the s-PDOS of VB. Since the contributions of d-PDOS to L3 XES
of Mg and Al are small [8] we ignored them. As can be seen from Figs. 1, 2 for
L3 XES, the one-electron approximation don’t describes the main features and the
shape of the experimental spectra. Experimental curves in low energy parts fall off
slower than theoretical spectra and they are distinguished by the presence of the
tighten “tails”. This behavior may be explained by the Auger decay of vacancies in
VB of the metals [5, 6]. In addition, the experi experimental spectra have a narrow
peak at the Fermi edge which is a result of a many-electron scattering of valence
55
Fig. 1. L3 XES of magnesium metal: Fig. 2. L3 XES of aluminum metal:
experimental band (dots), one-electron experimental band (dots), one-electron
approximation (dash line), many- approximation (dash line), manyelectron spectrum (solid line).
electron spectrum (solid line).
electrons on the core-hole potential (MND singularity [1,2]). The introduction of
the many-electron corrections significantly improves an agreement with the experimental data. Many-electron theory well describes the behavior of XES in the
vicinity of Fermi energy (peak intensity, position, width) and in the low-energy
part (shape and length of the tail).
To verify described computation scheme we apply it to calculations of K- emission bands of these metals. Since classic MND theory predicts MND singularity
is absent in K XES of simple metals, the coincidence between MND spectra and
one-electron spectra in the final state approximations have to be high. As it can
be seen from Figs. 3, 4 for K XES of both metals, in contrast to L3 XES, MND
Fig. 3. K XES of magnesium metal: ex- Fig. 4. K XES of aluminum metal: experimental band (dots), one-electron ap- perimental band (dots), one-electron approximation (dash line), many-electron proximation (dash line), many-electron
spectrum (solid line).
spectrum (solid line).
and one-electron spectra coincide with each other and with the experimental
bands. Obviously, for K XES taking Auger effect in the VB into account is not
required.
To account for Auger effect in VB we fit the parameter β for magnesium and
aluminum metals in the expression for γ(E). We have got follow values 0.038
56
and 0.060 eV-1 for Al and Mg L3 emission band, respectively. The width of the
valence level γ(E) is presented
in Fig. 5.
It can be seen the probability of the Auger process in VB
of magnesium is larger than in
aluminum for the states with the
same energy. However, since
the width of VB of Al metals
significantly larger than one
of magnesium, Auger effect
noticeable for the L3 XES of
aluminum. Note, this estimation
is correct only for s-states since Fig. 5. The Lorentz width of VB levels of metallic
the contributions of d-PDOS to magnesium and aluminum.
L3 XES of Mg and Al are small [8].
We have mentioned before that taking the Auger effect for K XES into account
is not required (see Figs. 3, 4). On the other hand there are no selection rules for the
symmetry of the VB states involved in Auger process. Although specific behavior
of p-DOS and large energy width of K level (more than 0.3 eV) in comparison
with L level (less than 0.01 eV) may mask “Auger-tailing” in the K XES to some
extent, described mechanism of Auger decay is still questioned.
Conclusion
To account for the influence of the many-electron effects of the core-hole dynamical screening by valence electrons on the shape of characteristic XES of metals
we combined MND theory with the ab initio method of the electronic structure of
crystals calculation. Auger effect in valence band was taken into account by the
phenomenological theory. Under this approaches we have calculated K and L3 XES
of Mg and Al crystals. A good agreement between theoretical and experimental
spectra is found across the whole spectral range of the bands.
References
1. Mahan G.D. // Phys. Rev. v. 163, p. 612 (1967).
2. Nozieres P., De Dominicis C.T. // Phys. Rev. v. 178, p.1097 (1969).
3. Grebennikov V.I. at al. // Phys. Stat. Sol. B v. 79, p. 423 (1977).
4. Wessely O. at al. // Phys. Rev. Let. v. 94, p.167401 (2005).
5. Landsberg P. // Proc. Phys. Soc. A 62, p. 806 (1949).
6. Nemoshkalenko V. Aleshin V. Theoretical foundations of X-ray emission
spectroscopy. – Kiev: Naukova Dumka, 1974.
7. Livins P., Schnatterly S.E. // Phys. Rev. B v. 37, p. 6731 (1988).
8. Ovcharenko R., Tupitsyn I., Kuznetsov V., Shulakov A.// Optics and
Spectroscopy 111, 940 (2011).
57
Spin structure in thin Au layers on W(110) and Mo(110)
M. Rusinova, A. Rybkina, I. Klimovskikh, E. Zhizhin and
A.M. Shikin
manja_ru@mail.ru
Scientific supervisor: Prof. Dr. Shikin A.M., Solid State
Electronics Department, Faculty of Physics, Saint-Petersburg
State University
Introduction
Effects of influence of spin-orbit interaction on electronic structure of surface
states have attracted last years increasing attention due to an opportunity of manipulation of spin of electron without application of magnetic field. It is crucial
for the emerging field of spintronics where the manipulation by spin of electron
instead of charge takes place. In this respect the investigations of spin-orbit interaction in non-magnetic materials with central inversion symmetry of wave functions
are especially interesting. The bulk continuum states for such systems are doubly
degenerated due to the time reversal and inversion symmetry conditions. However,
due to the breaking of the inversion invariance at the surface the spin degeneracy
for the surface states is lifted. According to the Rashba model [1] developed for
the two dimensional electron gas the spin-orbit interaction leads to the separation
of free-electron-like state into two parabolic branches with opposite spin polarization. This is illustrated in Fig. 1, where the two parabolas are each shifted by k0
relative to the Г-point of the Brillouin zone in opposite k||-directions resulting in
the momentum depending energy splitting. These parabolas are completely spin
polarized (blue color mark – the spin “up”, and red color – the spin “down” states),
with the spin polarization which is completely located in the plane of the electron
gas and arranged parallel to the circular constant energy contours. Quantitatively this
effect can be described with Rashba parameter αRB = f(αA, αV) which determinates
the spin splitting. According to the tight-binding model [2] it depends on the inner
atomic(αA) and surface potential barrier (αV) potential gradient, respectively.
However there is another possibility of spin-orbit coupling that doesn’t obey
the Rashba model for two-dimensional electron gas. Recently the specific spin
polarization of bulk continuum states in the surface region has been found. This
effect was firstly described by Krasovskii for a semi-infinite crystal, where in the
bulk continuum the bands for two spins are not split [3].
Far from the crystal surface, where the potential is periodic, the wave function
is a superposition of the incident and reflected Bloch waves. The surface barrier
produces the spin dependent phase shift between the incident and the reflected
waves that leads to the finite spin polarization in the continuum spectrum. Since
the lattice-periodic oscillations of spin density persist to infinity in the depth of
crystal while the beating term decays in the depth, the spin polarization of bulk
continuum states is spatially localized in the surface region.
58
a
b
Fig. 1. (a)-Schematic representation of dispersion of a nearly free two-dimensional
electron gas due to the Rashba effect;(b)-observation of Rashba-type spin splitting for SARPES data.
Schematic representation of observation such spin polarization is shown in
Fig. 2. Significant difference from the Rashba effect for surface states is that the
spin polarization of 3D states is not a consequence of the energy splitting of pure
spin states.
The similar type effect of spin-orbit b splitting and polarization was
a
observed for thin metal (Ag, Au) layers
on W(110) in [4] where it was
influenced by the inner atomic potential gradient (i.e. by atomic number) of the
substrate material.
The aim of this work is the investigation of the substrate-induced effects of
spin-orbit interaction developed in thin 3 monolayer Au films on W(110) and
Mo(110) for Au sp and d quantum-well states (QWS), where both Rashba type
spin-splitting of two-dimensional states and spin polarization of bulk continuum
states can be observed. Experiment was carried out using angle- and spin-resolved
photoemission spectroscopy with the application of synchrotron radiation at
BESSY (Helmholtz-Center Berlin).
Fig. 2. (a)-Schematic representation of finite spin polarization of bulk continuum
states;(b)-observation of spin polarization for SARPES data.
The similar type effect of spin-orbit splitting and polarization was observed
for thin metal (Ag, Au) layers on W(110) in [4] where it was influenced by the
inner atomic potential gradient (i.e. by atomic number) of the substrate material.
The aim of this work is the investigation of the substrate-induced effects of
spin-orbit interaction developed in thin 3 monolayer Au films on W(110) and
Mo(110) for Au sp and d quantum-well states (QWS), where both Rashba type
spin-splitting of two-dimensional states and spin polarization of bulk continuum
states can be observed. Experiment was carried out using angle- and spin-resolved
photoemission spectroscopy with the application of synchrotron radiation at BESSY
(Helmholtz-Center Berlin).
59
Results and Discussion
Figs. 3a and 3b show the dispersion dependences for the Au sp and d quantumwell states developed in 3ML Au films deposited on the W(110) and Mo(110),
respectively, which were measured in the ГН direction of the Brillouin zone by
a
b
Fig. 3. Photoemission peak intensity map vs binding energy and parallel electron
momentum for 3 ML Au/W(110) – (a) and for 3 ML Au/Mo(110) – (b) surfaces,
both measured in the ГH direction of the surface Brillouin zone using the photon
energy Һν=62 eV.
the angle-resolved photoemission spectra. Corresponding spin-dependent photoemission spectra are presented in Fig. 4 for both systems for different polar angles
relative to the surface normal. White arrows in Fig. 3 denote the momentum values
which were used for measurements of the spin- and angle-resolved photoemission spectra in Fig. 4. The presented dispersion dependences are characterized by
the branch of the surface state located near the Fermi level (marked as SS) and
the series of the branches of the dispersion dependences of the Au sp and d QWS
located in the energy region of 0 - 2.5 eV and 2.5-7.0 eV, respectively, observed
for both systems.
The sp-QWS for both Au/W(110) and Au/Mo(110) systems show the freeelectron-like parabolic dispersions in the film plane. According to the Rashba
model the sp-QWS dispersion dependences are expected to be shifted from each
other by k0. The most pronounced quantum-well state in Fig. 3a is marked as
QWS1. The spin splitting for the branch QWS1 for Au/W(110) system is sizeable
enough to be qualitatively demonstrated experimentally. It is marked in Fig. 4 by
blue and red arrows.
The dispersion dependences for d-states differ greatly from that observed for
the sp QWS. They are more flat-like. The assignment of spectral maxima to spin
“up” (blue lines) and spin “down” (red lines) in Fig. 3 is derived from the SARPES
spectra shown in Fig. 4.
As it has been mentioned before the spin splitting of the quantum-well states is
pronouncedly observed for the Au-layer on the W(110) substrate. The state marked
as QWS1 is clearly seen to be separated into two parabolas with completely spin
polarization of opposite signs. The size of the splitting increases with k||. In contrast,
60
a
b
Fig. 4. Spin- and angle-resolved photoemission curves for 3 ML Au/W(110) - (a)
and for 3ML Au/Mo(110) – (b) measured for different values of parallel momentum
k||. (a)-(b) - Photocurrent of spin “up” is shown by blue lines and spin “down”
by red lines.
there is no discernible spin splitting for the 3 ML Au/Mo(110) system. It means
that the Rashba-type spin-splitting takes place for these states and it is induced
by high atomic number substrate. This confirms the conclusion that the value of
splitting of the sp-QWS is mainly determined by the substrate and is influenced
by the interaction with substrate surface atoms.
At the same time Rashba-like spin splitting of the d-states can be observed
neither for tungsten nor for molybdenum substrate. The quantum d-states are appeared to have specific spin polarization. The branch 1 of the d-states for 3ML of
Au on W(110) crosses the spin split sp-QWS at normal emission at energy 2.5 eV.
It is seen from Fig. 4a that the spin polarization of the d QWS has a tendency to
change sign at the Г-point of the BZ, i.e. spin “up” for –k|| and spin “down” for
+k||. Another two branches 2 and 3 have also asymmetric spin polarization. One
can see from the Fig. 4a that the sign of spin polarization of both these branches
coincide with the sign of the first one.
From the Fig. 4b it is clear that spin polarization of d-states doesn’t change
sign at thе Г-point of the BZ i. e. the spin polarization of d-states for 3ML Au/
Mo(110) is not asymmetric. From other experimental data it is known that the
degree of the spin polarization depends on energy of the photon beam. One can
suppose that the polarization of photocurrent from d-states can also contribute to
the net spin polarization of the states. In order to make it clear the more experiments are required.
61
Conclusion
To conclude, two different types of spin polarization of states occur in nonmagnetic materials. The first one is the Rashba polarization of the sp-quantumwell states that is the consequence of spin-orbit splitting. Instead of viewing the
band splitting as an energy separation, one can also regard it as two bands that are
shifted in momentum (see Fig. 1a). This momentum splitting provides the phase
difference in the spin field effect transistor proposed by Datta and Das and causes
the necessary spin precession [5]. It is important that there is no need to apply
any external magnetic field or use ferromagnetic materials for the effect of spin
splitting to take place.
Another effect is spin polarization that doesn’t follow the spin-orbit splitting
of pure spin states. In this case the branch that indicates some state, doesn’t have
its spin-orbit split counterpart. It is significant that the polarization is to change
sing to the opposite due to the time reversal symmetry.
As one can understand from the comparison SARPES date for Au/W(110) and
Au/Mo(110) systems the substrate atomic number is crucial for both these types of
spin polarization. Since the effects depend mainly on the substrate characteristic,
the material of deposited film is not required to have high atomic number.
1.
2.
3.
4.
5.
References
Bychkov Y.A., Rashba E.I. // JETP Lett. 39, 78 (1984).
Petersen L., Hedegård P // Surf. Sci. 459, 49 (2000).
Krasovskii E.E., Chulkov E.V. // Phys. Rev. B 83, 155401 (2011).
Shikin A.M. et. al. // Phys. Rev. Lett. 100, 057601 (2008).
Datta S., Das B. // Appl. Phys. Lett. 56, 665-7 (1990).
62
NEXAFS study of various graphite fluorides
Zhdanov Ivan
zhdanov.IA.spbu@gmail.com
Scientific supervisor: Prof. Dr. Vinogradov A.S., Department of
Solid State Electronics, Faculty of Physics, Saint-Petersburg State
University
Introduction
Due to their numerous applications of graphite fluorides (CnF), such as electrodes of primary lithium batteries, solid lubricants, reservoirs for storage of strong
oxidant and fluorinating agents (BF3, ClF3, etc), they are extensively studied since
several decades. By this time it is known that their physical and electrochemical
properties depend strongly on the C-F bonding character in the fluorocarbon matrix
[1]. Therefore, the main aim of the present work was to characterize by means of
X-ray spectroscopy a series of CnF (n=1, 2, etc) samples synthesized in various
ways with a different fluorine content and a different character of chemical bonding
between carbon and fluorine atoms.
The investigated samples were prepared by the group of Prof. M. Dubois
(Clermont Université, France). Graphite fluoride C2F was synthesized by direct
fluorination of powdered graphite at 350 °C. Monofluoride graphite CF was prepared under different conditions of the fluorination: CF-400 (at the temperature
of the direct fluorination tF = 400 °C), CF-530 (at the temperature of the direct
fluorination tF = 530 °C), CF-550-IF5 (fluorination in mixture F2+HF+IF5 at room
temperature 25 °C, followed by post-fluorination at tF = 550 °C). White graphite
fluoride has a maximum fluorine content 62.4 wt.%. The spectra of the powdered
"white" graphite fluoride (WGF), the crystal is highly ordered pyrolytic graphite
(HOPG) and nanodiamond (ND) used in this paper as a reference, were taken
from [2, 3].
Results and Discussion
All measurements have been performed at the Russian-German beamline at the
BESSY II [4] and at the D1011 beamline of the electron storage ring MAX-2 [5].
The samples were prepared in air. The powders of the materials under study were
rubbed into a scratched surface of a pure substrate (metallic copper plate). The C1s
and F1s NEXAFS (Near-Edge-X-ray-Absorption-Fine-Structure) spectra of CF,
C2F, C2.5F as well as of reference samples (HOPG, ND, WGF) were obtained in the
total electron yield mode (TEY) by detecting a sample current and in the partial
electron yield (PEY) mode by detecting the electron yield using a multichannel
plate with retarding potential of -150V. In the course of measurement no effects
of sample charging were noticed.
The C1s absorption spectra of graphite fluoride C2F, measured by TEY and
PEY are compared between themselves in Fig. 1. It is clearly seen that the spectra
are virtually identical. This fact indicates on the uniformity of the fluorinated
63
samples at different depths, as
TEY and PEY methods have
different surface sensitivity and
*
*C
*
B1 B
probing depth – 10 and 0.5 nm,
* *
D*1 D*2
E F
respectively. Worse statistics
B *1
signal in the PEY spectrum is
TE Y
*
E* F
associated with a lower number
A
of electrons compared to the
PE Y
1
one recorded by TEY method.
A
In Fig. 2 C 1s absorption
2
spectra of graphite fluorides
280 285 290 295 300 305 310 315 320 (CF, C F, C F) are presented
2
2.5
Photon Energy, eV
together with those of crystal
Fig. 1. Comparison of the C 1s absorption spectra of pyrolytic graphite HOPG,
of C2F taken in TEY (1) and PEY (2) modes.
WGF and nanodiamond [2,
3]. All spectra were obtained
B
A
C
1s
absorption
spectra
C
D
F
E
TEY and normalized to the
HOPG
*
A'
*
*
B* C
D*
same level of the continuous
E
F
C 2.5F
A
B *1 B * C * D* D*
absorption at the photon energy
* *
1 2
F
*
E
B1
C 2F
*
* *
of 320 eV.
D*1 D2
* B C
A
*
B1
E* F
CF
*
As is known, the decisive
*
*
C
*
A
B
D1 D2
role in forming structure of the
B *1
E * F*
WGF
*
C1s absorption spectrum of
A
*
B* C
D
HOPG plays a benzene C6 ring
E * F*
ND
of graphene monolayer [2, 3].
a
A
Within these representations,
285
290
295
300
305
310
315 peaks A and B-C in the crystal
Photon E nergy, eV
spectrum are associated with
Fig. 2. C 1s absorption spectra for graphite fluo- the dipole-allowed transitions
rides C2.5F, C2F, CF, WGF, nanodiamond and py- of C1s electrons into empty
rolytic graphite HOPG [2, 3].
states of π and σ symmetry
of the conduction band. They are formed from π2pz and σ2px,y states of carbon
atoms oriented perpendicular and parallel to the plane of the carbon layer (graphene) respectively. These states are essentially quasi-molecular states and, as a
consequence, localized mainly within a single carbon hexagon [2, 8]. Therefore
the structure of C1s absorption spectra should be sensitive to the local environment of the absorbing atom. The carbon atoms located in one graphene layer of
HOPG have a triangular coordination which is explained by sp2-hybridization of
the valence C2s and C2px,y states.
A characteristic feature of the C 1s spectrum of the nanodiamond is a separate E*-F* band at photon energies of 305-307 eV which, along with the lack of
π resonance, reflects the sp3 hybridization of the valence 2s and 2p states of the
carbon atom.
C*
C 1s absorption spectra of C 2F
D*1 D*2
Absorption(Total Electron Yield)
Absorption(Total Electron Yield)
B*
64
In going from the spectrum of HOPG to the graphite fluoride spectra significant
changes in the structure of the spectra are observed along the HOPG-C2.5F-C2F-CF
series: a decrease of relative intensity of the π-resonance (band A) and significant
restructuring of the σ region resulting in formation of new B1*-F* bands (graphite
fluoride) instead of B-F bands (HOPG). In the C2.5F spectrum the relative intensity
of A resonance is reduced by about a factor of 2 and the resonance is shifted to the
low-energy side by ~ 0.35 eV.
Then in going from C2.5F to C2F, the intensity of A is further reduced by an
order and rather narrow and well-defined absorption bands B1*-F* are formed in σ
region of C 1s spectrum. In particular, an isolated σ band E*-F* appears at photon
energies of ~ 305 eV.
A subsequent significant decrease of π-resonance in intensity spectrum of CF
points to the destruction of π electron subsystem in the benzene ring of graphene
layer. As a result, empty states of CF are only of σ symmetry. The following interesting result is the formation of the E*-F* band (at hν ~ 305-307 eV). A similar highenergy σ band, along with the lack of π-resonance is considered as a characteristic
feature of C1s-spectrum of diamond with tetrahedral coordination of carbon atoms
(sp3-hybridization) (Fig. 2) [2]. Based on this experimental fact we can assume
that fluorination of graphite to CF occurs via the attachment of fluorine atoms to
carbon atoms perpendicular to the
graphene layer owing the covalent
C2pz-F2p bonding. As a result, the
sp2 hybridization of the valence
states of carbon atoms in HOPG is
transformed to the sp3 hybridization and thus graphene layers are
corrugate.
In moving from the spectrum
of CF to that of C2F NEXAFS at
the C 1s edge does not change
significantly (Fig. 2). The difference between C1s-spectra
consists only in the fact that the
relative intensity of the narrow
band B1* in the spectrum of C2.5F
is about 2 times less than in that
of CF. Fig. 3 represents the model
structures for CF and C2F [1]. It
is clearly seen that the number of
C-F bonds in C2F is two times less
than in the CF.
When moving to a spectrum Fig. 3. Fluorocarbon Layer Stacking in partly
C 2.5F the relative intensity of fluorinated Graphite (C2.5F)n and Covalent
the resonance A increases sig- Graphite Fluorides (C2F)n and (CF)n [1].
65
Absorption(Total Electron Yield)
Absorption(Total Electron Yield)
nificantly and structure
B 1 *-F* becomes more
B
D
*
diffuse and less contrast
*
F
C 2.5F
*
B 1 B * C*
*
D*1 D2
compared to that of the
E*
2
F
C 2F
B *1 * C * *
spectrum of C 2 F. The
*
B
D1
D2
2
*
B *1 * C *
E*
appearance in the C1s*
F
*
B2
CF-400
D1
D2
E*
F*
spectrum C2.5F resonance
*
B1 *
CF530
*
A, although less intense
B 2 C * D*
D
*
1
2
E
F * CF550-IF5 than in HOPG, is obviB *1 * *
*
*
B2 C
D1 D2
ously related to the coE*
F*
WGF existence in the layers of
fluorinated carbon atoms
with sp 3 -hybridization
680
685
690
695
700
705
710
and non-fluorinated carPhoton E nergy, eV
bon atoms with sp2-hybridization of the valence
Fig. 4. F 1s absorption spectra of fluorides C2.5F, C2F, 2s-and 2p-states of the
CF and white graphite fluoride WGF.
atom carbon [9] (Fig. 3).
In this case C-F bond showing weakening covalency because hyperconjugation,
which is manifested in the low-energy shift of the band A and B1* compared to
their position in the CF and C2F.
It is also interesting to compare the F1s absorption spectra for fluorinated
graphite (CF, C2F and C2.5F) (Fig. 4). It is clearly seen that the behavior of the F1s
absorption spectra is similar to that of C1s spectra: a strong change in the structure
in going from C2.5F to C2F and the small one along the C2F-CF series. In particular,
the absorption in the F1s spectrum of the C2.5F starts earlier than in those of the
C2F and CF. This fact
C 1s absorption spectra
*
correlates with the lowC
B *1 B
D*1 *
energy shift of the B1*
D2
band in C1s spectrum
*
*
B *2
of C2.5F and can be asF
E
*
B * C D*1 D*2
sociated with a weaker
WGF
*
B * C D* *
*
covalent C-F bonding
F*
E
1 D
A
CF-400
2
in C2.5F.
*
*
B C
*
*
*
E* F
In conclusion we
D
D
1
2
A
CF530
consider C1s absorp*
*
E F
B *2
tion spectra for samples
CF550-IF5
A
of graphite fluoride CF,
obtained in different
A
conditions of fluorina285
290
295
300
305
310
315 tion of graphite (CF400,
Photon E nergy, eV
CF530, and CF550-IF5).
Fig. 5. C 1s absorption spectra for graphite fluoride CF A comparison of these
synthesized in different ways.
spectra with C1s spec66
*
F 1s absorption spectra
C*
*
trum of "white" graphite fluoride WGF (Fig. 5) shows that the spectra graphite
fluorides agree well with that of WGF. Coincidence of the fine structure of the
F1s spectra of monofluoride graphite obtained in different conditions indicates the
proximity of the stoichiometry of the samples.
The differences are observed in spectra intensity of π-resonance: in graphite fluoride CF-530 and CF-550-IF5, resonance A virtually disappears, while in
graphite fluoride CF-400 and in WGF the intensity of the resonance A is more
substantial, but also small. In addition, C1s-absorption spectrum of CF-400 has a
lower contrast of fine structure in the σ-field with respect to the spectra of other
graphite fluorides that may be associated with less order of materials, i.e. most of
it is imperfection.
Thus, NEXAFS spectroscopy is shown to be an efficient method for characterization of chemical bonding for different graphite fluorides.
This work was supported by the Russian Foundation for Basic Research (project
no.12-02-00999 and no.12-02-31415) and the bilateral Program “Russian-German
Laboratory at BESSY”.
References
1. Giraudet J., Dubois M., Guerin K. et al. // J. Phys. Chem. B 2007, 111,
14143-14151.
2. Brzhezinskaya M.M., Vinogradov N.A., Mouradian V.E. et al. // Solid State
50, 565 (2008).
3. Brzhezinskaya M.M., Vinogradov A.S., Krestinin A.V. et al. // Solid State
52, 819 (2010).
4. Fedoseenko S.I., Iossifov I.E., Gorovikov S.A. et al. // Nucl. Instrum. Meth.
Phys. Res. A 470, 84 (2001).
5. Nyholm R., Svensson S., Nordgren J., Flodström A. // Nucl. Instr. and Meth.
A 246, 267, 1986.
6. Lukirskii A.P., Brytov I.A. // Solid State 6, 43 (1964).
7. Gudat W., Kunz C. // Phys. Rev. Lett. 29, 169 (1972).
8. Comelli G., Stohr J., Robinson C.J., Jark W. // Phys. Rev. B 38,
7511(1988).
9. Sato Y., Itoh K., Hagiwara R. et al. // Carbon 42(15) (2004) 32433249.
67
E. Applied Physics
Impact of radioactive fallout from Chernobyl and
Fukushima on the environment of Leningrad region
Myorzlaya Anastasia
stummeworte@mail.ru
Scientific supervisor: Dr. Sergienko V.A., Department of Nuclear
Physics, Faculty of Physics, Saint-Petersburg State University
Introduction
The explosion in Chernobyl (1986) nuclear power station was the worst radiation
catastrophe in history of atomic energy, the recent accident on nuclear power plant
occurred in Fukushima (2011). As a result of these two accidents released radionuclides from discarded nuclear reactor materials spread around the world. Residual
radiation affects on the level of pollution of the environment of St. Petersburg and
Leningrad region. So the aim of the work was study of the influence of radiation
precipitation from the accident at the nuclear power plant on radiation contamination of the territory of Saint-Petersburg and Leningrad region on the example of
the accident at the Chernobyl NPP and NPP Fukushima.
The method of investigation
Soon after the Chernobyl accident the investigation was carried out to notify the
pollution of Europe by gamma-spectrometric survey from planes using scintillation
detectors. Although this method gives uncertain data of radionuclide activity in the
study area, it allows obtaining an overall picture of the contamination. Thus, according to research in the Leningrad region was found 137Cs from Chernobyl accident
in Kingisepp and Lomonosov districts with an estimated activity 1–5 Ci/km2.
Another method of investigation of the contaminated territories is the collection
of samples of some objects of an environment (in this case, topsoil) and measuring them in the laboratory using semiconductor Ge(Li)-detector. This method of
registration of radioactivity requires a longer time, but gives much more reliable
information.
The studied radionuclides
Artificial radionuclides can track their movement and accumulation in different
environmental objects and can indicate the sources of the pollution. The degree
of pollution in the study area is determined by the activity of these radionuclides
in the environment.
In order to characterize the studied accidents radionuclides, which released
in the accidents, were selected from the radionuclides released in this accidents.
Isotopes of caesium 134Cs and 137Cs is optimally done for the consideration of the
impact of accidents on Fukushima and Chernobyl as they have half-decay periods
close to the time that past from the moment of the accidents (T1/2(134Cs)=2 years,
T1/2(137Cs)=30 years).
As the spectrometer is not graduated, we will measure the relative radioactivity. So the ratio of the activity of caesium to the activity of potassium NCs will be
70
calculated and considered, as 40K is contained in large quantities in any soil sample.
Using the known average radioactivity of potassium RK in Bq/kg the average radioactivity of caesium RCs can be counted taking into account the background.
,
A −A
N Cs =
Cs
extCs
A K40 −AextK40
RСs =RK
E K IK
ECs I Cs
N Cs
,
where ACs, AК40 - the areas under the graph of the spectrum peaks of Cs and K;
AextCs, AextК40 – amendments to the background (the areas of the peaks Cs and K);
E и I – the effectiveness of the registration and the intensity of the radiation of
gamma-quanta of this isotope detector. RK40=148 Bq/kg.
Fig. 1. Maps of Leningrad region and Saint-Petersburg with marked collecting
samples and their relative radioactivity of 137Cs.
Samples in the polluted zones according to the old map have 137Cs activity of
about 15 Bq/kg, however, some pleases outside of the contaminated areas were
detect with radioactivity of 137Cs more than 40 Bq/kg. That is, soil analysis showed
a significant expansion of the radioactive spots of 137Cs.
It is important to note that the radioactivity of 134Cs, which can identify the
Fukushima trace, doesn’t exceed 0.3 Bq/kg. Thus Leningrad region is not appreciably affected by the contamination from Fukushima.
71
Radioactive fallout
Pollution of Europe by 137Cs is diverse and rather complicated. The wind movement of radionuclides can not give such form of contaminateв spots, so one has
to assume that such dissemination has occurred by absorption of radionuclides in
the cloud and precipitation on the surface.
The amounts of fallen precipitation on the territory of Leningrad region and
in Saint-Petersburg soon after the accident at the Chernobyl nuclear power plant
were considered.
Number of tenths of millimeters of fallen rain on the territory of Leningrad
region (Fig. 2) is consistent with the edited map deposited radioactivity: the number
of caesium in the soil is directly proportional to the amount of precipitation.
Fig. 2. The amount of precipitation after the Chernobyl disaster in the Leningrad
region (tenths of mm).
In order to completely ensure the assumption, movement of the radioactive
plume from Chernobyl was analyzed. In general, the view of rainfall looks like
the picture of the deposition of caesium in the soil. As long as this method of
determining the number of deposition of radionuclides gives the data, that are
consistent with the map of deposition of caesium, obtained for Leningrad region
using Ge-detector, it can correct the data, getting by spectrometric survey from
the air. That is, some deviations of precipitation from the activity of caesium on
this territory also point out the inaccuracy of that method.
Wet precipitation are not fortuitous event, they considerably depend on some
external factors. The determining factor is the landscape, but they are also influenced by the proximity of the cities, plants and factories and other factors, that
break the natural environment.
The average annual rainfall remains constant for the place from year to year,
if there are no drastic changes of the ecosystem.
Precipitations in Leningrad region were considered from 1965 to 1995 every
5 years. During this time, the average annual values were changing, but in general
72
Fig. 3. The amount of precipitation after the Chernobyl disaster in Europe
(tenths of mm).
all this annual precipitations were changing together and in proportional proportion
relative to St. Petersburg they remained constant.
In the places of maximum precipitation the probability of deposition in
the ground of larger quantities of caesium increases. In densely populated
Fig. 4. The average annual precipitation in the Leningrad region (tenths of mm).
cities precipitation of rainfall after the accident on the nuclear power plant and
consequently depositions in the soils should be prevented. This can be achieved
by removal of rain clouds.
73
Conclusions
• The Chernobyl accident significantly effected on the pollution of SaintPetersburg and Leningrad region;
• The spot of contamination after the accident at the Chernobyl NPP is
larger than it was established earlier;
• The accident at Fukushima did not have a significant impact on the
environment of Leningrad region;
• There is a directly proportional dependence between the number
deposited caesium in the territory and the amount of the cyclone precipitation,
that came from the place of the accident;
• The most likely places, which may be exposed by radiation from the
next accidents, were determined and also the method of reducing the impact on
the cities of accidents was proposed.
References
1. Sapozhnikov U.A., Aliyev R.A., Kalmykov S.A. Radioactivity of the environment. - M.: Binomial: laboratory of knowledge, 2006.
2. Saharov V.K. Radioecology. M. 2006.
3. Smith J.T., Beresford N.A. Chernobyl – Catastrophe and Consequences. UK,
Chichester, Praxis Publishing Ltd. 2005.
4. Leningrad region. Ecological map. SPb, «Discus Media». 2009.
5. National Climatic Data Center. Data documentation for dataset 9813. Daily
and Sub-daily Precipitation for the Former USSR. Version 1.0. National Climatic
Data Center, 151 Patton Ave., Asheville, NC 28801-5001, USA. 2005.
74
Innovative non-chemical etching technology
Yang Cheng Wei
yangkovsky@gmail.com
Scientific supervisor: Dr. Sukhomlinov V.S., Department of Optics,
Faculty of Physics, Saint-Petersburg State University
Introduction
“Etching” is a key step in the planer process in nowadays industry, such as semiconductor and MEMS industries. In nowadays etching process it is necessary to use
many kinds of chemicals, including strong acid, which lead to pollution and impossibility of application to etch organic or bio materials. In order to eliminate these drawbacks,
we present a new etching technology, which uses only physical processes.
Nowadays etching technology and its drawbacks
Nowadays etching technology for planer process includes the following main
processes:
1. Photoresist coating: to coat photoresist on the surface of wafer;
2. Photoresist drying: to dry photoresist;
3. Exposure: to illuminate the wafer with a mask between the wafer and the light
source. On the mask there is necessary pattern, the light passes through the pattern
and “transfers” the pattern on the photoresist on the wafer;
4. Development: to put wafer with photoresist in some chemical to react with
photoresist. Parts of photoresist, which was illuminated, have different reaction.
As a result, after development, the property of the pattern is different from that of
resident parts of photoresist;
5. Etching: put the wafer in some chemical to remove the pattern or resident
parts;
6. Remove photoresist: to use some chemical to remove photoresist. Then one
can obtain the wafer with necessary structure on its surface.
Mention that in steps 1, 4, 5, and 6 chemicals are necessary. The use of many
kinds of chemicals leads to following drawbacks:
1. Pollution: It is very difficult and expensive to eliminate so many kinds of
chemicals. And usually such factories are placed in countries, where ecological
laws are still not very strict. As a result, “in practice” those chemicals are not
eliminated, but just “thrown out”, and lead to pollution to water and land in the
region of factories.
2. Lower benefit: due to the rise of environmental awareness, it is more and more
difficult to just “throw out” polluting chemicals. To eliminate these chemicals or
to pay the fine for making pollution both will lead to additional cost, which means
lower benefit.
3. Impossible to etch organic and bio materials: thanks to diversity of organic and
even bio materials, people may make many new products with new functions not
owned today, especially for medical purposes. However, organic and bio materi75
als can’t survive under nowadays etching process. For example, strong acid will
destruct organic and bio materials. Impossibility of applying etching process to
organic and bio materials leads to difficulty of their mass production.
There are some non-chemical etching technologies now, for example, to etch
material by laser beam, ion beam, or electron beam. However, they all have difficulties for mass production.
In this work we present a new non-chemical etching process, which allows one
not only to make prototypes, but to mass product.
Presented etching process
In Fig. 1 is shown the scheme of presented etching technology. Main concepts
of the process are as following:
Fig. 1. Scheme of presented etching
method. The upper part is a plasma
camber, while the lower part is the platform to carry wafer.
Fig. 2. The mask with necessary pattern
is a part (the wall) of the plasma chamber in Fig. 1.
1. The upper one is the chamber of plasma, while the lower one is the platform
to carry the wafer. The mask with necessary pattern (Fig. 2) is a part (the wall) of
the chamber of plasma.
2. When the plasma is formed in the chamber, ions will almost normally impact
to the mask and some of them will pass through the pattern. Thanks to special type
of plasma, the ion density can be high enough to provide high etching rate.
3. With optimized distance between the mask and the wafer, which is obtained
by considering gas pressure inside and outside the plasma chamber, ions passing
through the mask will almost normally impact to the wafer to etch it. As a result,
the pattern is “transferred” to the wafer and the wafer with necessary structure on
its surface is obtained.
Estimations of performance
According to our prior estimations:
1. The maximum etching rate is in the order of 1 μm/min, which is similar to
chemical etching rate.
76
2. Thanks to replacing 6 steps of nowadays chemical etching process to only 1
step, the necessary time for any etching process is decreased, too, which means that
the equivalent etching rate of our method may be faster than estimated above.
3. Physical limitation of etching accuracy of this etching process is about 5nm,
which supports 20nm process.
Expected superiorities
Presented etching technology has following superiorities:
1. Environmental friendly: There is no chemical used during the etching process,
which means almost zero pollution during the process. This process uses only
electric energy, whose pollution is much easily to be eliminated than chemicals.
2. The most important superiority of this method is the possibility to etch organic
and bio materials, which is easily destructed in nowadays etching process.
3. Combining with diversity of organic or bio materials, presented etching method
allows one to design and to mass-product many novel micro systems based on
organic and bio materials, with less pollution during the process. These superiorities are extremely important for medical applications.
4. Instead of many kinds of necessary specialists to guarantee the production line
work (for example, specialists of exposure, specialists of etching, specialists of
chemistry), presented etching process demands less kinds of specialists. This helps
to decrease personnel cost to handle the production line.
Plan of future work
The preparation of experiments is undergoing now by cooperating with company
“OOO IVS”. The equipment for experiment is under construction, and company
“OOO IVS” will provide plasma generator specially designed for our purpose.
Thanks to strong scientific background of Department of Optics in faculty
of Physics in Saint-Petersburg State University about plasma-surface interaction
and behavior of plasma itself, many possible problems of this etching method has
been predicted.
The first step of experiment is to confirm all predicted superiorities and problems, indeed, the etching rate and accuracy.
The second step of experiment is to optimize the process, for example, the
necessary plasma parameters and others.
77
Optimization of flow deceleration by MHD interaction
Yang Cheng Wei
yangkovsky@gmail.com
Scientific supervisor: Dr. Sukhomlinov V.S., Department of Optics,
Faculty of Physics, Saint-Petersburg State University
Introduction
This work relates to the technology to control flow by using magneto hydrodynamic (MHD) interaction.
Here we take care about the MHD interaction of gas flow with ionized gas
around a cylinder with current passing through its axis to form non-uniform
magnetic field, and whose axis is perpendicular to the flow direction (Fig. 1). The
effect of flow deceleration by MHD interaction is concerned. Such a situation is
interested in aviation application.
At first we found analytical approximation of distribution of Lorenz force
inside MHD zone in order to understand the physical processes inside the MHD
zone and to find ideas to optimize MHD flow deceleration.
Then we further considered the power consumption of ionizing system and
connected the power consumption with the effectiveness of deceleration.
Results of analytical
approximation
In our previous works [1, 2], by
solving equation of continuity of current inside MHD zone, the analytical
approximation for distribution of
electric field inside the MHD zone
was obtained for conductive and
non-conductive cylinder under the
assumption that the electron mobilFig. 1. Situation we considered. A cylinder ity and electron density inside the
in the transverse flow from left to right. A MHD zone are constants. From
current passes through the axis of cylinder the distribution of electric field in
to dorm magnetic field B. The ionized gas the MHD zone one can obtain the
is limited between radius r1 and r0. We con- distributions of current density and
sider 2 kinds of cylinder: conductive and Lorenz force. Then we define the
nonconductive.
parameter
r1
An ≡
∫ F V dr
r r
r0
(V0 B0 )
2
( r1 − r0 )
(1)
This is the dimensionless total work of Lorenz force in unit time, unit cross
section along decelerating line (φ=π) so can be considered as effectiveness of
78
flow deceleration. Pay attention that along decelerating line the work is negative.
In Fig. 2 shown the relation of value of An with relative width of MHD zone r1/r0
from 1 to 5, at Hall parameter 20,7, and 1 for conductive (dashed lines) and nonconductive (solid lines) cylinders.
0
20
7
An
− 0.05
1
− 0.1
1
2
3
4
5
r1/r0
Fig. 2. Relation of value of An with relative width of MHD zone at Hall parameters 20, 7, and 1. Solid lines correspond to non-conductive cylinder, while dashed
lines – conductive cylinder.
From here one can see that for flow deceleration, at first, non-conductive cylinder
is better than conductive one; second, Hall parameter decreases the decelerating
effect; third, for given parameter there is some intermediate relative width of MHD
zone, where the effectiveness is best.
The reason is as following: Lorenz force is a result of interaction of z-component
current, which comes from interaction of ionized flow with magnetic field, with
magnetic field. When there is radical current, it will interact with magnetic field
to decrease that z-component current, which leads to Lorenz force. For case of
non-conductive cylinder, current can’t pass through the surface of cylinder so that
near by the surface there is no radical current, while for case of conductive surface,
current can pass through the surface of cylinder so that near by the surface there is
radical current to decrease Lorenz force. Similar, the drawback of large Hall parameter is also because Hall effect leads to radical current. The influence of relative
width of MHD zone is more complicated: when the relative width is large, there
is more space to form radical current, and magnet field decreases with increase
of radius? Both of them lead to decrease Lorenz force. When the relative width is
too small, the radical velocity of the flow is small due to existence of cylinder, so
the MHD interaction is small, too. As a result, there is some intermediate relative
width, where the effectiveness of flow deceleration is best.
Consider the power consumption of ionizer
In previous work we further consider the influence of power consumption of
ionizer. For aviation application due to restrictions of volume and weight of onboard
79
systems, the power of ionizer can’t be too large. As a result it is reasonable to find
optimized parameters under given power consumption.
According to our previous work [3], the stationary electron density can be
represented as:


1 − AB A − B C − 1
(
)(
)(
)
1+ B
 (2)
n = n0 
+
2
1
+
A


1 + A) C (2 A + A + 2 )+ 3 A + B (C − 1)(1 − 2 A) 
(


where
n0 = q0 / Wi A =
kd N o2 + ν a
n0 β ei
B=
kd N o2
n0 β ei
C=
β ii
β ei
where Wi - energy cost to ionize the gas, kd - speed constant of destruction of negative
ions, νa - frequency of attachment of electron to molecule, βei, βii - speed constant
of electron-ion dissociative recombination and ion-ion dissociative recombination,
and q0 is the energy density deposited in the gas.
By this way, once the ionized region and output power of ionizer are given,
we know the energy density q0 and so as the stationary electron density. Using this
electron density in the term of conductivity in the analytical approximation, we can
connect the power consumption of ionizer with effectiveness of flow deceleration
by MHD interaction.
In Fig. 3 shown the relation of value
r1
A ≡ ∫ FrVr dr r0
to relative width of MHD zone from 1 to 5. Here A is total work of Lorenz
force to the flow in unit time, unit cross section along decelerating line, its
dimension is W/m2.
Fig. 3. Relation of value of A with relative width of MHD zone at different altitude
for case of conductive and non-conductive cylinders B0=2Tesla, q0 =1W/m3.
80
One can see that the best effects happen at some intermediate relative width
of MHD zone, and the value of optimized width is similar to that of our previous
work without consideration of power consumption. The reason of existense of
intermediate relative width is similar, but with a little rifference: at large relative
width, besides the reason in our previous work, the energy density deposited in
the gas is lower due to larger ionized zone with given output power. Moreover, the
effectiveness of flow deceleration is better at higher altitude because the higher is
the altitude, the higher electron density can be sustained by given output power.
Also we can see that non-conductive surface is better for flow deceleration. In
Fig. 4. shown th relative difference of value of A for conductive and non-conductive
cylinder
A
−A
error =
conduct
Anoncon
noncon
at different altitude. One can se that the influence of material of surface increases
with increase of altitude.
Fig. 4. Relation of error =
1 to, at different altitude.
Aconduct − Anoncon
to relative width of MHD zone from
Anoncon
Conclusion
We can make the following conclusions for optimization of maximum decelerating the flow by MHD interaction:
a) nonconductive surface is better than conductive one;
b) Hall effect decreases the MHD effect;
c) it is enough to locally ionize the gas instead of wholly ionization;
d) the optimized relative width(the ratio of outer radius of MHD zone to inner
radius) of MHD zone is about 1.5~2, no matter the power consumption of ionizer
is considered or not;
e) At high altitude, due to lower lose of plasma, higher electron density can be
sustained with given power consumption, the effectiveness of flow deceleration
is better.
81
References
1. Sheikin E.G., Yang Cheng Wie. // ЖТФ, 2013, т. 3, Вып. 1, с. 54-63.
2. Yang Cheng Wei, Sheikin E.G. Possibilities of MHD effect on the locally
ionized flow past a circular cylinder. 9th Workshop “Thermo chemical process in
plasma aerodynamics”, Saint-Petersburg, 2-6 July 2012.
3. Sheikin E.G. // Technical Physics, 2007, Vol. 52, No. 5, p. 537-545.
82
F. Optics and Spectroscopy
Destruction of polyelectrolyte microcapsules modified
with fluorescent dyes by laser irradiation
Marchenko Irina
iramarchenko85@mail.ru
Scientific supervisors: Prof. Plotnikov G.S., Department of General
Physics and Molecular Electronics, Faculty of Physics, Moscow
State University, Dr. Bukreeva T.V., Laboratory of Bioorganic
Structures, Institute of Crystallography RAS
Introduction
Polyelectrolyte microcapsules, fabricated by alternative adsorption of oppositely
charged macromolecules on colloidal particles, are prospective technological objects due to their monodisperse size distribution, simplicity of controlling their permeability and a wide variety of the wall materials. Inclusion of dye molecules into
the polyelectrolyte shell of capsules could provide sensitivity to the laser radiation
[1]. Irradiation of modified capsules by laser with wavelength corresponded with
the absorption band for the dye can result in effective excitation of dye molecules
and energy transfer to the capsule shell. It can lead to rearrangement and destruction of the shell. This method for remote release of the encapsulated material can
be used for inducing the drug action in particular place of an organism.
Results and Discussion
Polyelectrolyte capsules were produced using the technique of layer-by-layer
adsorption of oppositely charged polyelectrolytes on colloidal particles (cores) [2].
Calcium carbonate microparticles with medium size 4 µm were used as cores for
formation of the capsules. As polyanion for shell formation we used polystyrene
sulfonate (PSS) and as polycation polyallylamine (PAH) or polydiallyldimethylammonium (PDADMAC). Two different fluorescent dyes Rhodamine 6G and
Fluorescein isothiocyanate (FITC) were included in the capsules shell. Rhodamine
6G has a positive charge in aqueous solution, so it was embedded into polyelectrolyte shell by electrostatic interaction with negative charged PSS layer. The capsule
shell with composition (PDADMAC/PSS)2Rh(PDADMAC/PSS)2 was obtained.
For inclusion of FITC in the shell two different approaches were used. In the first
case, polyallylamine chemically bound with the dye FITC-PAH was included as one
layer of the shell. In this case the shell had composition (PAH/PSS)2/FITC-PAH/
PSS/(PAH/PSS)2. In the second case, the dye was adsorbed on the shell after the
layer of PAH and the shell with composition (PAH/PSS)2/PAH/FITC/PSS/(PAH/
PSS)2 was formed. The dye adsorption seems to be associated with intermolecular interactions with the polyelectrolytes of the shell e.g., with the formation of a
hydrogen bond between groups of the FITC and PAH molecules.
Measurements of absorption spectra of the obtained capsules showed that
capsules with Rhodamine 6G had absorption peak at wavelength 540 nm and
capsules with FITC - at 505 nm for adsorbed dye and at 515 nm for chemically
84
Fig. 1. Confocal fluorescent microscopy images of polyelectrolyte capsules with
dye molecules included in the shell: a) - capsules with Rhodamine 6G, b) - capsule
with chemically bound FITC and c) - capsule with adsorbed FITC.
bound dye (Fig. 2). To determine whether the capsules could be destroyed by
laser radiation, we subjected their suspensions to intensive laser irradiation with a
wavelength in the absorption band of the both dyes. The solutions with capsules
were irradiated for 5 min by the second harmonic of an LF114 Nd:YAG laser with
a wavelength of 532 nm, a beam diameter of about 7 mm, a pulse energy of 500
mJ, a pulse duration of 10 ns, and a pulse repetition rate of 10 Hz. Size distribution of the capsules before and after laser irradiation was calculated using the
correlation spectroscopy of the scattered light. Light scattering data of capsules
enables to determine the mean radius and the mean width of size distribution of
the capsules.
Fig. 2. Absorption spectra of capsules with dye molecules in the shell: a) - capsules with Rhodamine 6G, b) - capsules with FITC: 1- chemically bound FITC,
2- adsorbed FITC.
Size distributions of capsules with Rhodamine 6G are shown in Fig. 3. For
capsules without dye the mean radius is 1700 nm and the mean width of distribution is 1200 nm (curve 1). For capsules with Rhodamine 6G in the shell these
parameters are 2700 and 3100 nm (curve 2). Irradiation of the capsule suspension
by laser in absorption band of the dye molecules practically had no effect on the
size distribution of capsules without dye and significantly changed it for capsules
with Rhodamine 6G in the shell (curve 3). In the latter case the mean radius is
600 nm and the mean width of distribution is 450 nm. These results give evidence
85
of the destruction of the considerable part of capsules by laser radiation. It is also
confirmed by microscopic observation of the capsules.
Fig. 3. Size distributions of the capsules before and after laser irradiation: 1 - capsules without dye before laser irradiation, 2 - capsules with Rhodamine 6G before
laser irradiation, 3 - capsules with Rhodamine 6G after laser irradiation.
Size distributions of capsules with FITC are shown in Fig. 4. Curve 1 in Fig. 3
shows the size distribution of the microcapsules with the dye molecules adsorbed
in the shell. Irradiation by the laser did not almost affect the size distributions of
capsules with chemically bound FITC molecules, however, it cardinally changed
it for the capsules with the adsorbed FITC molecules (Fig. 3, curve 2). In the latter
case the average size of the scattering particles in the suspension is 480 nm. These
results show that laser irradiation led to
the destruction of the main part of the
capsules with adsorbed dye.
Such destruction can be caused by
effective absorption of the incident
radiation by the dye molecules and
following transfer of energy to matrix,
surrounding the molecules. It is possible
when oscillatory frequencies of dye and
polymer molecules coincide [3]. In this
case transfer of oscillatory energy by
inductive-resonance mechanism occurs.
The critical radius of the energy transport
Fig. 4. The size distributions of the cap- in such molecular systems is up to 5 nm.
sules with adsorbed FITC before (1) and If we take into account that the adsorbed
after (2) laser irradiation.
molecules of the dye and their clusters are
nonuniformly situated inside the polymer capsules, the above described process
can lead to nonequilibrium local heating of the polymer matrix, the breaking of
bonds and, as the result, destruction of the capsules. Failure to destroy capsules
containing chemically bound molecules of the same dye can be related to the
following reason: such molecules are more uniformly distributed in the polymer
layer, which can be seen from their absorption spectra, which are narrower as
86
compared to the spectra of adsorbed molecules (Fig. 2). The effective dissipation
of the photoexitation energy takes place over the vibrational levels of the polymer
molecules modified with the dye. As a result, rapid thermalization of the energy of
photoexcited molecules takes place over the entire capsule volume and nonuniform
heating and destruction do not occur. It can be important that the line of the laser
excitation falls at the edge of a narrower absorption spectrum of chemically bound
molecules, which lowers the efficiency of their excitation.
Conclusion
It was shown that irradiation of capsules with Rhodamine 6G and FITC molecules in the shell by laser in the range of absorption band of the dyes led to destruction of the capsules. Difference of effect of laser irradiation on capsules with
chemically bound and adsorbed dye molecules was shown. Thus embedding of dye
molecules into polyelectrolyte capsules makes possible to destroy the capsules in
a local area. It can be effectively used to release encapsulated drugs and reagents
in predetermined area of an organism.
References
1. Skirtach A.G., Antipov A.A., Shchukin D.G., Sukhorukov G.B. // Langmuir
20, p. 6988 (2004).
2. Suhhorukov G.B., Donath E., Davis S. et al. // Polym. Adv. Technol. 9, p. 759
(1998).
3. Plotnikov G.S., Zaitsev V.B. Physical Principles of Molecular Electronics. (in
Russian).- Moscow: Mosc. State Univ., p. 164 (2000).
87
Diffraction efficiency measurements of the Holoeye
Pluto SLM fed by blazed-profile pattern
Alexander Sevryugin, Konstantin Mikheev
sevr_sasha@mail.ru
S c i e n t i f i c s u p e r v i s o r : P r o f . Ve n e d i k t o v V. Yu . ,
St.-Petersburg State Electrotechnical University “LETI”
At the moment, thin dynamic holograms (DHs) find numerous applications
in solving various problems of applied optics, for instance, dynamic holographic
correction of distortions in optical and laser systems, processing of optical information, etc. It is known, however, that the diffraction efficiency (DE) of a thin
holographic grating recorded by the conventional “direct” method, i.e., as a pattern of interference of two laser beams, to the +1st and −1st orders of diffraction
cannot exceed 33% for a sinusoidal profile and 40% for a rectangular grating
(square wave). Correspondingly, the use of such a hologram as a corrector yields
high energy losses in the optical system. This problem can be solved by means of
holographic grating profile asymmetrization. In particular, for a phase transmission
grating which profile is formed by right triangles and the phase modulation depth
is 2π, its DE to the +1st or −1st order can reach 100%.
However, diffraction efficiency of currently popular electronically-addressed
spatial light modulators (EA SLM) is less than theoretical 100% when driven by
blazed-profile pattern. Apart from diffraction into higher orders caused by the lattice structure of the LCOS display itself, which leads up to about 40% losses, DE
of the Holoeye Pluto SLM is stated to be about 75% using 8-level blazed profile
and about 83% for 16-level blazed profile with 0 order intensity taken as a reference (100%).
To measure the real DE of the Pluto SLM by HOLOEYE Photonics AG (http://
www.holoeye.com/), which we possess, a simple experimental setup was build.
The Pluto SLM was illuminated by the collimated plain beam of He-Ne laser (633
nm) and fed by blazed-profile signal with 10 pixel period using Holoeye SLM
Application Software. This formed а grating with desired phase amplitude. The
output beams were focused by converging lens onto Lasercheck power meter with
an external diaphragm installed on it to pick out the 0 and ±1st orders of diffraction
and measure their intensities. Our Pluto SLM was calibrated prior the experiment
by modifying gamma curves and by limiting liquid cells (LC) driving voltage to
achieve 2π maximum phase shift. Our Lasercheck power meter provided us with
±5% accuracy, 0,01 μW resolution and 9,99 μW minimum full-scale power.
In Fig. 1 you can see ideal intensity versus saw-tooth phase grating amplitude
curves for different diffraction orders. Using the data obtained in this experiment
we plotted graphs of the intensity versus depth of modulation (Fig. 2, 3) for the
0 and ±1st orders respectively. Zero on the horizontal axis of the Fig. 2 and Fig.
3 corresponds to the 0 phase grating amplitude, and ±128 corresponds to the 2π
88
phase grating amplitude with different approach – (0 + ∆) and (2π - ∆), where ∆
is phase grating amplitude.
The diffraction efficiency was found to be 72,5% with good linearity , which
came close to our expectations.
Fig. 1. Ideal intensity curves.
Fig. 2. “0” order intensity.
89
Fig. 3. “1st” order intensity.
References
1. Holoeye Pluto Manual, v1.2 03/2011.
2. Venediktov V.Yu. // Optoelectronics, Instrumentation And Data Processing,
Vol. 48 No. 2 (2012).
3. Mark T. Gruneisen, Lewis F. DeSandre. // Optical Engineering, Vol. 43, No. 6,
(2004).
90
Application of Kerr effect for remote sensing of electric
fields over storm clouds
Egor V. Shalymov, Alina V. Gorelaya
ShEV1989@yandex.ru
Scientific supervisor: Prof. Dr. Venediktov V.Yu., St.-Petersburg
State Electrotechnical University “LETI”
Introduction
Quantitative knowledge of the magnitude as well as of the spatial and temporal
structure of electric fields above a thunderstorm, both prior to and immediately after
a lightning discharge, is crucially important for understanding the electrodynamic
effects of tropospheric weather on the upper atmosphere. At present such fields can
only be measured with local methods, either at ground level or by balloon-borne
instruments launched near or within the cloud. Such measurements do not allow accurate estimation of the overall structures inside and outside the cloud. The launch,
operation, and recovery of balloon- or aircraft-based experimental apparatus also
pose logistical problems, especially in the thunderstorm context. Using remote
sensing techniques can be overcome the problems described above.
Currently, remote sensing based generation of higher harmonics or the Stark
effect seem prospective. However, these methods require a special, powerful, and
above all expensive artificial sources.
We herein describe a recently developed method of remote sensing of thunderstorm electric fields by electrically-induced birefringence (Kerr) effects on
naturally polarized background skylight and as well as possibility of its practical
realization.
Sensing electric field using Kerr effect
A recently developed remote sensing method for measurement of near
thunderstorm electric fields by electrically induced birefringence (Kerr) effects
on natural sky polarization, providing detection of electric fields near or above
thunderclouds [1].
The Kerr effect is an electrically-induced birefringence that occurs when an
electric field partially distorts the electron structure of a material. The birefringence
results in a difference in index of refraction for waves polarized parallel and perpendicular to the applied field given by
Δnk=nKE2,
where n is the index of refraction, K is the Kerr constant, and E is the applied
electric field.
This results in difference in a phase shift
φk=2πdΔn/λ0,
where d is the path length and λ0 is the vacuum wavelength.
The effect is small in air and directly proportional to the density, with
K ≈ 2.3×10-25 (m2/V2) at sea level [2]. For light of wavelength λ0=500 (nm), we
91
have φk[rad] ≈ 3×10-9d[km]E[kV/m]n/n0, where n/n0 is the atmospheric density relative to
sea level, φk is measured in radians, d in km, and E in kV/m as indicated. Typical
static electric fields within a thundercloud are approximately E ≈ 100 kV/m), with
length scales d ≈ 1 (km), giving a nominal sea level phase shift of φk0 ~ 3×10-5n/n0
(e.g., φk ≈ 10-5 at 10 km altitude) [3].
The light source for this measurement is required to be stable and of a known
polarization state. Right-angle Rayleigh scattering of sunlight from air molecules
in the atmosphere, i.e. the clear blue sky in a band seen in directions perpendicular
to the sun, meets these requirements (see Fig. 1). The degree of linear polarization in this band can approach 100% (reaches 80% - 90%). The degree of linear
polarization is reduced by multiple scattering in the atmosphere and from indirect
illumination (by scattering off clouds, for example).
Light from
Sun
Linear polarization light
Rayleigh
scattering
Fig. 1.
Polarization of the sunlight scattered at an angle close to 90º to change when
passing close or over storm clouds. Light which is falling to the ground from the
sky from areas, adjoining on storm clouds, is partially elliptically polarized. The
possible geometry of measurements is shown in Fig. 2.
Region of
interest
Polarized band
To sun
Instrument
Fig. 2. Geometry of measurements.
Though small, Kerr effect can be measured by observing the polarization state
of scattered light. Besides this effect in atmosphere there are other phenomena
can cause birefringence. Strong air currents also can change natural sky light
polarization.
92
Birefringence caused by air flow
Polarization state of light scattered by atmosphere can changing under influence
of strong air currents [4]. This effect is observed in gases under following conditions: there are molecules with an anisotropic polarizability (eg, linear molecules);
there is preferential orientation of the molecules (full orientation is prevented by
the Brown motion); there is a deformation of preferential orientation, which is
caused by aerodynamic forces at relative displacement of adjacent layers of gas,
that is, if there is wind velocity gradient.
The above conditions are carried out within the Earth's atmosphere. Main
components of air (N2 and O2) are linear molecules. Strong air currents produce
preferential orientation of the molecules and deform it.
The value of the birefcaused by the air flow
Δnf=βG,
where β is coefficient of air flow birefringence, G is velocity gradient of wind.
Results of measurements β and description instrument to determine the values
of β have been published in [4]. Scheme of instrument is shown in Fig. 3.
Fig. 3. Scheme of instrument.
Linearly polarized laser beam L, passing the plate λ/4, becomes circularly
polarized. A circularly polarized beam propagating in direction of K and perpendicularly to the wind direction V. Under the effect of the stream birefringence the
light becomes elliptically polarized. Wind speed gradient data G collected from
4 pairs of propeller anemometers equally spaced along the path. Each pair being
fixed on the same post, the lower A2 and upper A1 anemometers had their axes at
0.25 m and 0.75 m above the ground while the light beam ran 0.5 m above the
ground. Ends 270 meter installation ellipsometer E which measures the polarization characteristics (Stokes parameters). Composition of the ellipsometer: wave
plate λ/4, an acousto-optic modulator and a photomultiplier.
According to the experimental data obtained in [4], values of the coefficient β
belong to the interval 3×10-14 – 5×10-14(s).
Air streams having impact on polarization state
Air flows along the way of scattered light affect its polarization. At sensing
the electric fields near the storm clouds on the polarization state can influence
atmospheric flows (see Fig. 4).
93
Birefringence
caused by air flow
Region of
interest
Polarized band
Atmospheric
flow
To sun
Instrument
Fig. 4.
Atmospheric flow is narrow strong stream with quasihorizontal axis, located
in the upper troposphere or stratosphere, characterized by large vertical and horizontal velocity gradients of wind (up to 0,01 s-1 or more) with the presence of one
or more maxima of wind speed.
Using the above relations, we can estimate the effect of atmospheric flow
on the polarization of scattered light passing through it. When β=5×10-14 (s) and
G=0,01 (s-1), value of birefringence: Δnf ≈ 5×10-16. For light with wavelength of
λ0=500 (nm) passing path of length l = 1 (km) through the airflow, nominal phase
shift: φ0f ≈ 6×10-6n/n0 (e.g., φf ≈ 2×10-6 at 10 km altitude). Thus, the phase shift
caused by the atmospheric flow, approximately 20% of the shift caused by Kerr
effect. In addition it should be noted that the contribution of air currents in the
phase shift value may increase with increase length of the path by which the light
crosses the stream.
Based on the above it is clear that effect of air flow on natural sky light polarization cannot be neglected. Application Kerr effect for remote sensing of electric
fields storm clouds in the form in which it is described in [1] is unlikely. However,
modifying this method may exclude the influence of the gradient of wind speed.
One of possible modifications is implementation of dual frequency measurements, i.e. the definition of the polarization states at different wavelengths. In this
case, probably, algorithmically exclude from results of measurements component
caused by air flows. Other possible variant is measurements during lightning discharges. In this case measurements are made before and directly after discharge.
After lightning discharge intensity of electric field of storm cell sharply falls. Value
of birefringence caused by Kerr's effect becomes negligible. State of polarization
of scattered light in this case is defined influence of air flow. Results measurements
state of polarization after lightning stroke are used for exception component of Kerr
effect from results measurements made before lightning discharge.
94
Conclusion
The aim of this work was to analyze possibility using Kerr effect for remote
sensing of electric fields thunderstorms. In this paper was reviewed by method
proposed in [1]. Some of the provisions of this article raised doubts. Therefore, we
have considered birefringence caused by the air currents. Approximate calculation of the phase shift induced by Kerr effect at thunderstorm (φk ≈ 10-5 rad) and
air flow (φf ≈ 2 × 10-6 rad) were presented. Since the values of the phase shifts
are comparable the effect of air flow on natural sky light polarization cannot be
neglected. Measuring the electric field of storm clouds using the method proposed
in [1] is impossible. However, modifying this method may exclude the influence
of the gradient wind speed.
References.
1. Carlson S.B.E., Inan U.S. // Geophysical Research Letters, Vol. 35, L22806,
doi:10.1029/2008GL035922, 2008.
2. Andrew J. Weinheimer // Journal of Atmospheric and Oceanic Technology,
3(1):175178, March 1985.
3. Kumada A., Iwata A., Ozaki K., Chiba M. Hidaka K. // J. Appl. Phys., 92,
2875–2879 2002.
4. Boyer G.R., Lamouroux B., Prade B. S. // J. Opt. Soc. Am., Vol. 68, No. 4,
April 1978.
95
G. Theoretical, Mathematical
and Computational Physics
Schwarzschild solution in R-spacetime
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 Schwarzschild metric in AdS-Beltrami spacetime. At first we write the AdS-Schwarzschild metric and then transform it into the
Beltrami coordinates [1, 2]. Furthermore the limit c → ∞ is considered to simplify
metric, and then the metric for the Schwarzschild solution in R-(Lorentz-Fock)
spacetime is presented [1]. We derive the equations of motion for massive particles
with the help of the Hamilton-Jacobi method [3]. Then we turn to the circular orbit
of the particle and find the time dependence of the radius.
1. Schwarzschild metric in AdS-Beltrami spacetime
There is one of important solutions in Einstein General Relativity with the
constant negative spacetime curvature R. It is called AdS-Schwarzschild solution.
We can represent this solution in the spherical coordinates system
xµ = ( x0 , r , θ , ϕ )
2

r
2M  2
dr 2
−
ds 2 = 1 − 2 −
dx
 0
r 

r 2 2M
 R
−
−
1

2
r
 R
2
2
2



− r 2d Ω2
(1)
2
where d Ω = dθ + sin θ dϕ .
Now we would like to introduce the coordinates and time transformations to
change a metric into the Beltrami coordinates [1, 2]:
x
r
r→
x0 → R tan 0 ,
(2)
R
In these coordinates the metric is as
dx 2 (xdx ) 2 Mh
ds = 2 − 2 4 − 4 dx02 −
h
Rh
rh0
2
2
1+
x
r2
cos 0
2
R
R
2

rx0 dx0  2M
 dr − 2 2  (3)
R h0 
 2Mh3  
rh 1 −
2 
rh0 

This solution is called the AdS-Schwarzschild metric in Beltrami coordinates
and for brevity called the AdS-Schwarzschild-Beltrami metric.
If we take the limit c → ∞, then metric (3) will become to
98
R 4  2 gt  2 R (tdr − rdt ) R 2 r 2
ds = 2 4 1 −
− 2 2 d Ω2  dt −
ct 
rR 
 2 gt  c t
.
c 2t 4  1 −

rR 

2
2
2
(4)
This is the Lorentz-Fock limit for the AdS-Schwarzschild-Beltrami metric and it
is called the Schwarzschild solution in R-(Lorentz-Fock) spacetime [1, 2]. In next
section we will solve the equations of motion for massive particle with the help
of the Hamilton-Jacobi method.
2. The equation of motion for massive particle in the Schwarzschild metric
in R-spacetime
It is known as that the Hamilton-Jacobi equation in General Relativity can be
written as [3]
∂S ∂S
(5)
g µν µ ν = m 2 c 2 , ∂x ∂x
Where g is the inverse metric to the Schwarzschild metric in R-spacetime,
and their elements can be expressed as


4 4
2 2
2

1
2 gt r
1 
ct
−c t
,
g rr =
1−
g 00 = 4
− 2

2 
μν
g 0r
R 1 − 2 gt
rR
3 3
ctr 1 ,
= 4
R 1 − 2 gt
rR
R
g θθ =


2 2
−c t
R2r 2
rR
,
g ϕϕ
R 1 − 2 gt 
rR 
2 2
−c t
= 2 2 2 .
R r sin θ
(6)
For convenience we consider the case θ=π/2 and our Hamilton-Jacobi equation
can be written out as
2


 t2
1
1 ∂S 
2 gt
∂S tr
t2
 2
 − 2 1−
+ 2
rR
r
2 gt ∂t R 1 − 2 gt ∂r  R
1−


rR
rR


2
 ∂S 
  −
 ∂r 
(7)
2
t 2  ∂S 
2
− 2 2
 =m .
R r  ∂ϕ 
This equation can be solved using the method of separation of variables. Present
the action as a sum of functions of individual variables [3]:
S = S1 (t ) + S 2 r + M ϕϕ .
(8)
(t)
99
We put the equation (8) in the equation (7), then a solution can be expressed as
M2
2 2
m R + ϕ 2
A
A2
1
x + M ϕ ,
−
S = − + ∫ dx 2
ϕ
2
x
t
R  xg 
g
1−
1 − 
x
x


(9)
where x=r/t, xg=2g/R and Mφ is an angular momentum of this system.
Now we can get two equations of motion from the two conditions. From the
first condition ∂S/∂A = const.
We get the equation of motion for the radial part:
R A
1
dx
=
.
(10)
2
2
t
m∫
 x 
M
 xg  A 
− 1 + 2 ϕ2 2  1 − g 
1 − 
2
x  m  m R x 
x 

From the second condition ∂S/∂Mφ = const .
We get the equation of motion for the angular part
Mϕ
ϕ = ∫ dx
x2
2

xg 
A2  M ϕ
2 2 
1
−
+
−
m
R





R 2  x 2
x 

.
(11)
Finally we turn to the problem of the motion in a circular orbit. To solve this problem it is need to rewrite the equation (10) in the differential form
1 dx
R
= − 2 A2 − U 2 ( x) ,
(12)
1−
where
xg dt
x
U ( x) = mR
At
2
 xg
1 −
x

M ϕ2 

 1 + 2 2 2  .
 m R x 
The circular orbit can be obtained from the conditions
dU
= 0 .
A =U ,
(13)
dx
There are two roots of the equation (13):
M2
r

3m 2 R 2 xg2
= 2 ϕ2 1 ± 1 −
( x± )circle =  
M ϕ2
 t ± circle m R xg 

 .


(14)
It is exactly that the point x+ corresponds to the stable circle orbit, and x- is vice
versa. We can put t=T+τ, where T is the cosmic time and τ is the time during our
era, and get the time dependence of the orbital radius:
100
 τ 
r± (τ ) = (r± )0 1 +  ,  T
where
(r± )0
M ϕ2T 
12m 2 g 2
=
±
−
1
1

2m 2 R 2 g 
M ϕ2
(15)

.


References
1. Manida S.N. Fock-Lorentz transformations and time-varying speed of light,
gr-qc/9905046 (1999).
2. Manida S.N. // Theor. Math. Phys.,169(2): 1643-1655 (2011).
3. Landau L.D., Lifshitz E.M. The Classical Theory of Fields 4-ed, 1980.
101
Optimization of calculations using Sector Decomposition
approach: 3-loops calculation of renormalization constants of
φ4- theory
Ivanova Ella
ellekspb@gmail.com
Scientific supervisor: Prof. Dr. Adzhemyan L.Ts., Department of
Theoretical Physics, Faculty of Physics, Saint-Petersburg State
University
Nowadays the method of renormalization group is generally accepted tool for
the study of second-order transitions and critical phenomena. It allows to justify
the critical scaling and one can make the calculation of the critical exponents
as an expansion in the formal small parameter ε. In this paper we calculate the
renormalization constants in the three-loop approximation of the most famous
models. These values are determined by multiple integrals, which are singularly
depended on ε That issue makes the integrals very difficult to find them numerically.
Renormalized action in the Feynman parameters appears in this way, corresponding
to 2 tails and 4 tails diagrams.
4 tails:
1
∫du ∫du
1..
0
∏u δ(u
∏Γ(l )
li −1
i
1
N
0
1
i
+ .. + um − 1) n
Au
D1d / 2
2 tails:
1
1
∫du1.. ∫duN
0
0
∏u δ(u
∏Γ(l )
li −1
i
i
1
+ .. + um − 1)
d
−1
2
1
ACu n
D
n- number of loops
d=(4-ε) – dimension of the space
n
n
εn
d
2
d
A=Г   Γ   2− n = Γ   2− n + ..
 2   2

εn
2
D= ∑ (ui1 ..uin )
C= ∑ (ui1 ..uin+1 )
As one can see one of the singularities on ε is allocated from Gamma functions, which appears in these integrals. Also we have the singularities within the
integrand. At this moment there is a problem of numerical integration. Sector
Decomposition method has been developing intensively during recent years. It
overcomes our difficulties with the allocation deductions at the singularities on ε,
102
which based on the representation of Feynman diagrams. We consider the method
of SD in details using a simple
example.
1
1
1
1
I = ∫dx1 ∫dx2 ∫dx3
d /2
(
x
x
+
x
x
1 2
2 3 + x1 x3 )
0
0
0
The integral is improper, and it is not taken in elementary functions. So we get
the zero in the denominator. Our range of integration is divided into sectors by the
method of SD. In each sector the singularity appears. If we make a substitution
of variables independently for each sector, we will allocate it. Let's discuss how
to do it. Consider a point in the region of integration. One can see that one of the
coordinates of this point will be greater than the other two, which means that the
integral can be written as the sum of three integrals.
I = I1 + I 2 + I 3
Let's consider (1,2) sector: we "make decomposition" by variable 1. x2 = x1 x2 ,
x3 = x1 x3.
With variables x1 , x2 , x3 integral is:
1
1
1
0
0
0
I1 = ∫dx1 ∫dx2 ∫dx3 x12
1
x14 −ε ( x2 + x2 x3 + x3 ) 2 − ε / 2
We see that we need to make another decomposition, for example by second
variable. x3 = x2 x3.
With variables x1 , x2 , x3 integral is:
1
1
1
0
0
0
I1 = ∫dx1 ∫dx2 ∫dx3
1
x12 −ε x2−1+ ε / 2 (1 + x2 x3 + x3 ) 2 − ε / 2
I = I12 + I13 + I 21 + I 23 + I 31 + I 32
At the end, we divide the region of integration into 6 sectors, in each of these
sectors there were the substitution of variables. Decomposition allocated singularities in the original integral and our operations allows us to express it through the
integral, convenient for numerical integration.
As a result:
Sector Decomposition method- is numerical method for extracting singularities
residues of diagrams
1) The integration region is divided into sectors, each containing a
singularity.
2) Сhange of variables (individual for each sector).
3) The number of decomposition corresponds to the number of loops in the
diagram.
4) The decomposition has removed the singularities in the original integral .
5) A large number of sectors.
The key practical interest of using this method is for integrals with high multiplicity. The main problem in these calculations is a huge number of sectors. In
the research papers of scholars there are several variants of so-called "strategy
of decomposition". The purpose of these variants is to minimize the number of
sectors.
103
The Department of Statistical Physics developed an approach which minimizes
the number of sectors by accounting symmetry of the Feynman diagrams that define
the renormalization constants.
Let's consider two examples: 2- and 3-loops diagrams. We write down the
denominators, which are corresponded to our diagrams.
D1 = u1u2 + u1u3 + u2 u3
I = I12 + I13 + I 21 + I 23 + I 31 + I 32 = 2( I12 + I13 + I 31 )
D2 = u1u2 u3 + u1u2 u4 + u1u3u4 + u1u3u4 + u2 u3u4 I = I123 + I124 + I132 + I134 + I142 + I143 + I 213 + I 214 + I 231 + I 234 +
+ I 241 + I 243 + I 312 + I 314 + I 321 + I 324 + I 341 + I 342 + I 412 + I 413 +
+ I 421 + I 423 + I 431 + I 432 = 4( I123 + I132 + I134 + I 312 + I 314 + I 341 )
After isolation of the singularities by SD in the first case we have a six terms,
and in the second we have 64. The time of calculation greatly increases in more
complicated diagrams. But if we consider symmetry, the number of terms will
decrease. The symmetry can be visually detected on this example.
Let's consider a more complicated example. This is a three-loop four-tails
diagram. One makes first decomposition. The symmetry is visible. But after this
operation we change our integrand. How to detect the following symmetry? In our
department developed a method to include all symmetries. One presents decomposition as color a line, on which we have decomposition. After we included all
symmetries instead of 96 terms, we have only four integrals.
104
D = u1u2 u4 + u1u2 u5 + u1u2 u6 + u1u3u4 + u1u3u5 + u1u3u6 + u1u4 u6 +
+u1u5u6 + u2 u3u4 + u2 u3u5 + u2 u3u6 + u2 u4 u5 + u2 u4 u6 + u3u4 u5 + u3u5u6
I = 24( I 241 + I 243 + I 246 + I 264 )
After all singularities were isolated, the new problem appeared. How can we
numerically integrate and do not lose the accuracy of the values?
We have integrals of two types.
1. One introduces a new parameter, and use that our ε greater than zero
1
∫u
−1+ εn
0
1
f (u ) du = ∫u −1+ εn ( f (u ) ± f ( 0))du =
0
1
1
0
0
= ∫u −1+ εn f ( 0) du + ∫u −1+ εn ( f (u ) − f ( 0)) du =
1
f (0)
∂ −1+ εn
+
u
f ( au ) duda
εn ∫0 ∫ ∂a
2. We use the fact that ε more than 1 (it is equivalent lambda regularization), and
after that analytically continue our result in the region, where ε is greater than 0.
105
1
∫u
−2 + εn
0
1
(
)
f (u )du = ∫u −2 + εn f (u ) − f (0) − uf ' (0) du +
0
1
+ ∫ ( f (0)u −2 + εn + u −1+ εn f ' (0))du =
0
1
= ∫∫
0
∂ −2 + εn
f (0) f ' (0)
u
(1 − a )∂ 2a f (au )duda +
+
∂a
nε − 1
nε
In this paper the effectiveness of the method SD is tested on calculation of
renormalization function of best-known model. The calculation was performed in
the three-loop approximation. The result, which was calculated with great precision, coincides with the previously known values.
Renormalization functions:
results of this work
known values
γφ
0.0834u2-0.062u3
0.0833u2-0.063u3
γτ
-u+0.833u2-3.503u3
-u+0.833u2-3.500u3
β
-2uε+3u2-5.66u3+32.56u4
-2uε+3u2-5.66u3+32.55u4
The Allocation singularities and the use of symmetry were made "by hand",
without using any computer programs, and only on the last step, the numerical
integration, used it . And the number of sectors in the decomposition was less
than in all known strategies. The number of nuances, which have appeared in this
work, would be considered in the higher orders of perturbation theory, where the
all operation would be made automatically.
The accuracy of the result gives us the confidence of successful further researches in case of the higher orders of perturbation theory.
106
Solving the time-dependent Dirac equation with the
B-spline basis set method
Ivanova Irina
ira.ivanova.v@gmail.com
Scientific supervisor: Prof. Dr. Shabaev V.M., Department of
Quantum Mechanics, Faculty of Physics, Saint-Petersburg State
University
Introduction
Collisions of heavy ions provide a unique tool for tests of relativistic and
quantum electrodynamics (QED) effects at strong fields [1].
In the pioneering work [2] it was presented that if the charge of a hydrogen-like
ion is larger than the critical value 173, energy of the ground 1s state reaches the
negative-energy Dirac continuum (Fig.1). But today only 120 chemical element
is synthesized. That’s why we cannot probe QED in supercritical Coulomb fields
for a single ion. The study of heavy ion collisions is necessary. When distance
between two colliding ions approaches the critical value ground-state level of
the electron dives into the negative continuum spectrum as for a single ion. For
example, in the U91+(1s)-U92+ collision the critical distance for the point nuclei was
found to be 36.8 fm [3].
Fig. 1. [1] Energies of the electron states in H-like ion as functions of ζ for extended nucleus, where ζ=αZ; plot is in units: ħ= c = me = 1.
In this way consideration of highly charged ion collisions is very important for
the investigation of QED effects at strong fields. For example, one of such effects
is a spontaneous electron-positron pair production. The present paper would be
helpful in understanding of the question how to catch QED effects by means of
calculation of different characteristics of the colliding process. In this work the
time-dependent Dirac equation in a central field for the U91+(1s)-U92+ collision has
been solved. The probability to find the electron after the collision in the ground
1s state, the probabilities of excitation and ionization have been calculated.
107
The problem and its solution
The time-dependent Dirac equation (in units: ℏ=1) is presented by

∂Φ (r , t ) ˆ 

i
= H Φ r , t , Hˆ = Hˆ 0 (r ) + Uˆ (r , t ).
∂t
( )
The Hamiltonian of the Dirac equation is considered as a sum of two parts.
The first term

Hˆ = c (αp ) + mc 2β + V (r )
c
0
is the Hamiltonian of the stationary Dirac equation (SDE):


Hˆ 0 φ (r ) = E φ (r ).
The second term Uˆ (r , t ) is a time-dependent potential.
When Û is absent, we have the SDE and its solution in a central Coulomb
field Vc(r) is four-component wave function

.
 1  G (r ) Ωκm (n )   r
φ (r ) = 
=
 , n
r  iF (r ) Ω −κm (n )
r
The separation of the radial variables leads to the radial equations which are
solved numerically. The solution of the radial SDE φ(r) is two-component wave
function:
 G ( r )
Hˆ r ϕ (r ) = Eϕ (r ), ϕ (r ) = 
.
 F (r )
The Hˆ is the radial Dirac Hamiltonian defined by
r
 2
 d κ
 mc + Vc (r ) c  − dr + r  
,
Hˆ r = 
  d κ

2
 c  dr + r  − mc + Vc (r )


φ(r) is approximated by the form
2n
ϕ (r ) = ∑ci ui (r ).
i =1
The two-component functions ui are assumed to be square integrable and linearly
independent. The use of these functions as a finite basis set constructed from B
splines is a principal part of the dual-kinetic balance (DKB) approach [4].
The total potential of this problem Uˆ (r , t ) + Vc (r ) is a central field, which is
the sum of the potential of the target nucleus and the potential of the projectile
nucleus (Fig. 2). The potential of the projectile is treated within the monopole
approximation which includes only the spherically symmetric part with respect to
the target. The target U91+(1s) is stationary at the origin. The projectile U92+ moves
along a straight-line trajectory with a constant velocityV, at the impact parameter
b, with energy E=6 MeV/u. The potential Uˆ (r , t ) is presented by
αZ
Uˆ (r , t ) = −
, Rmax (t ) = max {r , R (t )}.
Rmax (t )
R (t ) =
(Vt )2 + b 2 ,
108
V = 2E .
The wave function ψ(r,t) is a solution of the radial time-dependent Dirac
equation
∂ψ (r , t )
i
= [ Hˆ r (r ) + Uˆ (r , t )]ψ (r , t )
∂t
which contains only the radial variables.
The expansion of ψ(r,t) ψ(r,t) can be
written in such a form:
Fig. 2. The straight-line trajectory of
the ions collision. R(t) is the distance
2n
ψ ( r, t ) = ∑C j ( t ) ϕ j (r ),
between the target U91+(1s) and the
j=1
projectile U92+.
2n
where the basis ϕ j (r )
is the eigenfunctions of the radial SDE. The coefficients
j =1
Cj(t) are defined employing the Crank-Nicolson propagation scheme at each time
step using matrix M which consists of H:

 ∆t  
C (t + ∆t ) = M  t +  C (t ),

2
−1
∆t  ∆t   
∆t  ∆t  
 ∆t  
+
=
+
+
M t
I i H t
I − i H t +  .



2  
2 
2   
2 
2 
The matrix elements of H are given by
ϕ k (r )|Hˆ r (r ) + Uˆ (r , t )|ϕ n (r ) = En δ kn + ϕ k (r )|Uˆ (r , t )|ϕ n (r ) .
Results
The probability to find the electron after the collision in the ground 1s state
has been calculated:
ψ r, t = C t ϕ (r),
{
}
1s
( )
1s
()
1s
2
2
P1s = ψ (r , t )|ψ1s (r , t ) = C1s (t → ∞) .
The obtained values of P1s for different number of splines show a good convergence properties of basis set calculations (Table 1):
P1s
0.8820
0.8828
0.8828
0.8828
n
40
60
70
100
Table 1. The value of P1s for different number of splines n (b=38 fm).
after the collision in the ground 1s state decreases together with the impact
parameter b.
109
Fig. 3. |C1s(t→∞)|2 as a function of the collision time for different values of the
impact parameter b.
The obtained results are in a good agreement with the previous calculations [5]
(Table 2):
b,fm
P1s
[5]
20
25
38
50
100
0.7059
0.7756
0.8828
0.9317
0.9894
0.7072
0.7768
0.8838
0.9325
0.9895
Table 2. The value of P1s for different values of the impact parameter b.
Also in this work the electron probability to excite into 2s state after the collision,
the probability to find the electron after the collision in a bound state and the probability of ionization have been calculated:
P2s=0.0511,
Pbound=0.9410,
Pion=0.0583.
All probabilities are computed for the impact parameter (b=38 fm).
Conclusion
In this paper values based on the finite basis set method (DKB approach) of
some characteristics of the U91+(1s)-U92+ collision have been obtained. It was shown
how to receive the probability to find the electron in the ground state, probabilities
of excitation and ionization. The method used here is similar to the procedure for
calculating magnitude of QED effects in such collisions.
References
1. Eichler J., Meyerhof W.E. Relativistic Atomic Collisions. Academic Press,
New York, 1995.
2. Zeldovich Y.B., Popov V. S. // Usp. Fiz. Nauk 105, 403 (1971) [Sov. Phys. Usp.
14, 673 (1972)].
3. Rafelski J., Müller B. // Phys. Lett. B 65, 205 (1976).
110
4. Shabaev V.M., Tupitsyn I.I., Yerokhin V.A., Plunien G., Soff G. // Phys. Rev.
Lett. 93, 130405 (2004).
5. Deineka G.B., Mal’tsev I.A., Tupitsyn I.I., Shabaev V.M., Plunien G. // Russian
Journal of Physical Chemistry B 6 224228 (2012).
111
Crossing the boundary between parity breaking medium and
vacuum by vector particles
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
In Quantum Electrodynamics the interest to possible Lorentz and CPT Invariance
Violation was raised up after the seminal paper [1] where the very possibility to
have a constant vector background generating Lorentz and CPT parity breaking in
the large scale universe was conjectured and falsified. The latter was employed to
modify QED supplementing it with the Chern-Simons (CS) parity-odd lagrangian
spanned on a constant CS vector. However, the experiments showed that on scales
comparable to the size of the Universe axion fields are not observed. Nevertheless
spontaneous Lorentz symmetry breaking may occur after condensation of massless
axionlike fields [2] at space scales comparable with star and even galaxies sizes,
in particular, near very dense stars and black holes [3].
Another interesting area for observation of parity breaking is a heavy ion
physics. In the heavy ion collisions local parity breaking may occur in colliding
nuclei due to generation of pseudoscalar, isosinglet [4] or neutral isotriplet [5],
classical background in finite volume with magnitude depending on the dynamics
of the process.
We start from the effective Lagrange density which describes the propagation
of a vector field in the presence of a pseudoscalar axion-like background,
where Aµ and acl stand for the vector and background pseudoscalar fields respectively,
is a dual field strength. We have included the mass term for vector fields to account for parity breaking effects in the formation of massive vector mesons in
heavy ion collisions. If we consider a slowly varying classical pseudoscalar
background of the kind,
112
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, we get
the system of field equations,
We considered two different physical scenarios: the spatial CS vector and the
time-like CS vector. The first variant can help us to describe some phenomena in
astrophysics, while the latter is a model for fireball in central ion collisions.
Results for a spatial Chern-Simons vector
For the spatial CS vector (ζµ = (0, -ζ, 0, 0)) we found, that in the parity-odd
medium there are three different polarizations with three different dispersion
laws [6],
where kµ = (ω, k1, k2, k3) is a four-momentum. And each polarization has it's own
reflection coefficient for particles, which are escaping the area,
In order to obtain some plots we should rewrite our transversal polarizations
laws in terms of invariant mass,
Fig. 1. The reflection coefficient from the boundary for photons escaping the parity-odd region. The kinematically forbidden range of values of the invariant mass
is shaded.
113
Fig. 2. The reflection coefficient from the boundary for vector mesons escaping
the parity breaking region. The kinematically forbidden range of values of the
invariant mass is shaded. For vector mesons ζ = 300 MeV is taken.
Now we may build the dependence of the reflection coefficient on the invariant
mass for the photons. As one can see from the graph (Fig. 2), in case of the spatial
Fig. 3. Reflection of the photons with linear polarization. Dotted line corresponds
to the reflection coefficient for the process without polarization change. Dash line
is for the process with the change of polarization.
CS vector and the spatial boundary, for |M| >> | ζ| most photons escape from the
parity-odd medium, while a significant part of the photons with |M| ~ |ζ| for (-)
polarization and with |M| << |ζ| for (+) polarization reflects from the boundary
and does not escape from the medium.
It is also useful to build a plot for the vector mesons,
For the reflection of vector mesons there is one interesting fact - there is a range
of values where the total reflection takes place. This fact means that in this range
no vector meson can pass through the boundary.
It is no less interesting to consider another case, when a particle moves from the
vacuum to the parity-breaking medium. We will discuss this case only for photons.
Indeed, if we know how photons reflect from the boundary of axion star, we may
find specific optical effects of this kind of astrophysical objects. It is well-known,
that photons have two linear transversal polarizations in vacuum. We may con114
sider a situation, when a ray of polarized photons falls across the boundary from
vacuum, and we find an interesting result – part of incoming photons reflects and
their polarization changes.
At values k1 < ζ photons cannot penetrate into the parity odd medium (this
process is kinematically forbidden). However, starting from k1 = ζ photons may
cross the boundary, that is why in Fig. 3 one can see a cusp of the curves at this
value. Thus, low-energy photons are completely reflected from the boundary; with
increasing energy some of photons may change their polarizations; and only when
k1 ≥ ζ a part of photons may enter to the CS medium.
Results for a time-like Chern-Simons vector
The case of time-like CS vector (ζµ = (ζ, 0, 0, 0)) and spatial boundary may
be useful for a description of processes occurring in heavy ion collisions because
it helps to understand what happens with particles inside the fireball. In this case
dispersion laws change,
or, if we write them in terms of invariant mass,
Fig. 4. Reflection coefficient for vector meson. For vector mesons ζ = 300MeV is
taken.
The dispersion relations in these variables are the same for both transversal
polarizations again, however their domains are different. The reflection coefficient
coincides with the previous case (eq. (8)). But now the coefficient depends on k⊥.
Hence in case of time-like CS vector and the spatial boundary we get volume plot.
We may build it, in Fig. 4 there are shown two views on one plot. As in the case
of spatial CS vector, here one can see that at some values of invariant mass and
transversal momentum there is a total internal reflection. So, vector meson born in
the fireball cannot leave it, because it undergoes a total reflection at these values.
All calculations can be found in [7].
115
Conclusion
In this paper we studied the model of physical phenomena related to the propagation of photons and vector mesons between parity breaking medium and vacuum.
Main results were obtained for the spatial CS vector. The relations are presented
which are suitable to calculate the passage or reflection of incoming or outgoing
particles from the domain for each polarization. In particular it was shown that
transversal polarizations undergo strong reflection (up to the total internal one) at
certain values of frequency. The analogous relations are found for the time-like
CS vector but with a spatial boundary. In addition, for the spatial CS vector it was
revealed that during the irradiation of parity odd medium by photons the linear
polarizations can mix, thereby an additional rotation of circular polarizations may
take place upon reflection from the boundary.
Acknowledgements
The work was partially supported by the non-profit foundation “Dynasty”. I am
grateful to Prof. Dr. A.A. Andrianov for productive and valuable collaboration.
References
1. Carroll S.M., Field G.B., Jackiw R. // Phys. Rev. D 41, 1231 (1990).
2. Andrianov A.A., Soldati R. // Phys. Rev. D 51, 5961 (1995).
3. Schunck F.E., Mielke E.W. Class. Quantum Grav. 20 (2003) R301.
4. Andrianov A.A., Andrianov V.A., Espriu D., Plannels X. arXiv:1010.4688v1
[hep-ph]; Phys. Lett. B 710 (2012) 230-235.
5. Andrianov A.A., Espriu D. // Phys. Lett. B 663, 450 (2008); Andrianov A.A.,
Andrianov V.A., Espriu D. // Phys. Lett. B 678, 16 (2009).
6. Andrianov A.A., Kolevatov S.S., Soldati R. // Journal of High Energy Physics,
11 (2011) 007 [arXiv:1109.3440].
7. Andrianov A.A., Kolevatov S.S. [arXiv:1212.5723] [hep-ph]
116
Instrumental measurements of rogue waves in the
southeast area of Sakhalin Island
I.S. Kostenko, A.V. Yudin, K.I. Kuznetsov, V.S. Zarochintsev
Irenka_k@rambler.ru
Scientific supervisor: Dr. Zaytsev A.I., Special Research Bureau
for Automation of Marine Researches, Far Eastern Branch of
Russian Academy of Sciences, Uzhno-Sakhalinsk.
The ocean plays important role in human life: minerals are produced in the
ocean shelf; goods are delivered by the seas and oceans to remote areas of the
country and in other countries; fishing and extraction of other biological resources
is carried out.
However, the organization of work in the waters of the seas and oceans involves
great risk of storms, hurricanes, typhoons, seiches in ports, etc. The main of them
are tsunami, storms and rogue waves.
Tsunamis are very long waves. They produced usually as a result of the earthquake. Some of powerful tsunamis in Pacific Ocean were: Kamchatka tsunami
1952, Chilean tsunami in 1960 and 2010, Japanese Tsunami 2011. In the open
sea tsunami has a small height – about 1-2 meters, but on the coast wave height
is increasing by several times. We can calculate nature of generation and tsunami
wave propagation through the ocean using sea-level data from the deep DART
gauges and mathematical models of simulation of tsunami propagation. Software
system NAМI DANCE is used for modeling tsunami in Special Research Bureau
for Automation of Marine Researches (Sakhalin, Russia) and in some other countries. Unfortunately it is not always possible to prevent the destruction of the coast
and the loss of life.
However abnormally large waves or rogue waves is unpredictable phenomenon.
These waves suddenly appear on the sea surface, towering over the rest of the waves
in two and more times and also quickly disappear. Sometimes this is a series of
several waves. There are many cases of ships destruction and their flooding after
meeting with rogue wave as the pressure of fallen water is huge. We know how
much damage abnormally large waves cause but still accurately determine the nature
and cause of it is difficult. The main feature of such abnormally large wave is its
high altitude and steepness in comparison with the surrounding waves [1, 3].
Despite the fact that the number of victims of rogue waves smaller than tsunami
it is impossible to predict the occurrence of such wave and therefore it is very difficult to protect themselves from it.
According statistics about half of the cases of abnormally large waves were
detected on coastal waters [2].
Offshore Sakhalin international companies extract oil and gas and there are
many promising areas that will be developed in the future. Therefore the study of
marine hazards, especially abnormally large waves, is an actual problem.
117
Special Research Bureau for Automation of Marine Researches conducts
research of wave regime in the southeast part of Sakhalin. Series of gauges are
located in the south-eastern part of Sakhalin Island.
Fig. 1. Location of gauges in the south-eastern part of Sakhalin Island.
In this work the data present from gauges which were located near Ostry Cape,
Svobodny Cape and Aniva Cape (Fig. 1). The main instrumental device for measurement of surface wave was autonomous station ARV-K12 (Fig. 2), which was
located at sea bottom (depth 15-20 meters) and measure bottom pressure. Using
this gauge we measured fluctuations of pressure at the sea bottom and then we get
amplitude of real waves after recalculation [4].
Fig. 2. Construction of autonomous station ARV-K12.
Fig. 3 shows records of abnormally large waves from the gauge near the
Ostry Cape.
118
Fig. 3. Records of sea level from gauge near Ostry Cape the 27-th of July 2006.
Of course such waves are dangerous for small and medium ships, up to complete crash of a ship.
Fig. 4 shows records of abnormally large waves from the gauge near the
Svobodny Cape.
19-th of
November 2011
(4,85 m)
a.
8-th of
December 2011
(4,48 m)
b.
24-th of
December 2011
(6,23 m)
c.
25-th of
December 2011
(6 m)
d.
Fig. 4. Records of abnormally large waves from the gauge near the Svobodny
Cape.
In Figs. 4a and 4b wave height is small. But in comparison with the rest of the
waves in this period they are much higher and have no frequency of occurrence,
and therefore fall under the concept of abnormally large waves.
119
In Figs. 4c and 4d shown the records of abnormally large waves in December
2011. During this period there was a storm in the sea.
In Fig. 5 shown records of sea level in the area of Aniva Cape. The sea in this
period was calm. Wave height is small. However, these waves are much higher
than the maximum other waves, so they can also be attributed to the abnormally
large waves.
22-th of June
2009
1-th of July
2009
Fig. 5. Records of sea level near the Aniva Cape.
Computational experiments were performed of abnormally large waves in the
south-eastern part of Sakhalin. It was found that the analysis of records of sea level
can be detected only from 2% to 6% of the freak waves that occur in the area.
The analysis of the measurements obtained from the records in the southeastern part of Sakhalin is performed regularly. They noted the appearance of
abnormally large waves about 2 times a day, with no regularity in the period of
its birth [5, 6].
Conclusion
Rogue waves as a phenomenon are studied not enough, many aspects of this
phenomena, causes and conditions of their generation, frequency of occurrence of
such waves in different parts of the ocean have not realized fully.
Necessity of development of modern coastal infrastructure dictates rigid requirements to quality of operational and statistical information on wave modes in
studying regions that is impossible without work on long high-precision experiments, numerical modeling and development of systems of monitoring. Continuous
monitoring of the marine hazards in port bays and other coastal areas represents
great scientific interest and has the expressed applied aspect which is connected
with safety in zones of sea activity. Marine hazards which disasters to Far East
coast of the Russian Federation (tsunami, storm, abnormal waves) were presented
in this work.
References
1. Kharif C., Pelinovsky E., Slunyaev A. Rogue Waves in the Ocean. Springer,
2009.
2. Nikolkina I., Didenkulova I. // Nat. Hazards Earth Syst. Sci., 11, 2913-2924,
doi: 10.5194/nhess-11-2913-2011, 2011.
120
3. Pelinovsky E., Shurgalina E., Chaikovskaya N. // Nat. Hazards Earth Syst. Sci.,
11, 127-134, doi:10.5194/nhess-11-127-2011, 2011.
4. Zaitsev A.I., Kostenko I.S., Leonenkov R.V., Kuznetsov K.I., Giniyatullin A.R.,
Panfilova Y.A. Organizing in situ measurements of surface waves in the coastal zone
of Sakhalin Island // Transactions of Nizhni Novgorod State Technical University
n.a. R.Y. Alekseev / NSTU n.a. R.Y. Alekseev. - Nizhni Novgorod, 2012. № 4
(97). – P. 74-82.
5. Zakharov V., Shamin R. // JETP letters, 91, 62–65, doi:10.1134/
S0021364010020025, 2011.
6. Zakharov V., Shamin R. // JETP Letters, 96, 66–69, doi:10.1134/
S0021364012130164, 2012.
121
Calculation of propagator asymptotics in logarithmic
dimensions for the models φ3 and φ4 by means of
renormalization group method
Artem Pismenskiy
artem5085@mail.ru
Scientific supervisor: Prof. Dr. Pis’mak Yu. M., Faculty of Physics,
Saint-Petersburg State University
In this work we apply the renormalization group equation [1]


∂
∂
 − p ∂p + β (g ) ∂g + 2γ (g ) − 2 f ( p, g ) = 0
(1)
for investigation of asymptotical behavior of propagators for quantum field theory
models in logarithmic dimension. In the equation (1), the following notations are
used: p is a momentum, g is a coupling constant, β(g) is beta-function, γ(g) is the
anomalous dimension of a field and f(p,g) is a propagator.
We study the asymptotic behavior of the propagator f(p,g), when the coupling
constant g is fixed and the momentum p goes to zero (infrared asymptotic) or to
infinity (ultraviolet asymptotic).
The solution of the equation (1) is the following [1]:
 g ( s , g ) 2γ ( x) 
f (s, g ) = f 0 s −2 exp  ∫
dx  ,
 g β( x) 
where s=p/μ – dimensionless momentum (μ is a scale parameter); f0 is constant, it
depends on g, but it does not depend on s; g ( s, g ) is an invariant charge, which
is defined by the equation [1]:
g (s, g )
ln s =
∫
g
dx
, g (1, g ) = g .
β( x )
(2)
The renormalization group equation allows us to get only one of two asymptotics. If |ln s| →∞ the integral in (2) diverges on the top limit. The invariant charge g̅
goes to one of zeros of the beta function g* (fixed point): β(g*). In the logarithmic
dimension the beta-function has the form:
β(g) = b2g2 + b­3g3 +…,
and we receive:
g
g
dx
dx
1
1
=∫
=−
+
+… .
2
β
(
x
)
b
g
b
b
x
+
…
2
2g
g
g 2
ln s = ∫
The fixed point of the beta-function is g*=0. Thus, we have g̅→0. The coupling
constant g is fixed. Therefore in the main approximation we obtain:
1
ln s = −
+… .
b2 g
122
Let’s consider the φ3-theory of a scalar field φ:
Lint = (λ/3!) φ3.
The logarithmic dimension is d=6. We investigate the theory with an imaginary
coupling constant λ. It holds [1]:
β(g) = b2g2 + b­3g3 + b­4g4 +…, γ(g) = c1g + c2g2 + c3g3 +…,
2
where g~λ <0. Because we have g̅ (1,g)=g<0, the invariant charge g̅ is always
negative and g̅ →0 in the asymptotic. The coefficient b2 is negative (b2<0). We
receive the infrared asymptotic: ln s→−∞.
The asymptotical behavior of the propagator for the φ3-theory is the following:
f (s, g ) = f 0 s −2 ln s
−
2 c1
b2
∞ n

ln m ln s 
1
W
g
.
+
(
)
 ∑∑ nm
ln n s 
 n =1 m = 0
The coefficients Wnn are universal, they do not depend on g and they are expressed
in terms of b2, b3 and c1:
 2c

Γ  1 + n
n
  b3 
1  b2
Wnn =
.
n!
 2c1   b22 
Γ
 b2 
The coefficients Wnm with m<n are not universal, they depend on g and they can
be calculated in framework of perturbation theory. We have computed the coefficients Wnm up to n=2:
W11 =
2b3 c1
,
b23
W10 =
2 (b3 c1 − b2 c2 ) 2c1
−
B (g ) ,
b2
b23
W22 =
b32 c1 (b2 + 2b3 )
,
b26
W21 =
2
2(2b32 c12 − b22 b3 c2 − 2b2 b3 c1c2 ) 2b3 b2 c1 + 2c1
−
B( g ) ,
b26
b24
W20 =
−b2 b32 c1 + b22 b4 c1 + 2b32 c12 − b22 b3 c2 − 4b2 b3 c1c2 + 2b22 c22 + b23 c3
+
b26
+
(
2 −2b3 c12 + b22 c2 + 2b2 c1c2
where B (g ) = −
b24
(
)
)B (g )+ b c + 2c B (g )
b
2 1
2
2
2
1
2
,
b
b2 − b b
1
− 32 ln (b2 g )+ 3 3 2 4 g + …
b2 g b2
b2
The numerical values for the coefficients of the beta-function and the anomalous
dimension are equal to:
3
125
1
13
b2 = − , b3 = −
, c1 = , c2 =
,
2
72
12
432
123
and the approximated result for the propagator in the φ3-theory is the following
125 ln(− ln s )
1/9 
f (s, g ) = f 0 s −2 (− ln s ) 1 +
+
 1458 ln s

 43 1  2 125  3
 1

+
+ 
+
+ … .
ln  | g | + … 


  ln s
2

 729 9  3 g 162
Now let’s consider the φ4-theory of an nφ-component vector field φ:
Lint = (g/4!) (φ2)2.
The logarithmic dimension is d=4. The coupling constant g is positive (g>0). We
have:
β(g) = b2g2 + b­3g3 + b4g4 + …, γ(g) = c2g2 + c3g3 + c4g4 +…
In the asymptotic it holds g̅ →+0. The coefficient b2 is positive (b2>0). We get the
infrared asymptotic: ln s→−∞.
The asymptotical result for the propagator has the form:
∞ n

ln m (− ln s )
f (s, g ) = f 0 s −2 1 + ∑∑Wnm (g )
.
ln n +1 s 
 n=0 m=0
The coefficients Wnn are also universal, they aren expressed in terms of b2, b­3 and c2:
2c b
Wnn = − 22n + 23 .
b2
The coefficients Wnm with m<n are presented as perturbation theory series in
the coupling constant g.
2c
W00 = − 22 ,
b2
W11 = −
W10 =
2b3 c2
,
b24
2c22 + b2 c3 − b3 c2 2c2
+ 2 B (g ) ,
b24
b2
W22 = −
2b32 c2
,
b26
W21 =
4b3 c22 + 2b2 b3 c3 4b3 c2
+ 4 B (g ) ,
b26
b2
W20 =
4b32 c2 − 4b2 b4 c2 + 6b3 c22 − 4c23 + 2b2 b3 c3 − 6b2 c2 c3 − 2b22 c4
−
3b26
4c22 + 2b2 c3
2c
2
B (g ) − 22 B (g ) ,
b24
b2
b
b2 − b b
1
− 32 ln (b2 g )+ 3 3 2 4 g + …
where B (g ) = −
b2 g b2
b2
−
124
The analytical forms for the coefficients of the beta-function and the anomalous
dimension are the following:
b2 =
nϕ + 8
3
, b3 = −
3nϕ + 14
3
, c2 =
nϕ + 2
36
, c3 = −
(nϕ + 2)(nϕ + 8)
432
.
The infrared asymptotic of the propagator in the φ4-theory is the following:

nϕ + 2
1 3 nϕ + 2 3nϕ + 14 ln (− ln s )
f (s, g ) = f 0 s −2 1 −
+
+
2
4
ln 2 s
 2 n + 8 ln s
2 nϕ + 8
ϕ
(
(
)
(
)(
)
)

 − n3 + 20n 2 + 152n + 216

nϕ + 2
ϕ
ϕ
ϕ
 1 + … ,
B
g
+
+
(
)
4
2

 ln 2 s

16 nϕ + 8
2 nϕ + 8



3 3nϕ + 14  nϕ + 8 
3
+
ln 
g + …
where B (g ) = −
2
 3

nϕ + 8 g
n +8
(
(
)
)
(
(
ϕ
)
(
)
)
Conclusion
Using the renormalization group equation we have calculated the infrared
asymptotic of the propagator in the φ3- and φ4 -theories. In the φ3-theory the
propagator in the main approximation is power function with logarithm, whereas
the main approximation of the propagator in the φ4-theory is power function.
Corrections in both cases are expressed through the logarithm and logarithm of
logarithm of momentum. The more terms of the beta-function and the anomalous
dimension we know the more accurate result can be obtained. Some terms are
universal, they are expressed through the coefficients of the beta-function and the
anomalous dimension. Other terms are presented as perturbation theory series in
the coupling constant g.
References
1. Vasil'ev A.N. The field theoretic renormalization group in critical behavior
theory and stochastic dynamics. -SPb, 1998.
125
Estimate of the dilepton invariant mass spectrum in
B+→ π+μ+μ- using data in B0 → π-ℓ+νℓ and heavy quark
symmetry
Rusov Aleksey
rusov@uniyar.ac.ru
Scientific supervisor: Dr. Parkhomenko A.Ya., Department of
Theoretical Physics, Faculty of Physics, Yaroslavl P.G. Demidov
State University
Introduction
The physics of B-meson plays a fundamental role both in the precision tests of
the Standard Model (SM) and in the search of New physics. In this connection it is
important to determine different characteristics of B-meson decays, such as branching
fractions (BFs), differential distributions, CP-asymmetries, etc. with high accuracy.
The interest in B-physics is greatly stimulated by the experiments at the B-factories,
BaBar and Belle, and the LHCb at the LHC, which provide a huge amount of new
and accurate experimental data concerning B-meson and b-baryon decays.
In this work we give a short overview of the experimental data and theoretical
estimates concerning the B0→π–l+νl and B+→π+l+l– decays, with the view of using
the former to make a precise prediction for the latter. In particular, we extract the
form-factors in B0→π–l+νl using data and heavy quark symmetry to calculate the
dilepton mass spectrum in B+→π+μ+μ– in the range 4m2μ ≤q2≤8 GeV2.
The B0→π–l+νl decay and form-factor f+(q2)
Measurements of B+→π+l+l– decay allow to extract the CKM matrix element
Vub and the f+(q2) form factor shape (see Eq.(1)). The decay rate of the process
B0→π–l+νl takes the form:
,
(1)
where GF is the Fermi coupling, mB is the B-meson mass, ℓ = e, μ is the charged
lepton flavor (masses mℓ of the light leptons are neglected), q = Pℓ + Pν is the total
four-momentum of the final-state lepton and neutrino with the four-momenta Pℓ
and Pν respectively, and
.
(2)
The function f+(q2) defines the form-factor of the B0-meson decay B0→π–l+νl ,
entering the B→π transition matrix element:
. (3)
Here, PB and Pπ are the four-momenta of the B- and π-mesons respectively and mπ
is the pion mass. In practice, only f+(q2) is measurable in B0→π–l+νl decays with
the light e or μ leptons, since the contribution of the f0(q2) form- factor to the decay
rate is suppressed by the factor m2l/m2B and the second term on the right-hand side
of (3) can be neglected.
126
From Eq.(1) one can see that the B 0 →π – l + ν l differential decay rate
The form-factor is a non-perturbative quantity and can be
calculated, for example, by the model-dependent Light Cone Sum Rules (LCSRs)
or in Lattice QCD (LQCD). There also exist theoretical predictions concerning
the form-factor shape which involve some phenomenological parameters. The
most popular ones are the Becirevic-Kaidalov (BK), the Ball-Zwicky (BZ), the
Bourrely-Caprini-Lellouch (BCL) and the Boyd-Grinstein-Lebed (BGL) parametrizations. In terms of the BGL parametrizations the form-factor f+(q2) takes the
following form [6]:
,
(4)
,
(5)
where the fitted parameters are ak (k = 0,1, … kmax), q0 is a free parameter,
P(q2)=z(q2, m2B*) is the so-called Blaschke factor, and ϕ(q2, q20) is an arbitrary
analytic function. The shape of form-factor is given by the values of ak.
Experimental data on B0→π–l+νl decay
The partial branching fraction (BF) of B0→π–l+νl has been measured by the
CLEO, BaBar and Belle collaborations. As the BF is proportional to |Vub|2f+2(q2),
experimental data allows to extract the form-factor shape accompanied by |Vub|.
In Fig. 1 we show the partial branching fraction of B0→π–l+νl measured by the
BaBar collaboration. The experimental data from the Belle collaboration [3] are
shown in Fig. 2. In all plots, the data points are placed in the center of each bin and
the error bars include total (statistical and systematic) uncertainties. One can see
that all these measurements are in excellent agreement with each other.
Fig. 1. Partial ΔB(q2)/Δq2 spectra in 12 bins of q2 for the B0→π–l+νl decay measured by the BaBar collaboration. The left plot corresponds to the data analysis
of 2011 [2] and the right one to the analysis of 2012 [4].
.
127
Fig. 2. Partial ΔB(q2)/Δq2 spectra in 13 bins of q 2 for the
B0→π–l+νl decay measured by
the Belle collaboration [3].
Extraction of the f+(q2)
form-factor shape
We follow the standard
procedure of deriving best-fit
values applying to the χ 2distribution function, which
takes the form:
,
(6)
where N is the number of experimental points, yi imply the experimental values
of partial branching fractions in bins of q2, σi are the corresponding uncertainties, F(xi; α1,…, αk ) imply theoretical estimates of the partial branching fractions
ΔB(q2)/Δq2 for the given parametrization:
,
(7)
where xi and ai are the center and the range of the i-th bin and ΓB is the total decay
width of the B-meson.
The best-fit values of the parameters α1,min…, αk,min ) α1,min…, αk,min are extracted
by the standard procedure of the χ2-function minimization. The results obtained by
using the four form-factor parametrization for combined experimental data obtained
by the BaBar and Belle collaborations are presented in Fig. 3. .
Fig. 3. Partial ΔB(q2)/Δq2 spectrum for the B0→π–l+νl decays and the f+(q2) formfactor shapes multiplied by |Vub|. Results are obtained by combining the experimental data of BaBar 2012 [4] and Belle 2011 [3]. The curves show the results
of the fit to the data: BK (thick dotted blue line)),BZ (thick dashed purple line),
BGL (thick dot-dashed yellow line), BCL (thick solid green line) parametrizations. The thick solid black curve corresponds to the form-factor shape extracted
from LCSRs [8].
128
The B+→π+l+l– decay in heavy quark symmetry limit
Let us introduce the effective weak Hamiltonian encompassing the 567 transitions [9]:
,
(8)
where Vtb and Vtd are the CKM matrix elements, Ci (i = 1, …, 10) are the Wilson
coefficients depending on the renormalization scale μ, Oi (i = 1, …, 10) are the
dimension-six operators at the scale μ. Exclusive decay B+→π+l+l– is described
in terms of matrix elements of the operators Oi between meson states, which are
expressed in terms of several independent form factors. For the process B+→π+l+l–
the non-vanishing matrix elements are [9]:
,(9)
where f+(q2), f0(q2) and fT(q2) are the B→π transition form factors.
The B+-meson is a bound state of the heavy b̅ - and light u-quarks, hence one
can apply the so-called heavy-quark symmetry (HQS), which is valid in the large
recoil limit (at small values of q2). Applying heavy-quark symmetry results in
reducing the number of independent form factors of the B→π transition from
three to one. As it is shown in [10], in the HQS limit (without taking into account
symmetry-breaking corrections) the relations between the three form-factors f+(q2),
f0(q2) and fT(q2) are the following:
(10)
As one can see, in this case there is only one independent form factor f+(q2), the
shape of which can be extracted from the analysis of the B0→π–l+νl process. The
decay rate of B+→π+l+l– in the HQS limit is simplified and takes the form:
where
(11)
and λ(q2) is defined in Eq.(2).
Using the f+(q2) form-factor shape extracted in terms of the BGL parametrization
and the combined BaBar and Belle data and the numerical values of the different quantities entering (11) from [1], we have obtained the following branching
fraction distribution (for q2≤8 GeV2) shown in Fig. 4. The numerical value of the
B+→π+μ+μ– partial branching ratio in the range 4mμ2≤ q2≤8 GeV2 is given below:
129
(12)
Fig. 4. The branching fraction distribution for the
B+→π+μ+μ– decay in the
HQS limit for q2 values in
the range 4mμ2≤ q2≤8 GeV2.
Green curves correspond
to errors in one σ-range.
Conclusion
We have made a short discussion concerning the B0→π–l+νl and B+→π+l+l–decays.
From the data analysis by the BaBar and Belle collaborations we have extracted
the f+(q2) form-factor shape in terms of four different form-factor parametrizations.
It was found that the least χ2 was achieved with the Boyd-Grinstein-Lebed (BGL)
parametrization [6].
Also the numerical value of the branching fraction for the B+→π+l+l– decay given
in [7] was reproduced and compared with the experimental one measured recently
by the LHCb collaboration [5]. To reduce the number of unknown independent
form factors of the B→π transition, the heavy-quark symmetry was applied and
the expression for the decay rate was obtained in the HQS limit which is valid only
for small q2 values but involves only one independent form factor f+(q2). Using
the results of the form-factor extraction and the HQS-limit expressions we have
obtained the prediction (in the framework of the SM) for the partial branching
fraction for q2 ≤8 GeV2, which is expected to be measured by the LHCb collaboration soon. Predicting the total branching ratio in the B+→π+l+l– decay involves
some approximations due to modeling the f0(q2) and fT(q2) form-factors' shapes in
the range of large q2.
References
1. Beringer J. et al. (Particle Data Group) // Phys. Rev. D86, 010001 (2012).
2. P. del Amo Sanches et al. (BABAR Collaboration) // Phys. Rev. D 83, 052011
(2011).
3. Ha H. et al. (The Belle Collaboration) // Phys. Rev. D 83, 071101(R) (2011).
4. Lees J.P. et al. // hep-ex/ 1208.1253v1.
5. LHCb collaboration. // arXiv:1210.2645v1 [hep-ex]
6. Boyd C.G., Grinstein B., Lebed R.F. // Nucl. Phys. B 461, 493 (1996).
7. Song Hai-Zhen et al. // Commun. Theor. Phys. 50 696 (2008).
8. Ball P. and Zwicky R. // Phys. Rev. D 71, 014029 (2005).
9. Ali A., Lunghi E., Greub C., Hiller G. // Phys. Rev., D 66, 034002 (2002).
10. Beneke M., Feldmann T. // Nucl. Phys., B 592, 3 (2001).
130
pA collisions at LHC in Modified Glauber model
Andrey Seryakov
seryakov@yahoo.com
Scientific supervisor: Prof. Dr. Feofilov G.A., Department of High
Energy and Elementary Particles Physics, Faculty of Physics,
Saint-Petersburg State University
Abstract
One of the main possible evidences of the quark-gluon plasma formation in
collisions of ultra-relativistic heavy nuclei at the Large Hadron Collider (LHC) is a
decrease of the total multiplicity of charged particles compared to the pre-dictions
of superposition of the independent nucleon-nucleon collisions. Usual-ly a simple
geometric Glauber model is used to define the relevant initial condi-tions of the
collisions. However, this model is not justified in case of a so-called "soft processes
of particle production".
We present the further developments, such as the repulsive nucleon core and
multiple nucleon-nucleon collisions of the modified Glauber model (MGM) [1]. A
broad class of available experimental data on the multiplicity of charged particles
in Au-Au collisions is analyzed. The analysis of the total multiplicity yield in PbPb collisions at the Large Hadron Collider and forecast for p-Pb col-lisions are
made based on the new version of the MGM. The main result of the model is the
prediction of "nucleon stopping" in collisions of proton and lead nucleus. In this
connection, we expect to observe a considerable (2 fold) de-crease of the average
number of nucleon participants and the average multiplicity of charged particles
compared to the predictions of the standard Glauber model.
Standard Glauber model (SGM)
Glauber model (SGM) [2] is widely used to compute the number of nucleonparticipant (Npart), the number of nucleon-collisions (Ncoll) and their dispersion
[3], as base for the analysis of data coming from multiple hadron production in
high energy nucleus-nucleus collisions (AA-collisions).
The collectivity effects in AA-collisions (a non-trivial growth of multiplicity
with the atomic number of colliding nuclei,) are usually studied by normalizing
the charged particle data yields using the number of Npart or Ncoll to the collision [3-7].
In the SGM the heavy ion collision of two nuclei is the superposition of individual nucleon-nucleon interactions. Nucleons in the nucleus are distributed
according to the Woods-Saxon distribution:
where R is the radius of the nucleus, 𝜌0−nucleon density in the center of nucleus,
A – atomic number, 𝛼 – diffusivity. We assume that trajectories of nucleons are
131
linear and don't deviate
from the initial in the collision. Each nucleon-nucleon
collision occurs at definite
fixed energy and, accordingly, at definite crosssection, which is taken
from the experimental data
on proton-proton collisions
[8]. It is possible to obtain
the number of Npart and
Ncoll at the given impact
parameter. But if we match
Fig. 1. Average total multiplicity of charged particles for each collision multiplicfor the most central AA collisions, related to a pair ity of charged particles from
of nucleon-participants vs. energy in the center of the experiments on pp collimass (GeV) the Glauber model (SGM) (dotted line), sions [9], we obtain a strong
the Modified Glauber model (MGM) (full line) with overestimation of charged
k=0.35 and experiments [9]. Point 2.76 TeV was ob- particle multiplicity (Nch)
tained by us in this work from the preliminary rapid- for heavy-ion collisions
ity distribution ALICE [10-11].
(Fig. 1).
Modified Glauber model (MGM)
In order to solve this problem in the paper [1] it was assumed that the nucleons
in nucleon-nucleon collisions lose a fixed part of the momentum (k) in the center
of mass system:
It means that after the first collision the nucleons will have lower momentum and
the following collisions will occur with lower energy and lower cross-section.
Therefore the value of the multiplicity of charged particles will also be smaller.
In addition in our work, we have introduced the nucleon core and secondary
nucleon collisions. Nucleon core is a repulsive sphere around the nucleon, which
allows us to avoid the non-physical configurations of the nucleus when two nucleons
can have equal coordinates. Secondary collisions are the nucleon collisions between
the nucleons of one nucleus, this effect does not play a serious role in AA collisions, but it is essential for pA, when the projectile proton transfers substantially
all of its momentum to the nucleus.
The value of parameter of momentum loss k=0.35 was defined by fitting the
experimental data of multiplicity of charge particles for the most central AA collisions, related to a pair of nucleon-participant at different energy (Fig. 1).
The centrality dependence of charged particle yields per pair of Npart was also
calculated in the MGM for AuAu collisions at 3 energies and PbPb collisions at
2.76 TeV with the same parameter k=0.35 and then compared to the experimental
data [9] (Fig. 2). Points at 2.76 TeV were obtained by us in this work by integrat132
ing the preliminary rapidity
distribution ALICE [10-11].
The observed “stability” of
multiplicity of the charged
particle per pair of nucleonparticipant for different
centrality classes is well
reproduced in our simulation. This means that this
scaling could be explained
by equilibrium of two factors: decrease the number
and energy of the nucleon- Fig. 2. Charged particles multiplicity per nucleonnucleon collisions.
nucleon collision vs. Npart the MGM calculations
Predictions for the
with k = 0.35 are compared to experimental data for
proton-nucleus collisions different energy [9]. Points 2.76 TeV were obtained
(pA)
by us in this work from the preliminary rapidity disAs we have shown tribution ALICE [10-11].
above the MGM calculations are effective using
only a single parameter k
for description of global
observable such as mean
charged particle multiplicity in AA collisions. This
makes it possible to construct predictions for pA
collisions at the LHC.
For a better understanding of the processes we
have calculated the average
number of nucleon participants on the impact param- Fig. 3. The average number of nucleon participants on
eter using the MGM and the impact parameter in pPb collisions at √SNN=5.02
the SGM models (Fig. 3). TeV using the MGM k=0.35 (round points) and the
Result shows that, contrary SGM (square points).
to the predictions of the SGM (there the proton collides with all the nucleons that
are in his way); the MGM predicts that the proton will stop already after three
collisions. This means that in fact in the experiment we have only two classes of
centrality: central and peripheral collisions.
Plateau (Fig. 3) formed by nucleon stopping in pA collisions affects all other
distributions. For example, it forms the characteristic peak in the distribution of
Nch (Fig. 4). It is clear that the nucleon stopping also reduces the average values of
Npart, Ncoll and Nch (table 1). Note that in case of pA collisions Npart=Ncoll+1.
133
The difference in the mean
multiplicity of charged
particles predicted by two
models is large (see table
1). This decrease of multiplicity in the MGM is
due to the lower energy of
secondary nucleon-nucleon
collisions.
Experimental results for
charged-particle pseudorapidity densities at midrapidity normalized to Npart as
Fig.4. Distributions of Nch for 100k events for pPb a function of energy in the
at √SNN=5.02 TeV using the MGM k=0.35 and the center of mass per nucleons
SGM.
pair is shown in Fig. 5 (see
references in [12]). Our
estimate for the (dNch/
dη)/<Npart>=3.7±0.2
in pPb collisions at
√S NN =5.02 TeV is also
shown in Fig. 5, it makes
a significant difference
with the result obtained
using the SGM (dNch/
dη)/<Npart>=2.3±0.2 [12].
In our calculations of (dNch/
dη)/<Npart> we took the
Fig. 5. Charged-particle pseudorapidity density at value dNch/dη=17.44±0.01
midrapidity normalized to the number of nucleon- from [12]. We consider that
participants for pA and AA collisions as a function of data by the MGM is more
energy in the center of mass per nucleons pair, com- reasonable, because the last
pared to inelastic pp collisions. Experimental data for one describes in correct way
pPb collisions normalized to the number of Npart from of multiplicity of charge
the SGM [12] are below the line of pp collision. Our particles for the heavy ion
estimate for the (dNch/dη)/<Npart>=3.7±0.2 in pPb collisions at wide range
collisions at √SNN=5.02 TeV is also shown.
of energies (see Fig. 1 and
Fig. 2).
Table 1. Comparison of predictions of
<Npart>
<Nch>
average values of Nch and Npart for
SGM
7.7 +/- 0.6
420 +/- 80
pPb collisions at √SNN=5.02 TeV using the MGM k=0.35 and the SGM.
MGM
4.75+/-0.25
175+/- 25
134
Conclusion
We consider additional improvements to the Monte-Carlo Modified Glauber
Model (MGM) [1] developed earlier. They include the account of the nucleon
core and of secondary nucleon-nucleon collisions. The mean value of the model
parameter k=0.35 of the average momentum loss in the nucleon-nucleon collision is
obtained by fitting a wide range of experimental data on the multiplicity of charged
particles (including the latest data available from the ALICE experiment).
The MGM application for a description of the proton-nucleus collisions predicts
the effect of nucleon stopping that dramatically affects all observable quantities. For
example the expected average multiplicity for pPb collisions at the LHC is found
to be about of 175 as against 425 particles predicted by the standard Glauber.
This makes a significantly different result of (dNch/dη)/<Npart>=3.7±0.2 in the
MGM for the charged-particle pseudorapidity density at midrapidity normalized to
the number of nucleon-participants for pPb at √SNN=5.02 TeV compared to (dNch/
dη)/<Npart>=2.3±0.2 for the SGM.
References
1. Feofilov G., Ivanov A. // Journal of Physics G CS, 5, (2005) 230-237.
2. Glauber R.J., in Lectures on Theoretical Physics, edited by W.E. Brittin and
L.C.Dunham ~Interscience, New York, 1959, Vol. 1, p. 315.
3. Bialas A., Bleszynski M., Czyz W. // Nucl. Phys. B 111 461, 1976.
4. Adcox K. et al (PHENIX Collaboration) 2003 Preprint nucl-ex/0307010.
5. Antinori F. et al (NA57 Collaboration) // Eur. Phys. J. C 18 57 2000.
6. Kharzeev D., Nardi M. 2001 Preprint nucl-th/0012025v3.
7. Kharlov Y. (ALICE Collaboration) Recent result from ALICE arXiv:1203.2420v1
[nucl-ex] 12 Mar 2012.
8. Antchev G. et al. (ТOTEM Collaboration) First measurement of the total protonproton cross-section at the LHC energy of √S = 7 TeV, EPL, 96 (2011) 21002.
9. Back B. et al. (PHOBOS Collaboration) Comparison of the Total ChargedParticle Multiplicity in High-Energy Heavy Ion Collisions with e+e− and pp/p(anti)
p Data // arXiv:nucl-ex/0301017v1 28 Jan 2003.
10.Aamodt K. et al. (ALICE Collaboration) Charged-Particle Multiplicity Density
at Midrapidity in Central Pb-Pb Collisions at √SNN = 2.76 TeV, PRL, 105, 252301
(2010).
11.Aamodt K. et al. (ALICE Collaboration) Centrality Dependence of the ChargedParticle Multiplicity Density at Midrapidity in Pb-Pb Collisions at √SNN = 2.76 TeV,
PRL, 106, 032301 (2011).
12.ALICE Collaboration, Abelev B. et al. Pseudorapidity density of charged particles in p-Pb collisions at √S = 5.02 TeV” arXiv:1210.3615v1 [nucl-ex] 12 Oct
2012.
135
Net charge fluctuations in AA collisions in the color
strings approach
Titov Arsenii
arsenii.titov@gmail.com
Scientific supervisor: Prof. Dr. Vechernin V.V., Department of High
Energy and Elementary Particle Physics, Faculty of Physics, Saint
Petersburg State University
Introduction
Net charge event-by-event fluctuations have been proposed as one of the indicators of the formation of quark-gluon plasma in high-energy nucleus-nucleus
collisions [1, 2]. The theoretical predictions of the value of the fluctuations [1‒3] are
not directly consistent with the experimental results from the STAR Collaboration
at the Relativistic Heavy Ion Collider (RHIC) [4]. At present the preliminary results
extracted from the data obtained by the ALICE Collaboration at the Large Hadron
Collider (LHC) have rather ambiguous interpretation in the framework of existing
models [5, 6]. In this connection, we try to describe the experimental data in the
framework of an alternative string-inspired model [7].
It is known that the soft part of multiparticle production at high energy is successfully explained by the string models. In these models a hadron production is
described in the framework of a two-stage scenario. At the first stage a certain
number of quark-gluon strings (color field tubes) stretched between the projectile
and target partons are formed. At the second stage quark-antiquark pairs are created from a vacuum by the color field transforming these strings into the chains
of observed hadrons. It is important that those processes are dominant, in which
in a chain the produced hadrons are ordered in rapidity with a constant density.
It means that in string models one has an approximate conservation of the charge
locally in rapidity. The deviation of the charge created in mid-rapidity interval Δy
from zero is due to the processes, when on the borders of this interval only one
charged particle (of the created pair of the particles with close rapidities) belongs
to the interval Δy.
In string models the positive charge (and the baryon number) of initial nuclear
protons is associated with their valence quarks and is concentrated at projectile
and target rapidities. A transfer of this charge into the mid-rapidity region Δy is the
second cause of the deviation of the charge from zero in this region.
Based on this picture of hadronic interactions we formulate a simple model of
the net charge fluctuations in nucleus-nucleus (AA) collisions.
Results and Discussion
The observable ν is determined as follows:
2
 N+ N− 
(1)
,
ν≡ 
−
 N + N − 
136
where ⟨N+⟩ and ⟨N–⟩ denote the event-averaged numbers of positive and negative
particles in the mid-rapidity interval Δy. If the particle distributions are independent
from each other and Poissonian, the quantity ν is equal to so-called νstat.
1
1
.
νstat =
+
(2)
N+ N−
The subscript “stat” stands for “statistical”. The “dynamical” fluctuations are the
difference between the above two quantities:
νdyn = ν − νstat . (3)
This measure has been studied by the STAR Collaboration at RHIC [4] and the
ALICE Collaboration at the LHC [6].
The quantity νdyn can be written as follows:
D (N + )− N + D (N − )− N −
N N − N+ N−
νdyn =
+
−2 + −
.
(4)
N+ N−
N+2
N−2
Here D(N+)≡⟨N+2⟩-⟨N+⟩2 and D(N–)≡⟨N–2⟩-⟨N–⟩2 are the variances of the numbers of
positive and negative particles, respectively. In order to calculate the mean values
⟨N+⟩, ⟨N–⟩, the variances D(N+), D(N–) and the covariance ⟨N+N–⟩ - ⟨N+⟩ ⟨N–⟩, we
adopt the formalism used for calculating of long-range (forward–backward) rapidity correlations in [8]. Namely, we introduce the probability P(N+, N–) to have N+
positive and N– negative particles in some rapidity interval Δy in a given event due
to a fragmentation of N strings, independent between themselves:
N
P (N + , N − ) = ∑w (N ) ∑ ∑ δ N n+ +…+ n+ δ N n− +…+ n− ∏ p ni+ , ni− . (5)
+
n1+ ,…, nN+ n1− ,…, nN−
N
N
1
−
N
1
i =1
(
)
Here w(N) is the probability of the formation of N strings in a given event, p(ni+,
ni–) is the probability to have ni+ and ni– particles from the decay of the i-th string
in the rapidity interval Δy in a given event. It is important to note that we consider
that the probability p(ni+, ni–) does not factorize into the product of two probabilities, i.e., the corresponding particle distributions are not independent from each
other. For simplicity, we assume that all strings, formed in AA collisions at given
conditions, are identical. Then the following relations take place at any i:
∑ ni+ p ni+ , ni− = n+ , ∑ ni+2 p ni+ , ni− = n+2 ,
(
ni+ , ni−
)
∑ n p (n
ni+
, ni−
−
i
∑n
ni+ , ni−
+
i
+ −
i i
)
, ni− = n− ,
(
(
ni+ , ni−
)
)
∑ n p (n
ni+
, ni−
−2
i
+
i
)
, ni− = n−2 , (6)
n p ni+ , ni− = n+ n− .
We define the correlator for one string as follows:
k≡
n+ n− − n+ n−
d+ d−
,
(7)
where d + ≡ n+2 − n+2 and d − ≡ n−2 − n−2 are the variances of the numbers of positive
and negative particles from one string. The stronger the correlation between posi137
tive and negative particles produced due to the fragmentation of a string the more
the absolute value of the correlator. If the particle distributions are independent
from each other, then the correlator is equal to zero.
The fluctuations in the number of strings are characterized by w(N):
∑w (N ) = 1, ∑Nw (N ) = N , ∑N 2 w (N ) = N 2 .
N
N
N
Here ⟨N⟩ is the mean number of strings, and D(N)≡⟨N2⟩-⟨N⟩2 is the variance of the
number of strings. Note that these fluctuations are strongly non-Poissonian [9].
Now we can calculate the mean values, the variances, and the covariance of
N+ and N–. Using (5) and (6), we have (for details see [7])
N + = ∑ N + P (N + , N − ) = Nn+ , (8)
N+ , N−
N− =
∑ N P (N
N+ , N−
−
+
, N − ) = Nn− . (9)
Similarly, we obtain
D (N + ) = N d + + D( N )n+2 , (10)
D (N − ) = N d − + D (N )n−2 , (11)
and, using also (7),
N + N − − N + N − = N k d + d − + D (N )n+ n− . (12)
At high energy one can assume that in the mid-rapidity region the mean number
n̅ + of positive particles from one string is equal to the mean number n̅ − of negative ones. It is reasonable to suppose that the corresponding variances are equal
to each other too. Let us denote n̅ ≡ n̅ += n̅ − and d ≡ d+ ≡ d–. Then we can rewrite
(8)–(12) as follows:
(13)
N+ = N− = N n , D (N + ) = D (N − ) = N d + D( N )n 2 , (14)
(15)
N + N − − N + N − = N kd + D (N )n 2 . One can note that at such assumptions ⟨Q⟩=0, where Q= N+ – N– is the net
charge, and
(16)
N ch = 2 N n , where Nch = N+ + N– is the charged particle multiplicity.
Substituting (13)–(15) in (4), we obtain
2
νdyn =
(17)
(ω (1 − k )− 1), N n
where ω ≡ d / n̅ is the scaled variance. Thus the measure νdyn is expressed through
the parameters characterizing the individual string and the mean number of strings
and does not depend on the variance of the number of strings, i.e., on the eventby-event fluctuations of this number.
The D measure of the net charge fluctuations used by Jeon and Koch [1] and
νdyn are related to each other:
(18)
N ch νdyn ≈ D − 4. By (16) and (17) we have
N ch νdyn = 4ω (1 − k ) − 4. (19)
138
This quantity does not depend on the parameters of the event-by-event distribution
of the number of strings.
In this paper, we assume that the positive (negative) particle production from
the decay of one string obeys the Poisson distribution. Then the scaled variance
ω=1 and, using (17) and (19), one gets
2k
(20)
νdyn = −
,
Nn
N ch νdyn = −4k . (21)
Now we address to dependence of these quantities on the width of the central
rapidity interval Δy. We suppose that all strings formed in AA collision give a
contribution to Δy, and, therefore, the mean number ⟨N⟩ of strings does not depend
on Δy. At small values of Δy one can consider that
n = n0 ∆y, (22)
where n̅ 0is the mean number of positive (negative) particles from one string per
unit rapidity. Motivated by the string fragmentation picture [10], we use the following approximation for the correlator:
k = 1 − e − ∆y / λ0 . (23)
Here λ0 is the correlation length in rapidity. Substituting (22), (23) in (20) and (21),
we obtain explicit functions of Δy:
2
νdyn (∆y ) =
e − ∆y / λ0 − 1 , (24)
N n0 ∆y
N ch νdyn = 4e − ∆y / λ0 − 4.
(25)
We can approximately replace the rapidity interval Δy by the pseudorapidity
interval Δη and use (24) and (25) for description of the experimental data obtained
in [4, 6].
It is clear that in our consideration the quantity νdyn(Δη)/ νdyn (2) studied in [4]
and the quantity ⟨Nch⟩νdyn/(⟨Nch⟩νdyn)Δη=1 studied in [6] depend only on one parameter
λ0, which, nevertheless, enables to describe the experimental data. In Fig. 1 and
Fig. 2 the data obtained in [4] and [6] and the lines calculated with (24) and (25),
respectively, are presented.
Conclusion
The correlation length λ0 decreases with increasing energy and atomic number
of colliding nuclei. In the framework of our model this can be interpreted as the
formation of more intensive color strings in AA collisions with the growth of both
the energy and atomic number.
We thank Grigory Feofilov and Urs Wiedemann for useful discussions. A.Titov
is supported by the Dynasty Foundation fellowship. Prof. Dr. Vechernin V.V. is
supported by the RFBR grant 12-02-00356-a.
(
)
References
1. Jeon S., Koch V. // Phys. Rev. Lett., v. 85, p. 2076 (2000).
2. Asakawa M., Heinz U., Müller B. // Phys. Rev. Lett., v. 85, p. 2072 (2000).
3. Shuryak E.V., Stephanov M.A. // Phys. Rev. C, v. 63, p. 064903 (2001).
139
4. Abelev B.I. et al. (STAR Collaboration) // Phys. Rev. C, v. 79, p. 024906
(2009).
5. Abelev B. et al. (ALICE Collaboration) // arXiv:1207.6068 [nucl-ex].
6. Jena S. for the ALICE Collaboration // PoS(WPCF2011)046.
7. Titov A.V., Vechernin V.V. // PoS(Baldin ISHEPP XXI)047.
8. Braun M.A., Pajares C., Vechernin V.V. // Phys. Lett. B, v. 493, p. 54 (2000).
9. Vechernin V.V., Nguyen H.S. // Phys. Rev. C, v. 84, p. 054909 (2011).
10.Artru X. // Phys. Rep., v. 97, p. 147 (1983).
Fig. 1. Dynamical fluctuations νdyn(Δη), normalized to their value for |η|< 1
(Δη = 2), as function of Δη. Data for AuAu collisions at sNN = 62.4, 200 GeV
sNN = 62.4, 200 GeV (0‒10% central(0‒5% centrality), CuCu collisions at
ity), pp collisions at
s = 200 GeV obtained in [4] and lines calculated in our
model.
Fig. 2.Nchνdyn, normalized to the value for |η|< 0.5 (Δη=1), as function of Δη .
Data for PbPb collisions at
sNN = 2.76 TeV (0‒5% centrality) obtained in
[6] and line calculated in our model.
140
I. Resonance Phenomena
in Condenced Matter
The improvement of method of preliminary sample
polarization for detection of NMR signals in low
magnetic field
Kupriyanov Pavel
p.kupriyanov@physicist.net
Scientific supervisor: Prof. Dr. Chizik V.I., Department of Quantum
Magnetic Phenomena, Faculty of Physics, Saint-Petersburg State
University
Introduction
Information on the quantity and type of substances can be obtained on the basis
of identification of parameters of the NMR signals: amplitude, relaxation characteristics and spectra. The experience shows that the registration of NMR signals is
possible even in the Earth's magnetic field (-0.5·10 -4 T1) [1, 2]. To compensate
the decrease of the NMR signals in weak magnetic fields Bo, including Earth's
magnetic field, the following techniques are used: (i) the increase of a sample volume, (ii) the preliminary polarization of nuclei with an additional magnetic field,
(iii) the Overhauser effect, (iv) the use of coils of complex geometry which allow
us to increase the ratio of a signal to external electromagnetic hindrances, (v) the
application of signal digital processing, (vi) the accumulation of NMR signals.
In this work some peculiarities of the polarization of nuclei with an additional
strong magnetic field are investigated. In particular, the polarization of nuclei with
an alternating magnetic field at a very low frequency F (but F >>1/T1, where T1
is the spin-lattice relaxation time) has been considered. It is convenient and useful to turn the additional magnetic field on the perpendicular direction relatively
external magnetic field.
Theory
It is necessary to save adiabatic conditions for the net magnetisation of sample
in the alternative magnetic field: magnetization could be in time with the vector of
a resultant magnetic field BΣ which is the sum of the vector low constant field B0
and the vector of strong alternating polarizing field B* (Fig. 1).
Fig. 1. The behaviour vector of resulting magnetic field BΣ.
It is obviously the velocity of changing of direction resultant field is the largest,
when the angle θ is equal to zero. Therefore the velocity at this moment will be
the test of adiabatic changing for net magnetisation.
'

dθ 
B*
B*
= arctg
⋅ sin Ωt  = arctg
⋅ Ω ⋅ cos Ωt
dt 
B0
B0

142
The maximum of velocity, when θ=0, is found by equating of a derivative
with zero:
πk
πk
cos Ωt = 0 => Ωt =
, k = 1, 2,.. => Ω =
2
2t
Consider the behaviour of projection of vector of BΣ on vector B*:
B(t) = B* sin Ωt
(1)
The behaviour of B(t) depends on B* and Ω. We may find maximal velocity of
changing B(t) by the equating of a derivative with zero: B'(t) = B*Ω·cos(Ωt) = 0.
When the angle is small cos(Ωt) ) ≈ 1 and B'(t) ≈ B*Ω, then B(t) ≈ B*·Ωt. The tangents of this small angle may change the argument. Therefore:
 dB* 
 dt  ⋅ t
*
d θ ΩB
0
θ≈
=>
=
<< γB0 ,
B0
dt
B0
 dB* 
d *

*
 dt  =  dt B sin(Ωt ) = ΩB cos(Ωt )
0
0
(
) ≈ ΩB
*
0
Thus, the condition of adiabatic changing of resulting magnetic field is:
γ B02
f m <<
⋅ ,
2π B*
where fm- boundary frequency, with which observe the adiabatic changing of net
magnetization of sample is observed. Estimate fm with conditions: B* = 50 G,
B0 = 0,5 G, ν 0 = 2200 Hz. The result is: fm«22 Hz for Earth magnetic field.
Equilibrium average nuclear magnetization
in alternating magnetic field
To answering the question how the equilibrium average nuclear magnetization behaves in the alternating magnetic field it is necessary to solve the Bloch
equation:
My
M
M − M0
dM
= γ [M, B ]− e x x − e y
− ez z
,
dt
T2
T2
T1
If conditions of adiabatic are satisfied, then one may select a coordinate system
in which the axis z watch for direction resulting field. Thus:
M − M 0 (t )
dM z '
= − z'
,
dt
T1
where
2
M 0 (t ) = const·BΣ , BΣ = B02 + 2 B* cos(Ωt ) .
To skip here reasoning and calculations, give the limit is presented to which the
net magnetization M* aspires under action alternating field BΣ:
2
M * ≈ M 0 + M 0* ≈ 0, 7 ⋅ M 0*
π
Experiment
The experiment itself was carried out in the magnetic laboratory “Starorusskaya”.
The values of amplitude of NMR-signal for various frequencies and amplitudes of
preliminary sample polarization were received:
143
(
)
Fig. 2. Curves of dependences amplitudes NMR-signal on frequency of polarization for four values of tension of generator of low frequency.
The results obtained confirm the formula (1): the adiabatic changing is influenced not only the frequency of alternating field but amplitude this field.
The studying transitive process
Alternating current in difference from a direct current may be shutting down in
various phases. It is leading to variouse characters of transitive process.
Fading fluctuations in a parallel oscillatory contour is described by differential
equation of the second order [2]
d 2U
dU
U
L+
R + = 0, (2)
2
dt
C
dt
where L – induction of the coil, C - condenser capacity, R – winding resistance
of the coil.
The solution of this equation looks like
R
− t
U (t ) = e 2 L (U1eiω0t + U 2 e − iω0t ),
where ω0 - own frequency of the oscillatory contour.
We have considered a transitive processes in oscillatory contour with direct or alternating current shutdown. Change of phase in which alternating current shutdown
leads to various entry conditions for the solution of the differential equation.
The simulation of the process was execute with the program “Maple”.
An electron circuit was devised which enables accurate control over phase of
shutdown alternating current. The experimental data confirmed the theory well.
The minimum of transitive process amplitude of alternating current with shutting down at certain phase doesn't surpass in size tension at oscillatory contour
144
Fig. 3. The transitive process in case direct current shutdown (a). Shutdown alternating current in various phases (b). Dash line is a trajectory of maximum of
transitive process.
Fig. 4. The electronic circuit for the observation of transitive process in oscillatory contour.
before shutting down. This fact was shown as with modeling, as in experiment
(Fig. 3b). Compare transitive process after shutting down direct current and alternating current with optimal phase. Last method is the better because amplitude of
transitive process decreases approximately by 80-90 times. In addition the time
of such transitive process is decreasing, consequently excited pulse may be used
earlier, and registration of NMR-signal may starting early time. It is important
when examine the substances with short times relaxation.
Referenses
1. Quantum Radiophysics: Magnetic Resonance and It’s Applications. Manual.
Under edition by V.I. Chizhik. Saint Petersburg State University. 2009. 1969 (in
Russian).
2. Chizhik V.I. Neclear Magnetic Relaxation. Manual. Saint Petersburg State
University. 2004. 1969 (in Russin).
3. Borodin P.M., Melnikov A.V., Morozov A.A., Chernishev Yu.S.: Nuclear
Magnetic Resonance in Earth. -Leningrad: Izd. LGU. 1967 (in Russian). 1969 (in
Russian) - 232 p.
4. Kugushev A.M. , Golubeva N.S. Radiotronics Bases, Moscow: Energiya. 1969
(in Russian).
145
Table of Content
A. Chemistry....................................................................................................... 5
Quantum-chemical investigation of 2,1-benzisoxasoles formation by aromatic
nucleophilic hydrogen substitution process
Adreeva K.V., Tsivov A.V., Orlov V.Yu.................................................................. 6
А Synthesis diaryl esters in the presence of oxides of iron (III) as the promoting
agent
Lyutkin A.S., Orlov V.Yu., Volkov E.M................................................................ 11
A study into thermodynamics and structure of smeared charges fluids: the
hypernetted-chain closure of the fluid state theory
Nikolaeva Alexandra.......................................................................................... 15
B.Geo- and Astrophysics................................................................................. 21
Ionosphere profile reconstruction with oblique sounding ionogram
Muldashev Iskander........................................................................................... 22
С. Mathematics and Mechanics...................................................................... 27
Consensus in Stochastic Systems with Uncertainties in Measurements with
Simulation in JADE
Amelina Natalia................................................................................................. 28
Existence of a global B-pullback attractor for a periodically forced mechanical
system
Maltseva Anastasia............................................................................................ 33
D. Solid State Physics....................................................................................... 39
Factor analysis of Raman spectra of whisker GaAs with sphalerite-wurtzite
structure
Chirkov Evgenii................................................................................................. 40
Study of electron spin-polarization relaxation in LiF
Nikita Kan.......................................................................................................... 44
Spin polarization of interface states in thin films of Bi on Ag/W(110)
I.I. Klimovskikh, M.V. Rusinova, E. Zhizhin, A.A. Rybkina, A.G. Rybkin and
A.M. Shikin ....................................................................................................... 48
146
DFT Calculations of the X-Ray Spectra of Wide Band Materials Including
Dynamical Screening and Auger Effects
R.E. Ovcharenko1, I.I. Tupitsyn1, E. Voloshina2, B. Paulus2,A.S. Shulakov1 ..... 53
Spin structure in thin Au layers on W(110) and Mo(110)
M. Rusinova, A. Rybkina, I. Klimovskikh, E. Zhizhin and A.M. Shikin.............. 58
NEXAFS study of various graphite fluorides
Zhdanov Ivan..................................................................................................... 63
E. Applied Physics............................................................................................ 69
Impact of radioactive fallout from Chernobyl and Fukushima on the environment
of Leningrad region
Myorzlaya Anastasia.......................................................................................... 70
Innovative non-chemical etching technology
Yang Cheng Wei................................................................................................. 75
Optimization of flow deceleration by MHD interaction
Yang Cheng Wei................................................................................................. 78
F. Optics and Spectroscopy............................................................................. 83
Destruction of polyelectrolyte microcapsules modified with fluorescent dyes by
laser irradiation
Marchenko Irina................................................................................................ 84
Diffraction efficiency measurements of the Holoeye Pluto SLM fed by blazedprofile pattern
Alexander Sevryugin, Konstantin Mikheev........................................................ 88
Application of Kerr effect for remote sensing of electric fields over storm
clouds.
Egor V.Shalymov, Alina V. Gorelaya.................................................................. 91
G. Theoretical, Mathematical and Computational Physics......................... 97
Schwarzschild solution in R-spacetime
Angsachon Tosaporn.......................................................................................... 98
147
Optimization of calculations using Sector Decomposition approach: 3-loops
calculation of renormalization constants of φ4- theory
Ivanova Ella..................................................................................................... 102
Solving the time-dependent Dirac equation with the B-spline basis set method
Ivanova Irina.................................................................................................... 107
Crossing the boundary between parity breaking medium and vacuum by vector
particles
Kolevatov Sergey.............................................................................................. 112
Instrumental measurements of rogue waves in the southeast area of Sakhalin
Island
I.S. Kostenko, A.V. Yudin, K.I. Kuznetsov, V.S. Zarochintsev........................... 117
Calculation of propagator asymptotics in logarithmic dimensions for the models
φ3 and φ4 by means of renormalization group method
Artem Pismenskiy............................................................................................. 122
Estimate of the dilepton invariant mass spectrum in B+→ π+μ+μ- using data in
B0 → π-ℓ+νℓ and heavy quark symmetry
Rusov Aleksey................................................................................................... 126
pA collisions at LHC in Modified Glauber model
Andrey Seryakov.............................................................................................. 131
Net charge fluctuations in AA collisions in the color strings approach
Titov Arsenii..................................................................................................... 136
I. Resonance Phenomena in Condenced Matter......................................... 141
The improvement of method of preliminary sample polarization for detection of
NMR signals in low magnetic field
Kupriyanov Pavel............................................................................................. 142
148