ACTA PHYSICA
DEBRECINA
TOMUS XLV.
REDIGIT
ZOLTÁN TRÓCSÁNYI
DEBRECEN, 2011
ACTA PHYSICA DEBRECINA
TOM. XLV.
2011.
ADIUVANTIBUS
DEZSŐ BEKE, ISTVÁN LOVAS, ÁGNES NAGY,
JÓZSEF PÁLINKÁS, KORNÉL SAILER
ET ILONA TAMÁSSY–LENTEI
REDIGIT
ZOLTÁN TRÓCSÁNYI
REDACTOR TECHNICUS
SÁNDOR NAGY
4010 DEBRECEN, HUNGARIA
DEBRECEN, 2011.
The coloured electronic version of the articles can be downloaded from our
homepage: http://acta.phys.unideb.hu/.
Acknowledgement
This issue of the ACTA PHYSICA DEBRECINA,
was supported in part by the Hungarian Academy of Sciences and by the
TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project.
This support is highly appreciated and gratefully acknowledged.
Készült a Debreceni Egyetem Egyetemi és Nemzeti Könyvtárának
sokszorosító üzemében 100 példányban
Felelős kiadó: Dr. Fábián István
University of Debrecen 2011
HU ISSN 1789–6088 Acta Physica Debrecina
Preface
With this issue of the Acta Physica Debrecina we would like to pay tribute
to our editor and long time teacher Prof. István Lovas on the occasion of his
80th birthday. Regretfully, after his retirement we meet him less frequently
these days, but his spirit is still constantly with us.
Dear Bátyó! We wish you good health and happiness for many more
years!
Debrecen, October 1, 2011
Zoltán Trócsányi
editor
CONTENTS
S. Charnovych, Gy. Glodán: Effect of the nanoparticle size on the
plasmon enhanced photo-induced changes in amorphous
chalcogenide-gold nanoparticle system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
T.Y. Elrasasi, L. Daróczi, D. L. Beke: Calculation of elastic energy
contributions at the function of the martensite volume fraction in
single crystalline Cu-11.5WT%Al-5.0WT%Ni shape memory alloy . . .
17
J. Farkas: Recent γ-spectrometry based half-life measurements
at ATOMKI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
T. Győrfi, P. Raics: Diffusion cloud chamber – To observe the
invisible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
I. Angeli and Cs. Huszthy: New methods for the determination of
charge radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
R. Janovics, M. Molnár, I. Světlík, I. Major, L. Wacker:
Advances in radiocarbon measurement of water samples . . . . . . . . . . . . .
58
A. Kardos: Calculation of charge asymmetry in top quark-pair
hadroproduction at NLO accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
I.Kuti, J.Timár, D.Sohler, B.M. Nyakó, L.Zolnai, Zs.Dombrádi,
E.S. Paul, A.J. Boston, H.J. Chantler, M.Descovich, C.Fox,
P.J. Nolan, J.A. Sampson, H.C. Scraggs, A.Walker, J.Gizon,
A.Gizon, D.Bazacco, S.Lunardi, C.M. Petrache, A.Astier,
N.Buforn, P.Bednarczyk, N.Kintz, K.Starosta, D.B. Fossan,
T.Koike, C.J. Chiara,R.Wadsworth, A.A. Hecht, R.M.Clark,
M.Cromaz, P.Fallon,I.Y. Lee, A.O. Machiavelli: Parity
determination of excited states of the 132 La nucleus. . . . . . . . . . . . . . . . . .
76
A. Derecskei-Kovacs, I. Halasz, B. Derecskei, I. Tamassy-Lentei:
On the accuracy of atomistic simulations of IR spectra
in aqueous solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
I. Major, M. Molnár, L. Haszpra, É. Svingor, M. Veres: Fossil
fuel CO2 assay by simultaneous atmospheric 14 C and CO2 mixing
ratio measurements in the city of Debrecen . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
Á. Nagy, K. D. Sen: Fisher information from the pair density . . . . . . . . . . .
105
F. Nagy, G. Hegyesi, I. Valastyán, J. Molnár: Monte Carlo
simulations of silicon photomultiplier output signal . . . . . . . . . . . . . . . . . . . .
111
L. Papp, L. Palcsu, Z. Major: Advancing the use of noble gases in
fluid inclusions of speleothems as a palaeoclimate proxy: method
and standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126
B. Parditka, Z. Erdélyi, D.L. Beke: Stress effects on the kinetics of
nanoscale diffusion processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
156
E. Rozsályi: Charge transfer in collision of C2+ ions with HCl molecule . . .
166
T.S. Biró, Z. Schram: The Unruh effect for fluctuating trajectories in
high energy collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
176
F. R. Soha, I. A. Szabó, L. Harasztosi, J. Pálinkás, Z. Csernátony:
Development of a balance measurement system for
biomechanical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
186
L. Stuhl, A. Krasznahorkay, M. Csatlós, T. Adachi, A. Algora,
J. Deaven, E. Estevez, H. Fujita, Y. Fujita, C. Guess,
J. Gulyás, K. Hatanaka, K. Hirota, H. J. Ong, D. Ishikawa,
E. Litvinova, T. Marketin, H. Matsubara, R. Meharchand,
F. Molina, H. Okamura, G. Perdikakis, C. Scholl, T. Suzuki,
G. Susoy, A. Tamii, J. Thies, B. Rubio, R. Zegers,
J. Zenihiro: High resoluion study of the relative dipole strength
distribution in 42 Sc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
197
Z. Szoboszlai, Gy. Nagy, Zs. Kertész, A. Angyal, E. Furu,
Zs. Török, K. Ratter, P. Sinkovicz, Á.Z. Kiss:
Characterization of atmospheric aerosols in different indoor
environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
207
T. Szücs: α induced cross section measurement on 169 Tm for the
astrophysical γ-process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
218
G. Timár, F. Kun, J. Blömer, H. J. Herrmann: Fragmentation
of plastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
228
R. Trencsényi: Investigation of armchair hexagon chains by exact
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
234
Zs. Vajta, Zs. Dombrádi: Optimization of detector performance
in radioactive beam experiments via Geant4 simulations . . . . . . . . .
241
ACTA PHYSICA DEBRECINA
XLV, 7 (2011)
EFFECT OF THE NANOPARTICLE SIZE ON THE PLASMON
ENHANCED PHOTO-INDUCED CHANGES IN
AMORPHOUS CHALCOGENIDE-GOLD NANOPARTICLE
SYSTEM
S. Charnovych1 , Gy. Glodán2
1
2
Department of Experimental Physics, University of Debrecen, Bem sq. 18/a,
4026, Debrecen, Hungary
Department of Solid State Physics, University of Debrecen, Bem sq. 18/b, 4026.
Debrecen, Hungary.
Abstract
We have demonstrated a size effect of the gold nanoparticles
(GNP) on the previously established plasmon enhanced process
in the amorphous chalcogenide- GNP system under laser irradiation. GNP were deposited on glass substrate and covered by
thermally evaporated chalcogenide layer. We have focused on
the fabrication of the GNP, the size dependence properties and
on the influence of the plasmon field on the optical and structural
changes due to the laser irradiation.
I. Introduction
Amorphous chalcogenide glasses are well known as optical memory materials where different photo-induced processes like photo-darkening and
bleaching, local expansion or contraction, changes of the reflectivity and
of the refraction index take place [1-4]. These are important for the development of materials for optical recording, sensors and memory devices.
We have made investigation of the optical recording in chalcogenide films
(As20 Se80 for and As2 S3 ) by the selected laser irradiation, to study the above
mentioned processes.
Ag, Au and Cu are widely used as basic materials for exciting surface
plasmon resonance (SPR) in the visible spectral range [5]. Visible light
can excite surface plasmons on planar metal surfaces at special conditions,
while the localized surface plasmon resonance (LSPR) is easily observed in
nanometer sized metallic structures [6]. These processes are widely used for
the development of ultrahigh-sensitive fluorescence measurements, sensors,
semiconductor devices [7-9] and even for optical data storage [10].
In our samples the created GNP on a silica glass substrate satisfied the
conditions of the SPR in the green-red spectral range, where the selenide
and sulfide glasses are the most sensitive [1, 2]. It is a well known effect that
the increase of the size and the change of the shape of GNP lead to the red
shift of the SPR [7-9]. We have investigated the size dependence of the GNP
on the plasmon enhancement.
Localized electromagnetic field near the nanoparticle can enhance the
electron-hole processes and alter the interatomic bonds and the structure of
the surrounding dielectric as well. We used this effect for the enhancement of
photo-induced changes and for the amplitude-phase optical recording based
on the amorphous chalcogenide films made of the most widely investigated
As20 Se80 and Ass S3 glasses.
II. Experimental
We started our experiments with fabrication of appropriate plasmonic element, suitable for optical measurements of photo-induced effects in a given
chalcogenide layer. First a thin (15-25nm) Au layer was deposited on the
glass substrate. The films were beaded by heat treatment at 550◦ C for 1-16
hours. The continuous layer was disintegrated into islands, then during Ostwald ripening the GNP were formed. Ostwald ripening is a thermodinamicallydriven spontaneous process which occurs because the larger particles are
more energetically favored than smoller ones. According to Fig.1., if r < r∗
(r∗ is the critical radius), the total Gibbs free energy increases, the particle is
not stable. If r > r∗ , then the total Gibbs free energy decreases, the particle
grows, which means that the bigger particles swallow the smaller particles.
Since there is no significant evaporation or dissolution onto the substrate
and we can assume that the process is surface diffusion controlled, r∗ can be
2γω
given as R∗ = kT
σ where γ is the surface tension, ω is the atomic volume,
8
Figure 1: Gibbs free energy dependence on the radius of the particles
σ is the supersaturation, k is the Boltzman-coefficient, T is the absolute
temperature [11].
As the result of the annealing we got samples with non-spherical GNP in
a comparatively narrow size distribution range. The average size of the GNP
was established with standard image analysing process on the SEM (Hitachi
S-4300) pictures (Fig. 2.) made by NI Vision Assistant.
Figure 2: SEM picture of the GNP on the silica glass substrate
9
Figure 3: Schematic diagram of the sample structure with GNP and chalcogenide layer.
The prepared samples with Au nanoparticles were covered by a thermally
evaporated chalcogenide layer (As20 Se80 or As2 S3 ). The thickness of the
layers was 700 nm. Fig. 3 shows a schematic picture of the structure of our
samples.
As we mentioned in the introduction the increase of the size or shape
variations of the metallic nanoparticles lead to the red shifts of the SPR. So
the SPR wavelength can be varied by the change of the size of the metallic
nanoparticles that we used in our experiments. Fig 4. shows that the sample
sensitivity in the spectral range strongly depends on the average size of the
GNP.
Our layers are sensitive in the red and green spectral range respectively,
therefore the investigated samples were irradiated with red (λ=633 nm, output power P=7mW) or green (λ=533 nm, output power P=17mW) laser
beam through a 1,2 mm hole in a mask, and the maximum light intensity
at the surface was near 800 mW/cm2 for the red laser irradiation and for
the green one too. The change of transmission versus irradiation time was
detected with power meter setup (ThorLabs PM100), the thickness of the
layer and its changes were measured by Ambios XP-1 profile meter. Optical transmission spectra were used for calculations of the refractive index
by Swanepoel method [12] which was supplemented with data on thickness
change under illumination. The optical transmission spectra of the prepared
10
Figure 4: Size dependence of the plasmon resonant wavelength on the created
GNP dimension.
samples were measured with Shimadzu UV-3600 spectrophotometer.
III. Results and Discussion
We have excited surface plasmons by illumination of the nanoparticles in
a complex samples (Fig.3) and observed the effect of the photo-induced
changes in the system. Optical darkening, changes of the refractive indexes
and of the absorption coefficient as well as the red shift of the fundamental
absorption edge have been observed at the initial stage of the illumination
in both types of samples (prepared as pure chalcogenide films or structures
with GNP). The change of the transmitted light intensity (I) relative to the
initial intensity (I0) with time (t) during illumination γ = ∆(I/I0 )/∆t has
been taken as a rate of the photoinduced change. It was established that
the rates of the changes were different: γ was larger in the case of As20 Se80
(6·10−3 s−1 ) than in the case of As2 S3 (γ=4,2·10−3 s−1 ). We observed the
same effect in Au-containing structures too, where the process rate in both
cases was faster as in the single layers, and depends additionally on the size of
the nanoparticles (see Fig.5). We have investigated structures with different
11
size of GNP (1 - 60, 2 - 75 and 3 - 90 nm) and with different corresponding
plasmon peak (1 - 550, 2 - 580 and 3 - 610 nm).
An unusual effect of photo-induced bleaching appeared in GNP chalcogenide structures: the effect was different from the simple darkening and saturation in a single chalcogenide layer. After the initial process of darkening
the transmission started to increase in the samples that contain GNP. It has
been established that the rate of decreasing and increasing of transmission
depends on the size of the nanoparticles and on the plasmon wavelength in
the given structure. The fastest and the most significant changes (thickness,
refractive index, absorption coefficient) have been observed in the samples
that contain GNP with average diameter 90 nm, and where the plasmon
resonant wavelength is the closest to the wavelength of the laser light (Fig.
5). We calculated γ in As20 Se80 – GNP system for different particle sizes
and they were the following: 1 – 2,6·10−2 s−1 , 2 – 2,2·10−2 s−1 , 3 – 1,8·10−2
s−1 , 4 – 17,3·10−2 1/s. The same was observed and calculated for the As2 S3
– GNP system:
1 – 11,3·10−3 s−1 , 2 – 7,3·10−3 1/s, 3 – 5,9·10−3 1/s, 4 – 4,2·10−3 1/s.
As we see from the obtained data, in the samples that consist GNP with
plasmon absorption peak closer to the wavelength of the illumination, the
changes are more efficient as in the system with GNP where the plasmon
peak is further from the illumination wavelength.
In the system with GNP after the initial decrease of the optical transmission under the continuous illumination started to increase and the bleaching
rates were GNP-size dependent in a similar way as in the case of initial
darkening i.e. the faster was the increase of the optical transmission in
the sample, where faster the darkening rate was. It was interesting also to
measure the possible thickness change of the layer, since the local expansioncontraction effects are complementary with other photo-induced processes in
chalcogenides, although the relations are not completely clear yet [13]. The
results are presented in Table 1 and Table 2.
The optical bleaching of the samples that included GNP can be explained
with the change of αd in the exponential dependence of the transmission.
The decrease of the thickness were established in the samples without GNP at
the same low intensity. Contrary, the decrease of the thickness was observed
12
Figure 5: a) Time dependences of the optical transmission on the irradiation
time in a single As2 S3 and As20 Se80 layer (4) and structures with GNP (sizes
60 (1), 75 (2) and 90 nm (3)).
13
in all experimental illumination conditions in the systems that include GNP.
The rate of the changes depends on the size of the GNP, so on the plasmon
frequency as well. The change of the thickness was the most significant for
the system that includes GNP with the illumination ferquency closest to
plasmon frequency.
Table 1. The resulting values of the layer thickness d measured in asdeposited and illuminated single As20 Se80 layer and in the same material with
GNP at the stage of saturation of changes. The deviations, ∆(+ increase, decrease) as well as the changes in % are presented too.
Table 2. The resulting values of the layer thickness d measured in asdeposited and illuminated single As2 S3 layer and in the same material with
GNP at the stage of saturation of changes. The deviations, ∆(+ increase, decrease) as well as the changes in % are presented too.
Explanation of the enhancement effects relates to the initial stage of the
photo-induced effects in chalcogenide glasses, which consists generation of
electron-hole pairs, creation or modification of defects like dangling bonds
or changed number of bonds of chalcogen [14, 15] as well as to the existence
of localised electric field of plasmons. The localized electric field should
enhance the above mentioned electron processes, influence the rate and the
14
final value of the structural changes up to the small shifts of atoms or further
diffusion and mass transport at the presence of additional driving forces (the
gradients of excitation intensity, polarization of the light beam).
To support the above mentioned experimental results and assumptions
about the role of surface plasmons in the photo-induced processes in chalcogenide glasses, we have analysed the conditions of plasmon generation and
their influence in the given system of GNP in a glass matrix.
To analyze the effect of the LSPR on the photoinduced changes in chalcogenide system we need to look after the necessary conditions of the surface
plasmon resonance. They are the following [9]:
1. Dielectric constant of the surrounding medium has to be positive.
2. Real part of the dielectric constant of the GNP has to be negative.
In our case the dielectric constant ε of the surrounding medium, in the
chalcogenide layers, equals to 8 for As2 S3 , and to 7,7 for As20 Se80 [16], which
fulfill the first condition of plasmon resonance. But there was no data about
the real part of the dielectric constant of the GNP in the literature. Mie
theory was used for the calculations of the dielectric constant of the GNP,
because in this scale where the size is greater than a few tens of nanometer
the quasi-statistic approximation is no longer valid and the interaction of an
electromagnetic wave with a nanoparticle must explicitly take into account
the spatial variations of the field over size of the object. The dielectric
constant of the GNP was calculated according to this theory [17]. We used
the equations in [18] to calculate the dielectric constant of the GNP for
different sizes.
The real part of the calculated dielectric constant of the GNP1 - 0.350,
GNP2 - 0,351, GNP3 - 0,352. So the second condition is also valid in our
systems and theoretically the surface plasmon polaritons were generated in
them. These allow us to estimate the penetration depth of the plasmon field
[18].
So for the As20 Se80 the penetration depth is 1 - 926 nm, 2 - 930 nm, 3 935 nm and for the As2 S3 it is 1- 751 nm , 2 - 748 nm , 3 - 745 nm. It means
that our films were sufficiently influenced by plasmon fields.
15
According to our experiments in the samples which are sensitive in the
green spectral range the average diameter of GNP is 30nm, in the red spectral
range the average diameter of GNP is 110nm. To be sensitive in the green
spectral range 15 nm Au was deposited on the glass substrate, then we
annealed it for 2 hours at 550◦ C. To be sensitive in the red spectral range
25nm Au was deposited on the glass substrate, then we annealed it for 4
hours at 550◦ C.
IV. Summary
Localised plasmon fields in a composite system GNP-chalcogenide glass enhance the laser stimulated transformations in this glass. The efficiency of
the process depends on the matching of SPR and glass sensitivity wavelength. We have established the fabrication process of GNP with optimized
parameters for the given types of light sensitive chalcogenide glasses.
Acknowledgements
Authors are grateful to Dr. S. Kokenyesi for the support and fruitful discussions. The work is supported by the TAMOP 4.2.1./B-09/1/KONV-20100007 project, which is co-financed by the European Union and European
Social Fund.
References
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(1995).
[2] V.M. Lyubin, in: A. Kartuzhansky, Ed. Khimia (Eds.), Nonsilver Photographic Processes, 1984 (in Russian).
[3] Mihai A. Popescu, Non-Crystalline chalcogenides,(Kluver Academic
Publisher, Durdrecht, 2000).
16
[4] Y.ăSakaguchi and K.ăTamura, J Mater Sci: Mater Electron 18, 459
(2007).
[5] Richard B M Schasfoort and Anna J Tudos, Handbook of Surface Plasmon Resonance, (RSC publishing, Cambridge, 1998).
[6] Hutter E, Fendler J. Adv. Mater. 16, 1865 (2004).
[7] Susie Eustis and Mostafa. A. El-Sayed Chemical Society Reviews, 35,
209 (2006).
[8] T.A.El-Brolossy, T.Abdallah, M.B.Mohamed, S.Abdallah, K.Easawi,
S.Negm, H.Talaat, Eur. Phys. J. Special Topics 153, 361 (2008).
[9] Amanda J. Haes, Richard P. Van Duyne, Analytical and Bioanalytical
Chemistry 379, 920 (2004).
[10] Ovshinsky, S. R., Strand. D., Tsu, D., U. S. Patent no. 7292521, (2006).
[11] Y. S. Kaganovskii, D. L. Beke, S. P. Yurchenko, Surface Science 319,
207 (1994).
[12] M.L. Trunov, P. Lytvyn, V. Takats, I. Charnovych, S. Kokenyesi, J.
Optoelectronics Adv. Mat. 11, 1959 (2009).
[13] Yu. Kaganovskij, D. L. Beke, S. Kokenyesi, Appl. Phys. Lett. 97, 061906
(2010).
[14] J. Hegedus , K. Kohary, S. Kugler, J. Non-Cryst. Solids, 352, 1587
(2006).
[15] J. Hegedüs, K. Kohary, D. G. Pettifor, K. Shimakawa, and S. Kugler,
Phys. Rev. Lett. 95, 206803 (2005).
[16] M. A. Popescu, Physics and Applications of Disordered Materials, INOE
Publishing House, Buchuresti, 2002.
[17] T. Mortier, Ph.D. Thesis, College van Dekanen, University of Twente,
The Netherlands, 1992.
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519, 4309 (2011).
17
ACTA PHYSICA DEBRECINA
XLV, 18 (2011)
CALCULATION OF ELASTIC ENERGY CONTRIBUTIONS
AS THE FUNCTION OF THE MARTENSITE VOLUME
FRACTION IN SINGLE CRYSTALLINE
CU-11.5WT%AL-5.0WT%NI SHAPE MEMORY ALLOY
T.Y. Elrasasi, L. Daróczi, D. L. Beke
Department of Solid State Physics, University of Debrecen, PO Box. 2, H-4010
Debrecen, Hungary
Abstract
This is a report of my (T.Y.El Rasasi) research activity in the
third PhD year. I have published already four papers related to
my theses [2,3,4,5] and here I give a summary of some new results
which will be presented in VIII Hungarian conference on Material Science, October 2011, and will be published in Materials
Science Forum 2012 [1]. Elastic energy contributions, using the
local equilibrium model of the martensitic transformation [6], as
the function of martensite volume fraction ξ were calculated from
our measurements published earlier [2] on phase transformation
of single crystalline Cu-11.5wt%Al-5.0wt%Ni shape memory alloy. The derivative of the elastic energy δE/δξ = e (E is the total
elastic energy stored/released during the austenite to martensite
(A→M) as well as M→A transformation) could be calculated
only irrespectively of the ∆ST0 term (T0 is the equilibrium transformation temperature and ∆S is the change of entropy). But,
since ∆ST0 is independent of ξ, the functions obtained reflect
the ξ dependence of e as well as E quantities. Furthermore,
from DSC curves measured at zero uniaxial stress (σ = 0) [2],
we constructed the ξ −T hysteric loops. Then the e (ξ) curves at
fix σ as well as T were calculated. The E values obtained from
the integral of e(ξ), fit well to the E(σ) as well as E(T ) curves
18
calculated from the strain-temperature and stress-temperature
curves measured also in [2].
I. Introduction
Shape memory alloys (SMA) have important technological applications as
functional smart materials (see e.g. [7-10]). These applications imply a deep
knowledge of the characteristics of the thermoelastic martensitic transformation, which provides the basis mechanism for shape memory behavior. Thus
deeper understanding of the elastic and dissipative energy contributions to
the transformation can lead to improved control of the shape memory function [11, 12].
Characteristics of the thermoelastic martensitic transformation have been
studied experimentally in CuAl(11.5wt%)Ni(5.0wt%) single crystalline shape
memory alloys under constant temperatures, T , as well as under constant
uniaxial stressed, σ, measuring the relative deformation as the function of
σ and T , respectively [2]. From the obtained hysteretic curves the stress/
temperature dependence of the martensite and austenite start and finish
temperatures/stresses as well as of the dissipative and elastic energy terms
(belonging to the start and finish values) were calculated on the basis of the
model published in [12-15]. Furthermore, in [2-5] the ξ-dependence of the
dissipative energy was also calculated but no such an analysis was carried
out for the e(ξ).
In this communication we calculated the e(ξ) curves and also their integral
in order to get E(T ) and E(σ) functions at fixed σ and T values, respectively.
II. Experimental [2]
Characteristics of the thermoelastic martensitic transformation have been
studied in CuAl(11.5wt%)Ni(5.0wt%) single crystalline shape memory alloys
under constant temperature, T , as well as under constant uniaxial stress, σ,
measuring the relative deformation as the function of σ and T , respectively
(see [2] for more details).
19
The entropy of transformation was estimated from the measured differential scanning calorimeter runs (DSC Perkin-Elmer DSC-7) at zero uniaxial
stress with 2 K/min heating and cooling rate, according to the relations [4]
Z Mf
Z Af
sQ↓ ∼
sQ↑
↓
↓
∆s =
(1)
= −∆s =
T
T
MS
AS
III. Relations used in the analysis of data
The model used is in fact a local equilibrium formalism and based on the
thermoelastic balance [9, 11] and starts from the following relation (see also
[13, 14]) for the change of the Gibbs free energy per mole
∂(∆G↓ )
= ∆g ↓ + e↓ (ξ) + d↓ (ξ) = 0,
∂ξ
(2)
where ∆g, e and d are the derivatives of the changes in chemical free energy,
∆Gch , elastic, E, and dissipative energy, D, by ξ. The down arrow indicates
that (1) is written for the austenite to martensite transition and a similar
relation holds for the reverse process with arrow up. ∆g can have the general
form as (for the sake of simplicity, the up and down arrow is omitted):
∆g = ∆u − T ∆s − σV εtr
(3)
if only the effects of uniaxial stress and temperature is considered. Here
∆s = ∆s↓ (= sM − sA = −∆s↑ (< 0)) is the entropy change per mole, εtr is
the transformation strain, V is the molar volume.
In [14] only the field dependence of εtr was treated and ∆u and ∆s were
taken to be constant. In principle the term, containing the stress, has tensor
character and, as a consequence, even if one considers uniaxial loading conditions (leading to scalar term in (2)) the field dependence of εtr is related
to the change of the martensite variant distribution with increasing field
parameters. Thus at zero σ values thermally oriented multi-variant martensite structure forms, while at high enough values of σ a well oriented array
i.e. a single variant structure develops. The definitions of the equilibrium
transformation temperature, To , and stress, σ0 , give
∆g = ∆u − To ∆s = 0 (at σ = 0),
∆g = ∆u − σV ε
20
tr
(at T = 0),
To (0) = ∆u/∆s,
σo (0) = ∆u/V εtr
(4)
and thus
σo (0) = To (0)∆s/V εtr .
(5)
In these relations εtr should correspond to that value which corresponds to
the martensite variant structure (and thus to a given η value) formed at zero
temperature during stress induced phase transformation.
It was shown in [8], that the condition (1), using relations (3)-(5), gives
the following expressions
T ↓ (ξ) = To (σ) − [d↓ (ξ) + e↓ (ξ)]/[−∆s]
T ↑ (ξ) = To (σ) + [d↑ (ξ) + e↑ (ξ)]/[−∆s],
(6)
where again the up and down arrows correspond to the cooling and heating
process, respectively. In fact the inverses of these functions (i.e. the ξ(T ↓ )
and ξ(T ↑ ) curves) are the lower and upper parts of the normalized ε − −T
hysteretic loops, respectively. Thus, making the usual assumptions e(ξ) =
e(ξ)↓ = −e(ξ)↑ and d(ξ) = d(ξ)↓ = d(ξ)↑ and taking the difference and
the sum of T ↓ (ξ) and T ↑ (ξ), the e and d quantities can be calculated as
the function of ξ at different σ levels. (Note that the e(ξ) function can be
calculated irrespective of the To (σ)∆s constant value [12,2]), if To (0) is not
known.). Similar relations can be written for the σ − ε hysteretic loops [13].
In (6) To (σ) is the stress dependent equilibrium transformation temperature
and is given by
To (σ) = To (0) − (1/∆s)σV εtr ,
(7)
which is the integral form of the of the Clausius-Clapeyron relation. Here εtr
should depend on the stress (bay means of η) at which the given martensite
variant structure develops in the temperature induced phase transformation.
From the analogous relation for the σ − ε hysteretic loops
σo (T ) = −(δs/V εtr )[T − To (0)],
(8)
with temperature dependent εtr belonging to the martensite variant structure (and the corresponding η) which develops during the stress induced
phase transformation at a given temperature.
Now taking the relations (6) (and the analogous ones for the σ − ε hysteretic loops) at the start and finish points (i.e. at ξ = 0, and ξ = 1) one
21
arrives at the following relations for the martensite and austenite start and
finish temperatures (Ms , Mf , As , Af ) and stresses (σM s , σM f , σAs , σAf ):
Ms (σ) = To (σ) − [do + eo ]/[−∆s]
Mf (σ) = To (σ) − [d1 + e1 ]/[−∆s]
Af (σ) = To (σ) + [do − eo ]/[−∆s]
As (σ) = To (σ) + [d1 − e1 ]/[−∆s],
(9)
as well as
σM s (T ) = σo (T ) − [do + eo ]/[−V εtr ]
σM f (T ) = σo (T ) − [d1 + e1 ]/[−V εtr ]
σAf (T ) = σo (T ) + [do − eo ]/[−V εtr ]
σAs (T ) = σo (T ) + [d1 − e1 ]/[−V εtr ].
(10)
Finally, it is possible, by using the DSC curve [16], to obtain the volume
fraction of martensite ξ as a function of temperature (both for cooling, T ↓
and heating T ↑ ) as the ratio of the partial and full area of the DSC curve
(AM s−T and AM s−M f , respectively):
Z T
Z Mf
dQ↑
dQ↑
AM s−T
↓
=
/
.
(11)
ξ(T ) =
AM s−M f
T
Ms
Ms T
Similar relation holds for the ξ(T ↑ ) curve (obviously in this case the above
integrals run between As and T as well as As and Af , respectively). For the
illustration see Fig. 1 which shows the DSC curve measured at zero stress
[2], and the area corresponding to the nominator of (11): the denominator
is the entropy of this transformation (see Eqn. (11)). Fig. 2 shows the ξ(T )
hysteretic curve calculated from curves on Fig.1.
Thus, the e(ξ) and d(ξ) energies can be calculated at σ = 0 according to
relations (6).
IV. Summary of the results of [2]
The value of the entropy change was calculated and it was negative for
A → M and positive for M → A: ∆s = −1.26J/Kmol [2]. (The molar
22
Figure 1: DSC curve measured at zero stress [2].
Figure 2: ξ(T ), loops calculated from the DSC curve.
23
Figure 3: ε versus T at four different stresses [2].
volume of our sample is V = 7.9 · 10−6 m3 /mol.). Note that this value of the
entropy is in a good agreement with those obtained for β to β 0 transformation
in alloys of similar composition [17].
Figure 3 shows some of the deformation-temperature hysteretic loops at
different constant uniaxial stress levels [2]. It can be seen that at low stresses
the total ε belonging to the transformation, i.e. εtr , is small and there is a
strong increase between 22 and 100 MPa.
Figure 4 shows the σ − ε hysteretic loops measured at different temperatures. From Fig. 3 and 4 the stress and the temperature dependence of
the start and finish temperatures and stresses, were calculated respectively
and they had linear temperature and stress dependence [2]. For illustration
here we show the stress dependence of the elastic energies eo and e1 respectively (Fig.5). The vertical axis of theses figure shows the To (0)∆s + eo or
the To (0)∆s + e1 quantities calculated from the sums of Ms (σ) + Af (σ) and
Mf (σ) + As (σ), respectively with the help of the quantities σV εtr [2].
V. Calculation of the temperature and stress dependence of the
total elastic energy, E
Now it is possible to calculate the start and finish temperatures at σ = 0
from the curves shown in Fig. 2, and using these to get the eo and e1 values.
24
Figure 4: σ versus ε at four different temperatures [2].
These point are already shown in Fig. 5 and it can be seen that fit very
well to the straight lines obtained in [2]. Fig. 6 shows the T ↓ (ξ) + T ↑ (ξ) =
2To (σ) + 2e(ξ, σ))/[−∆s) vs. ξ functions calculated from the ξ − T curves
at different stresses. It can be seen that the curve at zero stress, calculated
from the DSC measurement, fits self-consistenctly to the others. Fig. 7
shows σ ↓ (ξ) + σ ↑ (ξ) = 2σo (T ) + e(ξ, T )/V εtr (σ) vs. ξ (see also [15]) at
different temperatures.
Figure 8 shows the integrations of the curves of Fig. 6 at different stresses
giving the total elastic energy, E, per cycle (irrespective of the constant
∆sTo value). Similarly, the integrations of the curves in Fig 7 are plotted
in Fig. 9 at different temperatures. Since the To (0)∆s quantity is negative
(∆s < 0), from the fact that the values on the vertical axis are negative
it can be concluded that E(0) < |To (0)|. Furthermore, it is clear from
the sign of the slopes in Figs. 8 and 9 that E decreases with increasing
stress and temperature. This suggests that with increasing fraction of single
variant martensite structure (see Figs. 3 and 4, indicating that εtr increases
with σ and T ) the elastic energy stored decreases. This is quite plausible
because the elastic energy can be larger for thermally induced (random)
multi-variant structure, because in this case the overlapping of elastic fields
of the differently oriented variants leads to more remarkable elastic energy
accumulation.
25
Figure 5: Stress dependence of the elastic energy terms, [2] belonging to
ξ = 0 and 1. For the explanation of the points shown at σ = 0, see the text
below.
Figure 6: T ↓ (ξ) + T ↑ (ξ) vs. ξ at different stresses.
26
Figure 7: σ ↓ (ξ) + σ ↑ (ξ) vs. ξ at different temperatures.
Figure 8: Stress dependence of the total elastic energy, E. The point at zero
stress is from the DSC data.
27
Figure 9: Temperature dependence of the total elastic energy, E.
VI. Conclusions and summary
1- The ξ − T hysteretic curve at zero stress was obtained from the DSC
curve. The elastic energy, calculated from the start and finish temperatures
of it, fits very well to the straight lines obtained from the start and finish
temperatures of the ξ − T hysteric curves at different σ 6= 0 levels.
2- The stress and temperature dependence of the total elastic energy per
cycle has been calculated from the T ↓ (ξ) + T ↑ (ξ) and σ ↓ (ξ) + σ ↑ (ξ) curves,
respectively. Both functions are linear in the range investigated with slopes
-1.30 J/mol Mpa and -1.04 J/mol K, respectively.
3- The total elastic energy, E, decreases with increasing stress and temperature in accordance with the increasing volume fraction of the well oriented
single variant structure.
Acknowledgments
This work was supported by the Hungarian Scientific Research Fund (OTKA)
project No. K 84065.
28
References
[1] T.Y. Elrasasi, L. Daróczi, D.L. Beke, to be published in Materials Science Forum 2012.
[2] T.Y. El Rasasi, L. Daróczi, D.L. Beke., Intermetalics 18, 1137 (2010).
[3] T.Y. El Rasasi, L. Daróczi, D.L. Beke, Materials Science Forum Vol.
659, 399 (2010).
[4] D.L. Beke, T.Y. El Rasasi, L. Daróczi, ESOMAT 2009, 02002 (2009)
DOI:10.1051/esomat/200902002 ľ Owned by the authors, published by
EDP Sciences, 2009.
[5] T.Y. El Rasasi, L. Daróczi, D.L. Beke, ACTA PHYSICA DEBRECINA
44, 149 (2010).
[6] Z. Palánki, L. Daróczi, D.L. Beke, Mater. Trans. A 46,, 978 (2005).
[7] T.W. Duerig, K.N. Melton, D. Stockel, C.M. Wayman (editors), Engineering aspects of shape memory alloys. (London: ButterworthHeinemann; 1990).
[8] A.R. Pelton, D. Hodgson, T.W. Duerig (editors), Shape Memory and
Superelastic Technologies. (Proceeding of SMST-94. USA: Asilomar;
1994).
[9] K. Otsuka, C.M. Wayman (editors), Shape memory materials (Cambridge, UK: Cambridge University Press; 1998).
[10] L. Delaey, Diffusionless transformations Chp.5 in R.W Cahn, P.Haasen
and E.J. Kramer (editors) “Materials Science and Technology – Comprehensive Treatment” Vol. 5. P. Haasen (ed.) Phase Transformations
in Materials, (Weinheim, VCH, 1991) p. 339.
[11] J. Rodriguez-Aseguinoza, I. Ruiz-Larrea, M.L. No, A. Lopez-Echarri, J.
San Juan, Acta Mater 56, 6283 (2008).
[12] Z. Palánki, L. Daróczi, C. Lexcellent, D.L. Beke, Acta Mater 55, 1823
(2007).
29
[13] Z. Palánki, L. Daróczi, D.L. Beke, Mater. Trans. A 46, 978 (2005).
[14] L. Daróczi, Z. Palánki, S. Szabó, D. L. Beke, Material Sci. and Eng. A
378, 274 (2004).
[15] D.L. Beke, L. Daróczi, Z. Palánki, C. Lexcellent, Proc. of Int. Conf.
on Shape memory and Superleastic Technologies, 2007 Tsukuba, Japan,
edited by S. Miyazaki (ASM International, Materials Park, Ohio) p.
607, 2008.
[16] A. Planes, J.L. Macquron , J. Ortin, Phil. Mag. Let 57 (6), 291 (1988).
[17] V. Recarte, J.I. Perez-Landazabal, P.P. Rodriguez, E.H. Bocanegra,
M.L. No, J. San Juan, Acta Mater 52, 3941 (2004).
30
ACTA PHYSICA DEBRECINA
XLV, 31 (2011)
RECENT γ-SPECTROMETRY BASED HALF-LIFE
MEASUREMENTS AT ATOMKI
J. Farkas
Institute of Nuclear Research of the Hungarian Academy of Sciences, H-4001
Debrecen, P.O.Box 51, Hungary
Abstract
The half-lives of 154m Tb, 133m Ce and 66 Ga have been measured recently at the Institute of Nuclear Research (ATOMKI).
These measurements were performed by utilizing the same technique: γ-spectrometry. Here I summarize the three measurements and present the results.
I. Introduction
Providing accurate nuclear data is an important issue both in nuclear and
applied sciences. As radionuclides are widely used in science and technology,
their most important parameters, their half-lives needs to be determined
more and more precisely. Precise half-life measurements have a tradition
in the Nuclear Astrophysics Group of ATOMKI [1, 2, 3]. In our case the
motivation is to decrease the uncertainty of the results of the astrophysically relevant nuclear reaction cross section measurements. Most of these
measurements have been performed via activation, when the decay of the
radioactive reaction product is followed. In order to determine the number
of the product nuclei, one has to know the precise value of the half-life.
In our most recent experiments the half-lives of the 66 Ga isotope as well
as the 154m Tb and 133m Ce isomeric states have been determined. In all
cases our results are more precise than the literature values, thus these are
suggested to be included in future nuclear data compilations. Here I only
describe the general steps of the half-life experiments. For the details we
refer to Refs. [4, 5, 6] (154m Tb, 133m Ce and 66 Ga, respectively) or Ref. [7].
II. Source production
Most of the targets were prepared by vacuum evaporation on Al backings.
In the case of the 154m Tb and 133m Ce measurements, enriched Eu2 O3 and
BaCO3 were evaporated. In the 66 Ga case natural isotopic composition zinc
were used, and alternatively thick copper discs were also irradiated.
The prepared targets were irradiated by ≈ 10 MeV α or proton beams.
The targets were fixed in the activation chamber shown in Fig. 1, which
was attached to the end of a beamline of the cyclotron of ATOMKI. Note
that the chamber is suitable for activation cross section measurements and
accordingly it is more equipped than a simple activation chamber. The target
stability was monitored by recording the RBS (Rutherford Backscattering)
spectrum of the target with the built-in ion implanted silicon detector of the
chamber.
III. γ detection
After the irradiation the sources were transported to the γ counting room.
HPGe (High Purity Germanium) detectors were used for the detection with
the associated electronics. The detector were shielded either with a 5 cm
thick lead wall or with a multilayered 4π low background shield. The spectra
were acquired with the Ortec MAESTRO software. Fig. 2 shows a typical
γ-spectrum.
IV. Data analysis
One peak of the 154m Tb (540.2 keV), three peaks of the 133m Ce (58.4 keV,
130.8 keV and 477.2 keV) and seven peaks of the 66 Ga (833.5 keV, 1039.2 keV,
1918.3 keV, 2189.6 keV, 2751.8 keV, 4295.2 keV and 4806.0 keV) decay were
followed. As a first step of the data analysis, linear background subtraction
32
Figure 1: A schematic view of the activation chamber. PHA: Pulse Height Analysis, MCS: Multichannel Scaling, RBS: Rutherford Backscattering.
was performed. In the case of the isomers the peaks were fitted with exponentially modified Gaussian functions after the background subtraction.
The area of the peaks were derived from the parameters of the fit. The area
of the 66 Ga peaks were determined by peak integration.
The peak intensities were plotted against time on a semi-logarithmic scale.
The acquired points were fitted with a line representing the exponential decay
law. The half-life of the decay can be extracted from the slope of the fitted
line. The result of such a fitting and its residuals can be seen in Fig. 3.
V. Results
The results of our measurements are summarized in Tables 1 – 3 and
Figs. 4 – 6. The half-life values based on our measurements have one order of magnitude less uncertainty than the literature values in the case of
the terbium and cerium, and the uncertainty of the half-life of 66 Ga is three
33
Figure 2: An example γ-spectrum, in which the most significant peaks of
decay are shown.
times less than the literature value.
References
[1] Zs. Fülöp et al., Nucl. Phys. A 718, 688 (2003)
[2] Gy. Gyürky et al., Phys. Rev. C 71, 057302 (2005)
[3] B. Limata et al., Eur. Phys. J. A Suppl. 27, 193 (2006)
[4] Gy. Gyürky et al., Nucl. Phys. A 828, 1 (2009)
[5] J. Farkas et al., Eur. Phys. J. A 47, 7 (2011)
[6] Gy. Gyürky et al., submitted to Appl. Radiat. Isotopes
[7] J. Farkas, Ph.D. dissertation, University of Debrecen, Hungary
34
133m
Ce
Figure 3: The result of the exponential decay curve fit of 14 one hour spectra of
Ce (Eα = 15 MeV, Eγ = 477.2 keV). The activity is given in counts per hour
(CPH).
133m
35
154m
Tb half-life measurement. The literature halflife value is t1/2 = 9.4 h ± 0.4 h. The given uncertainties are statistical only. The
suggested half-life value is t1/2 = 9.994 h ± 0.039 h.
Table 1: The results of the
Source no.
1
2
3
4
5
Figure 4: The results of the
154m
t1/2 /h
9.993 ± 0.106
10.008 ± 0.033
9.984 ± 0.019
10.008 ± 0.029
9.994 ± 0.006
Tb measurement for various samples. The recommended value and its uncertainty are shown as horizontal lines.
36
Table 2: The results of the 133m Ce half-life measurement. The literature
half-life value is t1/2 = 4.9 h ± 0.4 h. The given uncertainties are statistical
only. The suggested half-life value is t1/2 = 5.326 h ± 0.011 h.
Ebeam /MeV
Eγ /keV
t1/2 /h
14.0
58.4
130.8
477.2
58.4
130.8
477.2
130.8
477.2
58.4
130.8
477.2
5.332 ± 0.048
5.29 ± 0.11
5.278 ± 0.022
5.339 ± 0.014
5.254 ± 0.047
5.314 ± 0.020
5.301 ± 0.017
5.320 ± 0.014
5.344 ± 0.012
5.362 ± 0.029
5.331 ± 0.012
14.5
15.0
15.5
133m
Ce measurement. The recommended value and
its uncertainty are shown as horizontal lines.
Figure 5: The results of the
37
Table 3: The results of the
66
Ga half-life measurement. The literature half-life
value is t1/2 = 9.49 h ± 0.07 h. The given uncertainties are statistical only. The
suggested half-life value is t1/2 = 9.312 h ± 0.023 h.
Source no.
1
2
3
4
5
6
7
Figure 6: The results of the
66
t1/2 /h
9.337 ± 0.020
9.345 ± 0.017
9.304 ± 0.014
9.321 ± 0.019
9.303 ± 0.009
9.302 ± 0.011
9.332 ± 0.021
Ga measurement. The recommended value and its
uncertainty are shown as horizontal lines.
38
ACTA PHYSICA DEBRECINA
XLV, 39 (2011)
DIFFUSION CLOUD CHAMBER – TO OBSERVE THE
INVISIBLE
T. Győrfi1 , P. Raics2
1
2
Eötvös József College, Baja
Institute of Experimental Physics, University of Debrecen
Abstract
Experimental education in nuclear physics at secondary schools
is less common compared to the other disciplines of physics. The
reason is mainly due to the shortage in appropriate devices of
affordable prices. However, the phenomena of radioactivity, nuclear physics are fundamental in the general comprehension of
microscopic behaviour of matter and many fields of important
applications. In this paper we would like to show some simple
examples of the utilization of a diffusion cloud chamber in education. These experiments and demonstrations may extend the
students’ thinking and enhance their interest in nuclear physics.
I. Introduction
For the demonstration of background and artificial radioactive radiations
as well as other nuclear processes a PHYWE made large diffusion cloud
chamber [1] was utilized at the Institute of Experimental Physics. This
device had been developed at the University of Göttingen for demonstrational purposes. Three more such tools are available recently in Hungary:
Paks Nuclear Power Plant, Palace of Miracles (Budapest) and University of
Miskolc.
Apart from presenting spectacular track formations only, our aim was to
perform real measurements with the cloud chamber [2]. The description of
Figure 1: View and scheme of operation of the diffusion cloud chamber.
the experiments below highlights also some difficulties caused by the actual
device as well as the deficiency in the skills required for such tools.
Quantitative evaluation of the experiments are supported by live broadcast, prerecorded videos and photos as well as by programs for calcualtions.
All of them are available on the internet [3].
II. Experimental
II.1 Construction and operation of the diffusion cloud chamber
There is a black metal plate with area of 45x45 cm2 in the lower part of the
cloud chamber which is cooled to -27 ◦ C (Fig. 1). The upper part is made of
a double walled glass box to provide good thermal insulation. Between the
glass plates there are wires having double function: heating the upper part
of the chamber to prevent any condensations there, and creating electric field
to extract ions. Along the inner glass cover an electrically heated channel
is continuously fed dropwise by Isopropyl alcohol (C3 H8 O). Alcohol vapour
is condensing on the basement as small droplets and a supersaturated layer
is formed over the liquid phase. The electrically charged particles passing
through this sensitive volume produce tracks from the cloud. The height of
the layer was determined by alpha-particles to be ∼10 mm.
The dead time of the chamber is caused by the disappearance of the
alcohol vapour from the track volume by condensation. A “black track” is
40
then shown in the surrounding cloud inhibiting the formation of a new track
as long as the supersaturation is reestablished again locally. Evaluation of
videorecords resulted in an average value for the dead time of ∼1 s.
II.2 Radiation sources
Utilization of natural sources of radioactivity from background radiations
are the simplest solution for demonstrations. Many experiments were performed with α-particles of energy 6.050 and 8.785 MeV of 212 Pb (10.6 h)
from a Th(B+C) emanating preparation. Tracks of alphas of 222 Rn, 241 Am,
235,238 U and 232 Th were also displayed. A 90 Sr/Y source was utilized for
production of β-rays. A 239 Pu-Be neutron source allowed to demonstrate
the (n,p) scattering.
II.3. Recording and evaluation of particle tracks
Still as well as motion pictures were recorded by different cameras. The
best results could be achieved by the classical video tape recorder posteriorly
digitalized for evaluation. Quantitative experiments with simple web cams
may suffer from the inaccurate timing and frame counting caused by different
compression methods. Track processing code was developed for the real-time
evaluation of length, thickness and position coordinates.
III. Experiments with the Diffusion Cloud Chamber
Particles can be identified by their specific ionization [4] which results
in thick, straight tracks for α-particles and thin meandering ones for β-rays
(electrons). Thin trajectories are characteristic for muons. γ-rays may be observed through electrons produced by photoeffect, Compton-scattering and
pair production. The length of the track is the range measuring energy. An
other way for particle discrimination is their different absorption in various
materials.
41
Figure 2: Background α- and β-radiation tracks (left) and a muon path
(right).
III.1 Observation of the natural background
Background radiation is shown in Fig.2 where α- and β-particles are decay
products of the radon and its daughters originating from U and Th. Cosmic
ray muons are also detectable if they fly in the horizontal direction. One of
the most important result of the chamber experiments is the separation of
the different background radiations.
III.2 Muon identification
For the identification of muons in the cloud chamber a telescope equipped
with two GM-tubes of 890 mm length and 40 mm in diameter were applied.
Coincidence (or “AND”) condition was electronically established between the
counters to select events when a particle crosses both detectors simultaneously. If this was met then a click sounds and the event has been counted.
The system was located parallel with the sensitive volume of the cloud chamber.
A video camera recorded the scene in the chamber continuously. A coincidence between the sound from the telescope and the event on the display
indicated a muon intercept through both devices. Since in the horizontal direction the thickness of the air as well as the other absorbers (walls) is high
and the efficiency of the two detectors were rather low the number of coincidences was very limited giving hardly acceptable statistics. The experiment
will be repeated in better conditions.
42
III.3 Deflection of β − -particles in magnetic field
What is interesting from the nuclear physics point of view is to examine
the continuous spectrum of β-patrticles. Experimental difficulty arose from
the geometry of the cloud chamber making impossible to apply electromagnet
from outside. The solution was to put rare earth metal magnet of strength
2.4 kG into the sensitive volume the available size of which was 1,2×5,0×0,45
cm3 , only.
A 90 Sr/Y source (28.6 y) was selected for the experiment having two
energy groups Eβmax of 0,546 and 2,284 MeV without γ-rays from the decay.
There were two problems which obstructed the demonstration easily perceptible. The magnetic field covered a small area of the chamber therefore
the deflection by the Lorenz-force was small. The two spectrum groups and
their continuous shape made it difficult to observe clearly the trajectories.
To verify the effect the direction of the magnetic field was reversed. Use of
also a positron source would make the demonstration more expressive.
III.4 The range of charged particles
The measurement of ranges makes it possible to determine the energy of
the particles. Unambiguous experimental circumstances might be produced
for α-particles, only. Table 1 summarizes the ranges from measurements with
source of Th(B+C) evaluated by our real time image processing program [2]
as well as by the Lince off-line software [5]. Calculated values are also listed
which have been produced by the SRIM 2010 simulation code [6] and an
empirical expression [4]. SRIM results are shown for “air” and an “air + 10%
propanol” composition at normal temperature and pressure. The measured
values evaluated by our software are in good agreement with the simulation
for the composite gas. The random error of the measurements is estimated
to be 10 %.
43
Table 1: Comparison of the ranges (mm) determined with different methods
III.5 Determination of half-life
For this experiment the nuclide 212 Pb from the Th(B+C) source was used.
The α-tracks have been observed for two days and the images were recorded
in every five hours for 5 minutes. Our image processing program counted
the tracks of α-particles in the given frames. The results were plotted as a
function of time in the Excel program. An exponential trendline was fitted to
the diagram and the half-life was determined. As a control we have counted
the particle tracks manually, too.
A further check with α- and γ-spectrometry as well as β-counting was also
applied for the measurement of the sample activity. The results are compared
to the literature value in Table 2. Values measured with the cloud chamber
have the highest deviations from the literature one. It can be explained by
the low statistics and the resulting huge statistical uncertainties.
Table 2: Comparison of the
44
212 Pb
half-life determined in different ways
Figure 3: Rutherford-scattering on Ag-foil by Th(B+C) collimated alphaparticles.
III.6 Observation of nuclear reactions
2011 is the year of the atomic nucleus in honour of the centenary of the
Rutherford-scattering. This experiment proved the existence of the nucleus
and a new scientific era started. We considered its demonstration very important with the help of the diffusion cloud chamber. The α-beam of the
Th(B+C) source used for the experiment had to be collimated to show the
deflection clearly. We have applied silver foil because it can be handled easier
in the cloud chamber.
The existence of the nucleus would be shown nicely by backscattering.
Angles greater than 90 degrees can not be studied in the recent geometry.
However, as a demonstration, it is important to see the significant deflections,
which prove the scattering process in Fig.3.
Applying high intensity Th(B+C) source the α-particles create a circular
formation of tracks (“halo”). Sometimes a longer and thinner track can be
4
17
1
seen which stems from a proton originating from the 14
7 N +2 He =8 O +1 p
17
nuclear reaction as seen in Figure 4. The very short path of the nucleus O
can not be seen on the records because of the large α-background.
Neutrons can not be seen directly in the cloud chamber. It is possible
indirectly, only, by means of charged particles or γ-radiation produced in
nuclear processes. We have used a 239 PuBe-source of non-monoenergetic
neutrons of about 5.5 MeV average energy in our experiment. The (n,p)
elastic scattering of the neutrons on the hydrogen nucleus of the chamber’s
filling gas may be observed in Fig.5.
45
Figure 4: Nuclear reaction
Th(B+C) source.
14 N(α,p)17 O
Figure 5: Tracks of protons hit by
induced by α-particles from the
239 PuBe
source neutrons.
IV. Summary
The large diffusion cloud chamber can be used for different phases in
the teaching of physics: grammar school, secondary school, university. In
primary education the cloud chamber can be utilized mostly as a demonstrational tool: material structure, background radiation.
In secondary education the cloud chamber is important in teaching nuclear physics, radioactivity (eg. determining half-life). Observation of the
scattering processes, production of new particles, the possibility of transmutation offer delightful experience for the students (eg. 14 N(α,p)17 O nuclear
reaction). With the collision of particles (eg. n-p scattering) students can
master classical mechanics knowledge. When evaluating the experiments
statistics and uncertainties come to light. Teaching physics at universities
the diffusion cloud chamber can be used for determining half-life, stopping
power and range of charged particles, energy spectra from tracklength distribution, observation of nuclear reactions, statistical behavior of nature.
46
Utilization of the expensive but outstanding large diffusion chambers in
education and civil knowledge distribution may be best performed centrally
with internet support for live experiments.
References
[1] Phywe series of publications: Visualisation of radioactive particles /
Diffusion cloud chamber (Laboratory Experiments Physics, PHYWE
SYSTEME GMBH, Göttingen, Germany) http://www.phywe.com/.
[2] T. Győrfi, Nuclear Physics in Education. Investigation and demonstration of Environmental Radioactivity, PhD-Thesis (Univ. of Debrecen,
2011.).
[3] http://fizika.ttk.unideb.hu/kisfiz/tavtanulas.
[4] T. Fényes, Atommagfizika I., (2nd ed., Debrecen University Press, 2009).
[5] S. L. dos Santos e Lucato, Lince - Linear Intercept v. 2.4, (Department
of Material Science, Darmstadt University of Technology, 1999).
http://www.mawi.tu-darmstadt.de/naw/nawstartseite/service/
software/sv_software.de.jsp.
[6] J. Ziegler, SRIM: The Stopping and Range of Ions in Matter.
http://www.srim.org/index.htm.
47
ACTA PHYSICA DEBRECINA
XLV, 48 (2011)
NEW METHODS FOR THE DETERMINATION OF CHARGE
RADII
I. Angeli and Cs. Huszthy
Institute of Experimental Physics, University of Debrecen,
H-4010 Debrecen, Pf. 105. Hungary
Abstract
The study of short-lived radioactive nuclei necessitates special instruments and methods. After a summary of the traditional methods to measure nuclear charge radii and charge distributions, new methods are described: electron-ion scattering
in-flight; X-ray transitions from trapped highly charged ions;
isotope shift of di-electronic recombination spectra; Lamb-shift;
“second generation” optical isotope shift.
I. Introduction
The recent development of accelerators, mass and isotope separators rendered it possible to produce beams of radioactive nuclei far from the valley
of stability. The exploration of their properties would be essential to the development of nuclear physics. New, surprising effects have been found away
from the line of stability, e.g. disappearance of old magic numbers and appearance of new ones, in radii too [1]. This means the re-arrangement of the
shell structure. Much effort has been done recently to develop instruments
and methods to investigate the properties of radioactive nuclei.
The advantages and limitations of the new methods can be understood
only in comparison to the old ones. Therefore, in Chapter II. a short description of the traditional procedures is given. In Chapter III. the new methods
are described. These methods are still under development; some of them
have already produced initial results, while others are only in the stage of
experimentation. It is important that from the expensive experimental data
as much information should be gained as possible. This effort led to the
recognition of the importance of theoretical evaluation. Indeed, in one case,
it is the modern few-body theory of atomic electrons that constitutes the
basis of a new method.
II. Traditional methods of measuring nuclear charge radii and
charge density distributions
II.1 Elastic scattering of fast electrons
This is the only method able to yield a charge density distribution function
ρ(r), which is important for comparison to results of theoretical nuclear
structure calculations. Fast electrons are scattered on the target, and the
differential scattering cross section σ(θ) is measured. This cross section is
compared to that calculated theoretically for a point-like nucleus. The ratio
contains the form factor, which - for a spherical charge distribution ρ(r) has a simple form:
Z
sin qr
q 2 hr2 i
F (q) = ρ(r)
dv = 1 −
+ −...
(1)
qr
3!
There is an important quantity q = ∆p/~ where ∆p is the momentum transfer.
At “moderate” energies (E ≤ 100 M eV , q 1/R), qr 1 and sin(qr)
can be approximated by the first few terms of its power series. Plotting the
values of F in the function of q 2 , the slope multiplied by 6 yields hr2 i. This is
a simple, model independent (MI) experimental determination of the second
moment.
At higher energies (E 100 M eV ) the small λ allows the determination
of the shape of the charge distribution.
49
II.2 Characteristic X-rays of muonic atoms
Most properties of the muon agree with those of the electron but its
mass is two orders of magnitude higher. The wave function of a muon on
the innermost orbit has a significant overlap with the nucleus. Therefore,
the energy of the muon depends strongly on the size of the nucleus. The
energy of γ-rays from the innermost 2p → 1s and 3d → 2p transitions is
measured. Before interpreting the experimental data, theoretical corrections
have to be calculated: vacuum polarization (VP), Lamb-shift (LS), etc. The
least accurate is the nuclear polarization (N P ) correction, which takes into
account the effect of virtual excitations of the nucleus. It is this uncertainty
that limits the possibilities of mounic atom X-rays method, rather than the
experimental ones. Interpretation of the experimental results: the transition
energy is determined by the Barrett moment:
Z
ρ(r)rk e−αr dv = hrk e−αr i
(2)
Muonic atom X-ray measurements are able to determine this moment with
an accuracy of order 0.1%.
II.3 Isotope shift of Kα -lines (KIS)
Because of the small mass of the electron, the overlap with the nucleus and so the shift of the electron state - is small. This small shift is disturbed
by the chemical surrounding. Therefore, and because of Z-dependent QEDeffects, a derivation of an absolute nuclear radius R from the energy of X-ray
transition is impossible. However, the difference δE between energies of
X-rays emitted by two isotopes of the same element, does not contain the
Z-dependent contributions and the effect of chemical environment, and can
be used to derive quantities characteristic to the difference of the two charge
distributions. The typical energy shift to be measured is much less than the
natural line width. Therefore, the measurement of KIS is extremely difficult.
Interpretation of KIS measurements: The energy shift is caused by: 1)
difference in the mass of the isotopes, and 2) difference in their charge distributions.
2
δEexp = δEM ass + δECoul ≈ δEN M S + δECoul
(3)
3
50
There are two contributions to δEM ass : a) Inserting the M -dependent reduced mass µ for the electron in transition, we have the normal mass shift
δEN M S ; in the case of elements suitable for this measurement (Z > 40)
this shift is small and can be calculated accurately. b) The energy of the
relative movement of all the Z electrons is different in atoms containing different isotope masses; this causes the specific mass shift δESM S . This shift
is approximately −1/3 × δEN M S . Subtracting δEM ass from the experimental shift, the difference contains the contribution caused by the difference of
charge distributions; this is called Coulomb Shift: δECoul .
Seltzer has shown [2] that the Coulomb shift can be expressed by the
differences of even moments of the nuclear charge distribution. This series
converges rapidly; Seltzer calculated the values of the Z-dependent coefficients Ci (Z)(i = 1, 2, 3). Dividing δECoul by C1 , the nuclear parameter
λ≡
δECoul
C2
C3
= δhr2 i +
δhr4 i +
δhr6 i + ...
C1
C1
C1
(4)
is obtained. To sum up, the experimental determination of the Kα isotope
shift is very difficult. On the other hand, the physical interpretation is simple,
– as compared to the optical case (OIS); see the next section.
II.4 Isotope shift of optical spectrum lines (OIS)
The frequency ν of an optical transition depends on the size of the nucleus if one of the electron states, e.g. ns has an overlap with the nucleus.
Measuring the same transition for two isotopes (A1 , A2 ) of the same element,
the frequency difference δν1,2 contains the difference δ1,2 hr2 i.
Evaluation and interpretation. The frequency shift δν is caused by the
different masses (Mass Shift) and the different charge distributions of the
two isotopes (Field Shift):
δν = δνM S + δνF S = δνN M S + δνSM S + δνF S
(5)
The Normal Mass Shift (N M S) can be calculated accurately using the reduced electron mass. The the Specific Mass Shift (SM S) depends on the
movement of the Z - 1 electrons, its estimation contains uncertainties, which
51
result in systematic errors in the evaluation. Subtracting the mass shift, the
remaining field shift can be written in the form of a product: δνF S = F (Z)×λ
where F (Z) depends on the electron shell. Because of the uncertainties in the
calculation of δνSM S and F (Z), the accuracy of δhr2 i is only 10% - although
the frequency shift is measured with 0.1% relative accuracy! Therefore, for
long series of isotopes, often the shifts for the pairs (A1 , Ai , i = 1, 2, .., N .)
relative to a reference isotope pair (A1 , A2 ) are given, where the Z dependent
systematic errors cancel.
III. New methods for the determination of nuclear charge radii
and charge density distributions
III.1 Scattering of fast electrons on beams of radioactive nuclei
The idea (see refs. in [3]) is simple: the measurement is performed in flight
by the collision of electron and radioactive ion beams. The key parameter
of these experiments is the luminosity:
L = ne nRI Ncol I
(6)
where ne is the number of electrons in an electron bunch, nRI is the number of radioactive nuclei in an ion bunch, Ncol is the collision frequency
of the bunches, and I is a geometrical overlap integral, which can be increased by producing shorter bunches. The necessary luminosity is of order
1027 cm−2 s−1 . In the case of very short lifetime of the radioactive nuclei, the
luminosity L can be so low that the angular distribution can be measured
only within the first diffraction minimum, thus preventing the determination
of the charge distribution function ρ(r). In these cases, only the rms radius
can be determined.
The MUSES project (RIKEN, Japan) [4] contained two rings: one for the
electrons, one for the radioactive ions. Later, this arrangement proved to
be too expensive, but the results of an early calculation is useful to quote
here: the determination of ρ(r) may be expected for about 850 nuclei close to
stability, and hr2 i1/2 can be derived for 400 additional - less stable - nuclei.
In the SCRIT (Self Confining Radioactive Isotope Target) project [5] only
an electron ring is used; the radioactive ions are guided from the ion source
52
into a trap. The electric field of the electron beam forces the ions towards the
axis, while a pair of positive electrodes ensures longitudinal localisation. This
instrument has been tested by stable 133 Cs+ ions. From these preliminary
measurements a luminosity L = 2.5 × 1025 cm−2 s−1 was derived.
The ELISE (Electron-Ion Scattering Experiment) project, (GSI Darmstadt, Germany) [6]. This consist of an electron ring and a new experimental
storage ring (NESR) for the radioactive ions. It is also planned that interactions of the radioactive nuclei with helium and hidrogen gas targets will yield
information on the nuclear matter distribution, while the electron scattering
will explore the proton distribution. Combining these two results helps to
give limitations on the symmetry energy parameters. The knowledge of these
nucleon distributions in proton-rich and neutron rich radioactive nuclei has
great importance for the derivation of nuclear equation of state. The luminositiy range expected is from L = 1027 cm−2 s−1 to 1029 cm−2 s−1 dependig
on the nuclei to be investigated.
III.2 Measurement of X-ray transitions from trapped highly charged ions
The principle is similar to those of KIS and OIS (Sections II.3 and II.4):
from the measurement of the difference in the transition energies for two
different isotopes, the difference δhr2 i is derived. However, the ions are
highly charged, so that only a few electrons remain around the nucleus,
which renders possible more reliable calculations of the electron contribution
to the shift δνSM S and of F (Z). Only transitions between the same principal
quantum number are observed: ∆n = 0, e.g. 2S1/2 → 2P3/2 ; this reduces
the above corrections further. Owing to the ion trap, small quantities of
rare, very expensive or radioactive ions can also be investigated. The first
successful experiments were performed on the transition 2S1/2 → 2P3/2 of
uranium isotopes 233 U and 238 U [7], and yielded the value δhr2 i with an
accuracy better than earlier experiments.
53
Figure 1: The dielectronic recombination spectrum of a hypothetical heavy
element. Spectra from two different isotopes (A1 , A2 ) are shown by continuous and broken lines, respectively.
III.3 Isotope shift of di-electron recombination (DR-IS) spectra
In this process an electron of Ekin energy is captured into a high Rydberg
state (n ≥ 20) of an atom, and a core electron is also excited. As both the
states of the core electron and that of the Rydberg state are quantized, this
process takes place for well defined Ekin values only. Measuring the rate of
the process as the function of Ekin , resonances are obtained for each Rydberg
state, shown in fig. 1.
If one of the core electron state is an S state, this overlaps with the
nucleus, its energy depends on the nuclear charge distribution. Taking the
DR spectra for two isotopes of the same element, from the shift of the peaks
for the same n the value of δhr2 i is derived. The method has been applied to
highly charged 147 N d57+ and 150 N d57+ isotopes of neodymium [8] through
the following process:
e− +A N d57+ (1s2 2s1/2 ) →A N d56+ (1s2 2p1/2 , nlj)∗∗ →A N d56+ + photons
(7)
In this experiment the ion beam was lead through a monochromatic, cool
electron beam, in a tube. When the ion beam came to equilibrium, the
54
electron velocity was changed, so that the electrons had a common, nonzero kinetic energy Ekin relative to the ions. Passing the tube distance, the
electrons were directed away by a toriod, while the ions - following practically
the original direction - were separated according to their charge in a strong
dipole magnet, and the ions with charge +56 were detected. Repeating
this for various electron velocities, the DR spectra for n = 18, 19, 20 and
21 were taken. There are plans for the future to extend these di-electron
recombination isotope shift (DR-IS) investigations to unstable nuclei.
III.4 Application of Lamb-shift to measure nuclear radii
The charge distribution of nuclei has an effect on the shift of electron
energies from the Dirac value (Lamb-shift). Present day technique renders
possible to measure the 1S Lamb-shift in hydrogen to a high accuracy. Introducing the nuclear size into the theoretical calculations as a free parameter,
from the fit of theoretical values to experiment the nuclear size is obtained.
This method has been successfully applied to the hydrogen, i.e. to the proton [9]. Fig. 2 shows the the different contributions to the Lamb shift of 1S
electron states in hydrogen-like heavy atoms as the function of the atomic
number Z. The contribution from the nuclear size increases rapidly with Z.
In view of the successful measurement for hydrogen, one can expect even
better possibilities for heavier nuclei. During a preliminary study at the
experimental storage ring (ESR, GSI), uranium ions were accelerated, ionized completely to U92+ , slowed down, recombined with a single electron to
the hydrogen-like U91+ , and its characteristic X-ray spectra measured. This
experimental progress is encouraging.
However, serious difficulties arise in the theoretical calculations. Although
in absolute value the higher order QED corrections are less then the SE and
V P contributions, their uncertainty is higher. For high Z values αZ is close
to unity and perturbation methods can not be used. Therefore, there is an
intensive activity in the field of non-perturbative methods. In medium and
heavy nuclei the calculation of the nuclear polarization correction (N P ) is
even more difficult; remember that it was this correction that limited the
accuracy of muonic atom measurements, see Sect. II.2. The uncertainty of
the calculations is difficult to estimate; some authors give 25%.
55
Figure 2: Contributions to the 1S Lamb − shif t of hydrogen-like atoms in
the function of Z. The vertical scale is divided by Z4 for better visibility.
III.5 “Second generation” optical isotope shift
As discussed in Sect. II.4, the accuracy of derivation δhr2 i values from
the measured δν is limited by theoretical difficulties in calculating specific
mass shifts δνSM S and electron factors F (Z), see eq. (5). The mass shift
is especially critical for isotopes of light elements. In the last decade, highprecision atomic structure calculations were combined with accurate isotope
shift measurements for two- and three-electron systems in low Z atoms. The
progress is demonstrated by fig. 3, which shows the calculated mass shift
values δνSM S for the isotope pair 6,11 Li during the past ten years.
A more detailed figure can be found in [10], where the history of this
“second generation approach” to OIS is described in detail together with
references, numerical results and theoretical discussions for the rms radii of
Li isotopes. It should be stressed that the success of this method is mainly
due to recent developments in the few-body theory of atomic electrons.
56
Figure 3: Development of mass shift calculations for
6,11 Li
isotope pair.
References
[1] I.Angeli, et al., J. Phys. G. 36, 085102 (2009).
[2] E.C.Seltzer: Phys. Rev., 188, 1916 (1969).
[3] T.Katayama, et al., Phys. Scripta, T104, 129 (2003).
[4] M.Wakasugi, et al.: Phys. Rev. Letters, 100, 164 (2008).
[5] T.Suda, et al., Phys. Rev. Letters, 102, 102501 (2009).
[6] A.N.Antonov, et al., Nucl. Instr. Meth. P.R.A., 637 (2011).
[7] S.R.Elliott, Phys. Rev. Letters, 76, 1031 (1996).
[8] C.Brandau, et al., Phys. Rev. Letters, 100, 073201 (2008).
[9] K.Melnikov, and T.van Rittbergen, Phys. Rev. Letters, 84, 1673 (2000).
[10] W.Nörtershäuser, et al., Phys. Rev. C 84, 024307 (2011).
57
ACTA PHYSICA DEBRECINA
XLV, 58 (2011)
ADVANCES IN RADIOCARBON MEASUREMENT OF
WATER SAMPLES
R. Janovics1 , M. Molnár1 , I. Světlík2 , I. Major1 , L. Wacker3
1
2
Institute of Nuclear Research, Hungarian Academy of Sciences, Debrecen,
Hungary
Department of Radiation Dosimetry Nuclear Physics Institute AS CR, Prague,
Czech Republic.
3
Institute for Particle Physics, ETH Hönggerberg Zürich, Switzerland.
Abstract
In this paper we present two very different and novel methods
for 14 C measurement of water samples. The first method uses
direct absorption into a scintillation cocktail and a following liquid scintillation measurement. Typical sample size is 20-40 litre
of water and overall uncertainty is ±2% for modern samples. It
is a very cost-effective and easy to use method based on a novel
and simple static absorption process for the CO2 extracted from
groundwater. The other very sensitive method is based on accelerator mass spectrometry (AMS) using gas ion source. With
a MICADAS type AMS system we demonstrated that you can
routinely measure the 14 C content of 1 ml of water sample with
better than 1% precision (for a modern sample). This direct 14 C
AMS measurement of water takes less than 20 minutes including
sample preparation.
I. Introduction
Sustainable groundwater management is one of the most important task
for the human civilization. Subsurface resources with very old and clean
water can be affected with much younger and possibly not enough clean
modern surface waters by the human activity. Water supply of water resources usually originates from the surface [1]. If water supply is rapid, the
contamination by younger water may lead to problems, however, if it is slow
the water yielding capacity may rapidly decrease leading to the premature
termination of the water resource. To trace these conditions the investigation of the 14 C age of the water resources can be applied. 14 C is generated
from 14N by the cosmic radiation in the atmosphere with an approximately
constant rate and decays with a half life of 5730 years. The generated 14 C
isotope gets dissolved by the precipitation in the form of CO2 molecule and
falls out from the atmosphere. A part of the precipitation fallen out filters
to the soil and gets to the water resources within a shorter or longer time
depending on the porosity of the rocks [2]. Therefore, changes in the radiocarbon activity of the water resources in time give information about the
connection of the water resources with the surface waters. Using radiocarbon activity data of the water resources it is easier to model the subsurface
flow conditions of the water.
Radiocarbon emitted from nuclear facilities may get to the groundwater
in the form of dissolved CO2 , therefore, to explore the possible infiltrations
the most effective way besides the tritium is to measure the 14 C activity of
the groundwater in the groundwater monitoring wells settled in the vicinity
of nuclear facilities. With the help of the radiocarbon activity data of the
groundwater a contamination propagation model describing the flow conditions of the area can be established making the localisation of the source of
the contamination much easier [3].
Radiocarbon activity in hydrological samples has been measured in the
ATOMKI since the 1980’s by gas proportional counter (GPC) [4]. The GPC
method is one of the most accurate methods even today, however, a large
amount of sample is needed, and the sample preparation and the measurement require a lot of time. Besides, GPC is nowadays very rarely applied
due to its difficulty and the troublesome manufacture of the device. If highprecision measurement is to be taken by liquid scintillation counting technique, benzene should be prepared from the carbonaceous sample. Benzene
production requires very sophisticated and time consuming sample preparation, and also needs a large amount of samples [5]. The most high-profile
59
and most expensive method to measure 14 C is accelerator mass spectrometry (AMS) requiring significantly shorter measurement time compared to
the decay counting methods. It needs lower amount of samples by orders
of magnitude than the previous two methods, however, sample preparation
is time consuming even for this method. As a first step, graphite should
be generated from the CO2 deliberated from the water sample. To produce
graphite there are more methods (hydrogen or zinc reduction, TiH4 reduction
in sealed tube etc.)[6], which are all relatively expensive and slow.
A cheaper and well-available LSC method or a simpler AMS preparation
method would be preferable, which would be faster and cheaper than the
methods based on graphite production. We developed 1-1 methods to solve
both problems. During the development of the methods intercomparisons
were performed with the conventional GPC technique in both cases.
II. Direct absorption LSC counting method
A new chemical sample preparation method for liquid scintillation 14 C
measurements was implemented in the ATOMKI modifying previous routines [7,8]. Several tests were executed with old borehole CO2 gas without
significant content of 14 C and also performed on samples of known 14 C activities between 29 and 7000 pMC, previously measured by GPC. The 14 C
activities of all prepared samples were measured by an ultra low background
LSC (TRI-CARB 3170 TR/SL, Perkin Elmer) including quenching parameter (tSIE).
We used pure CO2 gas for LSC measurements after the same purification
procedure in the same chromatographic system as for radiocarbon dating
in our GPC. The carbonate containing samples were treated by acid (85%
H3 PO4 ). Than the obtained CO2 gas was purificated in a chromatographic
system [9]. Then the CO2 gas was frozen into a special plastic gas-storage
bag (Plastigas, Linde) equipped with a freezing finger for the quantitative
gas input. We connected the gas-storage bag to the vacuum line designed
for CO2 absorption (Fig. 1).
The line consists of the following parts: the CO2 sample in a Plastigas
(PG) equipped with a freezing finger (FF) and a fine-adjustable teflon valve
60
Figure 1: Schematics of CO2 absorption line.
(VG) for CO2 flow regulation. The bubbler (Hg) filled with Hg ensures
the visibility of the gas flow rate and precludes the intrusion of absorbent
vapour into the Plastigas. A 20-mL low-potassium glass vial with 11 mL
R
of Carbosorb - E
absorbent (CS) is connected to the quick vial-connector
(VF). The vial during the absorption is cooled about – 10 ◦ C degree using a
cooling flask (CF). The initial pressure reduction is ensured by an evacuated
puffer glass bulb (PB) with teflon valve (VB).
The valve of the Plastigas (VG) was slightly opened after evacuated the
system. Than the CO2 was continuously absorbed in the Carbosorb until
it became saturated. The amount of CO2 absorbed in Carbosorb was determined by direct weighing of the glass vial with the absorbent before and
after the absorption. After the absorption 11 mL PermaFlour-E cocktail
was added into each sample vial. Consequently, the samples were counted
by TriCarb 3170.
To test the sample preparation technique we performed absorption treatment on 8 samples (Table 1.). These samples were previously measured by
61
GPC technique.
Table 1: LSC vs. GPC intercomparision with different C-14 activity
(pMC) samples.
The measured pMC values with LSC technique were very similar to the
values of GPC method. To test the time-stability of the mixtures we repeated
the measurements of a higher activity sample during a long term (99 day).
There was no significant ageing effect. We concluded that the measurements
could be performed a longer time after the sample preparation. To estimate
the uncertainty of the blank sample preparation we prepared six parallel
blank (C-14 free) samples. The measured background count rates for the six
blanks were very similar. The average count rate was 1.869 ± 0.020 cpm. The
preparation of blank samples has a good reproducibility, any uncontrolled
contamination was not observed.
III. AMS technique
We investigated the applicability of a new method to prepare samples
from dissolved inorganic carbonate (DIC) in ground-water for radiocarbon
62
AMS analysis. This method doesn’t require graphite generation and a small
volume of water sample is enough for the radiocarbon measurement. The
procedure is very similar to pre-treatment of carbonate contained sample
preparation for stable isotope measurement with gasbench technique. We
applied a MICADAS type accelerator mass spectrometry (AMS) with gas
ion source for C-14 analysis.
The radiocarbon content of water was sat free with phosphoric acid and
then the headspace gas was rinsed vials. The developed pretreatment method
does not require sample preparation under vacuum, which significantly reduces the complexity. Reaction time and conditions can be easily controlled
as the CO2 content of a water sample is extracted by acid addition in a He
atmosphere using a simple septum sealed test tube. A double needle with
flow controlled He carrier gas is used to remove CO2 from the test tube (Fig.
2). The CO2 extraction yield is less than 80% mainly because a large portion
of the gas remains in solution in accordance with Henry’s law. CO2 is then
trapped on zeolite without using liquid N2 freezing.
The new method can be combined with an automated graphitization system like AGE from ETHZ [10] to have a fully automated water preparation
line for AMS graphite targets. In this case, about 5-12 ml of water is needed
for an AMS sample.
The greatest advantage of the new groundwater pretreatment method is
the possibility to connect the extraction line directly to an AMS system
using gas ion source interface (Fig. 3). The preparation is vastly reduced
compared to the other AMS methods and principally allows fully automated
measurements of groundwater samples with an auto-sampler.
With our MICADAS AMS system we demonstrated that the 14 C content
in 1 ml of water could be routinely measured with better than 1% precision (for a modern sample). The AMS measurement of one water sample
including sample preparation takes only about 20 minutes.
The method was tested on real groundwater samples and C-14 reference
carbonate materials from IAEA as well. We used IAEA carbonates as reference because C-14 reference water samples do not exist and the new AMS
method can be applied not only for dissolved inorganic carbon but also for
not dissolved carbonates in the very same way. As water sample transfer
63
Figure 2: CO2 transfer method using He carrier gas flushing through a double
needle (Thermo, Gasbench) in a septum sealed test tube (Labco, UK).
64
Figure 3: Layout of the connection of ground-water preparation line to the
existing ETHZ AMS gas ion source interface [11].
using classical plastic medical syringe from the sample holder into the septa
vial has the possibility to get atmospheric CO2 contamination into the samR
ple we tested the process blank level using high purity deionsied (MilliQ
)
water added to not dissolved IAEA C1 carbonate to have a better estimation on the processed blank level. Groundwater samples (1 to 4), whose age
spreads from very old to modern, were taken from real groundwater wells.
The groundwater samples were measured by the conventional GPC method
as reference (see ∗ results in Table 2). The radiocarbon activity of the CO2
gas was measured from 60 litre water in the case of the GPC method and
from 1 ml water in the case of AMS GI method. Table 2. shows the radiocarbon data of the samples measured by the two different methods.
Reproducibility of AMS GI measurement is excellent as shown by the 3
IAEA-C1, IAEA-C2 and 6 Groundwater-1 parallel measurements. The blank
level is very low (< 1.0 pmC). The processed blank (C1 carbonate+MilliQ
water) gave just a slightly higher level above the pure IAEA C-1 blank C-14
results (plus 0.1-0.6 pmC, depending on the carbon content of the sample)
which also could come from the trace carbon in the used MilliQ water. The
IAEA C-2 reference with known C-14 level has shown excellent match with
65
the expected value. The presented intercomparision between our conventional GPC and the new AMS GI methods was also satisfying. Just the
one sample (Groundwater-1) with the lowest C-14 level (∼ 20 pmC) gave a
slightly bigger difference than the certain measurement errors. It means that
it is very important to investigate more the sample storage effect on AMS
samples and other error sources during a C-14 analyses of groundwater even
for the conventional GPC method.
Table 2: Radiocarbon data of groundwater test and carbonate reference
materials measured by AMS gas ion source vs. reference values (IAEA reference data or water with known C-14 level as measured by conventional GPC
method). All reported AMS GI results are background subtracted except
IAEA-C1 results as they represent the background level.
66
IV. Conclusion
A new LSC sample preparation method for liquid scintillation 14 C measurements was implemented in the ATOMKI. The developed CO2 absorption
method is fast, inexpensive and simple. The corresponding limit of 14 C dating is 31200 year. The combined uncertainty of the described determination
is about 2% in the case of recent carbon.
A very sensitive and high throughput preparation technique for AMS
measurement was also developed. The new AMS gas ion source water C14 analyses require only 1 ml of water sample. The whole measurement
needs only 20 min of each sample. A MICADAS AMS system with online
sample preparation line is able to measure 14 C content from 1 ml water.
The precision of measurement is better than 1% (for modern samples). The
preparation is vastly reduced compared to the other AMS methods and principally allows fully automated measurements of groundwater samples with
an auto-sampler.
The presented two new methods can be suitable for 14 C measurements
and dating of hydrological, and environmental samples as well. The new
AMS facility in ATOMKI (Debrecen, Hungary) using an EnvironMICADAS
AMS system with gas ion source has a great potential in groundwater 14 C
analyses.
References
[1] S. M. Bruker, H. M. Haitjema, Modeling Steady State Conjunctive
Groundwater and Surface Water Flow with Analytic Elements, Water
Resources Research, 32/9 (1996).
[2] Z. Chen, Estimate of recharge from radiocarbon dating of groundwater
and numerical flow and transport modelling, Water Resources Research,
36/92607-26202000.
[3] É. Svingor, M. Molnár, L. Palcsu, Monitoring system with automatic
sampling units in the surrounding Paks NPP, Czechlovak Journal of
Physics 56, 133-139 (2006).
67
[4] E. Hertelendi, É. Csongor, L. Záborszky, J. Molnár, G. Dajkó, M.
GyÓrffi, S. Nagy, A counter system for high-precision 14 C dating, Radiocarbon 31/1, 399-406 (1989).
[5] G. Belluomini, A. Delfino, L. Manfra, V. Petrone, Benzene synthesis for
radiocarbon dating and study of the cathaliyst used for acetylene trimerization, The International Journal of Applied Radiation and Isotopes
29/7, 453-459 (2002).
[6] X. Xiaomei, S. E. Trumborea, S. Zhenga, J. R. Southona, K. E. Mcduffeea, M. Luttgena, J. C. Liu, Modifying a sealed tube zinc reduction method for preparation of AMS graphite targets: Reducing background and attaining high precision Nuclear Instruments and Methods
in Physics Research Section B: Beam Interactions with Materials and
Atoms 259/1, 320-329 (2007).
[7] Woo HJ, Chun SK, Cho SY, Kim YS, Kang DW, Kim EH, Optimization of liquid scintillation counting techniques for the determination of
carbon-14 in environmental samples, Journal of Radioanalysis and Nuclear Chemistry 239/3, 649-655 (1999).
[8] N. Horvatinčič, J. Baresič, Brončik, B. Obelič, Measurement of low 14 C
activities in a liquid scintillation counter in the Zagreb Radiocarbon Laboratory, Radiocarbon 46/1, 105-116 (2004).
[9] É. Csongor, I. Szabó, E. Hertelendi, Preparation of counting gas of proportional counters for radiocarbon dating, Radiochemical and Radioanalytical Letters 55, 303 (1982).
[10] L. Wacker, M. Němecb, 1 AND J. Bourquina, A revolutionary graphitisation system: Fully automated, compact and simple, Nucl. Instr. and
Meth. B 268, 931 (2010).
[11] M. Ruff, L. Wacker, H. W. Gäggeler, M. Suter, H-A. Synal, S. Szidat,
A gas ion source for radiocarbon measurements at 200 kV, Radiocarbon
49/2, 307-314 (2007).
68
ACTA PHYSICA DEBRECINA
XLV, 69 (2011)
CALCULATION OF CHARGE ASYMMETRY IN TOP
QUARK-PAIR HADROPRODUCTION AT NLO ACCURACY
A. Kardos
University of Debrecen, Department of Experimental Physics, Debrecen, Hungary
Abstract
We present the calculation of the charge asymmetry factor for top quark√
pair production at the TeVatron ( s = 1.96 TeV) at next-to-leading order
(NLO) accuracy in perturbative QCD. We perform the calculations with our
PowHel framework. We compare our predictions to recent measurement by
the CDF collaboration and find agreement within the respective incertainties
of the measurement and predictions.
I. Introduction
High-energy physics entered a new era with the start of LHC due to the
unprecetended energy range. Nevertheless, there are certain aspects of particle physics which can be better tested with a proton-antiproton collider,
such as the TeVatron. One such observable is the charge asymmetry [1, 2]
measured in heavy-quark-pair production∗ . This observable plays an important role in searching for new physics beyond the Standard Model (SM)
because it is sensitive to new color states, heavy vector bosons, scalars or
even gravitons.
The new CDF measurements [5] claim a significant deviation from the
SM predictions, therefore, we decided to reproduce the SM predictions with
our framework [6, 7] treating t t production at NLO accuracy.
∗
To some extent this asymmetry can also be studied at a p p collider [3, 4].
e−
e+
γ
µ−
e−
µ−
µ+
e+
µ+
γ
Figure 1: Two contributing Feynman-diagrams to e+ e− → µ+ µ− γ.
II. Physical origin of charge asymmetry
In order to show explicitly the presence of charge-asymmetric terms in
differential cross sections we decided to illustrate the problem through a
QED process, namely e+ e− → µ+ µ− γ. Two sample Feynman-diagrams are
shown on Fig. 1. In this case the charge asymmetry arises because the γ
couples to µ+ with e† , but with −e to µ− . The charge-asymmetric terms are
generated in the interfering terms of the squared matrix element (SME).
To show this explicitly we calculated the SME, then we introduced the
quantity
2
2 +
−
+
−
−
+
−
+
A = M(pe1 , pe2 , pµ3 , pµ4 , pγ5 ) − M(pe1 , pe2 , pµ4 , pµ3 , pγ5 ) =
(
8e6 y13
1 =
2y35 + y24 + y34 +
2y13 y24 + 2y14 y23 + y12 y35 +
ŝy12 y34 y35
y15
)
+ y14 y25 + y23 y45 + y24 y35 − (1 ↔ 2) − (3 ↔ 4) + (1 ↔ 2, 3 ↔ 4) ,
(1)
where yij = 2pi · pj /ŝ and ŝ is the partonic center-of-mass energy. Thus
we made the difference between two SME’s where the only difference arises
from the external kinematics: in the second term the momenta for muon
and antimuon are exchanged. The non-vanishing of this quantity leads to
the charge asymmetry. In this calculation we neglected the electron and
†
70
In our convention the electric charge of an electron is −e.
muon mass, since these masses are small and the phenomenon can be seen
without those terms as well.
III. Charge asymmetry in heavy quark production
In the previous section we explicitly showed the presence of charge asymmetry associated with a QED process. In the QCD case the origin of this
asymmetry can be found not in the electric but the color charge as gluons
do not have electric change, but they do have color charge, while the quark
and antiquark have color and anticolor, respectively. This is the basis of
charge asymmetry associated to heavy quark production. The interested
reader can find the QCD analog to Eq. 1 in [2] that looks much simpler than
Eq. 1, which is the mere consequence of the feature that QCD is an almost
supersymmetric theory [8].
Experimentally the study of this asymmetry is possible only when we can
identify heavy quarks through their decay products. In measurements one
uses the heaviest quark, the top, because it is the heaviest one and it almost
always decays into a W boson and b quark, and the latter can be tagged‡ ,
thus we can isolate those events where a top-quark pair is produced.
IV. Results
To measure the charge asymmetry we defined two quantities
AFB1 =
Nt (y > 0) − Nt (y < 0)
,
Nt (y > 0) + Nt (y < 0)
(2)
AFB2 =
Nt (y > 0) − Nt (y > 0)
,
Nt (y > 0) + Nt (y > 0)
(3)
where Nt (y > 0) means the number of those events where the top quark has
a positive rapidity. We defined these two definitions because the calculation
‡
The b quark lifetime is sufficiently long that its decay produces a well-separated secondary vertex in the detector, thus with a high-resolution apparatus we can isolate those
events where the b is produced in a top decay.
71
and the measurement of this observable is delicate from the statistical point
of view, since a difference is presented in the numerator that shows slow
convergence. Using these observables for the charge asymmetry observables
(CAO’s) we found agreement, but the second showed better convergence
behavior.
We calculated the CAO’s at NLO accuracy, using an up-to-date PDF
set, CTEQ6.6M [9], with a 2-loop running αs with a corresponding ΛMS
=
5
226 MeV. The top mass was chosen to be mt = 172.9 GeV.
We analyse the charge asymmetry in t t production when the tree-level
process is charge symmetric. Thus the effect is only present in the radiative
correction, the observable for charge asymmetry only bears LO accuracy.
Whence we expect strong dependence on the renormalization and factorization scales, µR and µF resepctively. We list our predictions for the charge
asymmetries at three different scale choices in Table 1.
µ
2mt
mt
mt /2
AFB1 (%)
2.88 ± 0.01
3.77 ± 0.01
5.47 ± 0.01
AFB2 (%)
2.87 ± 0.01
3.77 ± 0.01
5.47 ± 0.01
Table 1: Inclusive charge asymmetries obtained at different scale choices
(µ = µR = µF ) and with a setup listed in the text.
We also calculated the differential CAO’s with respect to the invariant
mass of the t t system. Our predictions for these together with the results of
the recent CDF measurements [5] are shown on Fig. 2 and on Fig. 3.
We collect our predictions with the corresponding experimental results
in Table 2. In this table we list our results for inclusive and differential
CAO’s with uncertainties coming from scale-dependence. At NLO accuracy
we found agreement with experiment within the resepctive uncertainties and
we find that the calculation for the two CAO definitions give identical predictions.
72
AFB1 [%]
35
30
25
20
15
10
5
0
-5
√
PowHel-NLO
CDF data
s = 1.96TeV
mt = 172.9GeV
µR = µF = mt
CTEQ6.6M
0
100 200 300 400 500 600 700 800 900
mtt̄ [GeV]
Figure 2: This plot shows the charge asymmetry as a function of the t t
system’s invariant mass calculated by using Eq. 2 and the measurement done
by CDF. In our calculation the band represents the scale uncertainty while
in the case of the measurement it illustrates the uncertainty associated with
the measurement itself.
incl.(%)
mtt < 450 GeV
mtt > 450 GeV
CDF
ANLO
FB1
ANLO
FB2
5.7 ± 2.8
−1 ± 3
21 ± 5
3.77+1.70
−0.89
0.48+0.19
−0.10
11.58+8.29
−3.31
3.77+1.70
−0.89
0.48+0.19
−0.10
11.58+6.77
−3.33
Table 2: Result of the CAO measurements and predictions from our calculations. The uncertainty shown for the CDF measurement is the total
uncertainty coming from the measurement while the uncertainties in our
calculations are come from the scale-dependence.
Conclusions
73
AFB2 [%]
35
30
25
20
15
10
5
0
-5
√
PowHel-NLO
CDF data
s = 1.96TeV
mt = 172.9GeV
µR = µF = mt
CTEQ6.6M
0
100 200 300 400 500 600 700 800 900
mtt̄ [GeV]
Figure 3: This plot shows the charge asymmetry as a function of the t t
system’s invariant mass calculated by using Eq. 3 and the measurement done
by CDF. In our calculation the band represents the scale uncertainty while
in the case of the measurement it illustrates the uncertainty associated with
the measurement itself.
In this paper we presented the charge asymmetry factors calculated with
PowHel in the case of t t production at NLO accuracy. We also calculated the
differential distribution of these respect to the t t system’s invariant mass.
In all cases we found that the experimental result is in agreement with our
predictions. As it is evident from our results the charge asymmetries suffer
large uncertainties from scale-dependence as expected, thus to make more
precise predictions, one should go beyond NLO accuracy.
In this paper we only analysed the charge asymmetry in an NLO calculation with stable top and antitop. We can elaborate further on this subject if
we decay our heavy particles and apply hadronization and parton showering
effects. We leave these aspects for a later publication.
74
Acknowledgements
This research was supported by the HEPTOOLS network. The author
would like to thank NCSR Demokritos for hospitality. The author is grateful to Costas Papadopoulos and Giuseppe Bevilacqua for useful discussions
and help with the original HELAC programs. The author is also grateful to
Joey Huston for providing him computational power at the Michigan State
University.
References
[1] F. Halzen, P. Hoyer, C. S. Kim, Phys. Lett. B195, 74 (1987).
[2] J. H. Kuhn, G. Rodrigo, Phys. Rev.
ph/9807420].
[3] P. Ferrario, G. Rodrigo,
[arXiv:0809.3354 [hep-ph]].
D59, 054017 (1999). [hep-
Phys. Rev.
D78,
094018 (2008).
[4] G. Rodrigo, P. Ferrario, Nuovo Cim. C33, 04 (2010). [arXiv:1007.4328
[hep-ph]].
[5] T. Aaltonen et al. [ CDF Collaboration ], Phys. Rev. D83, 112003
(2011). [arXiv:1101.0034 [hep-ex]].
[6] A. Kardos, C. Papadopoulos, Z. Trocsanyi, [arXiv:1101.2672 [hep-ph]].
[7] M. V. Garzelli, A. Kardos, C. G. Papadopoulos, Z. Trocsanyi, Europhys.
Lett. 96, 11001 (2011). [arXiv:1108.0387 [hep-ph]].
[8] S. J. Parke, T. R. Taylor, Phys. Lett. B157, 81 (1985).
[9] P. M. Nadolsky, H. -L. Lai, Q. -H. Cao, J. Huston, J. Pumplin,
D. Stump, W. -K. Tung, C. -P. Yuan, Phys. Rev. D78, 013004 (2008).
[arXiv:0802.0007 [hep-ph]].
75
ACTA PHYSICA DEBRECINA
XLV, 76 (2011)
PARITY DETERMINATION OF EXCITED STATES OF THE
132 La NUCLEUS
I. Kuti1 , J. Timár1 , D. Sohler1 , B. M. Nyakó1 , L. Zolnai1 ,
Zs. Dombrádi1 , E. S. Paul2 , A. J. Boston2 , H. J. Chantler2 ,
M. Descovich2 , C. Fox2 , P. J. Nolan2 , J. A. Sampson2 , H. C. Scraggs2 ,
A. Walker2 , J. Gizon3 , A. Gizon3 , D. Bazacco4 , S. Lunardi4 , C. M.
Petrache4,5 , A. Astier6 , N. Buforn6 , P. Bednarczyk7 , N. Kintz7 ,
K. Starosta8 , D. B. Fossan9 , T. Koike10 , C. J. Chiara11 ,
R. Wadsworth12 , A. A. Hecht13 , R. M. Clark14 , M. Cromaz14 ,
P. Fallon14 , I. Y. Lee14 , A. O. Machiavelli14
1
Institute of Nuclear Research, Debrecen, Hungary
OLL, Department of Physics, University of Liverpool, Liverpool, UK
3
Institut des Sciences Nucléaires, IN2P3-CNRS, Grenoble, France
4
Dipartimento di Fisica and INFN Sezione di Padova, Padova, Italy
5
Dipartimento di Fisica, Università di Camerino, Camerino, Italy
6
Institut de Physique Nucléaire de Lyon, IN2P3-CNRS, Villeurbane, France
7
Institut de Recherches Subatomiques, Strasbourg, France
8
Dept. of Chemistry, Simon Fraser University, Burnaby, Canada
9
Dept. of Physics and Astronomy, SUNY, Stony Brook, New York, USA
10
Graduate School of Science, Tohoku University, Sendai, Japan
11
Dept. of Chemistry and Biochemistry, Univ. of Maryland, Maryland, USA
12
Dept. of Physics, University of York, York, UK
13
Wright Nuclear Structure Laboratory, Yale University, New Haven, USA
14
Nuclear Science Division, LBNL, Berkeley, California, USA
2
Abstract
Medium- and high-spin states of 132 La were populated using
fusion-evaporation reactions 100 Mo (36 S,p3n) and 116 Cd(23 Na,
2p5n), and they were studied with the EUROBALL and GAMMASPHERE detector arrays. Unambiguous multipolarity assignment was made for ∼35% of the observed transitions using
linear polarisation analysis and determination of internal conversion coefficients. The derived multipolarities enabled us to
determine parities of many excited states of 132 La. Methods and
results of this study are discussed in the present paper.
I. Introduction
Investigating nuclei with possible chiral structures requires thorough experimental examination of the excited states of the studied nuclei, like a
properly built level scheme, unambiguous determination of the spins, parities and accurate configuration assignments. Determination of the exact
multipolarities of γ transitions between the excited states is a good tool to
assist in the spin-parity assignment. For the chiral-candidate nucleus 132 La,
two experiments were carried out using fusion-evaporation reactions 100 Mo
(36 S,p3n) and 116 Cd(23 Na,2p5n), which were studied with the EUROBALL
and GAMMASPHERE detector balls, respectively. Using the data obtained
in these experiments, earlier studies deduced unambiguous parity only for
the low-energy part of the yrast band [1]. In order to determine unambiguous spins and parities of the excited states of 132 La, multipolarities of the
depopulating γ transitions have been deduced. Multipole orders of the γ
transitions were determined using angular correlation (DCO) analysis, and
presented in a previous paper [2]. The electric or magnetic nature of the
γ-rays with sufficiently large energies were determined on the basics of their
measured linear polarization values. The polarization sensitivity for the lowenergy γ-rays were too small, therefore the multipolarities of these γ transitions were deduced from internal conversion coefficients determined from
transition-intensity balances.
II. Experimental methods
In order to obtain information on the γ-ray multipolarities of 132 La, angular correlation information was extracted from the data collected in the
GAMMASPHERE experiment [2]. The electromagnetic character of the
transitions cannot be determined from the DCO ratios. Therefore we deduced the electromagnetic characters by measuring the linear polarization of
the γ rays, using data obtained by the EUROBALL detector system. The
77
probability of Compton scattering in different directions is dependent upon
the linear polarisation of the scattered γ ray, hence this phenomenon can be
used to determine the electric or magnetic properties [3, 4]. For this purpose,
the four-element clover detectors of the EUROBALL array, placed close to
90o relative to the beam direction (Fig. 1.) were used as Compton polarimeters [5]. Two matrices were constructed from γγ-events; single hit energies
in any detectors were placed on one axis while the added-back double-hit
scattering event energies (the sum of the energy deposited in one of the crystals due to the Compton-scattering of the gamma ray and the photo-peak
energy of the scattered gamma ray deposited in the neighbouring crystal)
were placed on the other axis. In the first matrix the scattering events took
place perpendicular, while in the second matrix parallel to the reaction plane.
The number of perpendicular (N⊥ ) and parallel (Nk ) scatters for a given γ
ray were obtained from spectra gated on the single-hit axis of the respective
matrix by transitions in coincidence with the given γ ray. The spectra of
parallelly and perpendicularly scattered gamma rays gated by the 161 keV
transition are shown on Fig. 2. Assuming that each clover crystal has equal
efficiency, an experimental linear polarization is defined as
P =
1 N⊥ − Nk
,
Q N⊥ + Nk
(1)
where Q is the polarization sensitivity for the clover detectors, which is
a function of the γ-ray energy [5]. N⊥ and Nk denote the number of events
scattered perpendicular and parallel to the reaction plane, respectively. P>0
is characteristic for stretched E1, E2 and non-stretched M1 transitions, while
P<0 characterizes the stretched M1 and non-stretched E1 transitions.
The polarization sensitivity considerably decreases below the γ-ray energy of approximately 150 keV, hence the linear polarization values could
not be determined for the low-energy transitions. Among other bands in
132 La, earlier studies found a low-energy band with four transitions [6], each
with energy below 160 keV (see lower-left part of Fig. 3). Accordingly, determination of the M1 or E1 character of these transitions was carried out
in the present work by deriving the internal conversion coefficients of the γ
rays from data obtained in the GAMMASPHERE experiment. In internal
conversion process the energy difference between the initial and final states
78
Figure 1: The EUROBALL detector system.
Figure 2: Spectra of parallelly and perpendicularly scattered gamma rays
gated by the 161 keV transition.
79
Figure 3: Part of the
132 La
level scheme.
is transferred to a bound atomic electron, which is emitted from the atom.
This process competes with γ ray emission in the decay of an excited nuclear state [7, 8]. For some states γ ray emission may be inhibited (e.g. by
high multipolarity transition or a low γ ray energy) and the competing internal conversion decay branch becomes significant. The α internal conversion
coefficient is defined as:
α=
Ielectron
.
Iγ
(2)
To determine the internal conversion coefficients, the relative coincidence
intensities of the studied γ transitions seen in multiple gated coincidence
spectra were examined. We have analized these coincidence intensities in
cascades involving the studied γ-rays. The gates were set in a way that in the
studied cascades the relative total (γ-ray + conversion electron) transition
intensities of the transitions were expected to be equal. Using the known
M1 or E1 character of a reference γ-ray in the cascade, the relative total
transition intensities of the other transitions can be calculated, and their α
internal conversion coefficients can be derived by measuring their relative
80
Figure 4: The obtained linear polarization values.
γ-ray intensities in these spectra.
III. Results
During the linear polarisation analysis we determined the electromagnetic
character of ∼40 of the 108 observed γ transitions. The results of the linear
polarization analysis are presented in Fig. 4. As it is visible in the figure, the
linear polarisation values are divided into two, well-separated groups above
and below 0. On the basis of the deduced values and our previous DCO
results [2] 19 transitions were found to have electric character, while 20 γ
rays showed magnetic behaviour.
Our earlier DCO measurements showed that the γ rays examined in the
studied low-energy band are streched dipoles [2], which determines the ∆I=1
spin difference between the initial and final states of the γ rays, but does not
give any information on the parities of the connected states. Applying the
method described in the previous section we can determine internal conversion coefficients and consequently the multipolarity of the γ rays of interest.
Knowing that the reference 135 and the 161 keV γ transitions which are
in coincidence with the examined γ rays, have M1 character [6], the relative
total transition intensity in the cascade can be calculated, and the α internal
conversion coefficients of the examined transitions can be deduced. The rel81
Figure 5: Experimental internal conversion coefficients of four studied transitions of 132 La (points with error bars). Curves show theoretical values, as
a function of γ-ray energy [9].
ative coincidence intensities of the examined transitions seen in double gated
coincidence spectra have been determined, setting the 294-(279+312+482)
keV, 160-(231+557) keV, 160-(414+557) keV and 130-(231+557) keV gates
for the 66.8, 96.5, 129.6 and 160.5 keV transitions respectively. In these
spectra the relative total intensities for the transitions should be equal.
Table 1: α internal conversion coefficients of low-energy γ
rays assigned to 132 La.
Eγ
66.8
96.5
129.6
160.5
α
2.84(54)
1.34(24)
0.45(15)
0.36(14)
αth (M 1)
3.45
1.22
0.53
0.29
αth (E1)
0.71
0.26
0.12
0.06
The obtained α internal conversion coefficients are given in Table 1, and
are presented in Fig. 5. together with the theoretical E1 and M1 curves taken
from Ref.[9]. The results show a good agreement with the M1 theoretical
values, hence the examined γ transitions of the 132 La have M1 multipolari82
ties.
IV. Summary
Study of the medium- and high-spin states of 132 La was carried out with
the EUROBALL and GAMMASPHERE detector arrays. In order to obtain
parity assignments for the excited states, we made multipolarity assignments
for ∼35% of the observed transitions on the basis of linear polarisation analysis and determination of internal conversion coefficients. This information
was inevitable in the study of the structure of chiral-candidate nucleus 132 La.
References
[1] J. Timár et al., Eur. Phys. J. A 16, 1 (2003).
[2] I. Kuti et al., Acta Physica Debrecina XLIV, 59 (2010).
[3] Starosta K. et al., Nucl. Instr. and Methods A 423, 16 (1999).
[4] Droste Ch. et al., Nuclear Instruments and Methods 328, 518 (1995).
[5] P. M. Jones et al., Nucl. Instr. and Meth. A 362, 556 (1995).
[6] V. Kumar et al., Eur. Phys. Journal A 17, 153 (2003).
[7] Rösel F. et al., Atomic and Nuclear Data Tables 21, 91 (1978).
[8] Jaeger, J., Proceedings of the Royal Society of London 148, 708 (1935).
[9] T.Kibedi et al., Nucl. Instr. and Meth. A 589, 202 (2008).
83
ACTA PHYSICA DEBRECINA
XLV, 84 (2011)
ON THE ACCURACY OF ATOMISTIC SIMULATIONS OF IR
SPECTRA IN AQUEOUS SOLUTIONS §
A. Derecskei-Kovacs1 , I. Halasz2 , B. Derecskei3 , I. Tamassy-Lentei4
1
Air Products Computational Modeling Center, 7201 Hamilton Blvd., Allentown,
PA 18195, USA
2
PQ corporation, Research and Development Center, 280 Cedar Grove Road,
Conshohocken, PA 19428, USA
3
Millenium Inorganic Chemicals, a Cristal company, Research Center, 6752 Bay
Meadow Dr., Glen Burnie, MD 21060 USA
4
Department of Theor. Physics, University of Debrecen, POB 5, H-4010
Debrecen, Hungary
Abstract
Vibrational spectroscopic methods including infrared (IR) and
Raman spectroscopy are widely used in qualitative and quantitative characterization of chemical systems. The computational
accuracy of vibrational frequency calculations has been extensively studied and continuously improved by the software developers and the computational community. Using the simulated
infrared spectra of ions derived from hydrated sodium monosilicate we demonstrate that the accuracy of the chemical model is
equally or perhaps even more important in some cases to reproduce even the qualitative features of the experimental spectra
of some ions in aqueous solutions. The models must explicitly
include a hydrate shell surrounding the ions and applying the
continuum solvent model alone does not change the qualitative
behavior of vibrational spectra.
§
Dedicated to Prof. I. Lovas on the occasion of his 80th birthday
I. Introduction
Quantum chemists often specify the level of their calculations in a two
dimensional coordinate system with the basis set size on the horizontal axis
and the treatment of electron correlation on the vertical axis. The ’best’
calculation is located in the far right top corner of this imaginary diagram.
The accuracy of the calculations is routinely improved by moving the computational method toward that direction (i.e. increasing basis set size and
improving the treatment of electron correlation). However, there are some
cases when even the highest level of quantum chemical calculations fails to
correctly reproduce the experimental observations; namely, performing very
accurate calculations on an insufficient model system still yields erroneous
results. The challenge is that improving the accuracy of the chemical model
system is less straightforward than improving the quality of the computational approach. To determine which are the ’must have’ aspects of the
system for the purposes of a simulation may become challenging and requires intimate knowledge of the chemical system under study. Solution
phase chemistry may be one example of such systems: in many cases the
solvent does not play any role; in many cases the accuracy of the model
can be improved by applying simple continuum solvent models; and in many
cases, explicit solvent molecules also have to be considered.
Simulated IR spectra are often used for chemical analytical purposes.
The goal usually is to identify the presence and amount of various species
in a chemical system and follow the changes in time as a chemical reaction
takes place (if the reaction kinetics allows). In the present work, we attempt
to elucidate the structures of alkaline silicate molecules dissolved in water
using this methodology and identify an appropriate chemical model system
for their description.
The size of the dissolved silicates varies from 1 to 13 [SiO4 ] tetrahedral
building block per component depending on the concentration, the type of
the alkaline ion, and the alkaline/silica ratio [1, 2]. Aqueous silicate solutions
are important in biology, geology, and a number of technical processes from
zeolite synthesis to crude oil drilling [3]; however, there are a lot of unknowns
about the complex network of reactions changing the precursor molecules
into the complex and evolving silicate solution.
85
Vibrational characterization of such solutions is a convenient way to identify some characteristic siloxane rings and Q0 , Q1 , Q2 , or Q3 connectivities
between their [SiO4 ] building blocks in the sub-nanometer sized, dissolved
silicate particles (Q4 has not been identified in solutions) [1-4]. However,
these structural assignments of vibrational spectra mostly rely on analogous
studies on solids such as zeolites and glasses. Since high resolution Si29 NMR
studies indicate that the siloxane rings of the small soluble silicate ions might
be quite different from those found in solids [5], confirmation of the vibrational spectra of such hypothetical structures by molecular models would be
highly desirable.
It will be shown in this paper that simple gas phase models are nonadequate to generate monosilicate models which have reasonably similar IR
spectra to the experimental spectra of the corresponding dissolved silicates
regardless of the level of approximation used for the model calculations. Even
when the well established COSMO (conductor-like screening model) method
[6] is used for modeling potential silicate molecules in their aqueous solution
as used by others [7], the calculated spectra remain even qualitatively inaccurate. Good spectral resemblance can be achieved only when numerous
H2 O molecules surrounding the silicate ions are explicitly included into the
calculations.
II. Methodology
The preparation of silicate solutions used in the experiments has been
described in detail elsewhere [8].
R
The DMol3 module [9] as implemented in Materials Studio
4.0 program
package by Accelrys was used to perform the model calculations. Specifically,
full geometry optimizations and vibrational frequency analysis were carried
out using the BLYP gradient corrected density functional in conjunction
with the DNP double-numeric basis set with polarization functions in the all
electron approximation using the Fine quality throughout. Frequency and
intensity benchmark calculations were carried out using Gaussian 03 [10].
86
Figure 1: Experimental FTIR ATR spectra of 3 M and 0.2 M aqueous
surNa2 SiO3 solutions versus BLYP/DNP model spectrum of H2 SiO2−
4
rounded with 14 H2 O molecules.
III. Results and discussion
Figure 1 shows the experimental FTIR spectra of a 3 and a 0.2 mol/L
metasilicate solutions which contain only Na2 H2 SiO4 monomers from which
about 32 and 82% of the Na+ ions are dissociated in the pH range of 13.0-13.6
as was demonstrated before along with further details of these spectra.[8] It
can be seen in Figure 1 that dilution has a strong impact on the experimental
spectrum: peak positions and relative intensities are both changing.
In order to improve our understanding of the vibrational spectra of dissolved silicates, we calculated the simulated spectra of related monosili−
2−
−
cates (H2 SiO2−
4 , H3 SiO4 , H2 SiO4 , Na2 H2 SiO4 , H4 SiO4 , NaH2 SiO4 ) at the
BLYP/DNP level. As it was shown earlier [19] none of the spectra could
reproduce the experimental observations.
First, we attempted to change the computational model using hybrid density functionals and larger basis sets as implemented in Gaussian 03 [10] to
carry out benchmark calculations. As described in detail elsewhere [19], neither the better quality functional nor the larger basis affected significantly
87
Figure 2: The model spectrum of H2 SiO2−
4 calculated at the BLYP/DNP
level in vacuum and using a continuum solvent model.
the calculated IR spectrum. Applying a continuum solvent model using the
method of COSMO (conductor-like screening model) [6] changes the peak
positions somewhat but fails the change the qualitative behavior of the spectrum as illustrated in Figure 2 for the H2 SiO2−
4 anion (see for example the
preservation of the strong band at 600-700 cm−1 completely absent in the
experimental spectrum).
Second, we turned to change the chemical model by including explicit
water molecules as a hydration layer. The solvent molecules were placed first
to the proximity of a hydrogen or oxygen atom of the H2 SiO2−
4 ion and the
system was relaxed to reach one of the many local minima; so the calculated
spectra represent only one out of many possibilities collectively present in
the statistically averaged experimental spectrum. Figure 3 illustrates that
an increasing number of H-bonded H2 O molecules around the H2 SiO2−
4 ion
shifts both the positions and intensities of the IR bands substantially.
In our hands, the best agreement with the experimental spectrum of
the highly dissociated 0.2 M metasilicate solution was found with 14 H2 O
molecules. Simple size considerations suggest that approximately 18-20 H2 O
are needed to form a full hydrate layer around the H2 SiO2−
4 anion. In our
simulations (even after generating several different initial geometries), when
88
Figure 3: The shifts in the position and intensity of one of the major peaks
in the simulated IR spectrum of H2 SiO2−
4 calculated at the BLYP/DNP level
as a function of the number of explicitly coordinated water molecules.
15-20 H2 O molecules were used, the ’excess’ solvent molecules preferably
formed the beginnings of a second hydration shell instead of directly coordinating to the central anion which was left partially uncovered as shown
in Figure 4. The partial second hydration led to a loss of accuracy relative
to the experiments. It is very likely that the second coordination shell will
have to be completed before the central anion is fully solvated which would
probably yield a more accurate spectrum; however, this model is beyond our
current computational capabilities.
The inevitable conclusion is that in spite of the increasing computational
demand, it is necessary to include explicit water molecules to obtain a simulated spectrum for the aqueous silicate molecules the goal is a qualitatively
accurate description of the experimental spectrum.
IV. Summary
The IR spectra of some species relevant to the dissociation of Na2 H2 SiO4
in aqueous solution were investigated at several different levels using density
functional theory. It was demonstrated that the chemical model system must
contain several explicit water molecules in order to reproduce the qualitative
89
anion surrounded by
Figure 4: Space filling representation of a H2 SiO2−
4
20 explicit water molecules showing some uncovered central anion and the
beginning of the formation of the secondary hydrate layer.
behavior of experimental IR spectra while using more complex computational
methodology or a continuum solvent model fails to significantly improve the
quality of isolated molecule calculations.
Acknowledgement
The authors thank Mr. Mukesh Agarwal of PQ Corporation for measuring
the experimental FTIR spectra and the MIC management for supporting the
use of hardware and software resources at MIC’s Research Center.
References
[1] Halasz, I., Li, R., Agarwal, M., Miller, N.,Stud. Surf. Sci. Catal. 170A,
800 (2007).
90
[2] Halasz, I., Agarwal, M., Li, R., Miller, N.,Zeolites and Related Materials:
Trends, Targets and Challenges. Proceedings of 4th International FEZA
Conference, Ed. by A. Gedeon, P. Massiani and F. Babonneau, 2008
Elsevier, in press.
[3] Falcone, J. S., Jr., Surfactant Sci. Ser. 131, 721 (2006).
[4] Halasz, I., Li, R., Agarwal, M., Miller, N., What can vibrational spectroscopy tell us about the structure of dissolved sodium silicates?, Microporous Mesoporous Materials, submitted.
[5] Knight, T., G., Wang, J., Kinrade, S. D., Phys. Chem. Chem. Phys. 8,
3099 (2006).
[6] Klamt, A.; Schüürmann, J. Chem. Soc., Perkin Trans. 2, 799 (1993).
[7] Trinh, T. T., Jansen, A. P. J., van Santen, R. A., J. Phys. Chem. B
110, 23099 (2006).
[8] Halasz, I., Agarwal, M., Li, R., Miller, N., Catal. Lett. 117, 34 (2007).
[9] Delley, B., J. Chem. Phys. 92, 508 (1990); 113, 7756 (2000).
[10] Gaussian 03, Revision C.02, Frisch, M. J., Trucks, G. W., Schlegel, H.
B., Scuseria, G. E., Robb, M. A., Cheeseman, J. R., Montgomery, Jr.,
J. A., Vreven, T., Kudin, K. N., Burant, J. C., Millam, J. M., Iyengar,
S. S., Tomasi, J., Barone, V., Mennucci, B., Cossi, M., Scalmani, G.,
Rega, N., Petersson, G. A., Nakatsuji, H., Hada, M., Ehara, M., Toyota,
K., Fukuda, R., Hasegawa, J., Ishida, M., Nakajima, T., Honda, Y.,
Kitao, O., Nakai, H., Klene, M., Li, X., Knox, J. E., Hratchian, H. P.,
Cross, J. B., Adamo, C., Jaramillo, J., Gomperts, R., Stratmann, R.
E., Yazyev, O., Austin, A. J., Cammi, R., Pomelli, C., Ochterski, J. W.,
Ayala, P. Y., Morokuma, K., Voth, G. A., Salvador, P., Dannenberg,
J. J., Zakrzewski, V. G., Dapprich, S., Daniels, A. D., Strain, M. C.,
Farkas, O., Malick, D. K., Rabuck, A. D., Raghavachari, K., Foresman,
J. B., Ortiz, J. V., Cui, Q., Baboul, A. G., Clifford, S., Cioslowski,
J., Stefanov, B. B., Liu, G., Liashenko, A., Piskorz, P., Komaromi,
I., Martin, R. L., Fox, D. J., Keith, T., Al-Laham, M. A., Peng, C.
Y., Nanayakkara, A., Challacombe, M., Gill, P. M. W., Johnson, B.,
91
Chen, W., Wong, M. W., Gonzalez, C., and Pople, J. A., Gaussian,
Inc., Wallingford CT, 2004.
[11] Jamieson, P. B., Dent Glasser, L. S., Acta Cryst. 20, 688 (1966).
[12] McDonald, W. S., Cruickshank, D. W. J., Acta Cryst. 22, 37 (1967).
[13] Lazarev, A. N., Vibrational Spectra and Structure of Silicates, Consultants Bureau, NY, London, 1972.
[14] van Santen, R. A., Vogel, D. L., Advances in Solid-State Chemistry 1,
151 (1989).
[15] Bouteiller, Y., Perchard, J. P., Chem. Phys. 305, 1 (2004).
[16] Halasz, I., Agarwal, M., Marcus, B., Cormier, W. E., Microporous Mesoporous Materials 84, 318 (2005).
[17] Halasz, I., Agarwal, M., Senderov, E., Marcus, B., Cormier, W. E.,
Stud. Surf. Sci. Catal. 158, 647 (2005).
[18] Lazarev, A. N., Molecular approach to solids, Elsevier, Amsterdam,
1998.
[19] Halasz, I, and Derecskei-Kovacs, A., Molecular Simulation 34, 937
(2008).
92
ACTA PHYSICA DEBRECINA
XLV, 93 (2011)
FOSSIL FUEL CO2 ASSAY BY SIMULTANEOUS
ATMOSPHERIC 14 C AND CO2 MIXING RATIO
MEASUREMENTS IN THE CITY OF DEBRECEN
I. Major1,2 , M. Molnár1 , L. Haszpra3 , É. Svingor1 , M. Veres4
1
Hertelendi Laboratory of Environmental Studies, MTA ATOMKI, Bem tér 18/c,
H-4026 Debrecen, Hungary
2
University of Debrecen, Debrecen, Hungary
3
Hungarian Meteorological Service, Budapest, Hungary
4
Isotoptech Zrt., Debrecen, Hungary
Abstract
A field unit was installed in Debrecen at the beginning of autumn 2008 to monitor the atmospheric fossil fuel CO2 content of
the city. To establish a reference level CO2 sampling was started
at a rural site of Western Hungary (Hegyhátsál). Comparing
the 14 CO2 level observed at that background site with the city
site, the excess atmospheric CO2 component originating from
fossil fuel was reported in a regional scale. With a maximum in
the middle of winter 2008, a definite fossil fuel CO2 peak (10-15
ppm) was observed in the air of Debrecen. Significant local maximum (∼20 ppm) in fossil fuel CO2 during Octobers of 2008 and
2009 was also noted. Stable isotope results were in agreement
with the 14 C based fossil fuel CO2 observations as the winter of
2008 and 2009 was different in atmospheric δ 13 C variations, as
well. The more negative δ 13 C value of atmospheric CO2 in the
winter of 2008 means more fossil carbon in the air than in the
winter of 2009.
I. Introduction
Classical emission-estimates of fossil fuel CO2 and other greenhouse gases
are based on bottom-up statistics (estimating emissions using detailed process or facility-specific data); however, the accuracy of these estimates is
a matter of permanent debate, since current bottom-up inventory data are
reported by governments, and has the potential to be biased, especially as
emissions are regulated in the future. In the case of fossil fuel CO2 , error
claims range from ±2% to more than ±15%, exceeding the reduction target
at the higher end [1]. An independent method to estimate trace gas emissions is the top-down approach, using atmospheric measurements, but CO2
concentration observations alone do not allow source apportionment.
Estimating fossil fuel CO2 in the atmosphere is, in principle, possible via
radiocarbon (14 CO2 ) measurements. CO2 from burning of fossil fuels, due
to their long storage time of several hundred million years, is essentially free
of 14 C. Radiocarbon observations in atmospheric CO2 has been explored as
an excellent tracer for recently added fossil fuel CO2 in the atmosphere [2-7].
Adding fossil fuel CO2 to the atmosphere, therefore, leads not only to an
increase in the CO2 content of the atmosphere, but also to a decrease in
the 14 C/12 C ratio in atmospheric CO2 [8]. From a 14 CO2 measurement at a
polluted sampling site, for example, near the ground level on the continent,
we can directly calculate the regional fossil fuel CO2 surplus (recently added
fossil fuel CO2 amount in air) if the undisturbed background 14 CO2 level is
known [4, 5].
In the Institute of Nuclear Research of the Hungarian Academy of Sciences
(Atomki) we have developed a mobile, field deployable observation station
for monitoring atmospheric fossil CO2 in polluted areas including continuous
CO2 mixing ratio measurement and integrated atmospheric CO2 sampling
for 14 C analyses [9]. For the first test run, the new field unit was installed in
the backyard of ATOMKI in the summer of 2008 to start urban atmospheric
fossil fuel CO2 monitoring in the city of Debrecen (East Hungary). To have a
clear measured reference level we started synchronised 14 CO2 sampling and
measurements in a rural site in Western Hungary, at Hegyhátsál.
94
II. Materials and Methods
II.1 Monitoring sites
Location selected for urban atmospheric CO2 observations is the city of
Debrecen (code hereafter: DEB) in East Hungary (47◦ 32’10”N, 21◦ 38’40”E).
Its area covers 462 km2 and it is only 85 m above sea level, which means
that it is situated in a small basin.ăDebrecen with its almost 205 thousand
inhabitants is the second largest city and industrial centre in Hungary. This
urban observation station is located close to the centre of the city. Sample
air intake is at 3 m above the ground level, about 200 m from the nearest
road. This location could give an average picture about the air of Debrecen
city.
The background CO2 sampling station at Hegyhátsál (300 km from Debrecen) (Figure 1), is synchronised with the Debrecen city observations. The
sample collection is there carried out on a , TV and radio transmitter tower
with a height of 117 m in a flat region of western Hungary [10]. This observation station is surrounded by agricultural fields (mostly crops and fodder
of annually changing types) and forest patches.
II.2 Atmospheric CO2 measurement and sampling
A mobile and high-precision atmospheric CO2 monitoring station was developed for the continuous, unattended monitoring of CO2 mixing ratio in
the near surface atmosphere based on an infrared gas analyser (IRGA). In
the Debrecen station, an integrating atmospheric CO2 sampling unit developed in the ATOMKI was also installed. A detailed description of this field
deployable monitoring station can be found in Molnár et al. [9].
The mixing ratio of CO2 is measured at 3 m above ground level by the
monitoring station. Analyses have been carried out using a non-dispersive
infrared gas analyser (IRGA) Ultramat 6F, which is a specialised model for
field applications by Siemens. The overall uncertainty of our atmospheric
CO2 mixing ratio measurements in Debrecen is < 1 ppm (< 0.3 % of the
measured value). This relative error is similar to the uncertainty of the
95
Figure 1: Shaded relief map showing the geographical location of the observation points Debrecen and Hegyhátsál inside Hungary, with the Hungarian
capital (dark gray patch in the centre) and the greater cities (> 100,000 inhabitants, dark gray dots) as well as the frequency distribution of the wind
direction (proportional to the length of the lines) at Debrecen and Hegyhátsál.
96
radiocarbon measurements of the atmospheric CO2 where the typical relative
error is usually 0.4-0.5%.
CO2 samplers installed in Debrecen and Hegyhátsál were developed in
ATOMKI to obtain integrated samples for the measurement of 14 C in the
chemical form of CO2 which is trapped in bubblers filled with 500 ml 3M
NaOH solution [11, 12]. The sampling period is 4 weeks, and the flow rate
of sampling is stabilized at 10.0 liter/h [13].
The 14 C activity concentrations of the collected CO2 samples were measured by gas proportional counting method [14, 15]. According the counting statistics, the standard deviation of a single ∆14 C measurement was
4-5%after one week measurement of each sample [16]. Reported ∆14 C data
were corrected for decay and δ 13 C as described by Stuiver and Polach [17].
II. 3Calculation of the fossil fuel CO2
For calculation of atmospheric fossil fuel CO2 component (CO2
Debrecen city the formula suggested by Levin et al [4] was used:
CO2
f ossil
= CO2
city
·
∆14 Cref BG − ∆14 Ccity
,
∆14 Cref BG + 1000
f ossil )
in
(1)
where CO2city , ∆14 Ccity and ∆14 Cref BG are the CO2 mixing ratio in the city
and radiocarbon results from the city and a reference background observation
station, respectively. There are more complex approaches described in the
literature [5] where the 14 C/C ratio difference between regional biogenic CO2
and the mean background CO2 are not neglected. However, due to the large
heterogeneity of the biosphere, the mean 14 C/C ratio of the regional biogenic
component is rather difficult to measure, and reliable estimations for this
required parameter were not available for Debrecen region for the studied
period.
97
Figure 2: Atmospheric CO2 mixing ratio variation as 1 week moving average and daily average temperatures between Sept 2008 and March 2010 in
Debrecen (station DEB). Two weeks’ data in Jan 2009 have been lost due to
a data handling error.
III. Results and Discussion
Atmospheric CO2 mixing ratio variation and daily average temperatures
in Debrecen city (station DEB) are presented in Figure 2 for the studied
period. Annual atmospheric CO2 variation between September 2008 and
March 2010 in Debrecen showed a more or less harmonic trend with summer
minimum and winter maximum but with significant jumps around OctoberNovember. The conventional heating period of 2008 and 2009 winters started
in the middle of October and finished about the end of March in Hungary.
There is a small seasonal variation in the background 14 CO2 observed at
the Hegyhátsál (HUN) station (Figure 3), which, to a large extent, is caused
by the input from the stratosphere but also a seasonal variation of the fossil
fuel CO2 component in background air at mid-northern latitudes [18, 19].
98
Figure 3: Monthly average atmospheric ∆14 CO2 variation in Debrecen city
(Deb_3m) and in the rural site of Hegyhátsál (HUN) at two elevations (10m
and 115m) during the studied period. Data for HUN_10m are available only
until October of 2009.
99
The similar monthly average 14 CO2 results at 10 m and 115 m in HUN
suggest that this station could not be strongly affected by local ground level
anthropogenic fossil fuel sources. A small difference between 10 m and 115
m observation heights appeared only in the October of 2009. According the
applied fossil fuel calculation model, the lower ∆14 C is, the higher fossil fuel
CO2 in the atmosphere is to be observed. When 14 C data are more negative
in Debrecen relative to the clear site of Hegyhátsál, a higher rate of fossil
fuel CO2 is detected in the city.
In Figure 4. the calculated fossil fuel CO2 amounts in the atmosphere of
Debrecen city relative to the clear reference site of Hegyhátsál are presented.
In this calculation, the maximum of ∆14 C that was observed at the two
different elevations in Hegyhátsál in a given month was used. Using these
data, a more conservative upper limit for the excess fossil fuel CO2 in the
city was obtained relative to the clear field in Hungary.
Using our Hungarian background 14 CO2 observations from the rural site
at Hegyhátsál (HUN) we could report atmospheric fossil fuel CO2 component
for the city of Debrecen in only a regional "Hungarian" scale. If the aim is
to give a more general picture and to compare Debrecen fossil fuel CO2 level
to the continental or global scale, a synchronized reference ∆14 C observation
is needed from a continentally or globally clear station, if it exists. For this
publication it was not available. The observed well-expressed fossil fuel CO2
peak with a maximum in the middle of winter 2008 (January) seems to be a
realistic result, as domestic heating is mainly based on fossil fuels in Hungary
and outside temperature minimum was also in the middle of the winter (first
two weeks of January). This maximum in January was about 10-15 ppm
fossil fuel CO2 in the urban air, like in Heidelberg city in Germany (1520ppm) reported by Levin et al. [4] and like in Krakow city (20ppm) in
Poland reported by Rozanski [20]. Furthermore, we observed a significant
maximum (∼20ppm) in fossil fuel CO2 during Octobers of 2008 and 2009
in Debrecen, as well. Explanation of October fossil fuel CO2 maximum
result and/or possible source identification needs further investigations on
meteorological and other possibly relevant conditions.
Not only strange October “jumps” but the practically missing “winter
maximum” in the winter of 2009 is also an interesting result. To clarify
this strange effect, the stable isotope content of the collected atmospheric
100
Figure 4: Atmospheric fossil fuel CO2 in the city of Debrecen (DEB) relative
to the data of Hegyhátsál (HUN) between Sept 2008 and March 2010.
CO2 samples was also studied. The δ 13 C stable isotope ratio of the carbon of
fossil fuels are significantly different (-25 – -30 %VPDB) than in the natural
atmospheric carbon-dioxide (∼ -9 %PDB). Stable isotope results are more
or less in agreement with our 14 C-based fossil fuel CO2 observations, as the
winter of 2008 and 2009 is different in atmospheric δ 13 C variations, as well.
The more negative δ 13 C in the winter of 2008 means more fossil carbon
in the atmosphere than in the winter of 2009. Although, using δ 13 C as
a more selective tracer of fossil fuel carbon would also be problematic, as
biogenic carbon has similar stable isotope composition (-20 – -25 %PDB)
like fossil fuels. The observed local δ 13 C minimum in March of 2009 and
the significantly different data from July of 2009 at the two elevations of
Hegyhátsál may show the effect of carbon emissions from the biosphere.
IV. Conclusion
A well-expressed fossil fuel CO2 peak (10-15 ppm) was observed with a
maximum in the middle of winter 2009 (January) in Debrecen air. This maximum in January was similar to the published maximum data for Heidelberg
101
Figure 5: Stable carbon isotope results (δ 13 C relative to VPDB) of the collected atmospheric CO2 samples from Debrecen and Hegyhátsál air (two
elevations at Hegyhátsál).
102
city (Germany) and those in Krakow (Poland). Furthermore, a significant
maximum (∼20ppm) was also observed in fossil fuel CO2 during Octobers
of 2008 and 2009. The explanation of this October fossil fuel CO2 maximum and/or the possible identification of its source requires more study on
meteorological and other possibly relevant conditions.
Acknowledgement
This research project was supported by Hungarian NSF (Ref No. OTKAF69029 and OTKA-CK77550), ATOMKI and Isotoptech Zrt. We would like
to thank Ms. M. Mogyorósi and Ms. M. Kállai for the careful 14 C sample
preparation and measurements.
References
References
[1] G. Marland, R.M. Rotty, (1984) Carbon dioxide emissions from fossil fuels: A procedure for estimation and results for 1950-82. Tellus
36(B):232-61.
[2] I. Levin, KO. Münnich, W. Weiss, Radiocarbon 22, 379-91 (1980).
[3] I. Levin, J. Schuchard, B. Kromer, KO. Münnich, Radiocarbon 31, 43140 (1989).
[4] I. Levin, B. Kromer, M. Schmidt, H. Sartorius, Geophys. Res. Lett.
30(23), 2194 (2003).
[5] JC. Turnbull, JB. Miller, SJ. Lehman, PP. Tans, RJ. Sparks, J. Southon,
Geophys. Res. Lett. 33, L01817 (2006).
[6] DY. Hsueh, NY. Krakauer, JT. Randerson, X. Xu, SE. Trumbore, JR.
Southon, Geophys. Res. Lett. 34, L02816 (2007).
[7] T. Kuc, K. Rozanski, M. Zimnoch, J. Necki, L. Chmura, D. Jelen,
Radiocarbon 49, 807-16 (2007).
103
[8] HE. Suess, Science 122, 415-7 (1955).
[9] M. Molnár, L. Haszpra, É. Svingor, I. Major, I. vetlík, (2010) Atmospheric fossil fuel CO2 measurement using a field unit in a Central European city during the winter of 2008/09., Radiocarbon (in press).
[10] L. Haszpra, Z. Barcza, PS. Bakwin, BW. Berger, KJ. Davis, T. Weidinger, Journal of Geophysical Research 106D, 3057-70 (2001).
[11] M. Veres, E. Hertelendi, Gy. Uchrin, E. Csaba, I. Barnabás, P. Ormai,
G. Volent, I. Futó, Radiocarbon 37, 497-504 (1995).
[12] M. Molnár, T. Bujtás, É. Svingor, I. Futó, I. vetlik, Radiocarbon 49,
1031-43 (2007).
[13] G. Uchrin, E. Hertelendi, (1992) Development of a reliable differential
Carbon-14 sampler for environmental air and NPP stack monitoring.
Final Report of the OMFB contract No. 00193/1991 (in Hungarian).
[14] É. Csongor, E. Hertelendi, Nuclear Instruments and Methods in Physics
Research B 17, 493-5 (1986).
[15] E. Hertelendi, É Csongor, L. Záborszky, J. Molnár, J. Gál, M. Györffi,
S. Nagy, Radiocarbon 31, 399-406 (1989).
[16] E. Hertelendi, (1990) Developments of methods and equipment for isotope analytical purposes and their applications. (in Hungarian) C.Sc.
thesis. Hungarian Academy of Sciences.
[17] M. Stuiver, H. Polach, Radiocarbon 19(3), 355-63 (1977).
[18] V. Hesshaimer, (1997) Tracing the global carbon cycle with bomb radiocarbon. Ph.D. thesis, Univ. of Heidelberg.
[19] JT. Randerson, IG. Enting, EAG. Schuur, K. Caldeira, IY. Fung, Global
Biogeochem. Cycles 16(4), 1112 (2002).
[20] K. Rozanski, (2009) Personal communication.
104
ACTA PHYSICA DEBRECINA
XLV, 105 (2011)
FISHER INFORMATION FROM THE PAIR DENSITY
¶
Á. Nagy1 , K. D. Sen2
1
Department of Theoretical Physics, University of Debrecen,
H–4010 Debrecen, Hungary
2
School of Chemistry, University of Hyderabad,
Hyderabad–500 046, India
Abstract
Fisher information defined with the pair density is presented.
As an example two-electron entangled artificial atom is studied.
A relationship between the Fisher information of the pair density
and the Fisher information of the density is analyzed.
I. Introduction
Fisher information [1] is playing an increasing role in several fields of
physics. The importance of Fisher information in quantum mechanics was
first noticed by Sears, Parr and Dinur [2]. The equations of nonrelativistic
quantum mechanics [3] were derived using the principle of minimum Fisher
information [4]. The time-independent Kohn-Sham equations and the timedependent Euler equation of the density functional theory were also derived
from the principle of minimum Fisher information [5, 6]. The Fisher information of single-particle systems with a central potential was also determined
[7] and that of a two-electron ’entangled artificial’ atom proposed by Moshinsky was also studied [8]. In a recent papers atomic Fisher information [9]
was investigated and phase space Fisher information was defined [10].
¶
Dedicated to Prof. I. Lovas on the occasion of his 80th birthday
Fisher information [1] is a measure of the ability to estimate a parameter
and is a measure of the state of disorder of a system or phenomenon. The
Fisher informational functional [1] is defined as
Z
Z
∂lnp(x|θ) 2
[p0 (x|θ)]2
IF (θ) = p(x|θ)
dx =
dx .
(1)
∂θ
p(x|θ)
p(x|θ) is a probability density function, obeying proper regularity conditions
and depending on a parameter θ. Take θ to be a parameter of locality:
p(x|θ) = p(x + θ) = p(γ) .
(2)
∂p(x|θ)
∂p(x + θ)
∂p(γ)
=
=
.
∂θ
∂(x + θ)
∂γ
(3)
Then
In this case Eq. (1) has the form
Z ∂p(x + θ) 2
IF (θ) =
/p(x + θ)dx.
∂(x + θ)
(4)
This is Fisher information per observation with respect to the locality parameter θ. As the expression does not depend on θ, we may set the locality
at zero:
Z
[p0 (x)]2
IF (θ = 0) =
dx .
(5)
p(x)
This Fisher information for locality is called intrinsic accuracy. It measures the ’narrowness’ of a distribution. For the variance of x holds the
Cramer-Rao inequality [4]
V arx ≥ I −1 .
(6)
For the normal distribution the Fisher information is equal to the inverse
variance. In that case relation (6) is an equality showing that a narrower
distribution has a larger Fisher information. In general x is vector-valued
and the expression (5) has the form
Z
[∇p(r)]2
IF (θ = 0) =
dr .
(7)
p(r)
106
II. Fisher information from the pair density
Generally we use a distribution function that depends on one or three
variables. But the Fisher information can be defined with a more general
distribution function. A Fisher information can be defined using the pair
density
Z
[∇1 Γ]2 + [∇2 Γ]2
IΓ =
dr1 dr2 .
(8)
Γ
For a two-electron system the pair density can be expressed with the wave
function as
Γ(r1 , r2 ) = |Ψ(r1 , r2 )|2
(9)
Now we change the variables to
1
R = (r1 + r2 )
2
(10)
r = r1 − r2 .
(11)
and
Then
IΓ =
Z
1
2
2 [∇R Γ]
+ 2[∇r Γ]2
dRdr.
Γ
(12)
III. Fisher information in a two-electron entangled artificial atom
We can calculate Fisher information coming from the pair density for the
Moshinsky atom. In the Moshinsky model two electrons with antiparallel
spins interact harmonically in isotropic harmonic confinement. The Hamiltonian has the form
H=
1
1
1
−∇21 + kr12 +
−∇22 + kr22 + Kr2 ,
2
2
2
(13)
K is the coupling constant.
107
The pair function then has the form
Γ(r1 , r2 ) = |ΨCM (R)ΨRM (r)|2
(14)
Substituting Eq. (14) into Eq. (8) we obtain
1
IΓ = I˜CM + 2I˜RM ,
2
(15)
IΓCM = 12k 1/2
(16)
IΓRM = 3k 1/2 (1 + 2K/k)1/2 .
(17)
where
and
Combining Eqs. (15), (16) and (17) we arrive at the result
i
h
IΓ = 6k 1/2 1 + (1 + 2K/k)1/2 .
(18)
Compare this result with the Fisher information obtained from the density
normalized to 1 [8]:
I% = 12k 1/2
(1 + 2K/k)1/2
.
1 + (1 + 2K/k)1/2
(19)
we are led to the inequality
IΓ ≥ 2I% ,
(20)
with equality if there is no interaction between the electrons, that is, K = 0.
The Fisher information obtained from the density is related to the Weizsäcker
kinetic energy [11] as
I% =
8
Tw .
N
(21)
For a two-electron ground-state system the Weizsäcker kinetic energy is equal
to the non-interacting kinetic energy : Ts = Tw . The relationship between
108
the Fisher information obtained from the pair density and the total interacting kinetic energy for a two-electron system is IΓ = 8T . The inequality
(20) can be written as
T ≥ Ts ,
(22)
with equality if there is no interaction between the electrons, that is, K = 0.
The difference IΓ − 2I% gives
IΓ − 2I% = 8Tc ,
(23)
where Tc = T − Ts is the kinetic energy correction, i. e. the difference of the
interacting and the non-interacting kinetic energies.
References
[1] R. A. Fisher, Proc. Cambridge Philos. Soc.22, 700 (1925).
[2] S. B. Sears, R. G. Parr and U. Dinur, Israel J. Chem. 19, 165 (1980).
[3] M. Reginatto, Phys. Rev. A 58, 1775 (1998).
[4] B. R. Frieden Physics from Fisher Information. A unification. (Cambridge U. P., 1998).
[5] R. Nalewajski, Chem. Phys. Lett. 372, 28 (2003).
[6] Á. Nagy, J. Chem. Phys. 119, 9401 (2003).
[7] E. Romera, P. Sánchez-Morena and J. S. Dehesa, Chem. Phys. Lett.
414 468 (2005).
[8] Á. Nagy, Chem. Phys. Lett.425, 157 (2006).
[9] Á. Nagy and K. D. Sen, Phys. Lett. A 360, 291 (2006).
[10] I. Hornyák and Á. Nagy, Chem. Phys. Lett. 437, 132 (2007).
[11] C. F. Weizsäcker, Z. Phys. 96 341 (1935).
109
The work is supported by the TAMOP 4.2.1/B-09/1/KONV-2010-0007
project. The project is co-financed by the European Union and the European
Social Fund. Grant OTKA No. K 67923 is also gratefully acknowledged.
110
ACTA PHYSICA DEBRECINA
XLV, 111 (2011)
MONTE CARLO SIMULATIONS OF SILICON
PHOTOMULTIPLIER OUTPUT SIGNAL
F. Nagy, G. Hegyesi, I. Valastyán, J. Molnár
Institute of Nuclear Research of the Hungarian Academy of Sciences, Debrecen,
Hungary
Abstract
The Silicon Photomultiplier is a silicon based photodetector
built from a matrix of Geiger Mode Avalanche Photodiodes. It is
a promising substitute of the Photomultiplier Tube with several
advantages.
The presented Monte Carlo simulator includes specific SiPM
processes such as recovery, triggering probability, dark pulse,
afterpulse, crosstalk and saturation. The simulation results show
good correspondence with measured and theoretically predicted
values.
I. Introduction
In recent years a lot of attention [1]-[3] has been focused on the development of Silicon Photomultipliers (SiPM) as a promising substitute of Photomultiplier Tube (PMT) photosensors. SiPMs have several advantages over
PMTs: for instance low operating voltage (<100V), high photon detection efficiency for visible and near infrared light, good single photon response time,
short rise time (<ns), low power consumption, insensitivity to magnetic field,
compactness and mechanical robustness.
The SiPM is a silicon based detector built from a matrix of microcells. A
microcell includes an Avalanche PhotoDiode (APD) and a quenching resistor
Figure 1: a. Microcell matrix, b. Connection of microcells
Rq (Fig. 1/a). The APDs operate in Geiger mode with a few volts above
the breakdown voltage, so each APD works as a photon counter. The role
of the quenching resistor is to limit a triggered avalanche in time and also to
restore the microcell to the initial state allowing the detection of subsequent
photons. The microcells are connected in parallel (Fig. 1/b), thus the analog superposition of binary microcell signals appears on the SiPM output.
For a large enough number of microcells and uniform light exposition with
relatively low intensity the signal of the SiPM is proportional to the exposing
light intensity.
Within the ENIAC-CSI (Central Nervous System Imaging) project of the
European Union (EU-FP7) our home institute, ATOMKI has undertaken to
upgrade our small animal PET [4] replacing the original position sensitive
PMTs with SiPM detectors. Modeling the basic processes in the SiPM and
using computer simulation for these processes play a key role in understanding the operation of the SiPM. We decided to write a Monte Carlo simulator
that delivers the electronic signal of the detector while tracking the processes in the SiPM (recovery, triggering probability, dark pulse, afterpulse,
crosstalk, saturation).
112
II. Processes in the SiPM
The SiPM is biased a few volts above the breakdown voltage. The difference between the bias and breakdown voltages is called the overvoltage (OV ).
Since individual microcells can be differently charged after an avalanche, the
actual bias voltage of a microcell is called the cell overvoltage (OVcell ).
II.1 Recovery
A photon impinging on a SiPM microcell generates an electron and a
hole. These carriers accelerate in the bias field generating further carriers
so finally an avalanche will be formed (the microcell “fires”). The increased
current generates a great voltage drop on the quenching resistor decreasing
the bias voltage, which finally terminates the avalanche and lets the microcell
recover to the initial state. The recovery time is an important parameter of
the SiPM. If during this time another photon arrives on the microcell the
resulting signal will be smaller causing a decreased gain and nonlinearity.
II.2 PDE and Triggering Probability
The impinging photon only starts an avalanche with a finite probability. The
Photon Detection Efficiency (P DE) is used to characterize this probability.
The PDE consists of three factors:
P DE = η ? QE ? Ptrig
(1)
The first is the geometrical fill factor (η), the ratio of the active and total
area of the detector. The second is the Quantum Efficiency (QE), which
gives the efficiency of a carrier couple generation. The third is the triggering
probability (Ptrig ), the avalanche generation probability for a carrier couple.
II.3 Darkpulses
Even without light the thermally activated carriers may also start avalanches.
These avalanches appear on the detector output as dark pulses. The dark
pulse rate is proportional to the area of the detector.
113
II.4 Afterpulsing
During the avalanche the carriers may be captured at crystal contaminations.
The capturing probability is proportional to the magnitude of the avalanche
and also to OVcell . These captured carriers escape later with a characteristic
time constant generating afterpulses. The afterpulsing process can be neglected for small overvoltages, but increasing the bias it becomes dominant
and from a critical OV the microcell can not recover anymore. Afterpulsing
may significantly increase the gain, but will also paralyze the microcells for
a longer time.
II.5 Crosstalk (CT)
During an avalanche in a microcell electroluminescence may occur and the resulting photon can fire an adjacent microcell causing another avalanche. This
phenomenon is called crosstalk. Starting from an avalanche the crosstalk
may spread to several microcells in cascade (double, triple, etc. crosstalk).
Crosstalk happens with a finite probability (crosstalk probability).
II.6 Saturation and Nonlinearity
The finite microcell number and the recovery time of the microcells implies
the saturation effect of the SiPM. This causes the nonlinearity of the detector
for relatively large light intensities. When the SiPM is exposed uniformly
with a short light pulse, the number of fired cells (Nf ired ) can be calculated
for given microcell and photon numbers (Ncell , Nph ):
N
−P DE N ph
cells
Nf ired = Ncells 1 − e
(2)
III. Model and Monte Carlo Simulation of the SiPM:
We have written a Monte Carlo simulator in C++ that contains the processes
mentioned in section II. And also includes the modeling of light exposure with
a scintillation crystal.
114
III.1 Exposure model
The modeled SiPM was uniformly exposed to a simulated scintillation crystal light pulse. In the model the incoming gamma photon instantly excites
N 0 states in the crystal. The decay of the excited states, that is the emission
of light photons takes place after a time period having an exponential distribution with the time constant of the crystal. A very short time constant
will also allow us to model a laser light flash.
III.2 SiPM model
We consider the SiPM as a 2D finite microcell matrix. For each microcell
the capacitance (Cd ), the actual voltage (Vcell ) and the number of captured
carriers are kept accounted. The recovery of the cell after an avalanche is
modeled by a recharging Cd capacitance through a Rq resistance with the
time constant τcell (= Rq ? Cd ) assuming that the avalanche discharged the
capacitance instantly.
A SiPM consisting of n microcells can be modeled as the electronic circuit
seen in Fig. 2. The figure shows the case when only one microcell fires. The
avalanche is symbolized by a switch. In the model we only consider the
capacitance of APDs (Cd ), the quenching resistors (Rq ) and the readout
resistor (Rout ). We neglected the capacitance of Rq and the resistance of the
switch. The electronic scheme determines the current pulse (Fig. 3) through
the readout resistor caused by an avalanche at t = 0 in a microcell:
Iout (t) =
OVcell Cd − τ t
e out
τout
(3)
where τout = (Rq + nRout ) · Cd In the simulation, from the three factors of
P DE (Eq. 1) only Ptrig is given as an input parameter and the other two
factors are considered with a decreased incoming photon number. Ptrig is
approximated by the function of Fig. 4. This assumes that each impinging photon triggers an avalanche above a given overvoltage (OVmax ). The
thermally activated carrier rate is used as an input parameter in the simulation. There is no difference between a thermally or photon activated carrier
in the avalanche generation. Thus, the presence of dark pulses is as if the
115
Figure 2: Electronic model of a SiPM
Figure 3: Current response of a single avalanche
116
Figure 4: Triggering probability function
detector was exposed to uniform light. This is exploited in the simulation.
In the afterpulsing process the number of escaping trapped carriers (k) in a
simulation time step follows the Binomial distribution. Then the k escaped
∗
particles create an avalanche with a probability of Ptrig
= 1 − (1 − Ptrig )k .
Fig. 5 shows a simulated waveform with afterpulses. In the simulation an
avalanche in a microcell may cause crosstalk in only one adjacent microcell.
Fig. 6 shows a simulated waveform including both afterpulses and crosstalk
(single, double CT).
III.3 Algorithm
Fig. 7 shows the scheme of the simulation algorithm. The main loop (Time
loop) is timing the simulation. The simulation can be scaled by the Time
Step parameter. In each time step the program will take into account the
avalanches caused by photon activated, thermally activated and escaping
trapped carriers. The detector output current is calculated by adding up
the currents from avalanches in all the microcells. Microcells are recharging
with a time constant of τcell , and at the same time the output current is decaying to 0 with τout . In one cycle of the Time Loop, N 1 and N 2 carriers are
generated from the scintillation crystal and from noise, respectively. Also,
the number of escaping trapped carriers will be calculated for all the micro117
Figure 5: Afterpulsing in a simulated waveform
Figure 6: Crosstalk and afterpulsing in a simulated waveform
118
Figure 7: Algorithm scheme of the simulation
cells. For each free carrier the triggering probability determines whether the
carrier will create an avalanche. If so, the Avalanche function will be called.
In the Avalanche function a given microcell discharges and the function
increases the output current and calculates the trapped carrier number according to the magnitude of the avalanche. The Avalanche function also
handles crosstalk. Here it may randomly choose an adjacent cell, and while
the triggering constraints are met the function will recursively call itself. So
this recursive function call will loop until the crosstalk “fades away”. In this
manner in one time step multiple crosstalk events may occur in cascade.
Table 1 contains all the input parameters of the simulation.
119
Table 1: Input parameters of the simulation
IV. Tests
To check the correctness of the simulation we have compared a couple of
measured and theoretically predicted data to the results of simulation.
IV. Test 1: Reverse Characteristics
In Fig. 8 the reverse characteristics at 20◦ C can be seen measured by ST Microelectronics [3]. Using the parameters given in Table 2 we have performed
a simulation for the marked region which has delivered the result indicated
by the solid curve in the figure.
IV. Test 2: Single Photoelectron Spectrum
Fig. 9 shows a simulated (by the ST Microelectronics [3]) and a measured
single photoelectron spectrum. The method of this measurement is the following. We light the detector with a low intensity laser pulse, to see the
magnitude of the response signals for 0, 1, 2, 3, etc. photons coming at
the same time. The only difference between the simulated and measured
120
Figure 8: Test 1 - Measured and simulated reverse characteristics
Figure 9: Test 2 - Simulated (on the left) and measured (on the right) single
photoelectron spectrum
121
spectrum appears at the so called pedestal peak (the peak belonging to zero
photoelectron). The measured pedestal is shifted to the right and also spread
more than the simulated one. The shift results from an offset of the real signal and the spread is from the electrical noise of the measurement equipment
that is not present in the simulation. The small tail of the simulated pedestal
comes from the dark pulses (Table 2).
IV. Test 3: Dynamic Range
We have simulated the dynamic range of SiPMs with three microcell numbers
(10x10, 20x20, 30x30). An increasing laser intensity was applied to the
detector and the integral of the response signals was recorded. Fig. 10
demonstrates the results of the simulation (symbols) and the solid curves
corresponding to Eq. 4 with the same parameters that were used in the
simulation (Table 2). Eq. 4 is a rescaled variant of Eq. 2 SiPM Signal
N
−P DE N ph
cells
SiP M Signal = α · Nf ired = α · Ncells 1 − e
(4)
where α = 4.8 · 10−4 and P DE = 1.
IV. Test 4: LYSO signal
The LYSO (Cerium doped Lutetium Yttrium Orthosilicate) scintillator crystal is widely used for a range of ray detection applications in nuclear physics
and nuclear medicine. In this test the scintillator time constant was set to
40 ns corresponding to the LYSO characteristic time (Table 2). In Fig. 11
simulated and measured LYSO signals can be seen.
122
Figure 10: Test 3 - Dynamic range for different microcell numbers
Figure 11: Test 4 - Simulated (on the left) and measured (on the right)
LYSO signal
123
Table 2: Input parameters for Test 1 to 4
V. Conclusions
The presented simulation includes specific SiPM processes such as recovery,
triggering probability, dark pulse, afterpulse, crosstalk and saturation. The
simulation results show good correspondence with the measured and theoretically predicted values (reverse characteristic , single photon spectrum,
dynamic range, LYSO signal). Our further aim is to perform more simulations related to timing and energy resolution (especially the “Time Over
Threshold” method [5]). These features are crucial in PET imaging.
VI. Acknowledgement
The project was supported by the ENIAC Joint Undertaking under grant
agreement # 120209.
124
References
[1] Paolo Finocchiaro et al., Part I: Noise, IEEE Transactions on Electron
Devices, vol. 55, no. 10, 2757-2764 (2008).
[2] Paolo Finocchiaro et al., Part II: Charge and Time, IEEE Transactions
on Electron Devices, vol. 55, no. 10, 2765-2773 (2008).
[3] Massimo Mazzillo et al., IEEE Transactions on Nuclear Science, vol. 56,
no. 4, 2434-2442 (2009).
[4] J. Imrek et al., IEEE-Nuclear Science Symposium and Medical Imaging
Conference, 2930 – 2932 (2007).
[5] F. Powolny, et al., Nuclear Instruments and Methods in Phisics Research
A 617, 232-236 (2010).
125
ACTA PHYSICA DEBRECINA
XLV, 126 (2011)
ADVANCING THE USE OF NOBLE GASES IN FLUID
INCLUSIONS OF SPELEOTHEMS AS A PALAEOCLIMATE
PROXY: METHOD AND STANDARDIZATION
L. Papp, L. Palcsu, Z. Major
Institute of Nuclear Research of the Hungarian Academy of Sciences, Debrecen,
Hungary
Abstract
Stable isotopes, and recently, noble gases in fluid inclusions
of speleothems are increasingly used to reveal past climate variations. Noble gases in groundwater have already been proven to
be an absolute climate proxy; that is, solubility temperatures can
be calculated from the noble gas concentration of groundwater
samples. Recently, strong efforts have been made to adapt this
technique for very tiny amounts of water such as fluid inclusions
of speleothems or even coral skeletons. New studies have shown
that noble gases dissolved in fluid inclusions can be extracted
and then measured by sensitive noble gas mass spectrometers.
In this paper we show how the overall measurement process has
been established in our research group. It includes the extraction
of water from carbonates, determination of water amounts by
the vapour pressure, mass spectrometric measurement of the released noble gases, calibration, and standardization with tiny, air
equilibrated water samples enclosed in copper capillaries. The
results of the performed calibration measurements (using well
known air aliquots) show that measurements of most noble gas
isotopes occur with a deviation of less than 2% or even better
in some cases. However, we have found tiny different noble gas
concentrations when measuring air equilibrated water samples
enclosed in copper capillaries. Pre-treatment of the capillaries
with helium purging can improve the noble gas signature of a
standard water sample. Measurements of real soda straw stalactites show that special attention has to be paid to the sample preparation, although in some cases reasonable temperatures
could be calculated from the obtained noble gas concentrations.
I. Introduction
Continental and marine carbonates are essential archives of past geological and climatological occurrences [1,2]. Elemental and isotopic compositions
involve important information with respect to the circumstances of the carbonate formation [3-5]. Recently, fluid inclusions of carbonates have become
a focus of palaeoclimate research [6-8]. Stable isotopic composition (δ 2 H,
δ 18 O) of the fluid inclusion water of speleothems is related to that of the
seepage water from which the carbonate has been precipitated [9-11]. The
changes of stable isotopic composition in the seepage water infiltrating the
cave can be reflected in the fluid inclusions. In palaeoclimate research, how
the temperature changed in the past is always in question [12]. In the seepage
water, the isotopic composition is related to temperature since the isotopic
composition of the meteoric precipitation (the source of the seepage) depends
on the temperature of a certain region [13,14]. However, there are other factors that influence the isotopic signature of the meteoric waters, such as the
main moisture sources, air mass circulations, and so on [15]. Therefore, it is
usually problematic to convert stable isotopic composition of fluid inclusions
directly to temperature. If the oxygen isotope contents of the carbonate
and fluid inclusion are examined simultaneously more coherent temperature
information can be obtained [16]. When the carbonate is being precipitated
from the seepage water an isotope partitioning is occurring between the carbonate and the remaining water. During this process a tiny amount of the
seepage water can be enclosed by the precipitating carbonate. Hence, the
elemental and isotopic composition of the fluid inclusion water is supposed
to be the same as that of the seepage water. The oxygen isotope composition
of the carbonate (δ 18 Oc ) is strongly related to that of the seepage water, that
is, fluid inclusion (δ 18 Of ) [17]. A temperature-dependent isotope fractionation effect determines how the isotopic composition of the two members of
this chemical reaction (carbonate and water) alters during the precipitation,
127
and finally what difference in isotope amounts is reached [18]. In principle,
if we measure the oxygen isotopic compositions of both the carbonate and
fluid inclusion, the isotope fractionation coefficient can be obtained, which
depends on the temperature. So far, the temperature dependency is not
known precisely enough, additionally it might vary spatially. Even in the
case of ideal circumstances, the temperature determination can be hardly
achieved with a precision better than 2 ◦ C [18-20]. In marine environments,
Sr/Ca ratios of coral aragonite can be used to reveal sea surface temperatures
[21]. As a few examples mentioned above, all these temperature proxies are
based on empirical or semi-empirical relationships. A new approach using
temperature-dependent gas solubilities might be a way that uses only physical laws, for example Henry’s law of solubility and gas partitioning models
[22]. When the seepage water reaches the top of the cave interior, gases
from the cave atmosphere (usually air) will dissolve in the water according
to the cave temperature and atmospheric pressure. While the compositions
of chemically active gases change over time due to chemical and biological
processes, noble gases are not affected. This noble gas pattern will then
be enclosed in the fluid inclusions when the carbonate layers interrupt the
connection of the fluid inclusion water to the ambient air. The so-called
noble gas temperature (NGT) can be calculated from the measured noble
gas concentrations. The potential of noble gases dissolved in water to reveal
solubility temperatures, for example NGTs, has been widely demonstrated in
the case of groundwater studies (Kipfer ref). Hence, there is a good chance
of obtaining relevant NGTs from noble gases extracted from fluid inclusion
of carbonates [24-26]. Since a lot of experience has been gained in making
noble gas measurements from large water samples (5-40 ml), including in our
laboratory, there is a good chance of adapting the technique for tiny water
amounts [27]. This paper describes how our first advancing steps towards
obtaining NGTs from fluid inclusions and tiny water amounts have been performed. It includes the extraction of water from carbonates, determination
of water amounts by the vapour pressure, mass spectrometric measurement
of the released noble gases, calibration, and standardization with tiny, air
equilibrated water samples enclosed in copper capillaries.
128
I.1 Experimental
To determine noble gas temperatures (NGTs) from fluid inclusions of carbonates, noble gas concentrations have to be measured. To do so, the water
content of the carbonate and its noble gas abundances dissolved in the inclusion water have to be first liberated and then measured. To extract the water
inclusions from the carbonate matrix, the most suitable treatment is to crush
the carbonate under vacuum [28]. The water released from the inclusions
is then collected in a cold-finger by freezing. The amount of the liberated
water is measured via its vapour pressure in a certain volume. The liberated
dissolved noble gases which were in the fluid inclusions are separated by a
cryo-system and then admitted into the static mode noble gas mass spectrometer sequentially. The calibration of the noble gas mass spectrometric
measurements is performed by means of well known air aliquots. Purification and separation of the calibration air aliquots are done in the same way
as those of the samples. The next chapters show all of the abovementioned
steps in more detail.
I.2 Extraction of fluid inclusions
The water content of natural speleothems varies between 0.1 and 10 mass
permil, with an average of 1 permil (Kendall and Broughton 1978). To
measure the noble gases dissolved in these fluid inclusions, first the water has
to be released from the carbonate matrix. We use vacuum crushing to break
up the carbonate, and hence the fluid inclusions will be released [28,29].
In order to finally obtain noble gas concentrations, the exact quantity of
water also has to be determined accurately. The precise measurement (better
than 2%) of such an extremely small amount of liquid requires sophisticated
preparation work and instrumentation. To this end, a carbonate crusher
system has been built consisting of a stainless steel crusher, a vacuum gauge,
and a cold-finger (Fig. 1). The crusher is a closed stainless steel tube with
an internal diameter of 3 cm. Prior to the crushing, the whole system is
evacuated for at least half a day. The rough vacuum is ensured by a rotary
pump, and the ultrahigh vacuum by a turbomolecular pump. The carbonate
sample sits in a stainless steel hemisphere, whose base is connected to the
top of the crusher with a thin chain. During the vacuum pumping, the
129
magnetic ball holds the hemisphere including the sample. The lower part of
the crusher is heated to 150 ◦ C, while the temperature at the sample is less
than 50 ◦ C [30]. Pulling the magnet towards the bottom of the crusher, the
hemisphere tilts down and the carbonate sample falls down to the bottom
of the crusher. A magnetic iron ball that can be moved by a magnet from
outside breaks up and crushes the solid sample. Taking the magnet away, the
ball falls onto the carbonate sample and crushes it. This stroke is usually
repeated 100 times. The reason why we use the hemisphere is to avoid
contact of the sample with the entire hot surface of the crusher during the
vacuum evaluation. However, Scheidegger et al. use a pre-heating step to
heat the samples up to 100-200 ◦ C prior to crushing [27]. At the bottom
of the crusher the carbonate powder takes over the temperature (150 ◦ C),
hence avoiding adsorption of the released water on the freshly broken calcite
surfaces [28]. The crusher is connected to a cold-finger held at -70 ◦ C. The
water is frozen in this cold-finger, while none of the noble gases are adsorbed
at this temperature. Previous analyses have shown that the water collection
efficiency of the speleothem crusher (the percentage amount of the water
frozen out relative to the amount of water put into the crusher) is 99.72 ±
0.40% when the collection time is twenty minutes.
I.3 Determination of the water amount
A suitable process to define the water amount in an ultra high vacuum
system is to measure the pressure of the water vapour frozen in the coldfinger (Kluge; Scheidegger). The frozen water is melted, and then the water
vapour pressure is measured with an active capacitance vacuum gauge (CMR
363 Pfeiffer Vacuum) with a precision of 0.2%. To increase the stability of
the pressure measurement the cold-finger is kept at 24 ± 0.3 ◦ C, while the
whole laboratory is conditioned to 24 ± 0.5 ◦ C. If the volume of the cold
finger (∼70 cm3 ) is too small to keep the whole amount of water in vapour
phase, additional volumes of ∼150 cm3 or ∼250 cm3 are connected to the
given volume.
The water determination process is calibrated by means of well known
water amounts (Kluge2008). Distilled water aliquots were sucked into glass
capillaries, and then both ends of the capillaries were flame-sealed. Determination of the water amounts was performed by precise weight measurements
130
Figure 1: Sample crushing and water determination system
of the empty glass capillary and the filled one. The difference is thought
to be the water weight, which has an uncertainty of 0.004 mg. A balance
with a precision of 0.002 mg was used for this purpose. The capillary filled
with distillate water was then put into the crusher and was evacuated for a
few hours. Afterwards, the glass capillary was broken by the magnetic ball,
and the released water was frozen into the cold finger at -70 ◦ C for twenty
minutes. The water was melted at 24 ◦ C, and the vapour pressure was then
measured by the vacuum gauge in all of the three volumes. Obviously, a
little air was entrapped at the very ends of the capillaries, but its contribution to the pressure could be neglected. By plotting the measured vapour
pressure values against the known water weights, we obtain the calibration
curves for all volumes involved. The calibration curve of the pressure gauge
was fitted on nine measurement data between the water masses of 0.466 mg
and 3.143 mg. Figure 2 shows the calibration line for the largest volume.
When fitting a linear function, the uncertainties of the weight and pressure
measurements have been taken into account. During a sample measurement,
the water vapour is released into all of the three volumes sequentially, and
131
Figure 2: Calibration curve for water determination via vapour pressure
thus the final weight of the sample water is calculated as an arithmetic mean
of the three measurements. For water amount determination, an alternative
approach could be to use the ideal gas law. In this case, all volumes have to
be known very precisely. By measuring the absolute pressure of the vapour
in these well known volumes, the water amount can be calculated using the
gas law [27]. Our approach is thought to have less uncertainty, because it
does not need additional volume determination, although calibration curves
have to be determined from previous experiments.
I.4 Purification and separation of the liberated gases
As mentioned above, gases of interest liberated from the inclusions are not
adsorbed in the cold-finger. Apart from a fraction that has been enclosed in
the volume of the cold-finger, the gases are released towards a cryo-system
for purification and separation (Fig. 3) [31-33]. The cryo-system is equipped
with a stainless steel empty trap and a charcoal filled trap. The argon,
krypton, and xenon fraction and of course the other chemically active gases
(N2 , O2 , CO2 , etc.) are adsorbed in the empty trap at 25K. Afterwards,
neon and helium get adsorbed in the charcoal trap at 8K. To avoid the cryotrapping of neon by the other gases in the empty trap, the empty trap is
132
heated to 50K, and then cooled back to 25K [34]. Hence, if some neon has
been covered by frozen gas layers, it can be released and collected in the
charcoal trap. The amounts of the helium isotopes are not measured, since
incidentally present radiogenic helium largely influences the concentration
of helium, and therefore it has no significance in the noble gas temperature
determination. To eliminate helium, the charcoal trap is heated to 38K,
and the helium desorbed from the trap is pumped away. Afterwards, neon
desorbed from the charcoal trap at 90K is introduced to the mass spectrometer. The argon, krypton, and xenon fraction (AKX) has to be purified from
the other gases. To do so, the whole gas is released from the stainless steel
empty trap at 150K, and injected into a getter trap (St 707, SAES). The
getter trapping takes 15 minutes and then the AKX fraction is admitted to
the mass spectrometer and measured simultaneously.
I.5 Mass spectrometric measurements
The mass spectrometric measurements of the noble gases are performed with
a VG5400 (Fisons Instruments, now Thermo Scientific) noble gas mass spectrometer, which is an all-metal, statically operated, 90◦ sector field mass
spectrometer with 54 cm extended geometry. During the measurements, the
ionizing efficiency is tuned for xenon. All stable neon isotopes and the two
most abundant isotopes of argon, krypton, and xenon are detected in the
mass spectrometer. The detection of neon (20,21,22 Ne), krypton (84,86 Kr),
and xenon (129,132 Xe) isotopes occurs with an electron multiplier (Balzers
SEV217) in ion counting mode, while the ion current of the argon isotopes
is measured by a Faraday cup. The evaluation of the mass spectrometric
measurements is done according to the peak height method. The measured
peak intensities are extrapolated to the time of sample admission to the
mass spectrometer. Thus, we can achieve a negligible contribution of double
charged 40 Ar to the 20 Ne peak.
I.6 Calibration
The measurements of samples are calibrated with well-known air aliquots in
the expected range. There are two air reservoirs in the preparation line. A
133
Figure 3: Schematic picture of the preparation line and the mass spectrometric system
reservoir of 14105 ± %cm3 containing dry air with a pressure of 20010 ±
2 %Pa is normally used for a larger noble gas amount and tritium measurement for calibration (see ”Air” in Fig. 3). From this vessel, 0.1236 ± 2
%ccSTP air was introduced to a newly built reservoir of 6370 ± 1 %cm3
resulting in 2.0009 ± 3 %Pa. This reservoir called “diluted air” is used to
prepare well known air aliquots in the range of 2.5·10−5 to 1.0·10−6 ccSTP
by means of a gas pipette of 1.259 ± 1 %cm3 . The calibration air aliquots
in this range are admitted into the inlet system and handled in the same
way as for a sample.
To improve the precision of the mass spectrometric measurements, we
apply the so-called fast calibration method in order to be able to follow the
changes of the ion source sensitivity [35]. This means that well known, tiny
amounts of neon and Ar-Kr-Xe mixture are admitted to the mass spectrometer, by-passing the inlet line, from two reservoirs directly after each mass
spectrometric measurement. We consider that if there is a difference in the
measured signals (peak heights of 20 Ne, 21 Ne, 22 Ne, 36 Ar, 40 Ar, 84 Kr, 86 Kr,
129 Xe, and 132 Xe) of the subsequent fast calibration measurements the same
difference had to be present during the sample (or the air calibration) mea134
surement. The final measured value of a sample or an air calibration is the
ratio of the measured signals of the sample (or air) and the subsequent fast
calibration. To perform neon fast calibration we filled a reservoir with pure
neon of atmospheric isotopic ratio at a very low pressure of about 10−5 mbar.
The elemental composition of the AKX fast calibration corresponds to that
of air equilibrated water. We do not actually know what the exact pressure
is in the reservoir, but we do not have to know it. As we are taking fast
calibration gas amounts from the reservoir, the entire pressure is decreasing.
This dilution is taken into account, knowing the precise values of the volumes
of the reservoir and the pipette.
The reproducibility of the calibration measurements is of major importance. To demonstrate how precise the calibration measurements are, linearity values have been plotted in Fig. 4. This is a suitable way to compare
the deviations for calibration measurements of different air aliquots. Linearity for noble gas isotope ’i’ is defined as the ratio of the measured signal
(Si ) normalized to the dilution corrected fast calibration signal (Fi∗ ) and the
volume of the calibration air aliquot (Vair ) (Eq. 1.)
Li =
Si
Fi∗
Vair
(1)
The results of the performed calibration measurements show that the measurements of most noble gas isotopes occur with a deviation of less than 2%
or even better in some cases (Fig. 4 and Table 1). The reproducibility of
40 Ar measurements is better than 0.6%, while those of krypton and xenon
isotopes are 0.9-2.2% and 0.8-2.0%, respectively (Table 1). Theoretically,
these precisions for noble gas concentrations obtained from sample measurements allow us to determine noble gas temperatures with an uncertainty of
less than ◦ C.
II Results
II.1 Measurements of air equilibrated water samples
To check the reliability of the whole measurement procedure standard water samples have to be measured in the same way as a sample is handled
135
Figure 4: Linearity values obtained from the calibration measurements
136
(IT-principle: identical treatment). As for standard samples, first we have
prepared air equilibrated water (AEW) in conditioned circumstances. Two
litres of de-ionized water were continuously stirred for several days while the
water temperature and air pressure were monitored. The average temperature and pressure over the last 24 hours were chosen to be those parameters
with which the equilibrium noble gas concentrations as expected concentrations were calculated. Kluge et al. enclosed AEW in glass capillaries using
flame sealing, but later they could not obtain the expected concentrations,
because the gas concentrations in the glass capillaries might have been modified due to either the heating effect of the flame sealing or the capillary force,
or both. Here we use another approach that is an analogue to groundwater
sampling with copper tubes closed by pinch-off clamps at both ends. We fill
copper capillaries with AEW. The outer diameter of the capillary is 1.23 mm,
while the wall thickness is 0.36 mm. Hence, the inner volume of the capillary
of 1 cm length is 2.04 mm3 . The copper capillary is fixed to a copper block
(Fig. 5a) by a special two-component vacuum glue (Araldite, AW 134, HY
994). The copper block behaves as a gasket as well. This assemblage is built
into a CF-flange equipped with a Swagelok VCR fitting (Fig. 5b).
Having completed the copper capillary assemblage, the AEW is allowed
to flow through the capillary. After a few millilitres of water have streamed
through, two couples of small clamps close up a certain section of the capillary. The capillary is then attached to the inlet line of the mass spectrometric
system. After one night of pumping the AEW sample is handled as a sample. To inject the water and the dissolved gases into the system the inner
clamp is removed and the squeezed copper is stressed by a special pair of
pliers. We tested how tightly the clamps can squeeze the copper capillary.
We found that there is no difference between background measurements with
and without a clamped, squeezed copper capillary. Therefore, we assume the
copper capillary can be used as a good container for AEW.
During the first AEW measurements, we obtained excess gas concentrations for the heavier noble gases (Ar, Kr, and Xe). Therefore, we performed
an experiment to improve the standard preparation. Four capillaries were
filled with AEW using the preparation technique mentioned above. Another
four capillaries were pre-treated before filling with AEW. We streamed helium through the capillary, which was simultaneously heated to 200 ◦ C for
137
Figure 5: a) Copper capillary glued to a copper block, b) capillary assemblage
with a CF-flange and a Swagelok VCR fitting
138
at least half an hour. When the capillary had cooled down to room temperature, we removed the helium flow and immediately filled the capillary with
AEW as described above. All of these eight AEW samples were measured in
the same way as a carbonate sample or a calibration sample measurement.
Before each of these measurements, individual background measurements
were done. This means we executed a complete measurement process without opening the copper capillary. All of these individual backgrounds were
then assigned to their AEW measurement for subsequent background correction. Table 2 shows the individual background measurements, while Table 3
shows the water amounts and the noble gas concentrations obtained from the
AEW samples. AEW samples numbered from 1 to 4 have been prepared in
the usual way, while the copper capillaries of AEW samples from 5 to 8 have
been flushed with helium and heated. Table 2 additionally shows elemental
ratios compared to atmospheric ratios of the different noble gases. Note that
these noble gas quantities in Tables 2 and 3 have different units (ccSTP for
background values and ccSTP/g for AEW samples). To substract individual
background values from concentrations of AEW samples, the background
values have to be converted to concentrations using the individual AEW
water weights.
In this measurement run, a relationship between elemental ratios and
noble gas abundances could be observed. Figure 6 shows noble gas and argon
ratios as a function of argon amount. It can be seen that the more argon
there is in the background, the higher the ratios. For heavier noble gases,
the explanation can be that after a long term evacuation of the system the
heavier noble gases, such krypton and xenon, can be slowly desorbed from
the walls of the entire system. Hence, in the background measurement they
will be enriched compared to argon. For Ne/Ar ratios, the picture is not
that obvious. Below 1.8·10−8 ccSTP of argon, the neon/argon ratio is lower
than 1. This can be explained by the different desorption velocities for neon
and argon. However, above 1.8·10−8 ccSTP of argon an enrichment of neon
occurred, as if more neon than argon had infiltrated the system. This effect
can only be explained with difficulty. If the system, for example the copper
capillary assemblage, contains a very tiny leak through which the streaming
velocities are different for the noble gases, elemental fractionation can occur
when the ambient air is penetrating the system through the leak. The highest
velocity might belong to neon; hence it can be enriched in the background.
139
However, the relationship between elemental ratios and abundances exists.
The reliability of the obtained noble gas concentrations for AEW samples
can be verified by a noble gas fitting procedure where the measured concentrations are compared to theoretical concentrations calculated from a gas
partitioning model [36]. Here we used three approaches: 1) an unfractionated
excess air model for the background-corrected noble gas concentrations; 2)
an unfractionated excess air model for the noble gas concentrations without
substracting the background; 3) a self-made model for background correction
using the relationship of elemental ratios in the individual backgrounds. The
unfractionated excess air model generates theoretical concentrations according to Eq. 2:
Cth,i (T, p, A) = Ceq,i (T, p) + A · zi
(2)
where Cth,i is the modelled, theoretical concentration of noble gas i (i: Ne,
Ar, Kr, Xe), Ceq,i is the equilibrium concentration, zi is the atmospheric
mixing ratio of noble gas i, T is temperature, p is atmospheric pressure, and
A is excess air amount. To find a modelled concentration close to the measured one, in the unfractionated excess air model we vary only parameter A,
while the known temperature and pressure are fixed. For the noble gas fitting procedure the excel file “Noblebook” of Aeschbach-Hertig were used [37].
The goodness of this fitting procedure is reflected in a chi-square test. In the
case of noble gas concentrations of AEW samples where the individual backgrounds have been substracted, Table 4 shows the fitting parameter, A, the
chi-square values, and the relative deviation of the modelled concentrations
from the measured ones for each noble gas. Where the AEW samples have
been prepared in the usual manner (AEW_1 to AEW_4), excess heavier
noble gas concentrations can be seen. Even if a few samples contain some
excess air (see parameter A), there is an additional excess mainly for heavier
noble gases: the heavier the noble gas, the more the excess (Table 4). Here,
we chose the excess air parameter (A) to be as much as the neon excess
is close to zero. For AEW_2, the neon concentration is found to be very
small; therefore the modelled concentration is always larger (∆Ne is -13%)
and even the excess air is set to zero. Anyway, all of the excesses for argon, krypton, and xenon are about 1-5%, 2-9%, and 7-14%, respectively. We
have found similar effects for AEW samples in previous measurement runs
not shown here. This finding confirms that there is an apparent enrichment
of heavier noble gases in the AEW samples. We suspect that it is due to
140
Figure 6: Noble gas ratios compared to the atmospheric ratio in the background
141
an artefact during the sample preparation. We can exclude the possibility
that the excess comes from the copper material during squeezing. In an
earlier measurement run, we tested how much noble gas came from a multiple squeezed copper capillary, and found it to be somewhat more than but
comparable to a normal background measurement. We also suspect that in
a very thin tube water streaming can behave eccentrically. Our presumption
is that when the water is flowing through the capillary a very thin air layer
might stick between the water and the inner copper wall, and concurrently
the air layer is being fractionated. If anything like this happens it might
be eliminated by helium flushing and/or heat treatment of the capillary. To
do so, four AEW samples have been filled into pre-treated copper capillaries
(AEW_5 to AEW_8). The measured noble gas concentration can also be
seen in Table 3. As for the previous AEW samples, we searched for the fitting
parameter A with respect to these last four AEW samples corrected for the
background substraction. The results can be seen in Table 4. The noble gas
fitting of one of the four AEW samples (AEW_6) seems to be perfect; all
relative deviations of the modelled concentrations from the measured ones
are less than 1%, while χ2 is 0.33. The other three samples do not show such
a nice behaviour. Although AEW_5 has a larger neon deficit, the heavier
noble gas concentrations can be acceptably described by the unfractionated
excess air model. Due to the neon deficit, the goodness of the fit is found
to be too poor: χ2 is 100. Looking at Table 4, the model cannot handle the
concentrations of the last two AEW samples (AEW_7 and AEW_8). The
deviations and hence χ2 are too high.
To improve the evaluation of the results from the noble gas measurements,
we attempt to use the pure noble gas concentration from which the individual
backgrounds have not been substracted. Hence, the unfractionated excess
air model (Eq. 1) is to handle the background incorporated in the concentrations. Table 5 shows the fitting parameters and deviations. One can see
that excess air amounts (A) are slightly larger than in the previous modelling (Table 4). Obtaining an excess air of 2.8 ccSTP/kg for AEW_2, the
neon deficit of 13% has diminished; obviously the background incorporated
is responsible for this. Thus, all of the four AEW samples where the copper
capillaries have not been pre-treated show the same manner: extra excesses
for the heavier noble gases. The situation for AEW_5 is better than in the
previous case. Although there is still a slight neon deficit (-4%), χ2 could go
142
down to 7.6, which means a reasonable value. It can be further seen that even
if we use background-incorporated noble gas concentrations, the unfractionated excess air model cannot describe AEW_7 and AEW_8. Neither does
a further attempt, where we try to take into account the special elemental
ratio of the background obtained from individual background measurements
(Fig. 6). As mentioned above, the ratios of noble gas abundances are correlated with the amounts. This relationship has been used to correct for
potential additional excess noble gases for the equilibrium component and
normal excess air. Equation 3 describes these new model concentrations,
where Bi (ArB ) refers to the abundance of noble gas i in the background
(i: Ne, Kr, and Xe) being a function of argon from the background (ArB ).
These three functions can be obtained from Fig. 6. Bi(ArB) is also a model
parameter; hence during the noble gas fitting it is varied until such values
are found with which the theoretical noble gas concentrations approach the
measured ones by an acceptable level.
Cth,i (T, p, A) = Ceq,i (T, p) + A · zi + Bi (ArB )
(3)
Table 6 shows the results of this latter noble gas fitting procedure. One
can see that the χ2 values are slightly better, showing that this backgroundcorrection model describes the measured concentrations in a slightly more
reasonable manner than the previous models do. The obtained argon amounts,
if any, in the background (ArB ) fall between 1 and 2% (0.6-6.9·10−6 ccSTP/g) of the measured argon concentrations (2-4·10−4 ccSTP/g), which
are undoubtedly realistic. However, for AEW_7 and AEW_8 the measured
concentrations can be explained by neither this nor the previous models.
Mistakes may have occurred in these somewhere during the whole measurement process.
All in all, in this measurement run we have performed eight measurements
of AEW enclosed in copper capillaries. Four of these eight AEW samples
show excess noble gases, mainly in the heavier ones. Pre-treatment of the
capillaries by helium flushing and heating might improve the AEW sample
preparation: although the measurement of two samples might have failed,
the measurements of the other two AEW samples have given reasonable
noble gas concentrations very close to the expected ones.
II.2 Noble gas measurements of fluid inclusion from carbonates
143
During our measurement runs performed so far, noble gases from fluid inclusion waters of carbonate samples (speleothems, coral skeletons) have been
also measured. Figure 7 shows neon and xenon concentrations of fluid inclusion waters from a few speleothem carbonates. To demonstrate the potential
of the method we chose here soda straw samples in which air filled inclusions
are often found to be of minor importance. In Fig. 7, the data points
are related to selected samples where excess air amounts are not too large.
Black squares represent neon-xenon concentrations for air equilibrated water (AEW), while the two grey lines represent additional excess air to the
AEW concentrations. Grey circles with error bars are the concentrations of
fluid inclusions in the samples. All of these soda straw samples have been
collected from two caves in Hungary. The mean annual cave temperatures
are equal to the mean annual air temperatures and are found to be 10.4 ±
0.5 ◦ C. Five of these nine points in Fig. 7 are located between the two excess
air lines. This means that these concentration pairs can be explained by a
binary mixing of equilibrium and excess air components [38]. Extrapolating
a data point back to the AEW concentrations, the equilibrium temperature
can be read out from the diagram. However, only two samples lie close to
the expected temperature (∼10 ◦ C); the other three give temperatures below
it; that is, they represent colder noble gas temperatures. In fact, there are
data points above the upper excess air line, for which the xenon concentration is higher than 3·10−8 ccSTP/g. In this case, neon concentrations are
also higher, but an excess of xenon can be clearly seen. We suspect that
this strange xenon excess is a sample pre-treatment artefact because of the
following consideration. Many carbonate samples contain air filled inclusions as well . Contrary to water-filled inclusions, air inclusions are located
at the grain boundaries. If we want to reduce the noble gas contributions
from air inclusion a sample pre-treatment step can be introduced, where the
carbonate is pre-crushed according to the prevailing grain size. Prior to the
measurement of the abovementioned samples with strange xenon excess, we
used this pre-crushing preparation. We crushed the sample and then sieved
it with a 560 µm sieve. As a result, we were able to obtain less excess air but
more xenon excess relative to neon. We think that during the pre-crushing
xenon could have been adsorbed on the freshly broken carbonate surface
from the ambient air and then desorbed from the sample in the preparation
line during the measurement process. Because heavier noble gases can be
adsorbed more easily than lighter ones, this effect can be seen in an excess of
144
Figure 7: Noble gas concentrations of fluid inclusions in soda straw
speleothems
the heavier gases such as xenon. Although the pre-crushing step can reduce
the contribution of air inclusions, special attention should be paid to the
adsorption. It seems it cannot be done in air atmosphere. Special conditions have to be used for pre-crushing, where no noble gases, or at least no
neon, argon, krypton, and xenon, can be found in the ambient gas mixture.
As has already been proposed, a pure helium atmosphere in a glove box is
recommended.
II.3 Conclusion
In this paper we have shown that noble gases from fluid inclusions can be extracted and measured by a noble gas mass spectrometer. The water amount
is measured via its vapour pressure in a certain volume. The accuracy of
such a water determination is less than 1% in the case of 1 µl of liquid water, which allows us to determine accurate noble gas concentrations. The
overall mass spectrometric measurement process is calibrated by means of
well known air aliquots in the range of 2.5·10−5 to 1.0·10−6 ccSTP. The
reproducibility of 40 Ar measurements is better than 0.6%, while those of
krypton and xenon isotopes are 0.9-2.2% and 0.8-2.0%, respectively. Theo145
retically, these precisions for noble gas concentrations obtained from sample
measurements allow us to determine noble gas temperatures with an uncertainty of less than 1 ◦ C. To verify the reliability of the measurement,
noble gases of air equilibrated water (AEW) samples have been measured.
In this measurement run we have performed eight measurements. Four of
these eight AEW samples show excess noble gases mainly in the heavier ones.
Pre-treatment of the capillaries by helium flushing and heating improved the
AEW sample preparation: although the measurement of two samples might
have failed, the measurements of other two AEW samples gave reasonable
noble gas concentrations very close to the expected ones. Noble gas concentrations obtained from soda straw stalactites showed extra excesses mainly
for the heavier noble gases that can be attributed to an artefact from sample
preparation, although noble gases of few samples sustain the potential of
the method based on temperature dependent solubility concentration to be
a useful palaeoclimate proxy.
Acknowledgement
L. Palcsu appreciates the financially support by the Marie Curie Reintegration Grant (No. 221946).
146
Table 1. Standard deviations of the calibration measurements
147
Table 2: Individual background values for the air equlibrated water (AEW)
sample measurements.
148
Table 3: Water amounts and noble gas concentrations of the air equilibrated water (AEW) samples. Copper capillaries from AEW_4 to AEW_8
have been pre-treated by helium flushing and heating.
149
Table 4. Results obtained from the noble gas fitting procedure for noble
gas concentrations corrected for individual backgrounds.
150
Table 5. Results obtained from the noble gas fitting procedure for noble
gas concentrations without substracting backgrounds.
151
Table 6. Results obtained from the noble gas fitting procedure using a
special background correction model.
152
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[38] Y. Scheidegger, S. Badertscher, R. Wieler, V. Heber, R. Kipfer, European Geosciences Union, 2007.
155
ACTA PHYSICA DEBRECINA
XLV, 156 (2011)
STRESS EFFECTS ON THE KINETICS OF NANOSCALE
DIFFUSION PROCESSES
B. Parditka, Z. Erdélyi, D.L. Beke
Department of Solid State Physics, University of Debrecen, Hungary, 4010
Debrecen, P.O. Box 2.
Abstract
General description of the interplay between the Kirkendall
shift and diffusion induced driving forces in diffusion intermixing
of binary systems was given in [1]. It was shown that, if the Kirkendall shift is negligible, a steady state Nernst-Planck regime is
established with diffusion coefficient close to the slower diffusivity. Deviations from parabolic kinetics are expected only before
or after this steady state stage. In this paper, we summarize our
results concerning how the stress field shifts the system from the
Darken to the Nernst-Planck regime, why stress relaxation is not
observed even though expected and why the diffusion kinetics is
not influenced by the developing stress field [2].
I. Introduction
In [1], we have given general considerations on the effect of different diffusion induced driving forces on the kinetics of diffusion intermixing. The
characteristic feature of these phenomena is related to the interplay between
the Kirkendall shift, as a special way of relaxation of the interdiffusion induced fields, and the driving forces (field gradients, e.g. stress field) still
remaining present in the sample. In some special cases both Kirkendall shift
and field gradient(s) can be present. Parabolic kinetics (with either Darkenor Nernst-Planck-type interdiffusion coefficient or in between) is obtained
if a steady state is maintained or non-parabolic kinetics can be expected if
both the Kirkendall-velocity and the field gradient(s) are time dependent.
During intermixing stress-free strain develops due to the net volume
transport caused either by the difference in the intrinsic diffusion coefficient
and/or the atomic volumes or by the abrupt change in the specific volumes
if a new phase grows. As a consequence, stress develops within a certain
characteristic time, leading to a steady state. The stress field may also relax
within a certain characteristic time [3,4]. Due to terms proportional to the
stress gradient in the atomic fluxes (see later Eq. (5)), the kinetics is obviously expected to differ from “pure diffusional” (Fickian or parabolic) type
in the transient stages. In the steady-state – when the stress field remains
roughly the same – if it can be developed, the kinetics can remain Fickian,
but the intermixing rates can still differ considerably from the stress-free
case.
It was shown recently [5] that stress effects may have an easily measurable
influence on the intermixing rate in nanostructures of spherical symmetry but
surprisingly the diffusion kinetics remains parabolic in time. The influence
on intermixing rate was interpreted as a switching between the Darken and
Nernst – Planck regimes caused by the diffusion-induced stress.
Due to the complexity of the problem, however, until now there has been
no systematic investigation into the effect of stress on the kinetics.
In this communication we analyse theoretically – for the sake of simplicity,
for planar model geometry – how the stress field shifts the system from
Darken to Nernst – Planck regime. We show that the development of the
Nernst – Planck regime is very fast, finished before any detectable shift of the
interface, and that stress relaxation is not observed, although this would be
expected since the time interval investigated was much longer than the stress
relaxation time. We explain why the diffusion kinetics is not influenced by
the developing stress field, even though this would be expected. [5]
157
II. Theory
Stephenson – in a one-dimensional, isotropic n component system – derived a set of coupled differential equations for the description of the resultant
stress development and by viscous flow, convective transport and composition evaluation [4,5]:
" n
#
X
DP
2E
3
= −
(Ωi ∇i ) + P
(1)
Dt
9(1 − v)
4η
i=1
n
X
3(1 − v) DP
∇v = −
(Ωi ∇i ) −
E
Dt
(2)
i=1
∇ρi
Dt
= −∇ji , i = 1, ..., n
(3)
where D/Dt denotes the substantial derivative, v is the velocity field required
to determine the spatial evolution of the system (Kirkendall velocity), P is
the pressure, t is the time, E is Young’s modulus and η is the shear viscosity.
Furthermore, Ωi , ρi and ji are the molar volume, the material density and
the atomic flux of component i, respectively. The atomic fraction ci instead
of ρi is, however, more convenient
P for describing the evolution of the system.
Using c = ρi /ρ, where ρ = ni=1 ρi is the total material density , Eq. (3)
becomes
n
Dci
1
ci X
= − ∇ji +
∇jk , i = 1, ..., n
(4)
Dt
ρ
ρ
k=1
The atomic flux is given by
jk = −ρ
Θi Di∗ ∇ci
+
ci Ωi
Di∗
∇P
RT
(5)
Here Di∗ is the tracer diffusion coefficient, R is the molar gas constant, T is
the absolute temperature and Θi is the thermodynamic factor. Note that in
this paper we restrict ourselves to an ideal binary system, i.e. Θi = 1.
Ignoring the stress effects, Stephenson’s model is equivalent to Fick’s
model: ignoring the second term Eq. (5) and (3) is just Fick’s second law.
It is well know – according to Boltzmann transformation [6,7] – that the
solution of Fick’s second equation results in the parabolic shift of planes
158
√
constant ci = x2p ∝ t or xp ∝ t. Thus Stephenson’s model is suitable
to investigate how the stress effects influence the diffusion kinetics, whether
anomalous diffusion kinetics can be observed thanks to developing–relaxing
stress fields.
Theoretically, the time evolution of the effect of diffusional stresses can
be classified into four different stages [3,4,13]: (i) t < tQss , (ii) tQss < t < tr ,
(iii) t ≈ tr and (iv) t tr. Here tQss is the time necessary to develop a
steady-state stress distribution and tr is the stress relaxation time of “pure”
Newtonian flow determined by the second term in Eq. (1). Supposing that
in a stressed sample the sum of the divergences of the atomic fluxes are
negligible, Eq. (1) becomes
DP
E
=−
P
Dt
6(1 − v)
and its solution is.
E
P = P0 exp −
t
6(1 − v)η
(6)
(7)
where P0 is the value of the pressure at the beginning of the observation of
the relaxation. The relaxation time is time necessary for the pressure (stress)
to decreases to eth part of its initial value:
tr = 6(1 − v)η/E
(8)
We solved the above system of Eqs. (1), (2), (4) and (5) numerically with
input parameters corresponding to the Si/Ge binary system (i = A, B in
Eqs. (1)-(15). However, not to be restricted to one specific case, we varied
the parameters across a wide range (the figures in bold correspond to the
Si/Ge system [6,8,9,10]): Young’s modulus was supposed to be compositiondependent E = cA EA + cB EB , where EA = 18.5, 185 and 1850 GPa, EB =
10.3, 103, 1030, 16.3, 163 and 1630 GPa; Poisson’s ratio: vA = vB = 0.27;
viscosity: ηA = ηB = 2 × 1012 , 2×1014 and 2 × 1016 Pas; molar volumes:
ΩA = 1.20 × 10−5 m3 mol−1 , ΩB = 1.36 × 10−5 m3 mol−1 ; T = 700 K. We
supposed exponentially composition-dependent diffusion coefficients: Di∗ =
0 /D 0 =1,2.4 and 10; moreover, m0 = m log e =
Di0 exp(mcA ) and that DA
10
B
0,4 and 7 (where e is the base of natural logarithm and m0 gives, in order
of magnitude, the ratios of the diffusion coefficients in the pure A and B
159
matrixes; for instance, m0 = 4 means that the A atoms jumps 10,000 times
faster in the A matrix than in the B) [11].
Note that, using these parameter values, tr falls in the range of (5−9)×106
s.
III. Results and discussion
Figure 1 illustrate two typical, markedly different composition and pressure profiles calculated for stress free initial conditions, i.e. only stresses of
diffusion origin were taken into account. Figure 1 correspond to symmetrical
(m’=0) and asymmetrical (m’=7) cases.
In Fig. 1b, since the diffusion is orders of magnitude faster in the B matrix
than in A, practically only A atoms can dissolve into the B matrix and B
atoms can hardly penetrate into the A matrix. Consequently, a stress peak
develops in the A side close to the interface and on the B side an almost
homogeneous stress field (with opposite sign) appears. This sharp stress
peak shifts with the moving interface (see also [12]).
Figure 2 shows the shift of a plane with constant composition (cA = 0.6)
as a function of time. It can be seen that the slope of the straight line is close
to 0.5 (parabolic kinetics) although the shift is still in the range of 0.1 nm.
This was also the case for all investigated input parameters independently
whether symmetric or asymmetric profiles developed.
Thus the first stage (t < tQss ) is extremely short: intermixing on the
scale of a few tenths of nanometer was enough to reach it. It is in agreement
with the experimental observation of [4]. This means that a stress gradient
in the central zone, where the composition falls, becomes quasi-stationary
extremely fast for composition independent diffusivities (m’ = 0, symmetric
diffusion) as can be seen Fig. 1a.
In case of composition dependent diffusivities (m’6=0, asymmetric diffusion) the stress profile becomes also quasi-stationary, however not only one
stress gradient develops in the diffusion zone but two (see Fig. 1b). Thus
markedly different influence of the stress profiles on the atomic fluxes would
be expected in the second stage (ts < t < trelax ). In this case a slowing
160
down is expected because already a stationary stress field has been developed. Although there is a small stress gradient outside the diffusion zone its
effect can be neglected in the symmetric case. However, for the asymmetric
case the conclusion is not so obvious, since there are two stress gradients
with opposite signs at one of the borders of the diffusion zone (see in Fig.
1b). Thus one of them decreases whereas the other increases the resultant
volume flow. The computer simulations showed that even in this case the
stress effects slow down the intermixing process, i.e. the gradient slowing
down the process played the dominant role.
We have also shown in [2] that in the stationary stage indeed the NernstPlanck limit has been realized, i.e. the corrections due to stress gradients
counterbalanced the initially different fluxes of A and B. This is why there
were no deviations from the parabolic kinetics, but of course the kinetic coefficients were slower in this regime. This is in accordance with observations
of [2] where it was observed that in Al/Cu/Al as well as Cu/Al/Cu thin
film triple layers, deposited on tips of 25 nm apex radius, the growth of the
reaction product in both cases was parabolic, but with remarkably different
rates. This was interpreted by a switching between Darken and NernstPlanck regimes, caused by the inhomogeneous stress inside the sample.
Finally in the stage t > tr significant stress relaxation is expected. However surprisingly, we hardly observe any decrease in the stress levels in the
limits of the time interval investigated, although it was much longer than the
stress relaxation time belonging to a pure Newtonian flow (eq.(8)). This is
so because in obtaining the relation (8), the divergences of the atomic fluxes
were neglected in (1). Figure 3 shows the composition profile as well as the
two terms on the right hand side of (1) inside the diffusion zone. It can be
seen that in the steady state regime the contribution of the divergences is
not negligible as compared to the term 3P/4η. Consequently the stress is not
only relaxing but always re-developing also for t > tr , and thus maintains
the steady state.
161
Figure 1: Time evolution of the atomic fraction of A (cA ) and pressure (P)
profiles for (a) composition independent (m’=0) and strongly composition0 /D 0 = 2.4,
dependent (m’=7) diffusion coefficients. Input parameters: DA
B
14
EA =185GPa, EB =103GPa, η = 2×10 Pas. The initial composition profile
was rectangular, the sample was stress free and the interface was at 0 nm in
both cases. Only the interface region is plotted.
162
Figure 2: Shift of the position of plane with composition cA =0.6. Input
parameters are the same as in Fig. 1.
Figure 3: Composition profile as well as the first and second term on the
right hand side of (1) at t/tr ∼
= 2. Input parameters: m’=0, D0A /D0B = 10,
EA =185GPa, EB =163GPa, η = 2 × 1012 Pas.
163
IV. Conclusions
It is shown that, if the Kirkendall shift is negligible, a steady state NernsPlanck regime is established with diffusion coefficient close to the slower
diffusivity, independently of the type of the diffusion induced field and also
independently of whether this is a single field or a combination of different fields (e.g. stress field and extra chemical potential of non-equilibrium
vacancies). Deviations from parabolic kinetics are expected only before or
after this steady state stage. It is illustrated that the setting of time of the
Nernst-Planck regime is very short: intermixing on the scale of few tenths
of nanometer is enough to reach it. It is also illustrated that this stage is
realized even in the case of asymmetric interdiffusion, when the stress distribution has a more complex form (having a sharp peak at the interface).
Surprisingly the steady state is longer than it would be expected from the
relaxation time of Newtonian flow: This is so because the composition profile
is not static but changes fast in the timescale of the stress relaxation, and
thus the stress re-develops continuously.
Acknowledgments
This work was supported by the OTKA Board of Hungary (Nos K67969,
CK80126) and by TAMOP 4.2.1./B-09/1/KONV-2010-0007 project (implemented through the New Hungary Development Plan co-financed by the European Social Fund, and the European Regional Development Fund). One
of the authors (Z. Erdélyi) of this paper is a grantee of the ’Bolyai János’
scholarship.
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[1] D.L. Beke, Z. Erdélyi, B. Parditka, Defect and Diffusion Forum Vols.
309-310, 113-120 (2011).
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(2011).
164
[3] D.L. Beke, I.A. Szabo, Z. Erdélyi,G. Opposits, Mater Sci Eng A Vol.
387-389, 4-10 (2004).
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Symp. Proc. Vol. 527, 99 (1998).
[5] G. Schmitz, C.B. Ene and C. Nowak: Acta Mater Vol. 57 (2009), p.
2673
[6] J. Philibert, Atomic Movements, Diffusion and Mass Transport in Solids
(Les Editions de Physique, Paris) 1991.
[7] H.Mehrer, Diffusion in Solids, Springer – Verlag, Berlin, Heidelberg,
2007.
[8] F.Spaepen, Journal of Magnetism and Magnetic Materials 156 (1-3),
407 (1996).
[9] S.M. Porkes, F. Spaepen, Applied Physics Letters 47 (3), 234-236
(1985).
[10] J.J. Wortman, R.A. Evans, Journal of Applied Physics 36 (1), 153-156
(1965).
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(2010).
[12] Z. Erdélyi and D.L. Beke: Phys. Rev. B Vol. 68, 092102 (2003).
[13] G. Stephenson, Acta Metallurgica 36 (10), 2663 (1988).
165
ACTA PHYSICA DEBRECINA
XLV, 166 (2011)
CHARGE TRANSFER IN COLLISIONS OF C2+ IONS WITH
HCl MOLECULE
E. Rozsályi
Department of Theoretical Physics, University of Debrecen,
H-4010 Debrecen, PO Box 5, Hungary
Abstract
The charge transfer of C2+ ions in collisions with the HCl
molecule has been studied theoretically using ab initio molecular methods followed by semi-classical dynamical treatment. A
simple mechanism driven by a strong nonadiabatic coupling matrix element has been pointed out for this process.
I. Introduction
In the action of ionizing radiation with biological tissue, damage has been
shown to be induced by the secondary particles, low-energy electrons, radicals, or singly and multiply charged ions, generated along the track after
interaction of the ionizing radiation with the biological medium [1]. In these
reactions, generally at relatively low energies, different processes have to be
considered: excitation and fragmentation of the molecule, ionization of the
gaseous target, and also possible charge transfer from the multicharged ion
toward the biomolecule. We have undertaken a theoretical study of charge
transfer process in collisions of C2+ ions with hydrogen halide molecules.
Two main interactions have been pointed out for the C2+ + HF collision
[2,3]. The work is now extended to the hydrogen chloride target.
II. Molecular calculations
The geometry of the system is described using the internal Jacobi coordinates {R, r, α} with the origin at the centre of mass of the target molecule,
as defined in Fig. 1. The collision of the C2+ ion towards the hydrogen atom
in the linear approach would correspond to the angle α = 0◦ .
Figure 1: Internal coordinates for the C2+ +HCl collision system.
State-averaged complete active space self-consistent field (CASSCF) multireference configuration interaction (MRCI) calculations have been performed using the MOLPRO code [4]. The ECP10sdf 10 core-electron relativistic pseudopotential has been used for chlorine [5] with the correlationconsistent aug-cc-pVTZ basis set of Dunning [6]. The same basis set has
been chosen for carbon and hydrogen atoms. The active space includes the
1s orbital of hydrogen, the n=2 and n=3(sp) orbitals for carbon, and the
n=3 orbitals for chlorine. The 1s orbital of carbon has been frozen in the
calculation. The optimized geometry of the 1 Σ+ ground state of HCl at
the CASSCF (Complete Active Space Self-Consistent Field) level of theory
is rHCl =2.40988808 a.u. in good agreement with the 2.4086 a.u. experimentall value [7]. The corresponding vertical ionization potential calculated
using MRCI (Multi Reference Configuration Interaction) methods is 12.5901
eV, in good agreement with the 12.748 eV experimental value obtained from
photoelectron spectra measurements [7,8]. The charge transfer process is
driven mainly by nonadiabatic interactions in the vicinity of avoided crossings [9]. The radial coupling matrix elements between all pairs of states of the
167
same symmetry have thus been calculated by means of the finite difference
technique:
gKL (R) = hψK |
∂
1
|ψL i = hψK (R)| lim |ψL (R + ∆) − ψL (R)i,
∆→0 ∆
∂R
(1)
which, taking account the orthogonality of the eigenfunctions |ψK (R)i and
|ψL (R)i for K 6= L, reduces to
∂
1
|ψL i = lim hψK (R)|ψL (R + ∆)i.
(2)
∆→0 ∆
∂R
The parameter ∆ has been tested and a value of ∆ = 0.0012 a.u. has
been chosen as in previous calculations [10] using the three-point numerical
differentiation method for reasons of numerical accuracy.
gKL (R) = hψK |
The interaction between 1 Σ+ and 1 Π states by means of rotational coupling has also been taken into account. The rotational coupling matrix
elements hψK (R)|iLy |ψL (R)i between states of angular moment ∆Λ = ±1
have been calculated directly from the quadrupole moment tensor from the
∂
∂
expression iLy = x ∂z
− z ∂x
with the center of mass of the system as the
origin of electronic coordinates [11].
Taking account of the 1 Σ+ symmetry of the C 2+ (1s2 2s2 )1 S + HCl(1 Σ+ )
entry channel, only C + (2 P o ) or C + (2 D) states could be involved in the
collision process. Effectively, C + (4 P ) states could lead only to triplet and
quintet states which cannot be correlated to the entry channel as spin-orbit
coupling is negligible. With regard to the different excited states of HCl+ ,
there are thus four 1 Σ+ states which can be correlated by means of radial
coupling, the entry channel and three charge transfer levels. We have also
to take into account the 1 Π states which can be correlated to the 1 Σ+ entry
channel by rotational coupling interaction.
168
Four 1 Σ+ states and three 1 Π states must thus be considered in this
process with regard to the different excited states of HCl+ and spin considerations:
C 2+ (1s2 2s2 )1 S + HCl(1 Σ+ )
C + (1s2 2s2 2p)2 D + HCl+ (2 Π)
C + (1s2 2s2 2p)2 P ◦ + HCl+ (2 Σ+ )
C + (1s2 2s2 2p)2 P ◦ + HCl+ (2 Π)
41 Σ+
31 Σ+ , 31 Π
2 1 Σ+ , 21 Π
1 1 Σ+ , 11 Π
Figure 2: Potential energy curves for the 1 Σ+ (solid line) and 1 Π (dashed
line) states of the C2+ +HCl molecular system at equilibrium, (α = 0◦ ):
1, C + (1s2 2s2 2p)2 P ◦ + HCl+ (2 Π); 2, C + (1s2 2s2 2p)2 P ◦ + HCl+ (2 Σ+ ); 3,
C + (1s2 2s2 2p)2 D +HCl+ (2 Π); 4, C 2+ (1s2 2s2 )1 S +HCl(1 Σ+ ) entry channel.
169
Figure 3: Radial coupling matrix elements between 1 Σ+ states of the
C2+ +HCl molecular system at equilibrium, α = 0◦ . Same labels as in Fig.
2.
The corresponding potential energy curves have been calculated in the
2.0-15.0 a.u. internuclear distance range. They are presented in Fig. 2
for the equilibrium distance in the linear geometry. The main feature is
a very strong avoided crossing between the 1 Σ+ entry channel and the
31 Σ+ {C + (1s2 2s2 2p)2 D + HCl+ (2 Π)} charge transfer level around R=11
a.u. This avoided crossing appears to be the leading nonadiabatic interaction in the present collision system. The other avoided crossings, between 11 Σ+ {C + (1s2 2s2 2p)2 P ◦ + HCl+ (2 Π)} and 21 Σ+ {C + (1s2 2s2 2p)2 P ◦ +
HCl+ (2 Σ+ )} exit channels, or between the 21 Σ+ {C + (1s2 2s2 2p)2 P ◦ + HCl+
(2 Σ+ )} and 31 Σ+ {C + (1s2 2s2 2p)2 D + HCl+ (2 Π)} levels are significantly
smoother and correspond to large energy gaps. They could certainly not
be determinant in the process. Such interactions may be visualized also
on the radial and rotational coupling matrix elements. The most important
features are presented on Fig. 3 and Fig. 4 respectively. The radial nonadiabatic coupling matrix element between the 31 Σ+ and the 41 Σ+ entry channel
show clearly a strong peak, 1.48 a.u. high, in correspondence to the very
strong avoided crossing between the potential energy curves. Such coupling
170
is more than three times higher than the other radial couplings. Besides,
the strong interaction between the 21 Π{C + (1s2 2s2 2p)2 P ◦ + HCl+ (2 Σ+ )}
and 31 Π{C + (1s2 2s2 2p)2 D + HCl+ (2 Π)} exit channels pointed out on the
potential energy curves leads to a sharp crossing between rot22 and rot32
correlated to the change of character of the Π wavefunction in the neighbourhood of the avoided crossing. On the contrary, the interaction between states
1 and 2 being very smooth for both 1 Σ+ and 1 Π symmetries, the rotational
coupling rot11 remains almost equal to 1 for all distances.
Figure 4: Rotational coupling matrix elements between 1 Σ+ and 1 Π states
of the C2+ +HCl molecular system at equilibrium, α = 0◦ . Same labels as
in Fig. 2. (rot11=< 11 Π|iLy |11 Σ+ >; rot22=< 21 Π|iLy |21 Σ+ >; rot32=<
31 Π|iLy |21 Σ+ >; rot34=< 31 Π|iLy |41 Σ+ >.)
III. Collision dynamics
The collision dynamics has been performed by means of the EIKONX
code [12] in the keV laboratory energy range. As straight-line trajectories are
satisfying for energies higher than 10 eV/amu [13], semi-classical approaches
171
Figure 5: Total and partial charge-transfer cross sections for the C2+ +HCl
system at equilibrium, α = 0◦ . Full line, transition to 1 Σ+ states; broken
line, transition to 1 Π states.
(sectot, total cross section; sec 43, partial cross section on {C + (1s2 2s2 2p)2 D +
HCl+ (2 Π)}; sec42, partial cross section on {C + (1s2 2s2 2p)2 P ◦ + HCl+ (2 Σ+ )};
sec41, partial cross section on {C + (1s2 2s2 2p)2 P ◦ + HCl+ (2 Π)}).
may be used with a good accuracy in this collision energy range. Electronic
transitions being much faster than vibration and rotation motion, the sudden approximation may be used and cross sections, corresponding to purely
electronic transitions, are determined by solving the impact-parameter equation as in the usual ion-atom approach, considering the internuclear distance
of the molecular target fixed in a given geometry. This relatively crude approach is anyway widely used in the field of ion-molecule collisions and has
shown its efficiency in the keV energy range we are dealing with [14,15].
This treatment was performed for the collision of C2+ on HCl molecule,
taking account of all the transitions driven by radial and rotational coupling
matrix elements. The translation effects have not been taken into account.
The partial and total cross sections in the linear C-H-Cl geometry α=0◦
172
are presented in Fig. 5. The total cross section presents a peak around
vcoll = 0.1 a.u. (Elab = 3 keV ) which is mainly due to the contribution of
the corresponding peak of the partial cross section sec43. As pointed out
from the potential energy curves, the charge transfer process appears clearly
dominated by one nonadiabatic interaction corresponding to the avoided
crossing between the entry channel and the highest 31 Σ+ {C + (1s2 2s2 2p)2 D+
HCl+ (2 Π)} charge transfer level which gives rise to the strong peak of the
partial cross section sec43. Such behaviour is completely different from the
mechanism observed in the C2+ + HF collision system. The shorter range
crossing between 31 Σ+ {C + (1s2 2s2 2p)2 D+HCl+ (2 Π)} and 21 Σ+ {C + (1s2 2s2
2p)2 P ◦ + HCl+ (2 Σ+ )} channels is also involved in the C2+ + HCl charge
transfer process, it leads in particular to a hump in the sec42 partial cross
section, but its contribution is largely lower than the strong interaction between the entry channel and the 31 Σ+ {C + (1s2 2s2 2p)2 D+HCl+ (2 Π)} charge
transfer channel. As already pointed out for C2+ +HF, some rotational effect may be observed also for this system as charge transfer channels may
be all correlated to the entry channel by means of rotational coupling. At
higher collision energies, the contribution of 1 Π exit channels is almost of the
same order of magnitude as that of the corresponding 1 Σ+ transfer states
coupled by radial coupling. On the contrary, the nonadiabatic radial coupling interaction rad34 completely dominates the mechanism of the process
at vcoll = 0.1 a.u. (Elab = 3 keV ).
IV. Conclusions
We have presented a theoretical treatment of charge transfer processes induced by collision of the C2+ projectile ions on the HCl molecule. A simple
mechanism driven by the nonadiabatic radial coupling interaction between
the entry channel and the highest 31 Σ+ {C + (1s2 2s2 2p)2 D + HCl+
(2 Π)} charge transfer channel is exhibited for this collision system. The total cross section exhibits a maximum for a collision energy Elab = 3 keV .
This mechanism is completely different from the behaviour of the C2+ +HF
collision system. Such comparative analysis shows that it is hard to extract general conclusions for a series of molecular targets, even of almost the
same electronic configuration. The mechanism is basically dependent of the
specific nonadiabatic interactions involved in each collision system.
173
Acknowledgments
This work was granted access to the HPC resources of [CCRT/CINES/
IDRIS under the allocation 2011- [i2011081655] made by GENCI [Grand
Equipement National de Calcul Intensif] as well as from COST actions
CM0702 CUSPFEL and MP1002 Nano-IBCT.
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175
ACTA PHYSICA DEBRECINA
XLV, 176 (2011)
THE UNRUH EFFECT FOR FLUCTUATING
TRAJECTORIES IN HIGH ENERGY COLLISIONS
T.S. Biró1,2 and Z. Schram2
1
2
MTA KFKI RMKI, P.O.Box 49, Budapest 114, H-1525, Hungary
Department of Theoretical Physics, University of Debrecen, P.O.Box 5,
Debrecen, H-4010 Hungary
Abstract
We determine the analogue to an Unruh effect in high energy collisions by considering an Euler-Gamma distribution of
stopping times due to a finite deceleration in proper time. This
paper is dedicated to Prof. István Lovas on the occasion of his
80th birthday.
I. The Unruh effect
Since the seventies we face the fact that a uniform accelerating observer
experiences a Planck distribution for a monochromatic light-wave. The
equivalent temperature due to this Unruh effect is proportional to the
acceleration[1, 2, 3].
Such a relation between temperature and acceleration can already be
established in the framework of special relativity. An observer with constant
acceleration, g, measures a thermal looking, Planck-like spectrum for the
intensity of the Fourier transform of a field, which is a monochromatic plane
wave for a static observer.
Let the static time coordinate be t and the comoving proper time, τ . The
equivalent temperature, T , is proportional to the acceleration, g:
kB T =
~g
.
2πc
(1)
On Earth, g ≈ 10 m/s2 , the thermal energy of kB T ≈ 10−29 eV is to be
compared to the usual room temperature of about kB T ≈ 2.5 · 10−3 eV.
In high energy experiments carried out in particle accelerators, on the other
hand, one deals with a deceleration stopping particles from almost light speed
to zero on a distance of approximately 0.3 fm, the proton radius. This value
leads to kB T ≈ 100 MeV [3], a magnitude of temperature, which can in fact
be associated to the spectra of observed hadrons[4, 5, 6, 7, 8, 9, 10, 11, 12].
Here we present a simple derivation of the Unruh effect based on the
description of the relativistic motion with constant acceleration. Later we
analyze a Fourier-transformation in terms of the comoving time. The fouracceleration vector, aµ = duµ /dτ , is Minkowski-orthogonal to the fourvelocity, uµ . We fix its length g by the relation aµ aµ =p−g 2 . The fourµ
2 2
acceleration with these constraints
p becomes a = (gw/c, g 1 + w /c , 0,µ0),
2
2
while the four-velocity, uµ = (c 1 + w /c , w, 0, 0), is normalized as: uµ u =
c2 . From this we arrive at the differential equation:
p
dw
= g 1 + w2 /c2 .
dτ
(2)
The general solution for starting at τ = 0 with the velocity w0 is given as
gτ
w0 w = c sinh
+ Arsinh
.
(3)
c
c
This can also be written as
w = w0 cosh
gτ
+
c
q
gτ
c2 + w02 sinh .
c
(4)
A further τ -integration leads to the description of the world-line of the source
of monochromatic radiation:
q
c
c
gτ
gτ
x = x0 +
c2 + w02 cosh
− 1 + w0 sinh ,
g
c
g
c
1q
w0 gτ
gτ
t = t0 +
cosh
−1 +
c2 + w02 sinh .
(5)
g
c
g
c
Considering now a massless plane wave,
1
φ(t, x) = √ eiω(t±x/c) ,
2ω
(6)
177
one needs to use the advancing and retarded phases
h
q
i
1
2
2
w0 ± c + w0 e±gτ /c − 1 .
t ± x/c =
g
(7)
We note that the interchange of g and w0 by their negatives also interchanges
t + x/c with t − x/c.
For a constant acceleration path we Fourier analyze the plane wave as a
function of τ :
+∞
Z
1
f± (ν) = √
e iω(t±x/c) e −iντ dτ
(8)
2ω
−∞
is the complex amplitude observed by the comoving observer. Its absolute
value squared gives the intensity distribution over the frequency ν.
The integral (8) can be expressed by Euler’s Gamma function by using
the variable z = A± e±gτ /c :
1
c
νc
f± (ν) = √ eiϕ± eπc/2g Γ(i ).
g
g
2ω
(9)
with some phase factor ϕ± . The absolute value squared becomes
|f± (ν)|2 =
1 2πc
(n(ν) + 1)
2ω gν
(10)
for positive ν/g ratio and
|f± (−ν)|2 =
1 2πc
n(ν)
2ω gν
(11)
for a negative ratio. Here the n(ν) occupation number follows from the
product formula for the Euler-Gamma function:
Γ(ix) · Γ(−ix) =
One obtains
n(ν) =
178
π
.
x sinh(πx)
1
e2πνc/g
−1
.
(12)
(13)
This is alike to a thermal black body radiation with the temperature
kB T = ~
g
.
2πc
(14)
It is interesting to realize that the continuation of the path with constant
acceleration to imaginary time, τ = iθ, leads to periodic paths in spacetime with the period T = ~β, satisfying gT /c = 2π, and therefore – in
precise agreement with the KMS relations discussed in thermal field theory
– the Unruh temperature kB T = 1/β = g~/2πc emerges as the physical
temperature of the quantum gas of photons. This periodicity is explored by
the Fourier-transformation and the complex arguments of Euler’s Gamma
function, discussed above.
II. Fourier transformation along a general trajectory
By using the rapidity variable, η the general result for an arbitrary trajectory can be written as a Fourier transform of an integral phase with the
actual Doppler factor:
1
f (ν) = √
2ω
Z+K R τ“ q c+v(θ) ”
i
ω c−v(θ) −ν dθ
dτ
e 0
(15)
−K
while v(τ ) being the three-velocity of the source along the trajectory xµ (τ ),
η=
1 c+v
ln
2 c−v
(16)
being the rapidity variable leading to a general Doppler factor of eη and
finally the acceleration is given by
g=c
dη
.
dτ
(17)
The familiar relativistic Doppler effect occurs for a constant η, the Unruh
effect arises for a constant g. Here we considered a finite spectral analysis
eigentime from −K to +K – instead of infinity. The results of the previous
section are obtained for g = const. and K → ∞.
179
The integral over θ can be converted to an integral over η by using the
defining equation (17) for the acceleration:
dθ =
c
dη.
g(η)
(18)
The τ -dependent phase in the spectrum is now given by
ϕ(τ ) = ω
Zτ
e
η(θ)
dθ = ω
η(τ
Z )
c η
e dη,
g(η)
(19)
η0
0
and the intensity of the radiation spectrally analyzed in the comoving system
between τ = −K and τ = +K by
+K
2
Z
1 i(ϕ(τ )−ντ )
.
e
dτ
I(ν) = |f (ν)|2 =
2ω (20)
−K
In this case the constituting equation is given by the acceleration – rapidity
relation, g(η). From this the time and space coordinates of the far, static
observer can be reconstructed by the following integrals
Z
R
cosh(η)
t = cosh η dτ =
cdη,
g(η)
Z
R
sinh(η) 2
x = sinh η cdτ =
c dη.
(21)
g(η)
It is even more straight to use the Doppler-factor, z = eη as integration
variable and the g(z) relation as describing the trajectory of the source. In
this case the complex amplitude for an infinite time spectral analysis is given
by
Z∞
c
dz
f (ν) = √
e iϕ(z)
(22)
zg(z)
2ω
0
with the changing phase defined by the indefinite integral
Z
dz
ϕ(z) = c (ωz − ν)
.
zg(z)
180
(23)
The proper time is related to the z-factor by the integral
Z
dz
τ =c
,
zg(z)
(24)
with the boundary condition that z(−∞) = 0 and z(+∞) = ∞. The coordinates for the far, static observer are given as
Z 1
dz
c
z+
t =
,
2
z zg(z)
Z dz
c2
1
x =
z−
.
(25)
2
z zg(z)
III. Distribution of stopping positions and the Unruh
temperature
As it can be seen the general Fourier transformation formula for the spectral analysis of a monochromatic radiation from an accelerating source contains the proper time integral in a Doppler factor corresponding to the phase
variable ω(t ± x/c).
Simple solutions are the constant velocity (original Doppler effect) and the
constant acceleration (Unruh) solutions. A general solution is numerically
feasible, but at the first sight the wealth of possibilities is overwhelming.
Therefore we consider now the opposite extreme: a fluctuating acceleration.
Since the inverse temperature, β = 1/kB T is given by β = 2πc/g, we
consider this quantity to be Euler-Gamma distributed according to the superstatistical approach to non-extensive thermodynamics [13, 14, 15, 16].
The distribution,
γ α α−1 −αβ
β
e
,
(26)
P (β) =
Γ(α)
is normalized to one, defined on non-negative β values only and leads to the
following average temperature
γ
hkB T i =
(27)
α−1
and relative spread
δT
1
=√
.
T
α−2
(28)
181
The average of β on the other hand is given by
hβi =
and its relative spread by
α
γ
(29)
δβ
1
=√ .
β
α
(30)
The superstatistical average of a Boltzmann-Gibbs factor – giving the high
temperature approximation to the Planck spectrum, too – is given as
D
E ~ν −α
−β~ν
.
(31)
e
= 1+
γ
Utilizing the result for the average temperature (27) and the notation 1/(α−
1) = q − 1 one arrives at the Tsallis distribution
D
e
−β~ν
E
=
~ν
1 + (q − 1)
hT i
−
q
q−1
.
(32)
In the pT spectra of particles stemming from relativistic heavy ion collisions
such spectra give an excellent description of experimental data up to several
GeV momenta [17]. The parameters hT i and q can be fitted to the hadronic
final state and – assuming quark coalescence – can be followed back to the
quark matter stage. Typical values for the hadronizing quark matter at
RHIC are hT i ≈ 160 MeV and q ≈ 1.22. These values correspond to α = 5.5
and γ = 720 MeV resulting in a chracteristic temperature spread of δT = 85
MeV.
The Unruh effect helps to translate these parameters to the acceleration
and to its fluctuation. And even more, since the acceleration varies, so
do the related trajectories of source. A relative spread in the acceleration
parameter, g, translates to a spread in the stopping position x at t = 0. We
estimate this effect in the following.
We consider an incoming source from a positive x = x0 with a negative
rapidity η = −η0 at the initial proper time τ = 0. The deceleration is
positive, g > 0. In this case the rapidity changes as
η(τ ) = −η0 +
182
2π
τ.
~β
(33)
The time and position are
~β
(sinh η + sinh η0 ) + t0 ,
2π
~β
x(τ ) = c
(cosh η − cosh η0 ) + x0 .
2π
t(τ ) =
(34)
The stopping is at the proper time τstop , defined by η(τstop ) = 0. We obtain
~β
η0 ,
2π
~β
=
sinh η0 + t0 ,
2π
~β
= c
(1 − cosh η0 ) + x0 .
2π
τstop =
tstop
xstop
(35)
Since η0 is given by the beam energy, the fluctuation in these quantities
event by event must be a consequence of the fluctuation in β, the Unruh
temperature, which is due to the fluctuation in the stopping deceleration.
However, for a given δβ, the uncertainty in the stopping time and position
is magnified by the beam-energy dependent factors in the above array of
equations (35). The relative fluctuations in the stopping quantities are expected to be in the same order of magnitude as those of the deceleration,
√
δβ/ hβi = 1/ α ≈ 0.42 in the quark matter at RHIC.
183
Acknowledgment
Enlightening discussions with Prof. C. Greiner at Frankfurt University
are gratefully acknowledged. This work has been supported by the TÁMOP
4.2.1./B-09/1/KONV-2010-0007 project, co-financed by the European Union
and European Social Fund. Support by the Hungarian National Research
Fund OTKA (K68108) is acknowledged. The work was partly supported
by the Helmholtz International Center for FAIR within the framework of
the LOEWE program (LandesOffensive zur Entwicklung WissenschaftlichÖkonomischer Exzellenz) launched by the State of Hesse.
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H. Satz, B. Sinha, Springer, 2009
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(2002).
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(2003).
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184
[10] STAR Collaboration (Yichun Xu et.al.), Nucl. Phys. A830, 701C
(2009).
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022002 (2010).
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C. Beck, Phys. Rev. Lett. 98, 064502 (2007).
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185
ACTA PHYSICA DEBRECINA
XLV, 186 (2011)
DEVELOPMENT OF A BALANCE MEASUREMENT
SYSTEM FOR BIOMECHANICAL APPLICATIONS
F. R. Soha1 , I. A. Szabó1 , L. Harasztosi1 , J. Pálinkás2 , Z. Csernátony3
1
University of Debrecen, Department of Solid State Physics
2
University of Debrecen, Department of Physiotherapy
3
University of Debrecen, Orthopedic Clinics
Abstract
A forceplate with electronics and data acquisition software
has been developed for posturography. We present the description of the measuring device and the results of calibration and
error analysis for static and dynamic loads. The applicability of
the forceplate is illustrated for kinematic studies of mechanical
and biomechanical systems. The motion of vibrating pendulums was studied with the platform. The resulting kinematic
curves are represented with the generalization of the Lissajouscurve family. Posturography measurements were performed with
male adults with four different conditions - legs open/closed and
eyes opened/closed position. Several stability measures were performed to select the most sensitive method.
I. Introduction
Upright standing is an unstable position, which is disturbed continuously
by internal and external perturbations. Without feedback control the postural stability cannot be maintained. Maintaining balance is an involuntary process, in which a wide range of sensory information (haptic sensors,
labyrinthine sensors, proprioceptors and visual sensors) are integrated at
various levels of the spine and brain neural networks to produce muscle activation patterns [1]. The quantitative analysis of the postural stability can
Figure 1: The geometry of the forceplate.
provide important information about the health of an individual. For example, the degradation of balance control in elder persons can predict an
increased risk of fall, which is a major danger.
The first electronic forceplate was constructed in 1952 by J. Scherrer.
Commercial type equipments are available for clinical applications [2]. However these are expensive, and can restrict the applicable signal processing
and data evaluation techniques. Force platforms should be examined for
precision and accuracy [3].
II. The forceplate
The force measurements are performed by three calibrated load cells instrumented with strain gauge. The load cells were manufactured and individually calibrated by Kaliber Corporation from Hungary, Budapest. The
nominal range of the force sensors are 0-100 kg.
The load cells were fixed to the corners of an aluminum frame of equilateral triangle shape as shown on figure 1. The side lengths of the triangle
187
are 63cm. The legs of the balance are spherical joints, allowing for stable
contact to the ground without horizontal forces. The surface of the balance
is a stiff wood plate. The frame was made from welded aluminum. The total
mass of the balance is 4.5 kg.
The force platform provides the value of the vertical ground reaction force
acting on the load as a sum of the forces on the load cells. From the balance
of torques on the plate one can calculate the point of action of this force.
This point is the center of the pressure distribution of the foot-plate contact
forces, usually denoted by COP
W
COPx
COPy
= F1 + F2 + F3
F2 + F3 /2
= a
F1 + F2 + F3
√
F3 3/2
,
= a
F1 + F2 + F3
(1)
where F1 , F2 , F3 are the forces measured by the load cells, and a is the side
length of the triangle.
In static equilibrium, the vertical projection of the center of mass (CM)
would coincide with the COP point. If the dynamics of a standing person is
assumed to be similar to an inverted pendulum, the trajectory of the COP
is related to a weighted temporal average of the CM [4].
The load cell signals were pre-amplified and connected to an A/D card
(National Instruments USB 6212). The digitalized data is transferred to a
personal computer via an USB port. The measurement control and data
evaluation software was written in LabVIEW code. The sampling rate of
the A/D converter was set to 50 kHz. For the reduction of the electric noise,
200 samples were averaged to provide a single measurement data. This leads
to a sampling rate of 2.5 kHz.
The forceplate measurement cycle starts with an unloaded condition,
where the signal levels corresponding to zero external force are determined.
The gains and the sensitivity of each sensor were calibrated by a standard
weight positioned at different places on the balance. After this procedure,
the sensor voltage signals are converted to force values and finally the weight
and COP coordinates are calculated according to (1).
188
When the balance is set to an inverted position, there is no load on the
cells. Under these conditions, the standard deviation of the total force is
reduced to 0.06N by the averaging of 200 points from the original value of
0.3N.
When the balance is standing on the ground, external vibrations “seismic
noise” is introduced. Without additional load, this increases the standard
deviation to 0.07 N. The effect of the external noise on the COP determination depends on the amount of weight put on the balance. Under small
loads (5kg) the standard deviation of the COP position was 0.2 mm. We
have found that the internal structure and the stability of the contact greatly
modifies the observed standard deviation for an object. External vibrations
excite internal vibrations, which lead to deviations of the central of gravity
(COG) and COP coordinates. With a stable loading weight of 35 kg the
standard deviation was found to be reduced to 0.01 mm. The COP coordinate shows sways of amplitude 6-8 mm according to the literature. Thus an
error of 0.01mm is less than one percent of the measured signals amplitude.
The Fourier spectrum of the noise is flat up to a 40 Hz cut-off frequency.
This is the bandwidth of the forceplate. The 2.5 kHz sampling rate represents a large oversampling. No signal from the sway dynamics is expected
to be above 40Hz. This means, that the balance is appropriate for the measurement of the dynamical processes during human motion.
The dynamic characteristics of the forceplate were tested using a spring
mass system as a load. The motion of the mass was recorded by a video camera, and the amplitude of the vibration frequency was determined from the
recording. The periodic acceleration of the mass leads to a periodic change
of the load on the supporting platform, from which the vibration amplitude
can be calculated, knowing the angular frequency and the mass. There is
an excellent correlation between the calculated and measured amplitudes as
shown on Figure 2.
189
Figure 2: Correlation of the calculated and measured vibration amplitudes.
III. Test with mechanical systems
As a first application, we have put a tripod on the balance. A springmass system or a pendulum was hanged down from the center of the frame.
The motion of the COP was recorded with the force platform under various
initial conditions. The mass was always started from off center positions.
Several examples of the trajectories are given in the upper row of Figure 3.
These trajectories can be understood as a superposition of damped vibrational and rotational motions. They can be well represented using a
generalization of the Lissajous-curves
X = Ae−t/T sin(ω1 t + φ1 ),
Y
COPX
COPY
= Ae−t/T sin(ω2 t + φ2 ),
= cos(Φ(t))Z − sin(Φ(t))Y,
= cos(Φ(t))Z + cos(Φ(t))Y,
(2)
In the XY coordinate system the motion is represented by a damped Lissaguecurve. This curve is slowly rotated relative to the reference frame of the
190
Figure 3: Measured COP trajectories for spring mass (a) and pendulum
systems (b,c).
Figure 4: Simulated COP trajectories for spring mass (a) and pendulum
systems (b,c).
balance: Φ(t) = Ωt. The curves on Figure 4 are generated with the above
formulas with proper initial conditions and parameters.
The spring mass system is shown on Figure 3a. The actual motion is the
result of the coupling of the vibration and the elliptical pendulum motion.
The spring had a large diameter, and was extended to several times of its
original length by the mass hanging on it. An elementary calculation or
the dimensional analysis shows, that the angular frequency of a string is the
same as the angular frequency of the pendulum with the same mass, if the
length of the pendulum is the same as the extension of the string under the
weight of the mass. Accordingly ω1 and ω2 was chosen to slightly different
191
values. A slow rotation is also present in the COP trajectory.
For the pendulum, ω1 and ω2 are the same. The twisting of the wire
could provide the torque, which leads to a slight rotation of the axis. The
two cases shown on Figure 3b and Figure 3c represents an elliptic and a
nearly circular orbit, which is damped in amplitude and slightly rotated.
The above examples show, that the force platform can be useful in the
analysis of the COP-COG dynamics in classical mechanics. The main advantage of the method is its simplicity, because there is no need to attach
any sensors or position measuring devices to the system under study.
IV. Stabilometry measurements
Posturography is the study of human balance with force platform measurements. Figure 5 shows the track of the posturogram of a standing person
with eyes open in a normal standing posture with the Anteroposterior direction (forward-backward) corresponding to the y axis, taken in 40 seconds.
The left side is the trajectory, and the right side shows the temporal graph
of the COP coordinates in the X mediolateral (ML) and Y Anteroposterior
(AP) directions. One can clearly see the difference between the AP and ML
sway amplitudes. The thick line represents a 15 degree polynomial (COP
trend) fitted to the data. The polynomial approximately represents a slow
drifting of an imaginary target position, around which the COP oscillates.
Shifting of the target position can take place within the range of support
without the loss of stability. This means, that the increase of the range of
motion during a longer period of time does not mean the loss of stability,
but just indicates the change of weigh distribution between the legs.
Several measures were performed to quantify the stability based on the
porsturography measurements, but no single parameter was found to give
consistent results under all circumstances [5]. As clear from the figure, the
displacements are different in the AP and ML and will be measured separately. The following quantities were used in this study:
192
VX , V Y :
The averaged velocity magnitude for X and Y directions
Xsd, YSd:
The standard deviation of COPx and COPy ,
Figure 5: Stabilogram for quiet standing posture with eyes opened.
Xrange, Yrange:
The difference of the maximal and minimal values.
X90%, Y90%:
The range of COPX and COPY values after removing
the 5% highes and 5% lowest values.
sXsd, sYsd:
Standard derivation of the COPX -COPtrendX and
COPY -COPtrendY
Stabilograms of 10 volunteers of ages between 25 and 55 males were
recorded. The subjects stood with eyes open (EO) or eyes closed (EC), and
placed their legs their regular standing position legs open (LO), and putting
the legs directly side by side in the legs closed (LC) conditions. These settings have provided four different balance conditions for the subjects.
Using all data, we have examined the correlation between the stability
measures introduced above. The R2 values for the fits are shown below
in table (1) for the two directions for each variable pairing. The x90% and
y90% values are highly correlated with the standard deviations Xsd and Ysd.
From these two dependent measures, the standard deviation was kept. The
Xrange and Yrange have a much lower correlation. The range will strongly
emphasize rare extreme deviations in the path, which leads to large scatter
for repeated trials. For this reasons, the Xrange and Yrange is not useful as
a stability measure.
193
Table 1. R2 values for straight line fits between the stability measures.
The selected stability measures were averaged under a given experimental condition for the 10 participants and the standard deviations were also
determined. These data are presented as bar charts for comparison. The
first row represents the X – AP direction and the second row is the Y – ML
direction. Similar trends can be observed on all of the quantities. The average velocity and the trend subtracted standard deviations have the smallest
scatter and present the largest changes with changing conditions.
In general, the stability is larger in the (ML) direction, but for legs closed
(LC) stance, the two directions shows similar levels of stability. The eye
closed (EC) condition leads to decrease in stability mainly in the AP direction
for LO stance. For LC stance the stability decreases in both directions.
From this preliminary study, we can conclude, that our forceplate is applicable to postugraphy studies. We have also presented two examples of the
stability measures, which can point the difficulties of the balancing tasks.
Acknowledgements
Financial support of the TÁMOP-4.2.1/B-09/1/KONV-2010-0007 grant
is greatfully acknowledged.
194
ht
Figure 6: Average values and scatter for stability measures under four different conditions for 10 male adults.
195
References
[1] J. Massion, M. H. Woollacot, Prog Brain Res. 143, 13 (2004).
[2] H. Chaudy, B. uklet, Z. Ji, T. Findles, Journal of Bodywork and Movement Therapies 15, 82 (2011).
[3] J. Browne, N.O’Hare, Physiological Measurement 21, 515 (2000).
[4] F. G. Borg, arXiv:physics/0503026v1 (2005).
[5] T. Doyle, R.U.Newton, A.F. Burnett, Arch.Phys.Med Rehabil 86, 2034
(2005).
196
ACTA PHYSICA DEBRECINA
XLV, 197 (2011)
HIGH RESOLUTION STUDY OF THE RELATIVE DIPOLE
STRENGTH DISTRIBUTION IN 42 SC
L. Stuhl1 , A. Krasznahorkay1 , M. Csatlós1 , T. Adachi2 , A. Algora3 , J.
Deaven4 , E. Estevez3 , H. Fujita2 , Y. Fujita5 , C. Guess4 , J. Gulyás1 , K.
Hatanaka2 , K. Hirota2 , H. J. Ong2 , D. Ishikawa2 , E. Litvinova9,11 , T.
Marketin9,10 , H. Matsubara2 , R. Meharchand4 , F. Molina3 , H.
Okamura2 , G. Perdikakis4 , B. Rubio3 , C. Scholl6 , T. Suzuki2 , G.
Susoy7 , A. Tamii2 , J. Thies8 , R. Zegers4 , J. Zenihiro2
1
Inst. of Nucl. Res. (ATOMKI), P.O. Box 51, 4001 Debrecen, Hungary
2
RCNP, Osaka University, Ibaraki, Osaka 567-0047, Japan
3
IFIC, CSIC-Universidad de Valencia, 46071 Valencia, Spain
4
NSCL, Michigen State University, East Lansing, Michigan 48824, USA
5
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
6
Institut für Kernphysik, Universität zu Köln, 50937 Köln, Germany
7
Istanbul Univ., Fac. of Sci., Phys. Dept., 34134 Vezneciler, Istanbul, Turkey
8
Institut für Kernphysik, Universität Münster, 48149 Münster, Germany
9
GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt
10
Physics Department, Faculty of Sci., Univ. of Zagreb, 10000 Zagreb, Croatia
11
Institut für Theoretische Physik, Goethe Universität, 60438 Frankfurt am
Main, Germany
Abstract
Experimental data from the Ca(3 He,t)Sc charge exchange reaction on the target 42 Ca at 420 MeV beam energy are presented.
The achieved energy resolution of 20 keV, and the measured angular distributions allowed the extraction of the dipole strength
for excitation energies lower than 15 MeV in this Sc isotope for
the first time. Detailed informations for many individual excited
levels were identified. The Pygmy Dipole Resonance (PDR) in
42 Sc was investigated.
I. Introduction
The aim of this letter was to study the fragmentation of the dipole
strengths into low-lying excited states in Sc isotopes. The (3 He,t) charge
exchange reaction was used to access the dipole strengths distribution as
suming a simple proportionality between the cross sections and the dipole
strength values. The work has been performed in a wide international collaboration.
During the last few years much interest has been devoted to the experimental investigation of electric dipole strength distribution, in connection
with the neutron-skin thickness [1] and with the so-called Pygmy Dipole Resonance (PDR) in even-even nuclei. In a macroscopic picture this resonance
is described as an out-of-phase oscillation of a neutron skin against an inert
core. Therefore, properties such as integrated strength and mean excitation
energy of the PDR should strongly depend on the N=Z ratio. The microscopic nature of the pygmy dipole resonance in the stable Ca isotopes has
been investigated by Hartmann et al. [2] in high resolution photon scattering
experiments for the first time. So far, no high resolution studies have been
performed, however, for the dipole strengths distribution excited in charge
exchange reactions.
II. The experiment
The experiment was performed at the Research Center for Nuclear Physics
(RCNP) at Osaka University, Japan. The 3 He beam at 420 MeV was provided through the cascade acceleration with the K = 120 AVF cyclotron
and the K = 400 RCNP Ring Cyclotron. The energy of the 3 He beam was
achromatically transported to the self supporting metallic 42 Ca target. The
properties of the target can be found in Ref. [5]. The typical beam current
was 25 nA.
The energy of tritons was measured with a magnetic spectrometer using
complete dispersion matching technique [3]. Outgoing tritons were momentum analyzed in the Grand Raiden Spectrometer (GRS) [4] at two angle
positions: 0◦ and 2.5◦ with an opening angle of ś20 mrad horizontally and
±40 mrad vertically defined by a slit at the entrance of the spectrometer.
198
The results of both settings were combined to achieve angular distributions,
by which the character of single transitions could be determined.
Since our reactions are (3 He,t), the angular momentum transfer can be 0,
1, 2 ~. To characterize the different multipoles, theoretical angular distributions for states with J π = 1+ , 3+ and 0− , 1− , 2− were calculated using the
distorted-wave Born approximation (DWBA).
III. The evaluation
We analyzed the spectra using the program package: Gaspan, which was
developed for the evaluation of gamma- and particle-spectra. We could fit for
the given energy range many peaks at the same time. The peaks were fitted
with Gaussians plus exponential tails. We used second order polynomials for
describing the background. The quality of the fit was good. The chi-square
test gave typically χ2F = 1.1.
The excitation energies of the IAS and 10 well known excited states were
used for determining the precise energy calibration.The spectra were studied
in eight distinct angular regions: 0◦ -0.5◦ , 0.5◦ -0.8◦ , 0.8◦ -1.2◦ , 1.2◦ -1.6◦ , and
1.6◦ -2◦ , 2◦ -2.5◦ , 2.5◦ -3◦ , 3◦ -3.5◦ . The intensities are determined for eight
distinct angular bins too. The angular distributions were determined for
each of the known levels, and also for new peaks. They were normalized to
the corresponding opening angles, whiches were determined by experimental
data. We obtained the angular distributions first of all for the peaks, which
spin and parity were well-known in the literature. According to their characteristic angular distributions three different groups can be recognized. In
order to deduce the spin and parity, an An parameter was introduced, which
is the ratio of the intensity of first region devided by the average of intensity
of the 3rd and 4th region. To determine the threshold of An , the distribution
of the parameter was investigated. We obtained two distinct groups, for J π
= 1+ , 3+ and for J π = 0− , 1− , 2− . Fig. 1. shows a few typical angular distributions for an 1+ and a few L=1 levels. The angular distributions of the
newly identified levels are very similar to those levels having the same multipolarity known from the literature. Due to the excellent energy resolution
of ∼15 keV, the spectra contains highly detailed information, which allows
identification of many individual levels up to about 16 MeV excitation.
199
Figure 1: Angular distribution of a newly identified 1+ and a well-know and
newly identified ∆L=1 levels in 42 Sc.
200
Figure 2: The dipole strength distributon of
42 Sc.
IV. Results and conclusions
We have found 150 new excited levels out of the 192 ones observed in the
We reproduced in 40 cases the values of the well-known excited levels.
We determined 79 new ∆L=1 states. Their energies were deduced with a
precision of 10-15 keV. Tabel I shows the energies of levels, the values of the
An parameters and the intensities of the newly identified ∆L=1 levels.
42 Sc.
The SDR strength distribution for Sc isotope was also determined. It is
shown on Fig.2. We identified some Pigmy like resonances around 9 MeV.
The spin dipole giant resonance (SDR) is expected to be around 22 MeV
excitation energy in the Sc isotopes. However, we could observe dipole
strengths at much lower energy in each of the isotopes. The dipole strengths
distribution especially in the lighter isotopes shows some interesting periodic
feature. It resembles to a soft, fragmented vibrational band withă hω=1.8
MeV [5].ă Pigmy dipole resonances might play a more enhanced effect in the
heavier isotopes.
201
In order to understand the experimental results relativistic RPA (RRPA)
calculations have been performed with NL3 [6] and DD-ME2 [7] interactions.
RRPA predicts low-lying strength with nearly periodic peaked structure
caused by the dipole isospin-flip and spin-isospin-flip transitions governed
by the pion and effective rho-meson exchange interactions. The obtained
strength, however, shows no enhancement of the strength at the lowest energies. Thus, it is expected that correlations beyond RRPA can be responsible
for the observed enhancement.
Acknowledge
The authors acknowledge the RCNP cyclotron staff for their support during the course of the present experiment. This work has been supported by
the Hungarian OTKA Foundation No. K72566 and by MEXT, Japan, under Grant No. 18540270 and No. 22540310 and by the LOEWE program
of the State of Hesse (Helmholtz International Center for FAIR) and the
Hungarian-Spanish collaboration program (ES-26/018).
References
[1] A. Krasznahorkay et al., Phys. Rev. Lett. 82, 3216 (1999). [2] T.
Hartmann et al., Phys. Rev. Lett. 93, 192501 (2004). [3] H. Fujita et al.,
Nucl. Instrum. Methods A 484, 17 (2002). [4] M. Fujiwara et al., Nucl.
Instr. Meth. Phys. Res. A422, 484 (1999). [5] L. Stuhl et al., Acta Physica
Polonica B 42 (2011) 667. [6] G.A. Lalazissis, J. König, and P. Ring, Phys.
Rev. C 55 (1997) 540. [7] G.A. Lalazissis et al., Phys. Rev. C 71, 024312
(2005).
References
[1] A. Krasznahorkay et al., Phys. Rev. Lett. 82, 3216 (1999).
[2] T. Hartmann et al., Phys. Rev. Lett. 93, 192501 (2004).
[3] H. Fujita et al., Nucl. Instrum. Methods A 484, 17 (2002).
[4] M. Fujiwara et al., Nucl. Instr. Meth. Phys. Res. A 422, 484 (1999).
202
[5] L. Stuhl et al., Acta Physica Polonica B 42, 667 (2011).
[6] G.A. Lalazissis, J. König, and P. Ring, Phys. Rev. C 55, 540 (1997).
[7] G.A. Lalazissis et al., Phys. Rev. C 71, 024312 (2005).
203
Table 1: Level energies, intensities and angular distribution coefficients
Eexp /keV / ∆Eexp /keV / An ∆An I /rel.u./ ∆ I /rel.u./
892
8
0.65 0.38
45
23
1065
4
0.00 0.00
12
2
1230
21
0.27 0.57
3
7
1362
8
1.61 1.12
17
10
1846
8
1.23 1.17
28
22
2913
6
0.17 0.03
63
17
3047
9
0.56 0.39
14
12
4141
6
0.66 0.13
54
13
4440
7
0.21 0.05
26
10
4722
7
0.58 0.17
79
30
4819
5
0.94 0.11
160
22
5037
7
0.36 0.11
28
13
5232
4
5279
7
0.71 0.28
27
13
5347
7
0.26 0.09
25
12
5459
6
0.33 0.08
60
19
5532
14
0.08 0.13
8
22
5627
8
0.29 0.19
24
23
5826
5
0.39 0.10
144
49
6132
5
0.83 0.25
122
42
6161
5
1.10 0.15
405
58
6381
8
0.20 0.12
51
49
6447
4
6601
5
0.54 0.15
94
30
6659
7
0.78 0.50
36
24
6721
7
0.67 0.52
168
232
6776
6
0.97 0.37
94
28
6930
6
0.55 0.26
45
25
7000
9
0.13 0.08
20
22
7091
7
0.78 0.31
94
41
7317
7
0.50 0.20
61
32
204
Eexp /keV /
7370
7574
7725
7804
7873
8168
8358
8384
8654
8797
8844
9028
9063
9091
9141
9188
9266
9294
9355
9390
9425
9598
9877
10667
10964
11092
11143
11179
11226
11288
11399
11735
11892
∆Eexp /keV /
5
5
8
13
6
5
6
5
6
6
7
7
6
7
6
6
7
6
7
6
6
7
7
7
7
7
7
7
6
8
14
9
11
An
0.83
0.53
0.44
0.14
0.57
0.85
0.79
0.80
0.33
0.85
0.90
0.47
0.86
0.63
0.76
0.80
0.72
0.77
0.60
0.74
0.40
0.76
0.60
0.89
0.46
0.82
0.79
0.93
0.86
0.46
0.36
0.80
0.51
∆An
0.19
0.18
0.27
0.18
0.16
0.23
0.39
0.24
0.10
0.34
0.32
0.19
0.25
0.25
0.21
0.19
0.28
0.23
0.18
0.18
0.09
0.19
0.08
0.21
0.16
0.19
0.19
0.21
0.15
0.17
0.45
0.24
0.18
I /rel.u./
1 166
148
49
22
214
328
200
480
300
696
432
188
508
296
345
672
280
524
304
524
540
252
542
244
292
285
275
307
358
117
52
412
232
∆ I /rel.u./
40
46
38
46
64
70
67
88
88
128
136
80
136
136
74
128
120
144
88
104
112
70
78
68
128
70
75
77
71
58
88
120
88
205
Eexp /keV /
12017
12213
12295
12648
12931
12988
13043
13226
13382
13467
13560
13621
13784
14323
14999
206
∆Eexp /keV /
22
9
11
14
15
14
15
15
15
9
8
9
10
20
15
An
0.31
0.89
0.49
0.84
0.99
1.02
0.71
0.64
0.71
0.97
0.74
0.62
1.19
∆An
0.29
0.28
0.17
3.16
4.60
4.09
3.77
3.05
3.88
14.50
8.52
7.44
17.70
I /rel.u./
84
500
212
364
83
184
92
164
132
113
175
146
74
∆ I /rel.u./
128
136
80
120
67
80
58
76
80
52
54
48
46
ACTA PHYSICA DEBRECINA
XLV, 207 (2011)
CHARACTERIZATION OF ATMOSPHERIC AEROSOLS IN
DIFFERENT INDOOR ENVIRONMENTS
Z. Szoboszlai1,2 , Gy. Nagy2 , Zs. Kertész1 , A. Angyal1,2 , E. Furu1 , Zs.
Török1 , K. Ratter3 , P. Sinkovicz3 and Á.Z. Kiss1,2
1
3
Institute of Nuclear Research of the Hungarian Academy of Sciences
(ATOMKI), H-4026 Debrecen, Bem tér 18/c
2
University of Debrecen, H-4010 Debrecen, Egyetem tér 1.
Department of Materials Physics, Roland Eötvös University, H-1117 Budapest,
Pázmány Péter sétány 1/A
Abstract
In this study indoor aerosol concentration levels in three different microenvironments of the Institute of Nuclear Research
(ATOMKI) were investigated. We determined coarse and fine
mass concentrations and elemental composition during two summer and two winter campaigns at each location (laboratory,
workshop, and library). We also defined the possible emission
sources of the indoor aerosols and studied the effects of the windows replacements of the buildings to the levels of the aerosol
concentrations. We demonstrated that the ion beam analysis
combined with electron microscopy is an excellent tool for studying indoor air pollution.
I. Introduction
Several studies reported that aerosol particles have negative impact on
human health [1,2]. The effects are dose dependent. Most of the human
exposure to particles takes place in indoor environment. A workplace is
typically such an environment where people are staying in a limited space
during long-term periods. Because of these facts it is crucial to investigate
the air pollution of the workplaces.
The present study is addressed to compare particle matter levels in three
indoor environments in the Nuclear Research Institute of the Hungarian
Academy of Sciences, Debrecen. We selected three totally different microenvironments for our study.
II. Sampling
Four 48-h long measurement campaigns were carried out in an outdoor
and in three indoor places of the ATOMKI: in the garden, in the laboratory
of the Van de Graaff (VdG) accelerator under the ground level, in the mechanical workshop on the ground level and in the library at the 4th floor.
The 4 sampling periods were the following: 25-27, February, 2009; 19-21,
May, 2009; 22-24, September, 2009; 17-19, February 2010. Samples were
collected only during working hours. All sampling days were working days.
We collected aerosols with 2-stage Nuclepore samplers equipped with 2.5
cm diameter Nuclepore polycarbonate filters. Coarse (aerodynamic diameter between 2.5 µm and 10 µm) and fine (aerodynamic diameter < 2.5 µm)
particles were collected on filters with of 8 µm and 0.4µm pore diameter, respectively. Portable membrane pumps developed in the ATOMKI were used.
The samplings were done with a flow rate of 3-4 l/min. Outdoor samples
were collected parallel to the indoor campaigns.
The library and the workshop were ventilated naturally through doors
and windows. However, all windows were replaced on the buildings of all
sites in August 2009. There is an old air blasting equipment in the VdG
laboratory, but most of the time it is not in use thus the laboratory was
ventilated through the corridors during the campaigns.
III. Analysis
The total mass concentrations were measured by gravimetric methods:
filters were weighed before and after sampling on a microbalance. The aerosol
filters were conditioned at least 24 h before weighing in the weighing box at
room temperature and approx. ∼50% relative humidity.
208
The aerosol samples were analysed by PIXE [3] with a 2 MeV proton
beam, in vacuum, at the 5 MV Van de Graaff accelerator in the IBA Laboratory of ATOMKI [4]. The beam spot size was 0.5 cm diameter and the
beam current was typically 40 nA. The irradiation time of one sample was
approximately 20 minute and the accumulated charge was 40 µC. The facility include a Canberra type Si(Li) X-ray detector with 30 mm2 active area
and with 20 µm Be window. A 24 µm thick mylar absorbent was used to
prevent the detector from the scattered protons.
Concentrations of the following elements were determined: Al, Si, P, S,
Cl, K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, As, Br, Sr, Cd, Ba and Pb.
The X- ray spectra were analysed using the PIXECOM program package
[5,6].
Moreover, elemental and morphological analysis was performed using a
FEI (Focused Ion Beam combined with a Scanning Electron Microscope)
Quanta 3D Dual Beam System [7] at the Department of Materials Physics,
Eötvös University, Budapest. This analysis was carried out on the samples
collected in the mechanical workshop.
IV. Results
IV.1 Particulate matter concentrations
Based on the PM concentrations data (Fig.1.) we found remarkable differences between the indoor places for PM10 (particles with aerodynamic
diameter < 10 m) values. The PM10 concentrations showed bigger variability at the workshop and at the laboratory while at the library it followed the
outer PM10 level. However the PM2.5 values were broadly equal with the
PM2.5 concentrations of the outer air. The fine PM concentrations showed
seasonal variability: higher concentrations were in winter than in summer.
Both the PM10 and the PM2.5 values exceeded the guidelines of the WHO
in several times. (The WHO guideline is 25 µg/m3 and 50 µg/m3 annual
averages for PM2.5 and PM10 respectively)
Since the PM concentration outdoor varied in a wide range we examined
the indoor-outdoor ratios in more detail instead of the indoor concentrations.
209
Figure 1: PM concentrations at the sampling places during the 4 campaign.
IV.2 Indoor-Outdoor ratios
The indoor/outdoor ratio represents [8] the relationship between the indoor and outdoor particle concentrations (Fig.2.) However, it is difficult
to draw uniform conclusions from the I/O ratio, because it may depend on
several factors such as outdoor concentration levels, deposition rate, indoor
particle source emission rate, penetration factor, deposition factor and indoor activities. The I/O ratios, averaged over the 4 campaign showed that
the coarse values were the lowest at the library (0.78) and were higher than 1
for the laboratory (1.38) and for the workshop (1.35). The fact, that the I/O
ratio in the library was lower than 1 can be explained by some factors. The
outdoor concentration of the coarse particle at the level equal with the 4th
floor is usually less than at the ground level. This means that smaller amount
of coarse particles can penetrate form outdoor to the library. In addition,
the activities - which favours to the resuspension and generation of coarse
particles, - were much less in the library than in the other two places. In
the case of the workshop and the laboratory higher I/O ratios was observed
which implies intensified activities and/or indoor emission rates. Since the
210
Figure 2: Indoor/outdoor concentration ratios of PM.
VdG laboratory has not direct connection (through doors or windows) to the
outer air the penetration rate might have been low.
The fine I/O ratio exhibited less variability between the measurement
campaigns and the ratio values were under 1 for the VdG laboratory (0.65)
and for the library (0.81). Nevertheless the fine I/O ratio was higher than
1 for each campaign in the workshop. The fact that each of the fine I/O
ratios did not reach the 1 value in the VdG implies that the fine particles
penetration rate is low due to the lack of the direct doors and windows. Since
the library is naturally ventilated through open windows the penetration
factors of the fine particles should be approximately equal to 1. This was
fulfilled in September 2009, when the weather was warm, and all the windows
were open. The windows were closed when the lower I/O fine ratios were
detected in the library.
211
IV.3 Indoor-Outdoor ratios of elemental concentrations
Coarse and fine I/O ratios of crust related elements are given in the
Table 1. It can be seen that the I/O ratios of the natural origin elements
were different from the I/O ratios of the PM. The I/O ratio of the Si, Al,
Ca, and Ti were approximately 1 or lower except in the workshop. The fact
that the Si, Al, and Ti I/O ratios were higher in the workshop was related to
the activities of the workers. Dominant I/O ratios of Fe and the Mn in the
workshop imply that there were some Fe and Mn emission sources inside.
In the VdG laboratory the I/O of Ca showed remarkably high values
in September 2009. Since the I/O of other crust related elements did not
became significant the resuspended dust raised by human activities could
not be the explanation. Since there was a damaged part of the wall in the
laboratory it is likely that the restoration work using cement were the source
of the Ca. This explains the higher Fe concentrations in this period since
FeS2 (pyrite) often found in cement.
We highlighted coarse and fine I/O ratios of some elements of anthropogenic origin on Table 2. In the VdG laboratory high Zn, S, and Cl levels were detected. The high Zn and S concentration confirmed the usage
of cement since beside the FeS2 Zn is also commonly found in cement in
non-negligible concentrations [9]. It is likely that the reason of the high Cl
concentration were the intense use of a chlorine-containing detergent.
Beside the high I/O ratio of Fe we found that some heavy metals such as
Zn and Cu showed considerable high I/O ratios in the workshop. In addition
immense concentration of W (fine: 111 ng/m3 , coarse: ∼318ng/m3 ) and
some amount of Co (fine: ∼16 ng/m3 , coarse:∼103 ng/m3 ) was also observed
both in the fine and the coarse fraction. These metal elements most probably
originate from the activities related turning and machining of metal tools.
Moreover the ratio of wolfram and cobalt concentration was approximately
equal (0.08-0.11) during the sampling campaigns (except in September when
the Co concentration was under the detection limit) implying the same origin.
Most probably particles with W and Co content were originated from the
abrasion of metal tools since tungsten-cobalt alloy is commonly used in such
tool materials [10].
212
Table 1: Indoor/outdoor concentration ratios of the natural origin elements.
Table 2: Indoor/outdoor concentration ratios of the anthropogenic elements.
In the library the I/O ratio of the anthropogenic elements was usually
213
Figure 3: Average indoor/outdoor ratios of Ti, K and S concentrations before
and after the windows replacements (w.r.).
lower than 1 except some cases. The larger I/O ratio of chlorine was attributed to the effect of detergents. In February 2009 the outstanding sulphur I/O ratio may be explained by the variability of the outer S level related
to the heating season. Probably in the periods before the February campaign
the sulphur level were high and penetrated into the library.
Based on the fine I/O ratios of the elemental concentrations we could
observe the effect of windows replacements in the library and the workshop.
For example we highlighted the Ti, K and S. This case we could assume
that there were no indoor sources of these elements and the resuspension of
fine particles was negligible. The average I/O ratio of the Ti, K and S were
shown on Fig.3.before and after the windows replacements. It can be clearly
seen that the I/O ratios of the observed elements is spectacularly decreased
after the windows replacements. Similar results were found for most of the
other elements too. This could be explained by the good insulation of the
new windows.
214
IV.4. Analysis with Scanning Electron Microscope
In order to get more information about the generation of the particles
sampled in the workshop individual particle analysis were carried out by
scanning electron microscopy. Backscattered electron image of particles containing W, Ti and Co can be seen on Figure 4. Since an alloy containing W,
Co and Ti is commonly used in tools like milling and turning machines these
particles might have been originated from the abrasion of the various tools.
This is confirmed by the fragmented morphology of a w and Co containing
particle shown on figure 5.
High amount of Fe were both in the coarse and in the fine fraction. On
Figure 6 a Fe rich particle (the brighter one) can be seen which morphology
show rotation symmetry. This implies that these kinds of particles might
have been originated from spindle processes. In the fine fraction we found
great amount of Fe particles with spherical shape (see on Fig. 7.). Based
on the morphology these particles may be arisen from high temperatures
processes like welding.
Figure 4: Backscattered electron
image of W-Co-Ti content particles. (These were the bright particles)
Figure 5: Secondary electron image
of a W-Co rich particle.
215
Figure 6: Backscattered electron
image of a Fe rich particle (the
brighter)
Figure 7: Secondary electron image
of a Fe particle.
V. Summary
In this paper we have investigated the coarse and fine mass concentrations
and the elemental composition of aerosols in 3 different microenvironments,
i.e. the library, the mechanical workshop and an accelerator laboratory of the
ATOMKI HAS in Debrecen. The PM10 concentration values varied between
21 and 106 µg/m3 . The smallest values were measured in the library, and it
showed strong correlation with the outdoor PM concentration. We observed
increased PM levels in the workshop which was related to indoor emission
sources and the intense activity of the workers. In this place high amount
of heavy metal containing particles were detected which originated from the
mechanical processes. In the library and the VdG laboratory we found some
indoor sources such as cleaning materials and renovation. We showed that
after replacing the windows with better isolating ones the penetration of
outdoor air decreased.
Acknowledgement
This work was supported by the Hungarian Research Fund OTKA and
216
the EGT Norwegian Financial Mechanism Programme (contract no. NNF78829) and the János Bolyai Research Scholarship of the Hungarian Academy
of Sciences.
This work was carried out on the research infrastructure of the Hungarian
Ion Beam Physics Platform.
The SEM analysis is supported by the European Union and co-financed by
the European Social Fund (grant agreement no. TAMOP 4.2.1/B-09/1/KMR2010-0003).
References
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Diaz- D.Sanchez, S.M.Tarlo, P.B.Williams, J. Allergy Clin. Immun.
114, 1116 (2004).
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Second edition, Marcel Dekker. Inc., 2001)
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[5] Gy. Szabó, and I. Borbély-Kiss, Nucl. Inst. and Meth. B 75, 123 (1993).
[6] I. Uzonyi, and Gy. Szabó, Nucl. Inst. and Meth. B 231, 156 (2005).
[7] http://www.fei.com/products/dualbeams/quanta3d.aspx.
[8] C. Chen, B. Zhao, Atmospheric Environment 45, 275 (2011).
[9] http://en.wikipedia.org/wiki/Cement#cite_ref-27.
[10] J.R. Davis, Tool Materials, (ASM International, 1995).
217
ACTA PHYSICA DEBRECINA
XLV, 218 (2011)
α INDUCED CROSS SECTION MEASUREMENT ON
FOR THE ASTROPHYSICAL γ-PROCESS
169 Tm
T. Szücs
Institute of Nuclear Research (ATOMKI), Debrecen, Hungary
Abstract
There are about 35 isotopes heavier then iron on the proton
rich side of the valley of stability, which can not be produced via
neutron capture reactions. The so-called astrophysical γ-process
is responsible for production of these isotopes. This reaction
chain consists of photodisintegration reactions on pre-existing
seed nuclei. In the heavy mass region (α,γ) reactions have big
impact on γ-process reaction flow, but almost no experimental
cross section data exists in the relevant energy region. We performed a measurement on 169 Tm to get experimental (α,γ) cross
section close to the astrophysicaly relevant energy range to check
and improve the theoretical predictions.
I. Introduction
Nuclei heavier than iron are produced by the three main processes called
s-, r-, and p-process [1]. The first two involve neutron capture reactions
and β-decays. The s refers to slow neutron capture, and the reaction flow
is running close to the valley of stability. The s-process has two sites: low
mass AGB stars for the main s-process [2] and massive stars for the weak
s-process [3]. The r refers to rapid neutron capture, in this case consecutive
neutron captures drive the material far away from the line of stability close
to the neutron drip line. It requires an explosive scenario, presently favoured
candidates are core collapse supernovae [4] and merging neutron stars [5].
There are about 35 isotopes (less than 1 % of total solar abundance beyond
iron) on proton-rich side of the valley of stability, which cannot be produced
via neutron capture reactions. The main fraction of these so-called p-nuclei is
produced in the γ-process [6]. It proceeds by photon-induced reactions starting from pre-existing s- or r-process seed nuclei. First (γ, n) reactions drive
the material towards the neutron deficient region, before charged-particle
emitting reactions like (γ,α) and (γ,p) become possible and divert the reaction flow towards lower masses.
However, astrophysical network calculations cannot fully reproduce the
observed abundances of all p-nuclei within one scenario and call for additional processes [7]. These reaction network calculations involve thousands
of stable and unstable isotopes and tens of thousands of reactions [8]. Only
the minority of these reactions is known experimentally, whereas the cross
sections for the largest fraction have to be inferred from statistical model
calculations.
The fact that self-consistent studies of the γ-process have problems in
synthesizing p-nuclei in the mass regions A < 124 and 150 < A < 165 may
result from difficulties related to astrophysical models as well as from systematic uncertainties of nuclear physics input. Therefore, an improvement
of nuclear reaction cross sections is crucial for further progress in γ-process
models, either by directly replacing theoretical predictions by experimental data or by testing and improving the reliability of statistical models, if
the relevant energy range (so called Gamow window) is not accessible by
experiments.
In case of the production of heavy p-nuclei (140 < A < 200) the reaction
flow is strongly sensitive to (γ,α) photodisintegration rates. If those reaction
rates are high, more material will contribute to the synthesis of lower mass pisotopes. On other hand, if reaction rates are lower, processing toward lower
atomic numbers is weaker, resulting in a relative enrichment in higher mass
p-isotopes such as 174 Hf, 180 W, and 190 Pt [9]. Consequently, to reproduce
the path of the γ-process, experimental data is highly needed in this mass
range.
In principle, photodisintegration cross sections can be determined directly
by photon induced reaction studies [10]. However, in such an experiment the
219
target nucleus is always in its ground state, whereas in stellar environments
thermally populated excited states also contribute to the reaction rate leading to large corrections of the ground state rate which can only be modeled
theoretically [11]. It has been shown that the influence of thermal population
is much less pronounced in capture reactions, therefore it is advantageous to
measure in the direction of capture and convert the measured rate to the rate
of the inverse reaction by applying the principle of detailed balance [12, 13].
II. Activation method
The simplest way to derive capture cross sections is the activation method,
as it was proven in several measurements in the A ≈ 100 mass range [14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24]. Though the activation method proves to be
successful to measure α-induced cross sections it has numerous limitations.
Those reaction can be measured where the product nucleus is radioactive, has
a reasonable half-life (typically from few hours up to a few days) and its decay
is followed by some detectable radiation. Moreover a detector with a good
signal-to-noise ratio is necessary for going to low energies. To compensate
the low cross sections, long half-lives or low γ-branchings a high efficiency
detector must be used. The laboratory background can be reduced using lead
shielding, but beam-induced background on impurities of the target and/or
the backing can still be present. Furthermore, if the reaction product decays
via cascades, in close geometry the true coincidence summing effect has to
be taken into account.
Unstable nuclei around the mass A < 150 on proton-rich side close to
the valley of stability typically decay by electron capture. Such an electron
capture is usually followed by X-ray emission. Since electrons are captured
by a nucleus mainly from inner atomic shells, the decay is followed by X-ray
emission when vacancies are filled by electrons from higher lying shells. This
provides an alternative application of the activation method: measurement
of cross section based on detection of characteristic X-ray radiation.
This approach has numerous advantages: the relative intensities are high
since electron capture is followed dominantly by emission of either a Kα1
or a Kα2 X-ray (the probability of Auger electron or Kβ X-ray emission are
typically an order of magnitude smaller). Furthermore, after the decay of one
220
product nucleus only one Kα characteristic X-ray is emitted, consequently
the target can be placed very close to the surface of the detector without
summing effect. Therefore the detector efficiency can be very high.
Disadvantage of the X-ray counting is, however, that it is not able to
distinguish between decay of the different isotopes of same element. If the
target consists of several isotopes of the same element, then different isotopes
of the final elements will be inevitably produced, separation of which by Xray detection is impossible. Therefore elements having only one single stable
isotope are preferred for this technique, or high isotopic enrichment is needed.
Even if the target is isotopically pure, it is still possible that, e.g., both
(α,γ) and (α,n) reaction channels are open (which is very often the case
for heavy p-isotopes in studied energy range). In this case two isotopes of
same element are produced which need to be separated. This can be done,
if the half-lives of the two isotopes are very different. Moreover, if one of the
two reaction products also emits γ-radiation with sufficiently high intensity,
combination of X-ray and γ-counting can help the separation.
III. Investigated reactions
We have measured (α,γ) and (α,n) reactions on 169 Tm at low energies.
The Gamow window for this reaction at a typical γ-process temperature of
2.5 GK is between 7.2 and 10.2 MeV [25]. Thulium is good material to demonstrate the X-ray counting method. It has only one stable isotope, therefor
enrichment is naturally guaranteed. However 169 Tm(α,n)172 Lu channel is
opened above Eα = 10.4 MeV, making the separation of the two reaction
products necessary. The half-life of 172 Lu is 6.7 ± 0.04 days, in the case of
173 Lu from (α,γ) channel is 500 ± 4 days. Thus, decay of the two isotopes
can be separated using the following procedure. Calculations show that close
above threshold the (α,n) channel becomes dominant, its cross section exceeds by orders of magnitude that of the (α,γ) channel. Taking also into
account the shorter half-life of 172 Lu, the decay spectrum shortly after irradiation will be totally dominated by the decay of 172 Lu. On the other
hand, if the counting is repeated after many half-lives of 172 Lu, its activity
decreases to a negligible level and the decay of the longer-lived 173 Lu can
be measured. Thus, with two countings of the same target separated by a
221
suitably long period both cross sections can be determined.
IV. The experiment
Targets have been made by evaporating metallic thulium onto thin aluminium foils. The mass of the foils had been measured before and after
the evaporation and from the difference (assuming that the evaporated layer
is uniform) the number of target atoms could be calculated. The absolute
number of target atoms and the uniformity have also been determined by
the PIXE method (Proton Induced X-ray Emission) [26] using the Nuclear
Microbeam Facility of ATOMKI. A 2 MeV proton beam provided by the Van
de Graaff accelerator was scanned over a surface of 500 µm × 500 µm at several different positions of the target. The precision of determination of the
number of target atoms was better than 3 % and the thickness was found
to be uniform within 3 %. The agreement between derived thicknesses are
within 4 %.
The 169 Tm targets have been irradiated with α beams from the MGC cyclotron of ATOMKI [27]. The investigated energy range between Eα = 11.5
and 17.5 MeV, was covered with 0.5 − 1 MeV steps with a beam current of
typically 2 µA. After beam-defining aperture, the chamber was insulated.
A secondary electron suppression voltage of −300 V was applied at the entrance of the chamber. Each irradiation lasted between 8 and 24 hours. The
collected charge in each irradiation was between 50 mC and 250 mC.
The X-ray countings have been carried out with a Canberra type GL2015R
LEPS detector. In order to reduce the laboratory background a multilayer
shield has been built around the detector including inner layers of aluminum,
copper and cadmium and a 5 cm thick outer lead shield. The absolute efficiency of the LEPS detector was determined using the 53.16 keV line of 133 Ba
and the 59.54 keV line of the 241 Am calibrated radioactive sources. At higher
bombarding energies (at and above 15 MeV) the γ-activity of the targets has
also been measured with a 40 % relative efficiency HPGe detector, while at
13.5 MeV and 14 MeV a 100 % relative efficiency HPGe detector in Ultra
Low Background (ULB) configuration has been used. The cross section of
the (α,n) reaction obtained from the measurements with the LEPS and the
HPGe detectors were found to be in agreement within 4 %. γ-lines belonging
222
to the 169 Tm(α,γ)173 Lu reaction were not observable in the γ-spectra taken
either with the ULB or the HPGe detectors.
Figure 1: (Color online) Schematic view of the X-ray counting setup.
223
V. Results
Tables 1 and 2 are summarising the derived cross sections and S-factors.
In those cases where two irradiations at same energy were carried out, consistent results have been obtained. At these energies the average cross section
values weighted by the statistical uncertainty are given in table. The quoted
uncertainty in EC.M. values corresponds to the energy stability and determination of the α-beam and to uncertainty of energy loss in the target.
Uncertainty of the cross section is quadratic sum of the following partial
errors: efficiency of the HPGe detector and LEPS (6 and 4 %, respectively),
number of target atoms (4 %), current measurement (3 %), uncertainty of
decay parameters (≤ 5 %) and counting statistics (0.5 − 7 %).
VI. Summary
In the present work the cross section of the 169 Tm(α,γ) and 169 Tm(α,n)
reactions were determined well below the Coulomb barrier with high precision (≤ 10 % total uncertainty) in spite of the very long half-life (500 and 6.7
d) of the reaction products by counting the characteristic X-ray radiation. It
has to be emphasized that this method is less sensitive to the usual obstacles
of activation technique such as low γ branching ratios or long half-lives.
224
Table 1: Experimental cross sections and S-factor of the 169 Tm(α,n)172 Lu
reaction.
Elab
EC.M.
σ
S-factor
[MeV]
[MeV]
[µbarn]
[1028 MeV barn]
11.5 11.21±0.06
0.57±0.06
65±7
11.85 11.55±0.06
1.49±0.16
53±6
12.2 11.90±0.06
4.1±0.5
46±5
12.5 12.19±0.06
8.0±0.7
37±3
13.5 13.16±0.07
96±8
26±2
14.0 13.66±0.07
261±20
18.6±1.4
15.0 14.63±0.08
1840±145
12.0±0.9
15.5 15.12±0.08
4471±366
9.5±0.8
16.0 15.61±0.08
8279±653
6.0±0.5
16.5 16.10±0.08 17075±1246
4.5±0.3
17.0 16.59±0.08 32707±2415
3.3±0.2
17.5 17.08±0.09 56436±4176 2.23±0.17
Table 2: Experimental cross sections and S-factor of the 169 Tm(α,γ)173 Lu
reaction.
Elab
EC.M.
σ
S-factor
25
[MeV]
[MeV]
[µbarn]
[10 MeV barn]
13.5 13.16±0.07 1.28±0.13 347±35
14.0 13.66±0.07
2.6±0.2
184±14
15.0 14.63±0.08
8.1±0.8
53±5
15.5 15.12±0.08 14.6±1.5
31±3
16.0 15.61±0.08 22.0±1.7
16.1±1.2
16.5 16.10±0.08
30±2
7.9±0.6
17.0 16.59±0.08
41±3
4.1±0.3
17.5 17.08±0.09
61±5
2.40±0.18
225
References
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(2000)
[4] Y.-Z. Qian Astrophysical Journal 534, L67 (2000)
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[6] S.E. Woosley, and W.M. Howard Astrophysical Journal Supplement Series 36, 285 (1978)
[7] M. Arnould, and S. Goriely Physics Reports 384, 1 (2003)
[8] M. Rayet et al. Astronomy & Astrophysics 227, 271 (1990)
[9] W. Rapp, J. Görres, M. Wiescher, H. Schatz, and F. Käppeler Astrophysical Journal 653, 474 (2006)
[10] C. Nair et al. Journal of Physics G: Nuclear and Particle Physics 35,
014036 (2008)
[11] P. Mohr, Zs. Fülöp, and H. Utsunomiya European Physical Journal A
32, 357 (2007)
[12] G. Gy. Kiss et al. Physical Review Letters 101, 191101 (2008)
[13] T. Rauscher et al. Physical Review C 80, 035801 (2009)
[14] E. Somorjai et al. Astronomy & Astrophysics 333, 1112 (1998)
[15] Gy. Gyürky et al. Physical Review C 64, 065803 (2001)
[16] S. Galanopoulos et al. Physical Review C 67, 015801 (2003)
226
[17] Gy. Gyürky et al. Physical Review C 68, 055803 (2003)
[18] Gy. Gyürky et al. Physical Review C 74, 025805 (2006)
[19] Gy. Gyürky, G. Gy. Kiss, Z. Elekes, Zs. Fülöp, E. Somorjai, T. Rauscher
Journal of Physics G: Nuclear and Particle Physics 34, 817 (2007)
[20] G. Gy. Kiss et al. Physical Review C 76, 055807 (2007)
[21] N. Özkan et al. Physical Review C 75, 025801 (2007)
[22] C. Yalçın et al. Physical Review C 79, 065801 (2009)
[23] Gy. Gyürky et al. Journal of Physics G: Nuclear and Particle Physics
37, 115201 (2010)
[24] G. Gy. Kiss et al. Physics Letters B 695, 419 (2011)
[25] T. Rauscher Physical Review C 81, 045807 (2010)
[26] S. A. Johnson Particle Induced X-ray Emission Spectroscopy (J. Wiley
& Son, 1995)
[27] http://www.atomki.hu/atomki/Accelerators/Cyclotron/mgc20.html
227
ACTA PHYSICA DEBRECINA
XLV, 228 (2011)
FRAGMENTATION OF PLASTIC MATERIALS
G. Timár1 , F. Kun1 , J. Blömer2 , and H. J. Herrmann3
1
2
Department of Theoretical Physics, University of Debrecen,
P.O. Box 5, H-4010 Debrecen, Hungary
Spezialwerkstoffe, Fraunhofer UMSICHT Osterfelder Str. 3,
46047 Oberhausen, Germany
3
Computational Physics IfB, HIF, ETH Hönggerberg,
8093 Zürich, Switzerland
Abstract
An experimental and theoretical study of the fragmentation of
plastic materials is presented. In the experiments Polyoxymethylen
particles of spherical shape were fragmented by impacting them
against a hard wall. The experiments revealed a power law distribution of fragment masses similar to heterogeneous brittle materials, however, with a significantly lower exponent. To understand the experimental findings a three dimensional discrete element model is introduced. The model discretizes the sample in
terms of randomly sized spheres which are connected by elastic
beams. To capture the effect of plasticity a novel type of failure
criterion is introduced complemented by a healing mechanism
of broken particle contacts under compression. Computer simulations show a good quality agreement with the experimental
findings.
I. Introduction
Fragmentation of heterogeneous materials is a very complex scientific
problem with an enormous technological importance [1, 2, 3] . From the
usage of explosives in mining through the comminution of minerals to the
liberation of grains in particle composites fragmentation processes play a crucial role which calls for a thorough understanding. During the last decade research efforts have been concentrated on the breakup of heterogeneous brittle
materials which is by now fairly understood [1, 2, 3, 4, 5, 6, 7]. Experimental
and theoretical investigations have revealed that the energy imparted to the
solid has to surpass a threshold value (critical energy) to achieve complete
breakup [4, 5, 6, 7, 8, 9]. In this fragmented state the mass (size) distribution of pieces follows a power law functional form with universal exponents
depending mainly on the effective dimensionality of the system [1, 3, 4, 6].
The branching-merging scenario of dynamically propagating cracks provided
a qualitative physical picture underlying the universality [1].
Industrial processes require also the fragmentation of polymeric materials
which exhibit ductile fracture. For polymers a complex deformation state
may arise before breakup which leads to a more complicated crack initiation
and propagation compared to brittle materials. In spite of its industrial
relevance and scientific importance, the breakup of polymeric materials is
still poorly understood. In the present project we carried out a detailed
experimental and theoretical investigation of the impact fragmentation of
polymers which revealed a broad spectrum of novel features.
II. Results
We carried out experiments by impacting spherical particles made of Polyoxymethylen (POM) against a hard wall. A single particle comminution device was used which accelerates the particles in a rotor up to the desired
speed. Our device ensures normal impact on a hard wall in an evacuated
environment. Particles of diameter d = 5 mm were fragmented varying the
impact velocity v0 in the range 30m/s - 180m/s. Experiments showed that
to achieve breakup of the POM particle the impact velocity has to exceed
a critical velocity vc which was estimated to be vc ≈ 60m/s. Below vc the
particles suffered large permanent deformation with a few cracks opposite to
the impact side, however, the sphere kept its integrity.
In the fragmented regime v0 > vc we evaluated the mass distribution of
fragments of POM spheres by scanning the fragments with an open scanner.
229
This way the fragment identification was reduced to searching white spots
on a black background in a black-and-white digital image. Figure 1 shows
that the measured mass distribution of fragments F (m) has a power law
functional form F (m) ∼ m−τ over three orders of magnitude followed by
an exponential cutoff due to the finite particle size. The most interesting
outcome of the experiments is that the exponent of the power law has a
unique value τ = 1.2±0.06 significantly smaller than the exponents obtained
for brittle materials τ = 1.8 − 2.1 [1, 2, 3, 4, 5, 6, 7].
Figure 1: The mass distribution of fragments obtained in the experiments
can be well fitted by a power law over three orders of magnitude. The slope
of the straight line is τ = 1.2. Computer simulations provide a reasonable
agreement with the experimental findings.
In order to understand the physical mechanism which gives rise to the
novel exponent, we carried out computer simulations of the impact process
using a three-dimensional discrete element model. In the model the spherical
sample is represented as a random packing of spheres connected by elastic
beams. In 3D the total deformation of a beam is calculated by the superposition of elongation, torsion, as well as bending and shearing in two different
230
planes. The unique deformation behavior and the fracture of plastic materials is captured by introducing two novel components in the model: The
beams break when they get over-stressed similarly to other DEM models,
however, breaking is allowed also under compression due to shear deformation. Additionally, we assume that broken contacts can get reactivated when
compressed over a sufficiently long time. This healing mechanism leads to
plastic behavior of the material with the possibility of permanent deformation [8, 9].
Final states of the impact process obtained by computer simulations are
presented in Fig. 2 for two different impact velocities v0 below and above the
critical velocity of fragmentation vc . It can be observed that below vc (Fig.
2(a)) the impact induces a large permanent deformation of the particle, but
no cracks appear. However, above vc complete breakup is achieved into a
large number of pieces (Fig. 2(b)). Plastic deformation implies that a large
fraction of the imparted energy is dissipated which cannot contribute to
fragmentation. It has the consequence that the fragments in Fig. 2(b) attain
a surprisingly low speed by the end of the breakup process. The fragment
mass distribution obtained by computer simulations are compared to the
experimental findings in Fig. 1, where a nice agreement is evidenced [8, 9].
Figure 2: Final states of computer simulations of the impact process at different impact velocities. In (a) the impact velocity v0 falls below the critical
velocity of breakup v0 < vc so that large permanent deformation remains
without any apparent damage. (b) Above vc fragmentation is achieved into
a large number of pieces. Fragments are highlighted by different colors.
III. Summary
231
We carried out experiments on the fragmentation of polymeric materials
by impacting spherical particles made of Polyoxymethylen against a hard
wall. The experiments revealed a power law behavior of the mass distribution of plastic fragments with a unique exponent different from the one of
heterogeneous brittle materials.
A 3D Discrete Element Model was used to obtain a deeper understanding
of the fragmentation of plastic materials. The sample was discretized in
terms of spherical particles connected by elastic beams. To capture the
effect of plastic deformation a healing mechanism is introduced for beams
at particle contacts under compression. The breaking rule of beams ensures
the dominance of shear in crack formation.
Computer simulations provide a good quality description of the experimental findings, i.e. at low impact velocities the sample suffers permanent
deformation but does not break. Fragmentation is achieved at high enough
velocities where the exponent of mass distribution has an excellent agreement
with the measured value. Our experimental and theoretical work demonstrate that the fragmentation of plastic materials define a novel universality
class of fragmentation phenomena [8, 9].
References
[1] J. A. Aström, Statistical models of brittle fragmentation, Adv. Phys. 55,
247 (2006).
[2] D. L. Turcotte, Factals and Fragmentation, J. Geophys. Res. 91, 1921
(1986).
[3] L. Oddershede, P. Dimon, and J. Bohr, Self-organized criticality in fragmenting, Phys. Rev. Lett. 71, 3107 (1993).
[4] H. Katsuragi, S. Ihara, and H. Honjo, Explosive fragmentation of a thin
ceramic tube using pulsed power, Phys. Rev. Lett. 95, 095503 (2005).
[5] F. Kun and H. J. Herrmann, Transition from damage to fragmentation
in collision of solids, Phys. Rev. E 59, 2623 (1999).
232
[6] F. K. Wittel, F. Kun, H. J. Herrmann, and B.-H. Kröplin, Fragmentation of shells, Phys. Rev. Lett. 93, 035504 (2004).
[7] F. Kun, F. K. Wittel, H. J. Herrmann, B.-H. Kröplin, and K. J. Maloy,
Scaling behaviour of fragment shapes, Phys. Rev. Lett. 96, 025504
(2006).
[8] G. Timár, J. Blömer, F. Kun, and H. J. Herrmann, New Universality
Class for the Fragmentation of Plastic Materials, Phys. Rev. Lett. 104,
095502 (2010).
[9] G. Timár, F. Kun, and H. J. Herrmann, Fragmentation of Plastic Materials, submitted to Phys. Rev. E.
233
ACTA PHYSICA DEBRECINA
XLV, 234(2011)
INVESTIGATION OF ARMCHAIR HEXAGON CHAINS BY
EXACT METHODS
R. Trencsényi
Department of Theoretical Physics, University of Debrecen
H-4010 Debrecen, Hungary
Abstract
Exact ground states for the armchair hexagon chains have
been studied, and the physical properties of the ground state
have been analysed, as well.
I. Introduction
Quasi-1D chains building up from hexagons periodically can classify to
the conducting polymers. The hexagons can join each other in many different
ways, so various types of chains can occur. In them we have already studied
in details the zig-zag hexagon chains [1], and in the latter period we have
investigated the armchair hexagon chains [2]. While in the case of the zig-zag
chains a primitive cell consists of only one hexagon, the primitive cell of the
armchair chains includes two hexagons. Hence, one must treat much more
parameters and equations, so the problem of armchair chains is much more
difficult and complicated than of the zig-zag chains. It was an interesting
challange, and that was one of the motivations to deal with the armchair
chains extensively and seriously.
The second reason of the motivations is that hexagon chains are very
important because they can be found in the nature high often. Among the
representatives of short chains of armchair type e.g. phenanthrene (C14 H10 )
or chrysene (C18 H12 ) can be mentioned. Both belong to the group of the
polycyclic aromatic hydrocarbons. Phenanthrene is constituted by three
attached benzene rings. For example, the natural opiates possess phenanthrene structure, furthermore phenanthrene in its pure form can be found in
cigarette smoke or in a crystal named ravatite, as well. Chrysene consists of
four fused benzene rings. It is a natural component of coal tar from which it
was first isolated. It can be also found in creosote which is a chemical used
to preserve wood. Nevertheless, chrysene is applied in the manufacture of
some dyes. Longer chains built up by many hexagons can be denominated
by the prefix “poly”, as e.g. polyphenanthrene or polychrysene.
In this paper we focus on the ground states of armchair hexagon chains.
In these quantum-mechanical systems can be present a large number of electrons, and among them short-range Coulomb repulsion exists. Therefore, the
correlation effects relating to the electrons can not be neglected during the
calculations. Consequently, in order to describe the system genuinely, one
needs to consider real correlation effects. The best way to have respect for
correlations is making use of exact methods. Our exact calculation technique
is based on the properties of positive semidefinite operators [3].
Henceforth, the paper is systematized as follows: Sec. II. presents the
studied system and the calculation method based on positive semidefinite
operators. Sec. III. discusses the deduced results regarding the ground
state. Finally, in Sec. IV. the correlation effects are examined in details.
II. The system and the used method
II.1 The Hamiltonian of the system
Fig. 1 shows schematicly the frame of the armchair hexagon chain. It can
be seen that one cell of the chain is built up by two neighbouring hexagons,
and accordingly contains eight sites.
The Hamiltonian, Ĥ = Ĥ0 + ĤU of the system is given by
XX
XX
X
Ĥ0 =
i n̂i,σ +
(ti,i0 ĉ†i,σ ĉi0 ,σ + H.c.), ĤU = U
n̂i,↑ n̂i,↓ , (1)
σ
i
σ
i,i0
i
where Ĥ0 provides the kinetic part, while ĤU represents the potential part
of the Hamiltonian. ĉ†i,σ creates an electron on the site i with spin projection
235
Figure 1: The armchair hexagon chain.
σ. n̂i,σ the particle number operator which counts the electrons appearing
on the site i with spin projection σ. In constructing Ĥ0 , besides the local onsite potentials i , we have taken into account only nearest and next nearest
neighbour hopping matrix elements denoted by ti,i0 . ĤU arises from the
electron-electron interaction, and by the Hubbard model it considers only
local Coulomb repulsion between the particles. Because of the repulsive
character of the interaction the coupling constant U must be positive, U > 0.
II.2 The applied method
Our calculation procedure is based on the characteristics of the positive
semidefinite operators. These operators are special because their spectrum
(the set of their eigenvalues) has a well-defined lower bound which is a known
number, namely the zero. Even this circumstance makes the possibility to
deduce the ground state wave function of the chain in exact terms. The substance of the method is decomposing the Ĥ Hamiltonian of the system into a
sum of positive semidefinite operators (ĤP ), i.e. Ĥ = ĤP + C, where C is a
constant. Then, acting with this transformed Hamiltonian onto an arbitrary
|Ψi wave vector from the Hilbert space, one obtains Ĥ|Ψi = ĤP |Ψi + C|Ψi.
Since the ground state (with minimal energy) is investigated in the system,
at this level one must choose the zero from the spectrum of ĤP , whereupon
the ĤP |Ψi = 0 condition holds. Thus, finding the most general wave function which satisfies the ĤP |Ψi = 0 requirement, that |Ψi will be the ground
state wave function of the system, i.e. |Ψi = |Ψg i, and in that case the C
constant will provide the Eg ground state energy of the system, i.e. C = Eg .
The decomposition of the Hamiltonian can be carried out in several different ways, since the form of the transformed Hamiltonian depends on the
concrete structure of the positive semidefinite operators appearing in the decomposition. In this manner, one can get to different domains of the phase
236
space. The obtained solution regarding the ground state will be valid only
for that domain of the phase diagram where the decomposition stands.
The above written method is very useful and advantageous because it
is absolutely independent from the dimension and the integrability of the
system.
III. The obtained results relating to the ground state
The ground state wave function of the system has the form
Y
†
|Ψg i =
D̂µ,σ
|0i,
(2)
µ
†
where |0i is the bare vacuum state, and D̂µ,σ
are linear combinations of
P
†
fermionic creation operators, i.e. D̂µ,σ
= i yi ĉ†i,σ , where yi are numerical
†
operators are socoefficients belonging to the sites of the chain. The D̂µ,σ
called extended operators which
means that every single site of the chain
P
†
operators act on the
gives a contribution to the i yi ĉ†i,σ sum. So the D̂µ,σ
†
complete extent of the chain. One D̂µ,σ
operator places one electron in the
system with σ spin. This electron can appear in principle on any sites of the
chain, and |yi |2 characterizes the probability that the electron emerges even
on the site i.
†
operators, the ground state is
Due to the extended nature of the D̂µ,σ
conducting. Our further calculations regarding the chemical potential shows
that charge gap is not present. This result confirms the fact that a real
conducting ground state exists in the system.
In addition, based on our calculations, we were led to the conclusion that
the ground state is a saturated ferromagnetic state. This statement follows
†
from the fact that the D̂µ,σ
operators in (2) create electrons in the chain
with the same spin index σ, thus each electron is invested with the same
spin projection. The formation of this ferromagnetic state arises from the
circumstance that the extended operators from (2) can act on only a narrow
strip determined by the geometry of the quasi-1D chain, i.e. all extended
operators are jammed into the same limited domain traced out by the chain.
237
†
Therefore, the D̂µ,σ
operators satisfy so-called connectivity conditions which
means that they are in contact with each other on certain sites. Accordingly,
if e.g. two operators touch each other, then two electrons could appear on
the same site. Consequently – via the interaction between the two electrons –
the total energy of the system would increase. But, to avoid this energy rise,
the system eliminates the double occupied sites in that way that the spin
indices of the electrons become correlated. So, all spin index will be fixed
to the same value, effectuating in this manner the saturated ferromagnetic
state.
These results are valid in the low concentration limit, on such a domain of
the parameter space which is defined by the rough value 1/16 of the charge
of system.
IV. Bulk states versus edge states ?
Fig. 2 depicts the external (edge) and the internal (bulk) sites of the
hexagon chain. External sites are those that have not any contact with the
other hexagons, and the internal sites are present at the touching points of
the hexagons.
Figure 2: The positions of the edge and bulk sites.
In preceding theoretical studies [4], which concentrated on hexagonal armchair ribbons (namely systems built up by juxtaposed hexagon chains in a
plane), the bulk and edge states were treated separately at the level of meanfield approximation. As a result, a ferromagnetic state was obtained which
238
was described as an edge state. So it was established that the bulk states
do not participate in creating the ferromagnetic behaviour in the ribbon.
On the grounds of these approximative results we have analysed the relation
of the bulk and the edge states at exact level. We have searched for the
answer for that question whether bulk states do not play really any role in
generating the ferromagnetic state ? In order to find the answers, first of all
one must calculate the emergence probabilities of the bulk and edge electron
states, then the correlation effects must be closer examined.
†
Based on (2), the D̂µ,σ
operators lead the electrons into the system.
Hence, the calculation of the emergence probabilities of particles described
†
on different sites happens by the help of the coefficients yi
by the same D̂µ,σ
as follows:
P
P
2
2
i∈IE |yi |
i∈IB |yi |
P
PE = P
,
P
=
,
(3)
B
2
2
i |yi |
i |yi |
where PE and PB the emergence probabilities of the edge and bulk electrons,
respectively. IE and IB denote domains for the edge and bulk sites in the
chain so that IE includes the external sites, while IB contains the bulk
sites. Our results show that the PE /PB ratio is of order PE /PB ∼ O(1)
†
operators which proves that the bulk and edge
by averaging on some D̂µ,σ
sites contribute in the same measure to developing the electron states in the
chain.
The ferromagnetic nature of the ground state – as written in Sec. III. –
is physically caused by the connectivity condition of the extended operators.
†
operators from (2) plays a
Accordingly, the overlap between different D̂µ,σ
very stressful part in the study of correlation effects. These overlaps can be
described by the
C(D̂i† , D̂j† )
=
|h0|D̂i D̂j† |0i|
Nc
(4)
functions, where Nc denotes the number of cells in the chain. Performing
the calculations by (4), one is led to the conclusion that in several C(D̂i† , D̂j† )
only edge states are present, but there are overlaps, as well, in which bulk
states exist solely. Since all extended operators live at the same time in the
system thus beside the edge states the bulk ones are at least so important
in the ground state.
239
Our exact results verify that in the evolving of the deduced itinerant
ferromagnetic ground state the bulk and the edge electron states can not be
dissociated, nothing can explain to treat them separately because neither is
favoured by any conditions.
References
[1] R. Trencsényi, E. Kovács, Z. Gulácsi, Phil. Mag. 89, 1953 (2009).
[2] R. Trencsényi, Z. Gulácsi, Eur. Phys. Jour. B75, 511 (2010).
[3] Z. Gulácsi, D. Vollhardt, Phys. Rev. Lett. 91, 186401 (2003).
[4] A. R. Akhmerov, C. W. J. Beenakker, Phys. Rev. B77, 085423 (2008).
240
ACTA PHYSICA DEBRECINA
XLV, 241 (2011)
OPTIMIZATION OF DETECTOR PERFORMANCE IN
RADIOACTIVE BEAM EXPERIMENTS VIA GEANT4
SIMULATIONS
Zs. Vajta and Zs. Dombrádi
Institute of Nuclear Research of the Hungarian Academy of Science, P. O. Box 51,
Debrecen, H-4001, Hungary
Abstract
As a preparation for search for new doubly magic nuclei in
nuclear reactions induced by radioactive beams GEANT4 simulations have been performed to study the components affecting
the resolution of the γ-ray detectors. Effects of the interaction of
the high energy γ-ray with the detector material, internal resolution of the detectors, Doppler broadening due to the finite size of
the detectors, slowing down of bombarding particle in the target
material and finite size of the target has been investigated. It
was found that the detector size and the target thickness are the
crutial factors which determine the quality of the spectra.
I. Introduction
In modern radioactive beam experiments high speed (u ≈ 0.2 − 0.5c)
radioactive ions interact with the target and the reaction products emit γrays in flight. Due to the high speed of the emitting ions the frequency of the
detected γ-ray depends on the speed of the ion and on the angle of detection
similarly to the case of the change of the wavelegth of the light waves emitted
by a moving object:
s
1 + uc
λo = λs
,
(1)
1 − uc
where λo and λs are wavelengths of observer and source, u is the relative
velocity of source and observer and c is the speed of light. This is called the
Doppler effect. The detection angle is well defined in the case of a point like
source and a point like detector, but having finite size detectors and target
γ-rays of different energy will inping into the detector resulting in a widening
of the γ lines. Similarly, due to the slowing down of the bombarding and
product nuclei in the target material a spread in the speed of the emitting
sources will be produced, which also adds to the Doppler broadening of the
peaks.
Since the radioactive beam experiments are very expensive, the optimal
conditions in efficiency, production rate and peak to background ratio determined by the resolution of the detectors have to be find off-line via simulation
of the experimental conditions. In the following we report on a search for
optimal conditions of detection of a high energy γ-ray emitted by a high
speed heavy ion.
II. The experimental conditions used in the simulation
In the simulation we assumed fully stripped ions
200 MeV*A energy.
with Z=50 and of
The target is liquid hydrogen with the following parameters: density =
0.0715 g/cm3 , atomic mass = 1.01 g/mole and atomic number = 1. The
shape of the target is a cylinder and its radius is 1.4 cm. The target thickness,
corresponding to the height of the cylinder, was varied during the simulation.
In the simulation the DALI2 γ-detector array of RIKEN Nishina Center [1] was used as the detecting device. The DALI2 array contains 160 NaI
crystals which almost completely (in nearly 4π solid angle) surround the target. The array consist of brick shape detectors with 8.3 cm*4.15 cm*2.25 cm
side lengthes each. The detectors are mounted on thin aluminum sheets in
a petal shape geometry shown in Figure 1.
Each detector unit is built up of three parts: the core of the detector,
made of a NaI crystal [2], and two covering layers. They are made of magnesium oxide and aluminum, respectively. The middle layer has a thickness
242
Figure 1: The geometry of the DALI2 γ-detector array with the liquid hydrogen target in the center of the array.
of 140 µm. It and the crystal are covered by an aluminum box, the wall
thickness of which is 120 µm.
In the simulation the aluminum frames which hold the detectors were
also included. In addition, a system of aluminum tubes through which the
target can be inserted into and removed from the experimental area were
also considered.
During the interacion of the target and beam nuclei γ rays of well defined
energy will be emitted. Due to the statistical processes in the detection the
detector has a Gaussian shaped response function with a finite resolution.
The detector resolution depends on the γ-ray energy (the higher the energy
the greater the broadening). In the case of the NaI scintillators the energy
dependence follows a square root law [4]. The energy dependence can be
tuned to the actual setup via one parameter (Pres ):
p
σ = Pres Eraw ,
(2)
where Eraw is the detected γ-ray energy, and σ is the width parameter of
the Gaussian line shape. Using the Pres parameter the resolution was tuned
243
to 6% of the γ-ray energy.
The Gaussian line shape in the simulation is achieved through a Monte
Carlo method. First a random real number A is generated between -1 and
1, than the following factor is calculated:
r
log(A2 )
F = −2
.
(3)
A2
F ensures that the energy deviation calculated below will have a normal
distribution. The energy deviation, ∆E is calculated as:
∆E = σF A,
(4)
where A together with the nominator in Eq. 3. decides the sign of ∆E.
Finally, ∆E is added to Eraw :
Esmo = Eraw + ∆E,
(5)
where Esmo is the γ-ray energy corrected for the resolution of the detecting
setup.
The different detectors are placed at different angles in the DALI2 arrangement. As a consequence, each detector sees γ-rays of different energy
due to the Doppler effect. Since this effect is well known, we can take it into
account and reconstruct the events.
The Doppler correction is performed in a relativistic way:
Edop = Esmo (γ(1 − βcor cos(θ))),
(6)
1
γ=p
,
2
1 − βcor
(7)
where γ is the Lorentz factor and β = v/c. βcor is the speed of the emitting
ion corrected for slowing down in the target.
244
Figure 2: Spectrum obtained with point like target and detector.
III. Results of the simulation with a point like target
III.1 Interaction of the gamma rays with the detector material
In the simulation 3 different interactions of the γ-rays with the detector
material were taken into account:: the photo effect, the Compton effect and
the pair creation, which is important for high energy gamma rays. Assuming
a point like detector, 3 sharp lines at the photo energy and at the single and
double escape peak positions will be produced. This is shown in Figure 2.
III.2 Effect of the finite size of the detector
In reality we have finite size detectors. As it was mentioned above, the
finite size of the detectors results in a Doppler broadening of the peaks. In
the simulation γ rays are emitted in all direction with isotropic distribution
using a random number generator (The relativistic corrections disturb this
distribution and produce a forward peaked distribution, which is handled
by the program.). γ rays of an appropriate solid angle are collected by
each detector, but the Doppler correction is performed for the center of this
solid angle. The farther the γ rays hit the detectors, the worse the Doppler
correction will be. The approximate knowledge of the angle of detection of
245
Figure 3: The same spectrum as above using the DALI2 detector array.
the γ ray results in an approximate Doppler correction, and as a consequence
into broadening of the photo and escape peaks. A realistic spectrum from a
thin target can be seen in Figure 3.
As we can see in Figure 3, the photo and the two escape peaks overlap with the Compton edge and form a very broad peak with a waving top
structure. Such a wide structure hardly can be used for spectroscopic studies. Due to the overlapping position of the different detector layers in the
DALI2 geometry a large quantity of the escaping radiations from one detector are detected by an other one. Thus, disclosing events where multiple
detectors are firing an anti-Compton like mode of operation of the array
can be achieved. A dramatic improvement of the spectrum can achieved by
requiring multiplicity 1 in the simulation. This is shown in Figure 4. It is
worth to mention that in the special case of having a single γ ray, although
we loose a lot of events, the efficiency of the photo peak detection remains
the same.
IV. Simulations with a finite size target
IV.1 Slowing down of the ions
Slowing down of the bombarding ion is a statistical process itself. The
246
Figure 4: DALI2 spectrum restricted to multiplicity 1.
slower the ion becomes, the more complicated the process is. Since we are
interested in passing of high speed ions through the target, instead of the
statistical simulation of the slowing down a simplified solution has been used
to keep the calculations within a reasonable time limit.
The energy loss was calculated using the LISE code [3] and is presented
in Figure 5 as a function of the target thickness. It can be seen that the
slowing down is a rather smooth process as long as the energy of the ion is
high enough, and the energy loss is quickly increasing when we are close to
the penetration depth.
In the simulation it was assumed that the reaction takes place with equal
probability at any depth in the target. The energy loss for the depth of
emission generated by a random number generator was determined from the
energy loss curves shown in Figure 5.
IV.2 Effect of the slowing down of the projectile
Increasing the thickness of the target, the bombarding ions are slowed
down more and more. Thus, the radiations emitted at different stages of the
slowing down process will suffer different Doppler shifts. In the simulation
the Doppler correction is performed for an average speed. For this reason,
247
Figure 5: Energy loss of Z=50 ions in liquid hydrogen target calculated for
different incident energies Ebeam .
the thicker the target is, the less precise will the correction be. The effect of
the finite target thickness is presented in Figure 6.
One can see in Figure 6 that the quality of the spectrum remains reasonable up to about 400 mg/cm2 target thickness. To achieve the highest yield
with 200 MeV/u ions this target thickness is the optimal.
IV.3 Effect of the target size
Increasing the size of the target can cause a further widening of the peaks,
since the reaction can take place away from the center of geometry of the
arrangement, and as a consequence, the angle of detection will be different
compared to the value assumed in the Doppler correction. This effect has a
visible effect only in the case of very long (longer than 20 cm) targets, which
is not the case in our arrangement.
V. Conclusion
Different factors affecting the resolution of a given γ-ray detection setup
such as the Doppler broadening due to the finite size of the detectors, slowing down of bombarding particle in the target material and finite size of
248
Figure 6: Calculated spectra for the beam energy Ebeam =200 MeV/u, and
target thicknesses 100, 400, 500 and 600 mg/cm2 .
the target has been examined. It was found that restricting the spectra to
multiplicity 1 events, spectra of reasonable quality can be obtained even for
high energy γ-rays. Although, the resolution of the NaI scintillators is rather
limeted, the width of the photopeak is determined not by this factor, but by
the Doppler broadening of the peaks due to the size of the detector elements.
To achieve the best possible resolution a new detector geometry has to be
found.
References
[1] S. Takeuchi et al., RIKEN Accel. Prog. Rep. 36, 148 (2003)
[2] W. R. Leo, Techniques for Nuclear and Particle
Experiments(Springer-Verlag, Berlin, Heidelberg,1994).
Physics
[3] O. Tarasov and D. Bazin, NIM B 266, 4657 (1995)
[4] M. Moszyński et al., IEEE Transactions on Nuclear Science 45, NO. 3
(1998)
249