In present article the pressure dependence of Debye-Waller factors in crystals has been investigated by using statistical moment method and anharmonic correlated Einstein model. These two methods provide similar results which indicate that the Debye-Waller factors of crystals decreases slightly under high pressure. Our numerical results for several crystals are compared to other theoretical and experimental values and showed a good agreement

Hieu, Ho Khac

Hung, Nguyen Van

Hung, Vu Van

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 20, No. 3 (2010), pp. 219-225
PRESSURE DEPENDENCE OF EXAFS DEBYE-WALLER
FACTORS IN CRYSTALS
VU VAN HUNG
Hanoi National University of Education
HO KHAC HIEU
National University of Civil Engineering
NGUYEN VAN HUNG
Hanoi University of Science
Abstract. In present article the pressure dependence of Debye-Waller factors in crystals has
been investigated by using statistical moment method and anharmonic correlated Einstein model.
These two methods provide similar results which indicate that the Debye-Waller factors of crystals
decreases slightly under high pressure. Our numerical results for several crystals are compared to
other theoretical and experimental values and showed a good agreement.
I. INTRODUCTION
EXAFS (Extended X-ray Absorption Fine Structure) spectroscopy has become a
powerful technique for determining local structures of noncrystalline as well as crystalline
materials. It is of great interest in EXAFS procedure to characterize the local atomic
environment of substances under the temperature T and pressure P dependence.
EXAFS data are analyzed by means of the cumulant expansion technique. In this
formulation, an EXAFS oscillation function χ (k) is given by [1]
χ (k) =
F (k)
kR2
e−2R/λ(k)Im
{
eiφ(k) exp
[
2ikR+
∑
n
(2ik)n
n!
σ(n)
]}
, (1)
where k and λ are the wave number and mean free path of emitted photoelectrons, F (k) is
the real atomic backscattering amplitude, ϕ (k) is the net phase shift, and σ(n) (n = 1, 2, 3...)
are the cumulants which can be extracted from experimental EXAFS data.
This EXAFS oscillation function Eq.(1) contains the second cumulant which is equal
to the mean square relative displacement (MSRD) or Debye-Waller factor (DWF) σ2
accounting for the effects of the thermal vibrations of atoms.
The temperature dependence of DWF in crystals (metals, quantum crystals,. . . )
has been studied by many authors [1,5]. However, the researches on pressure dependence
of DWF in crystals are still limited.
The purpose of this paper is to investigate the pressure dependence of DWF in
crystals by using the anharmonic correlated Einstein model (ACEM) [5] and the statistical
moment method (SMM) [6–9]. Numerical calculations have been carried out for metal Cu
220 VU VAN HUNG, HO KHAC HIEU, AND NGUYEN VAN HUNG
(fcc), quantum crystal Kr (fcc) and semiconductor Si (dia). The calculated results are
found to be in good agreement with the other theories [2–4] and experimental values [3,4].
II. FORMALISM
II.1. Statistical Moment Method
We will derive the pressure versus volume relation of crystals limiting only quadratic
terms in the atomic displacements. The pressure versus volume relation of lattice is given
by [8, 9]
Pv = −a
[
1
3
∂U0
∂a
+ θx cothx
1
2k
∂k
∂a
]
, x =
~ω
2θ
, θ = kBT, (2)
where P denotes the hydrostatic pressure and v is the atomic volume v = VN of a crystal
having volume V and N atoms. Using Eq.(2) one can find the nearest neighbor distance
a at pressure P and temperature T . However, for numerical calculations it is convenient
to determine firstly the nearest neighbor distance a (P, 0) at pressure P and at absolute
zero temperature T = 0K. For T = 0K, Eq.(2) is reduced to
Pv = −a
[
1
3
∂U0
∂a
+
~ω0
4k
∂k
∂a
]
· (3)
For simplicity, we take the effective pair interaction energy in crystal systems as the
power law, similar to the Lennard-Jones potential
φ (r) =
D
m− n
[
n
(r0
r
)m
−m
(r0
r
)n]
, (4)
where D and r0 are determined by fitting the experimental data (e.g., cohesive energy and
elastic modulus). The potential parameters of some crystals are shown in Table 1 [10,11].
Using the effective pair potential of Eq.(4), it is straightforward to get the interaction
energy U0, the quantity k, and the parameter γ in the crystal as
U0 =
D
n −m
[
mAn
(r0
a
)n
− nAm
(r0
a
)m]
,
k =
1
2
∑
i
(
∂2φ
∂u2iβ
)
=
Dnm
2a2 (n −m)
[
(n + 2)A
a2ix
n+4 −An+2
] (r0
a
)n
− Dnm
2a2 (n−m)
[
(m+ 2)A
a2ix
m+4 − Am+2
] (r0
a
)m
= m0ω
2
0 ,
(5)
PRESSURE DEPENDENCE OF EXAFS DEBYE-WALLER FACTORS IN CRYSTALS 221
γ = Dnm12a4(n−m)

 (n + 2) (n+ 4) (n + 6)
(
A
a4
ix
n+8 + 6A
a2
ix
a2
iy
n+8
)
−18 (n+ 2) (n + 4)Aa
2
ix
n+6 + 9 (n + 2)An+4

 ( r0
a
)n
− Dnm
12a4(n−m)

 (m+ 2) (m+ 4) (m+ 6)
(
A
a4ix
m+8 + 6A
a2ixa
2
iy
m+8
)
−18 (m+ 2) (m+ 4)Aa2ixm+6 + 9 (m+ 2)Am+4

 (r0
a
)m
,
(6)
wherem0 is the mass of particle, ω0 is the frequency of lattice vibration, An, Am, A
a2
ix
n , A
a2
ix
m
are the structural sums for the given crystal.
From Eqs. (3, 4, 5, 6) we obtain equation of state of crystals at zero temperature
as
Pv =
Dnm
6 (n−m)
[
n
(r0
r
)m
−m
(r0
r
)n]
+
1
a
~
4
√
m0
√
Dnm
2 (n −m)
×
{
(n+ 2)
[
(n + 2)A
a2ix
n+4 −An+2
] (
r0
a
)n − (m+ 2) [(m+ 2)Aa2ixm+4 − Am+2] ( r0a )m}√[
(n+ 2)A
a2
ix
n+4 − An+2
] (
r0
a
)n − [(m+ 2)Aa2ixm+4 − Am+2] ( r0a )m
(7)
Using Maple or Matlab program and the values of parameters D and r0 determined
by the experimental data [10, 11] (Table 1), we found the values of the nearest neighbor
distance a (P, 0) at T = 0K and pressure P. Using the obtained results of nearest neighbor
distance a(P, T ), we found the values of parameters k(P, 0), γ(P, 0) at pressure P and
T = 0K.
The thermally induced lattice expansion y0 (P, T ) at pressure P and temperature T
is given in the below formula using the force balance criterion of the fourth order moment
approximation as [6, 9]
y0 (P, T ) =
√
2γ (P, 0) θ2
3k3 (P, 0)
A (P, 0), (8)
where
A (P, T ) = a1 +
γ (P, 0)2 θ2
k (P, 0)4
a2 +
γ3 (P, 0) θ3
k (P, 0)6
a3 +
γ (P, 0)4 θ4
k (P, 0)8
a4, (9)
a1 = 1 +
x coth (x)
2
, a2 =
13
3
+
47
6
x coth (x) +
23
6
x2 coth2 (x) +
1
2
x3 coth3 (x) ,
a3 = −
[
25
3
+
121
6
x coth (x) +
50
3
x2 coth2 (x) +
16
3
x3 coth3 (x) +
1
2
x4 coth4 (x)
]
,
a4 =
43
3
+
93
2
x coth (x)+
169
3
x2 coth2 (x)+
83
3
x3 coth3 (x) +
22
3
x4 coth4 (x)+
1
2
x5 coth5 (x) ,
x =
~ω (P, 0)
2θ
;ω (P, 0) =
√
k (P, 0)
m0
; θ = kBT.
222 VU VAN HUNG, HO KHAC HIEU, AND NGUYEN VAN HUNG
Then, one can find the nearest neighbor distance a (P, T ) at pressure P and tem-
perature T as
a (P, T ) = a (P, 0) + y0 (P, T ) . (10)
The parallel mean square relative displacement (MSRD) to a good approximation
corresponds to the second cumulant:
σ2 =
〈[
~R. (~ui − ~u0)
]2〉
=
〈
u2i
〉
+
〈
u20
〉− 2 〈uiu0〉 . (11)
The first two terms on the right-hand side are the uncorrelated mean square dis-
placement (MSD), while the third term is the parallel displacement correlation function
(DCF).
Using the expression of the second order moment [9], we obtain the MSD
〈
u2i
〉
= 〈ui〉2 + θA1 + θ
k
(x cothx− 1) , (12)
where
A1 =
1
k
[
1 +
2γ2
k4
(
1 +
x cothx
2
)
(x cothx+ 1)
]
,
〈uiu0〉 ≈ 〈ui〉 〈u0〉 ,
〈
u2i
〉 ≈ 〈u20〉 .
(13)
From Eqs. (5, 6, 8, 10) we derive the second cumulant of crystals at pressure P and
temperature T as
σ2 (P, T ) ≈ 4γ (P, 0)
2 θ3
k (P, 0)5
(
1 +
x cothx
2
)
(x cothx+ 1) +
2θ
k (P, 0)
x cothx (14)
II.2. Anharmonic Correlated Einstein Model
The anharmonic correlated Einstein model [5] is now widely used in EXAFS data
analysis [1]. In the present approach we apply this theory to the calculation of the cumu-
lants where the effective interaction potential is given by
φeff (x) ≈ 1
2
keffx
2 + k3x
3 + ... = φ (x) +
∑
j 6=i
φ
(
µ
Mi
.Rˆ12Rˆij
)
(15)
where x is the deviation of instantaneous bond length between two atoms from equilibrium,
keff is the effective spring constant, and k3 is the cubic anharmonicity parameter and Rˆ
is the bond unit vector. The correlated Einstein model may be defined as an oscillation
of a pair of atoms with masses M1 and M2 (e.g., absorber and backscatterer) in a given
system. Their oscillation is influenced by their neighbors given by the last term in the
left-hand side of Eq.(12), where the sum i is over absorber (i = 1) and backscattering
atom(i = 2), and the sum j is over all their nearest neighbors, excluding the absorber and
backscatterer themselves whose contributions are described by the term φ (x).
For the calculation of thermodynamic parameters, we use the further definition
x = r − r0, a = 〈r − r0〉 and y = x − a [1, 5] as the deviation from the equilibrium value
PRESSURE DEPENDENCE OF EXAFS DEBYE-WALLER FACTORS IN CRYSTALS 223
of x at temperature T to rewrite Eq. (15) in the y-dependence as
ϕeff (y) =
1
2
keffy
2 + δϕeff (y) , (16)
where δφeff (y) is the anharmonic contribution of the effective pair potential.
By using quantum statistical method we derived the 2nd cumulant or DWF of crys-
tals as
σ2 = σ20
1 + z
1− z , σ
2
0 =
~ωE
2keff
, (17)
where σ20 is zero-point contribution to σ
2, kB is the Boltzmann constant and
ωE =
√
keff
µ
, µ =
M1M2
M1 +M2
, z = exp (θE/T ) , θE =
~ωE
kB
.
To build the pressure dependence of DWF expressions of crystals we start with the
definition of the Gruneisen parameter γG
γG = −∂ lnωE
∂ lnV
. (18)
This parameter γG depends on volume which, in turn, depends on pressure. With
a slight modification of a formula found in Refs. [12, 13], the simplest parametrization of
the volume dependence for crystals can be written as
γG (V (P ))
V (P )
=
γG (V0)
V0
= const, (19)
where the subscript 0 indicates zero pressure. Integration of Eq. (18) and using Eq. (19)
we obtained the Einstein frequency at pressure P and temperature T as
ωE (V (P )) = ωE (V0) exp
{
γG (V0)
[
1− V (P )
V0
]}
,
V (P )
V0
=
a3 (P, T )
a3 (0, 0)
. (20)
This fraction can be calculated by using any appropriate isothermal equation of
state. For simplicity, we used the data of a (P, T ) derived from SMM to calculate this
fraction.
Using the above results, we obtain the pressure dependence of the effective spring
constant expression as
keff (V (P )) = µω
2
E (V (P )) . (21)
Eqs. (20, 21) are then inserted into those of the ACEM which express the EXAFS
DWF in crystals as a function of pressure
σ2 (P, T ) =
~ωE (V (P ))
2keff (V (P ))
1 + exp [−β~ωE (V (P ))]
1− exp [−β~ωE (V (P ))] ; β = 1/kBT. (22)
224 VU VAN HUNG, HO KHAC HIEU, AND NGUYEN VAN HUNG
III. DISCUSSION OF NUMERICAL RESULTS
Now we apply the expressions derived in previous section to numerical calculations
for metal crystal (Cu), quantum crystal (Kr) and semiconductor crystal (Si). The m-n
potential parameters of Cu, Kr, and Si have been listed in Table 1 and the Morse potential
parameters of Cu are D = 0.3429eV , α = 1.3588A˚−1 [14]. They will be used for further
calculations.
Table 1: Lennard-Jones potential parameters D and r0 for Cu, Kr and Si.
Crystal m n r0
(
A˚
)
D/kB (K)
Cu 5.5 9.0 2.5487 4125.7
Kr 6.63 16.42 3.993 237.16
Si 6.0 12.0 2.2950 3269.1
0 5 10 15 20 25 30
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P(GPa)
σ
2 Kr
(P
)(Å
2 )
 
 
ACEM(m−n)
SMM
Monte Carlo simulation [2]
Exp [2]
Exp[3]
Loubeyre´s results[3]
Fig. 1. The pressure dependence of
DWF of Kr
0 5 10 15 20 25 30
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P(GPa)
u
2 Kr
(P
)(Å
2 )
 
 
MC simulation [2]
ACEM (m−n)
SMM
Fig. 2. The pressure dependence of
MSD of Kr
0 2 4 6 8 10
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
x 10−3
P(GPa)
σ
2 Cu
(P
)−σ
2 Cu
(0)
(Å
2 )
 
 
ACEM (Morse)
ACEM (m−n)
SMM
Debye Model [1]
Model 2 [1]
Fig. 3. The pressure dependence of
DWF of Cu
0 2 4 6 8 10 12
−3
−2
−1
0
x 10−4
P(GPa)
σ
2 Si
(P
)−σ
2 Si
(0)
(Å
2 )
 
 
ACEM (Morse)
SMM
Fig. 4. The pressure dependence of
DWF of Si
Fig. 1 shows the pressure dependence of our calculated second cumulant or DWF of
Kr crystal compared to those calculated by Monter-Carlo simulation [2] and Loubeyre [3]
as well as to the experimental values [3,4]. The pressure dependence of our calculated MSD
of Kr crystal is presented in Fig. 2 in comparison to Monte-Carlo simulation result [3].
It is shown that our calculated MSRD and MSD are found to be in good agreement with
PRESSURE DEPENDENCE OF EXAFS DEBYE-WALLER FACTORS IN CRYSTALS 225
the experimental values [3, 4] and with the other theoretical results [3, 4]. Fig. 3 presents
the pressure dependence of the second cumulant or DWF of Cu crystal, calculated using
the SMM and ACEM with Lennard-Jones and Morse interaction potentials, respectively.
The calculated results of both methods are found to be in good agreement with those of
Debye model [2] and the model for calculation of the moments of the nearest neighbor
distance distribution from an expansion to third order of the potential energy [2]. Fig.
3 shows that the SMM values are considerably higher those calculated using the ACEM
with the Lennard-Jones potential. It indicates that the Morse potential is the best choice
for Cu crystal when using ACEM to investigate the thermodynamic properties of Cu. Fig.
4 shows the pressure dependence of the DWF of Si crystal calculated using SMM with
Lennard-Jones potential.
IV. CONCLUSIONS
In this work a formalism for the statistical moment method and the anharmonic
correlated Einstein model using Lennard-Jones and Morse potentials to investigate the
pressure dependence of EXAFS Debye-Waller factors has been developed. Our new devel-
opment is the derivation of analytical expressions of the pressure dependence of EXAFS
second cumulant or Debye-Waller factors in crystals and carrying out numerical calcula-
tions for Cu, Kr, Si.
The good agreement of our calculated results using these two methods as well as
their agreement with experimental and other theories values shows the advantage and
efficiency of these theoretical for the calculation and analysis of the pressure dependent
cumulants and other parameters of anharmonic EXAFS of crystals.
ACKNOWLEDGEMENTS
This work is supported by the research projects No. 103.01.09.09 and No. 103.01.26.09
of NAFOSTED.
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Received 30 September 2009.


Pressure Dependence of EXAFS Debye-Waller Factors in Crystals

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Pressure Dependence of EXAFS Debye-Waller Factors in Crystals

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