By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors
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The document discusses the application of the anharmonic correlated Einstein model to derive expressions for cumulants and thermodynamic parameters in cubic crystals using new structure factors. It highlights the importance of incorporating anharmonic effects, particularly at high temperatures, to achieve accurate calculations of x-ray absorption fine structure (XAFS) spectra. The study presents analytical formulas for cumulants and demonstrates their alignment with structural contributions from various cubic crystal configurations.
![By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new
structure factors
JPAR
By using the anharmonic correlated einstein model to
define the expressions of cumulants and
thermodynamic parameters in the cubic crystals with
new structure factors
Nguyen Ba Duc
Chancellor of Tan Trao University, Tan Trao University, Tuyen Quang City, Vietnam
Email: hieutruongdhtt@gmail.com, Tel.: +84 273890012
By using potential effective interaction in the anharmonic correlated Einstein model on the
basis of quantum statistical theory with phonon interaction procedure, the expressions
describing asymmetric component (cumulants) and thermodynamic parameters including
the anharmonic effects contributions and by new structural parameters of cubic crystals
have been formulated. These new parameters describe the distribution of atoms. The
expansion of cumulants and thermodynamic parameters through new structural
parameters has been performed. The results of this study show that, developing further the
anharmonic correlated Einstein model it obtained a general theory for calculation
cumulants and thermodynamic parameters in XAFS theory including anharmonic
contributions. The expressions are described through new structural parameters that agree
with structural contributions of cubic crystals like face center cubic (fcc), body center cubic
(bcc).
Keywords: Anharmonic XAFS, cumulants, thermodynamic parameters.
INTRODUCTION
In the harmonic approximation X-ray Absorption Fine Structure spectra (XAFS), the theoretical calculations are
generally well appropriate with the experimental results at low temperatures, because the anharmonic contributions
from atomic thermal vibrations may have been neglected. However, at the different high temperatures, the XAFS
spectra provide apparently different structural information due to the anharmonic effects, and these effects need to
be evaluated. Furthermore, the XAFS spectra at low temperatures may not provide a correct picture of crystal
structure. Therefore, this study of the XAFS spectra including the anharmonic effects at high temperatures is
crucially needed. The expression of anharmonic XAFS spectra often is described by:
,
!n
ik2
ikR2expeIm
kR
]k/R2exp[
kFk
n
n
n
ki
2
(1)
where )k(F is the real specific atomic backscattering amplitude, )k( is total phase shift of photoelectron, k is
wave number, is mean free path of the photoelectron, and
n
,...3,2,1n are the cumulants to describe
asymmetric components. They all appear due to the thermal average of the function
ikr2
e
, in which the asymmetric
terms are expanded in a Taylor series around value rR with r is instantaneous bond length between
absorbing and backscattering atoms at T temprature and then are rewritten in terms of cumulants.
At first, the cumulant expansion approach has been used mainly fitting the XAFS spectra to extract physical
parameters from experimental values. Therefore, some procedure were formulated for the purpose of analytic
calculation of cumulants, and the anharmonic correlated Einstein model which has been given results is in
Journal of Physics and Astronomy Research
Vol. 1(1), pp. 002-006, September, 2014. © www.premierpublishers.org, ISSN: 2123-503Xx
Review](https://image.slidesharecdn.com/pdf-141010055346-conversion-gate02/85/By-using-the-anharmonic-correlated-einstein-model-to-define-the-expressions-of-cumulants-and-thermodynamic-parameters-in-the-cubic-crystals-with-new-structure-factors-1-320.jpg)
![By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new
structure factors
J. Phys. Astron. Res. 003
To compare the above expressions in Table 1, we although see different structures of cubic crystals in which have
special common factors, we call these factors as new structure factors 21 c,c , the parameters calculated
statistically is in Table 2.
Table 2. New structural parameters of cubic crystals
Structure c1 c2
s.c 3 1
fcc 5 6/5
bcc 11/3 18/11
The 3k parameter is identical with any structures, the expressions of effk , )y(UE thermodynamic parameters for
the structural cubic crystals are generalized according to new structural parameters as following forms:
2
E32
2
1eff akcDck ; 4/y5aycDyU 3
1
2
E (7)
To derive the analytical formulas for cumulants through new structural parameters for the crystals of cubic structure,
we use perturbation theory [5]. The atomic vibration is quantized as phonon, and anharmonicity is the result of
phonon interaction. Accordingly, we express y in terms of annihilation and creation operators
aˆ , aˆ respectively:
aˆaˆy 0
; E
0
m2/ ; naˆaˆ
, (8)
and use the harmonic oscillator states n| as eigenstates with eigenvalues En nE , ignoring the zero-point
energy for convenience. The
aˆ , aˆ operators satisfy the following properties 1aˆaˆaˆaˆaˆ,aˆ
;
1n1nnaˆ
; 1nnnaˆ . The cumulants are calculated by the average value
mm
yTr
Z
1
y , ,...3,2,1m , Hexp ,
1
B )Tk(
, where Z is the canonical partition function,
with is the statistical density matrix, and Bk is Boltzmann’s constant. The corresponding unperturbed quantities
are )(TrZ 00 and 00 Hexp . To leading order in perturbation EU , 0 with is given by:
H ; 000 H (9)
we obtained:
'd'U
~
e E
0
H0
; 00 H
E
H
E eUeU
~
.
If we put unperturbed quantities equal to zero, we have:
n 0n
n
E00
z1
1
znexpTrZ ,
where
T//EE
eez
is the temperature variable and determined by the BEE k/ is Einstein
temperature. Now we are using above expressions to calculate analytics of the cumulants.
+ The cumulants even order:
nyne
Z
1
yTr
Z
1
yTr
Z
1
y mn
n0
m
0
0
m
m
m E
ch½n
With 2m we have calculation expression of the second cumulant
nyne
Z
1
y 2n
n0
22 E
. (10)
Using matrix 1n2naˆaˆaˆaˆnnyn
2
0
2
and substituting into (10) and applying the mathematical
transformations, and according to form (7) we have expression of second cumulant which is rewritten through 1c
structural parameter:
z1
z1
Dc2
y 2
1
E22
. (11)
+ The cumulants odd order:](https://image.slidesharecdn.com/pdf-141010055346-conversion-gate02/85/By-using-the-anharmonic-correlated-einstein-model-to-define-the-expressions-of-cumulants-and-thermodynamic-parameters-in-the-cubic-crystals-with-new-structure-factors-3-320.jpg)
![By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new
structure factors
J. Phys. Astron. Res. 005
Figure 1. The graphs illustrate temperature dependence of second (Fig.a) and third (Fig. b)
cumulants by present theory and compared to experiment values
DISCUSSION AND CONCLUSIONS
Developing further the anharmonic correlated Einstein model we obtained a general theory for calculation cumulants
and thermodynamic parameters in XAFS theory including anharmonic contributions. The expressions are described
through new structural parameters that agree with structural contributions of cubic crystals like face center cubic
(fcc), body center cubic (bcc), and results published before [8]. The expression in this work is general case of
present procedure when we insert the magnitudes of ,c,c 21 from Table 2 into the calculation of the thermodynamic
parameters and above obtained expressions of cumulants. The results of the numerical calculations according to
present method for cumulants are in an agreement with experimental values for Cu crystal (Table 3) and illustrated
by graphs in Figure 1. This is noted that the experimental values from XAFS spectra are measured at HASYLAB
(DESY, Germany).
With the discovery of the XAFS spectra, it provides the number of atoms and the radius of each shell, the XAFS
spectroscopy becomes a powerful structural analysis technique. However, the problem still remained to be solved is
the distribution of these atoms. The factors ,c,c 21 introduced in the presented work contains the angle between
the bond connecting absorber with each atom and the bond between absorber and backscatterer, that is why they
can describe the nearest atoms distributions surround absorber and backscatterer atoms. Knowing structure of the
crystals and the magnitudes of ,c,c 21 from Table 2 we can calculate the cumulants and then XAFS spectra. But for
structure unknown substances we can extract the atomic number from the measured XAFS spectra, as well as,
extract the factors ,c,c 21 according to our theory from the measured cumulants like Debye-Waller factor to get
information about atomic distribution or structure.
The thermodynamic parameters expressions described by second cumulant or Debye-Waller factor is very
convenient, when second cumulant
)2(
is determined, it allows to predict the other cumulants according to Eqs.
(21), (24), consequently reducing the numerical calculations and experimental measurements.
ACKNOWLEDGMENT
The author thanks Prof. Sci. Ph.D Nguyen Van Hung for useful discussions and for authorizing the author to use
some results published.
REFERENCES
Nguyen QB, Bui BD, Nguyen VH (1999). Statistical Physics, The publisher National University, Hanoi.
Beni G, Platzman PM (1976) “Temprature and polarization dependence of extended x-ray absorption fine-structure
spectra” Phys. Rev. B (14): 1514.
Born M, Huang K (1954). Dynamical Theory of Crystal Lattices Clarendon Press., Oxford.
Crozier, E. D., Rehr, J. J., and Ingalls, R. (1998). X-ray absorption edited by D. C. Koningsberger and R. Prins,
Wiley New York.
Feynman RP (1972). Statistics Mechanics, Benjamin, Reading.](https://image.slidesharecdn.com/pdf-141010055346-conversion-gate02/85/By-using-the-anharmonic-correlated-einstein-model-to-define-the-expressions-of-cumulants-and-thermodynamic-parameters-in-the-cubic-crystals-with-new-structure-factors-5-320.jpg)
By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors
- 1. By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors JPAR By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors Nguyen Ba Duc Chancellor of Tan Trao University, Tan Trao University, Tuyen Quang City, Vietnam Email: hieutruongdhtt@gmail.com, Tel.: +84 273890012 By using potential effective interaction in the anharmonic correlated Einstein model on the basis of quantum statistical theory with phonon interaction procedure, the expressions describing asymmetric component (cumulants) and thermodynamic parameters including the anharmonic effects contributions and by new structural parameters of cubic crystals have been formulated. These new parameters describe the distribution of atoms. The expansion of cumulants and thermodynamic parameters through new structural parameters has been performed. The results of this study show that, developing further the anharmonic correlated Einstein model it obtained a general theory for calculation cumulants and thermodynamic parameters in XAFS theory including anharmonic contributions. The expressions are described through new structural parameters that agree with structural contributions of cubic crystals like face center cubic (fcc), body center cubic (bcc). Keywords: Anharmonic XAFS, cumulants, thermodynamic parameters. INTRODUCTION In the harmonic approximation X-ray Absorption Fine Structure spectra (XAFS), the theoretical calculations are generally well appropriate with the experimental results at low temperatures, because the anharmonic contributions from atomic thermal vibrations may have been neglected. However, at the different high temperatures, the XAFS spectra provide apparently different structural information due to the anharmonic effects, and these effects need to be evaluated. Furthermore, the XAFS spectra at low temperatures may not provide a correct picture of crystal structure. Therefore, this study of the XAFS spectra including the anharmonic effects at high temperatures is crucially needed. The expression of anharmonic XAFS spectra often is described by: , !n ik2 ikR2expeIm kR ]k/R2exp[ kFk n n n ki 2 (1) where )k(F is the real specific atomic backscattering amplitude, )k( is total phase shift of photoelectron, k is wave number, is mean free path of the photoelectron, and n ,...3,2,1n are the cumulants to describe asymmetric components. They all appear due to the thermal average of the function ikr2 e , in which the asymmetric terms are expanded in a Taylor series around value rR with r is instantaneous bond length between absorbing and backscattering atoms at T temprature and then are rewritten in terms of cumulants. At first, the cumulant expansion approach has been used mainly fitting the XAFS spectra to extract physical parameters from experimental values. Therefore, some procedure were formulated for the purpose of analytic calculation of cumulants, and the anharmonic correlated Einstein model which has been given results is in Journal of Physics and Astronomy Research Vol. 1(1), pp. 002-006, September, 2014. © www.premierpublishers.org, ISSN: 2123-503Xx Review
- 2. By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors Duc 002 agreement with experimental values. The important development in this procedure is that model has been calculated into the interaction between absorbing and backscattering atoms with neighboring atoms in a cluster of nearest atoms at high temperatures. The potential interaction between the atoms becomes asymmetric due to the anharmonic effects and the asymmetric components were written in terms of the cumulants. The first cumulant is net thermal expansion, the second cumulant is Debye-Waller factor, and the third cumulant is description phase shift of anharmonic XAFS spectra. Based on the above initial illustration, the main purpose of this work is to formulate the cumulant expressions and write thermodynamic parameters as a general form through the new structure parameters by using the anharmonic correclated Einstein model. FOMALISM Because the oscillations of a pair single bond between of absorbing and backscattering atoms with masses 1M , 2M , respectively, is affected by neighboring atoms, when taking into account, these effects via an anharmonic corelated Einstein model, effective Einstein potential is formed as follow: 2,1i ij ij12 i E Rˆ.Rˆ M U...)x(UU (2) where Rˆ is the unit bond length vector, is reduced mass of atomic mass 1M and 2M , the sum according to j,i is the contribution of cluster nearest atoms, )x(U an effective potential: 3 3 2 eff xkxk 2 1 xU …, 0rrx (3) where r is spontantaneous bond length between absorbing and backscattering atoms, 0r is its equilibrium value, and effk is effective spring constant because it includes total contribution of neighboring atoms, and k3 is cubic anharmonicity parameter which gives an asymmetry in the pair distribution function. The atomic vibration is calculated based on quantum statistical procedure with approximate quasi - hamonic vibration, in which the Hamiltonian of the system is written as harmonic term with respect to the equilibrium at a given temperature plus an anharmonic perturbation, with axy , x)T(a , 0y , we have: 2 eff 2 0EE0E 2 yk 2 1 2 P H;yUaUH)(U 2 P H , (4) with a is the net thermal expansion, y is the deviation from the equilibrium value of x at temperature T. Next, the use of potential interaction between each pair of atoms in the single bond can be expressed by anharmonic Morse potential for cubic crystals. Expanding to third order around its minimum, we have: ...xx1D)e2e(DxU 3322xx2 E (5) where is expansion thermal parameter, D is the dissociation energy by DrU 0 . From expressions (4), (5) we have potential effective interaction Einstein generalize as: yUyk 2 1 aUU E 2 effEE , ayx (6) Substituting Eq. (5) into (3) and using Eq. (6) to calculate the second term in Eq. (3) with 2/M ( 1M = 2M = M ), sum of i is over absorber )1i( and backscatterer )2i( , and the sum of j which is over all their near neighbors, excluding the absorber and backscattered themselves, because they contribute in the xU , and calculation of )Rˆ.Rˆ( ij12 with lattice cubic crystals like s.c, fcc and bcc crystals, we obtain thermodynamic parameters like effk , 3k and )y(UE in Table 1. Table 1. The expressions of thermodynamic parameters for cubic crystals Factor s.c crystal fcc crystal bcc crystal 3k 4/5 3 D 4/5 3 D 4/5 3 D effk 4/513 2 aD 2/315 2 aD 3/22/45111 2 aD )y(UE 4/53 32 yayD 4/5 32 yayD 4/53/11 32 yayD
- 3. By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors J. Phys. Astron. Res. 003 To compare the above expressions in Table 1, we although see different structures of cubic crystals in which have special common factors, we call these factors as new structure factors 21 c,c , the parameters calculated statistically is in Table 2. Table 2. New structural parameters of cubic crystals Structure c1 c2 s.c 3 1 fcc 5 6/5 bcc 11/3 18/11 The 3k parameter is identical with any structures, the expressions of effk , )y(UE thermodynamic parameters for the structural cubic crystals are generalized according to new structural parameters as following forms: 2 E32 2 1eff akcDck ; 4/y5aycDyU 3 1 2 E (7) To derive the analytical formulas for cumulants through new structural parameters for the crystals of cubic structure, we use perturbation theory [5]. The atomic vibration is quantized as phonon, and anharmonicity is the result of phonon interaction. Accordingly, we express y in terms of annihilation and creation operators aˆ , aˆ respectively: aˆaˆy 0 ; E 0 m2/ ; naˆaˆ , (8) and use the harmonic oscillator states n| as eigenstates with eigenvalues En nE , ignoring the zero-point energy for convenience. The aˆ , aˆ operators satisfy the following properties 1aˆaˆaˆaˆaˆ,aˆ ; 1n1nnaˆ ; 1nnnaˆ . The cumulants are calculated by the average value mm yTr Z 1 y , ,...3,2,1m , Hexp , 1 B )Tk( , where Z is the canonical partition function, with is the statistical density matrix, and Bk is Boltzmann’s constant. The corresponding unperturbed quantities are )(TrZ 00 and 00 Hexp . To leading order in perturbation EU , 0 with is given by: H ; 000 H (9) we obtained: 'd'U ~ e E 0 H0 ; 00 H E H E eUeU ~ . If we put unperturbed quantities equal to zero, we have: n 0n n E00 z1 1 znexpTrZ , where T//EE eez is the temperature variable and determined by the BEE k/ is Einstein temperature. Now we are using above expressions to calculate analytics of the cumulants. + The cumulants even order: nyne Z 1 yTr Z 1 yTr Z 1 y mn n0 m 0 0 m m m E ch½n With 2m we have calculation expression of the second cumulant nyne Z 1 y 2n n0 22 E . (10) Using matrix 1n2naˆaˆaˆaˆnnyn 2 0 2 and substituting into (10) and applying the mathematical transformations, and according to form (7) we have expression of second cumulant which is rewritten through 1c structural parameter: z1 z1 Dc2 y 2 1 E22 . (11) + The cumulants odd order:
- 4. By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors Duc 004 m 0 m ml m yTr Z 1 yTr Z 1 y Î (12) With 3,1m we have expression to calculate first cumulant and third cumulant. Transformation following matrix correlative with y and 3 y , we have: 2/1 000 1nnn1n1naˆaˆn1nyn , (13) 2/33 0 3 0 3 1n3nn1n31nn31nyn , (14) 2/13 0 3 3n2n1n33nyn (15) - The first cumulant (m=1) ny'n'nycaycDn 'nn ee Z 1 y 3 3 3 1 2 'nn EE 'nn 0 )1( EE with 1n'n and from Eqs (12, 13) and transform, we have: , z1 z1 c3ac k2 D z1 z1 c3ac D y 2 031 eff E E 2 2 031 2 0 E 2 because 0y and approximate 2 1eff Dck , the transformation and reduction we obtained first cumulant )2( 1 2 1 E1 c4 15 z1 z1 Dc8 15 a (16) - The third cumulant (m=3) ny'n'nUn EE ee Z 1 y 3 E 'nn 'nn EE 0 33 'nn . (17) From Eqs. (7, 17), we have: ny'n'nycn'naycn 'nn ee Z D y 33 31 'nn EE 'nn 0 2 3 EE (18) Using Eqs. (14, 15), the calculation of Eq.(18) with 1n'n , 3n'n , respectively, and note that matrix only affect 3 y and according to Eqs. (7, 8), we determine the third cumulant: )2( 2 2 2 1 E 2 2 323 1 2 E3 z1 zz101 Dc4 15 z1 zz101 Dc8 15 (19) The results of the numerical calculations according to present method for cumulants are in agreement with experimental values for Cu crystal (Table 3). The Figures 1 illustrates good agreement of the second, and third cumulants in present theory with experiment values. Table 3. The comparision of the results of 2 and )3( calculated by present theory with experimental data for Cu crystal at different temperatures. T(K) )A( 22 )A( 33 Present Expt. Present Expt. 10 0,00298 0,00292 - - 77 0,00333 0,00325 0,00010 - 295 0,01858 0,01823 0,000131 0,000130 683 0,01858 0,01823 - -
- 5. By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors J. Phys. Astron. Res. 005 Figure 1. The graphs illustrate temperature dependence of second (Fig.a) and third (Fig. b) cumulants by present theory and compared to experiment values DISCUSSION AND CONCLUSIONS Developing further the anharmonic correlated Einstein model we obtained a general theory for calculation cumulants and thermodynamic parameters in XAFS theory including anharmonic contributions. The expressions are described through new structural parameters that agree with structural contributions of cubic crystals like face center cubic (fcc), body center cubic (bcc), and results published before [8]. The expression in this work is general case of present procedure when we insert the magnitudes of ,c,c 21 from Table 2 into the calculation of the thermodynamic parameters and above obtained expressions of cumulants. The results of the numerical calculations according to present method for cumulants are in an agreement with experimental values for Cu crystal (Table 3) and illustrated by graphs in Figure 1. This is noted that the experimental values from XAFS spectra are measured at HASYLAB (DESY, Germany). With the discovery of the XAFS spectra, it provides the number of atoms and the radius of each shell, the XAFS spectroscopy becomes a powerful structural analysis technique. However, the problem still remained to be solved is the distribution of these atoms. The factors ,c,c 21 introduced in the presented work contains the angle between the bond connecting absorber with each atom and the bond between absorber and backscatterer, that is why they can describe the nearest atoms distributions surround absorber and backscatterer atoms. Knowing structure of the crystals and the magnitudes of ,c,c 21 from Table 2 we can calculate the cumulants and then XAFS spectra. But for structure unknown substances we can extract the atomic number from the measured XAFS spectra, as well as, extract the factors ,c,c 21 according to our theory from the measured cumulants like Debye-Waller factor to get information about atomic distribution or structure. The thermodynamic parameters expressions described by second cumulant or Debye-Waller factor is very convenient, when second cumulant )2( is determined, it allows to predict the other cumulants according to Eqs. (21), (24), consequently reducing the numerical calculations and experimental measurements. ACKNOWLEDGMENT The author thanks Prof. Sci. Ph.D Nguyen Van Hung for useful discussions and for authorizing the author to use some results published. REFERENCES Nguyen QB, Bui BD, Nguyen VH (1999). Statistical Physics, The publisher National University, Hanoi. Beni G, Platzman PM (1976) “Temprature and polarization dependence of extended x-ray absorption fine-structure spectra” Phys. Rev. B (14): 1514. Born M, Huang K (1954). Dynamical Theory of Crystal Lattices Clarendon Press., Oxford. Crozier, E. D., Rehr, J. J., and Ingalls, R. (1998). X-ray absorption edited by D. C. Koningsberger and R. Prins, Wiley New York. Feynman RP (1972). Statistics Mechanics, Benjamin, Reading.
- 6. By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors Duc 006 Hung NV, Duc NB, Vuong DQ, (2001), “Theory of thermal expansion and cumulants in XAFS technique”, J. Commun. in Phys (11): 1-9. Hung NV, Rehr JJ (1997). “Anharmonic correlated Einstein-model Debye-Waller factors” Phys. Rev. B (56): 43. Hung NV, Vu KT, Nguyen BD (2000). “Calculation of thermodynamic parameters of bcc crystals in XAFS theory” J. Science of VNU Hanoi (XVI) pp. 11-17. Accepted 01 September, 2014. Citation: Nguyen Ba Duc (2014). By using the anharmonic correlated einstein model to define the expressions of cumulants and thermodynamic parameters in the cubic crystals with new structure factors. Journal of Physics and Astronomy Research, 1(1): 002-006. Copyright: © 2014 Nguyen Ba Duc. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.