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International Journal of Modern Physics B
Vol. 22, No. 29 (2008) 5155–5166
c World Scientific Publishing Company

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ANHARMONIC EFFECTIVE POTENTIAL, LOCAL FORCE
CONSTANT AND EXAFS OF HCP CRYSTALS: THEORY AND
COMPARISON TO EXPERIMENT

NGUYEN VAN HUNG∗ and TONG SY TIEN
Department of Physics, University of Science, VNU-Hanoi,
334 Nguyen Trai,Hanoi, Vietnam
∗
LE HAI HUNG
Institute of Engineering Physics, Hanoi University of Technology,
1 Dai Co Viet, Hanoi, Vietnam
RONALD R. FRAHM
Bergische Universit¨
at-Gesamthochschule Wuppertal, FB: 8-Physik,
Gauß Straße 20, 42097 Wuppertal, Germany
Received 30 April 2008
Anharmonic effective potential, Extended X-ray Absorption Fine Structure (EXAFS)
and its parameters of hcp crystals have been theoretically and experimentally studied.
Analytical expressions for the anharmonic effective potential, effective local force constant, three first cumulants, a novel anharmonic factor, thermal expansion coefficient
and anhamonic contributions to EXAFS amplitude and phase have been derived. This

take into account the anharmonic contributions to the mean square relative displacement (MSRD). This procedure provides good agreement with experiment, but
the expressions for the anharmonic factor and for the phase change of EXAFS due
to anharmonicity contain a fitting parameter, and the cumulants were obtained
by an extrapolation procedure from the experimental data.14,15 For calculation of
anharmonic effects, it is very important to calculate interatomic potentials.17,21,24
A procedure for calculation of Morse pair potential for crystals of cubic structures
is available,26 and its parameters have been extracted from experimental EXAFS
data.27,28 However, for calculation of thermodynamic properties of materials including anharmonic effects, the pair potential may not be suitable, and an effective
interatomic potential17,24 containing the pair potential is necessary.
This work is devoted to theoretical and experimental study of the anharmonic
EXAFS and its parameters of hcp crystals, an interesting structure. Our development presented in Sec. 2 is the derivation of analytical expressions for the anharmonic interatomic effective potential, effective local force constant, correlated Einstein frequency and temperature, three first cumulants, where the second cumulant
is equal to the Debye-Waller factor (DWF), anharmonic contributions to EXAFS
amplitude and phase of hcp crystals. This model includes only near-neighbor interactions between absorber and backscattering atoms and their immediate (or first
shell) neighbors instead of a single-bond model.12 The cumulants contained in the
derived expressions can be calculated by several procedures.5–7,12,13,17,18 In this
work, the quantum statistical approach with the anharmonic correlated Einstein
model,17 has been used. This model avoids full lattices dynamical13 or dynamical
matrix18 calculations, contributing to the extraction of physical parameters from
the EXAFS data,27,28 to the investigation of local force constants of transition metal
dopants in a nickel host,22 compared to Mossbauer studies, and to theoretical approaches to the EXAFS.19 Anharmonic factor has been derived again expressed by
the second cumulant or DWF, and the fitting parameter is avoided. The procedure
for calculation of Morse potentials for crystals of cubic structure26 has been generalized to calculation of those for hcp crystals. They characterize the interaction
of each pair of atoms, and are contained in the effective potentials, in the cumulants and in the other EXAFS parameters. The EXAFS and its parameters for hcp
crystals Zn and Cd at 77 K and 300 K have been measured at HASYLAB (DESY,
Germany), represented in Sec. 3, and its physical parameters have been obtained
by a fitting procedure. Here the experimental data are analyzed and compared to
the calculated values. Moreover, unnegligible anharmonic effects appearing in the
experimental and calculated EXAFS parameters have been evaluated in detail.




where F (k) is the real atomic backscattering amplitude; Φ is the net phase shift;
k and λ are the wave number and the mean free path of the photoelectron, respectively; R = r with r as the instantaneous bond length between absorber and
backscattering atoms and σ (n) (n = 1, 2, 3, . . .) are the cumulants.
2
The total mean square relative displacement (MSRD) σtot
(T ) or anharmonic
DWF at a temperature T is given as the sum of the harmonic σ 2 (T ) and anharmonic
2
σA
(T ) contributions
2
2
σtot
(T ) = σ 2 (T ) + σA
(T ) ,

2
σA
= β0 (T )[σ 2 (T ) − σ02 ] ,

β0 (T ) = 2γG

∆V
,
V

(2)

where γG is Gr¨


1 ˆ0 ˆ
xR · Rij
2

,

(5)

where the first term on the right concerns only absorber and backscatterer atoms,
the remaining sums extend over the remaining neighbors, and this relation is used
for calculation of the effective potential for monatomic hcp crystals based on its
atomic structure.


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Applying the Morse potential Eq. (4) to Eq. (5) and comparing it to Eq. (3) for
the case of correlated atomic vibrations, the effective local force constants are now
expressed in terms of the Morse potential parameters as
k0 = 5Dα2 .

(6)


10

2
= µωE
,

θE =

ωE
,
kB

µ=

MA MS
MA + M S

(9)

from which we obtain the correlated Einstein frequency ωE and temperature θE ; kB
is Boltzmann constant; µ is reduced mass of absorber and backscatterer with masses
MA and MS , respectively; and the perturbation δV due to the weak anharmonicity
is given by
δV (y) ∼
= 5Dα2 ay −

3
αy 3 .
20


ωE
,
2keff
a
ˆ|n =

(12)
√

n − 1|n − 1 ,

(13)


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Anharmonic Effective Potential of HCP Crystals

5159

and use the harmonic oscillator state |n as the eigenstate with the eigenvalue
En = n ωE , ignoring the zero-point energy for convenience.
Using the above results for correlated atomic vibrations and the procedure depicted by Eqs. (11)–(13), as well as the first-order thermodynamic perturbation
theory and considering the anharmonic component Eq. (10) of the potential, we
derived the cumulants.
The second cumulant or MSRD is expressed as

Int. J. Mod. Phys. B 2008.22:5155-5166. Downloaded from www.worldscientific.com


=

n

n=0

1
,
1−z

z = e−θE /T .

(15)

Applying Eqs. (13) to calculate the matrix element in Eq. (14), we obtain the
second cumulant
1+z
ωE
σ 2 (T ) = σ02
, σ02 =
, z = e−θE /T .
(16)
1−z
10Dα2
Using Eq. (11) to evaluate the traces, the remaining odd moments are given by
ym =

5Dα2
Z0


σ0

+ 10z + z 2
(3)
= σ0 [3(σ 2 /σ02 )2 − 2] ,
(1 − z)2

=

9α 2
σ ,
20 0

(3)

σ0

=

3α 2 2
(σ ) ,
10 0

(18)
(19)

and the thermal expansion coefficient
αT =
α0T

=
(3)
(1 − z)(1 + 10z + z 2 )
σ

σ (1) σ 2
3
,
=
(3)
6 − 4(σ02 /σ 2 )2
σ

(21)

where the first equation coincides with the one of Ref. 12, and the second equation
with the result of Ref. 17 for the other crystal structures.


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To calculate the total MSRD or anharmonic DWF Eq. (2), an anharmonic factor
has been derived
3α 2

3

(23)

We obtained from Eq. (1), taking into account the above results, the temperature
dependent K-edge EXAFS function including anharmonic effects as
χ(k, T ) =
j

2
2
2
S02 Nj
Fj (k)e−(2k [σ (T )+σA (T )]+2Rj /λ(k)) sin(2kRj +Φj (k)+ΦjA (k, T )) ,
kRj2

(24)
S02

where
is the square of the many-body overlap term, Nj is the atomic number of
each shell, the remaining parameters were defined above, the mean free path λ is
defined by the imaginary part of the complex photoelectron momentum p = k+i/λ,
and the sum is over all atomic shells.
2
It is obvious that in Eq. (24), σA
(T ) determines the anharmonic contributions
to the amplitude characterizing attenuation, and ΦA (k, T ) the anharmonic contributions to the phase characterizing the phase shift of the EXAFS spectra. They
are calculated by Eq. (2) and Eq. (23), respectively. At low temperatures, these
values approach zero so that our anharmonic theory becomes the harmonic one,

0.1653

1.7054
1.7000
1.9069
1.9053

2.7931
2.7650
3.0419
3.0550

39.5616
39.0105
48.7927
48.0711

2.6917
2.6729
2.2798
2.2628

205.6101
204.1730
174.1425
172.8499


November 20, 2008 16:18 WSPC/140-IJMPB



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5161

4

6

8

10
-1
k(Å )

(a)
Fig. 1.
300 K.

12

14

16

Zn
0.3
-1


(b)

Fig. 2. (a) Calculated Morse potentials and (b) anharmonic effective potentials compared to
harmonic components. They all agree well with experimental results.

is attenuated and shifted to the right [Fig. 1(a)], its Fourier transform magnitude
is attenuated and shifted to the left [Fig. 1(b)] when the temperature changes from
77 K to 300 K. Similar properties of the measured EXAFS for Cd have also been
obtained. This temperature dependence of the EXAFS of hcp crystals need to be
analyzed and compared to the developed theory. In order to perform it, we apply
the expressions derived in the previous section to numerical calculations and compare the results to experimental data for Zn and Cd. Morse potential parameters of
Zn and Cd have been calculated by generalizing the procedure for cubic crystals26
to the one for hcp crystals. The experimental Morse potential and other EXAFS
parameters have been obtained from the experimental EXAFS data by a fitting
procedure. Effective local force constants, correlated Einstein frequencies and temperatures have been calculated using the obtained Morse parameters. The results
are written in Table 1. They are used for calculation of the anharmonic EXAFS


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and its parameters. Figure 2 shows the calculated Morse potential (a) and anharmonic effective potentials (b) for Zn and Cd compared to experiment and to their

high temperatures, as these properties were shown in theory 12,17 and in experiment7 for the other crystals. They are slightly different for different materials only
below the Einstein temperature. Hence, we can conclude that these cumulant relations have the same properties for all crystals as the standards for evaluation of
cumulants in EXAFS technique. Figure 5 shows the anharmonic contributions to
the EXAFS phase according to Eq. (23) at 77 K, 300 K and 500 K. This contribution is negligible at 77 K, but valuable at 300 K and 500 K, especially at high

(a)

(b)

Fig. 3. (a) Temperature dependence of total MSRD compared to the harmonic one and (b) anharmonic contribution for Zn and Cd. They agree well with experiment at 77 K and 300 K.


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Anharmonic Effective Potential of HCP Crystals

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5163

contributions
EXAFS
phaseEq.
for(17)
Zn for
at Zn and Cd compared to
Fig. 4. Temperature
dependence of to
cumulant


-3
-4
-5
0

5

10

15

20

-1

k(Å )
Fig. 5. Calculated k-dependence of anharmonic contributions to EXAFS phase for Zn at 77 K,
300 K and 500 K.

k values. For calculation of anharmonic EXAFS of Zn and Cd, we modified code
FEFF1 by adding our anharmonic contributions. For XANES, the multiple scattering is important, but for EXAFS, the single scattering is dominant,31 and the main
contribution to EXAFS is given by the first shell.7 This is why for testing theory,
only the calculated EXAFS for the first shell for single scattering has been used
for comparison to experiment. The calculated EXAFS and their Fourier transform
magnitudes for Zn at 77 K, 300 K and 500 K for the first shell and for single scattering are illustrated in Fig. 6. They reflect all properties of temperature dependence
of the experimental results (Fig. 1). They show significant changes of amplitude and


November 20, 2008 16:18 WSPC/140-IJMPB

0
-5
-10
-15
5

10

15

77K
300K
500K

0.2

0.1

0.0

20

0

1

2

3


0

Zn, Present theory.
1st shell, single scattering.
3Å-1 < k < 13.5Å-1

0.3

Zn, 300K

0.14

3Å-1 < k < 13.5Å-1

0.12
Expt.
Present theory
FEFF

0.10
0.08
0.06
0.04
0.02
0.00

Cd, 77K

0.25


1

2

3

4

5

6

R(Å)

(b)

Fig. 7. Fourier transform magnitudes of EXAFS for Zn (a) at 300 K and for Cd at 77 K (b)
calculated by present anharmonic theory compared to experiment and to those calculated by
FEFF code.1

phase as the temperature increases, e.g., the EXAFS is attenuated and shifted to
the right, the peak of its Fourier transform magnitude is attenuated and shifted to
the left. Figure 7(a) shows a good agreement of our calculated anharmonic EXAFS
of Zn at 300 K with experiment and its difference from the one calculated by the
harmonic FEFF,1 but at low temperature 77 K, where the anharmonic contributions are negligible, as it was shown in Figs. 3(b) and 5, then the Fourier transform
magnitudes calculated by the present anharmonic procedure and by the harmonic
FEFF code both agree well with experiment [Fig. 7(b)]. Hence, our anharmonic
theory is advantageous and shown to be in agreement with experiment both at low
and high temperatures, while the harmonic one can obtain this agreement only at
low temperatures where the anharmonicity is negligibly small, but disagreement

The cumulant relations αT rT σ 2 /σ (3) and σ (1) σ 2 /σ (3) for hcp crystals have the
same form and the same properties as for the other crystal structures, so that they
can be proposed to be the standards for cumulant evaluation in EXAFS procedure.
The EXAFS and Fourier transform magnitudes for Zn and Cd, measured at HASYLAB (DESY, Germany) provide the necessary physical parameters for comparison to those of the developed anharmonic theory. They show their thermodynamic
properties in temperature dependence and anharmonic effects.
The derived analytical expressions for EXAFS parameters in the present anharmonic theory provide the results satisfying all their fundamental properties,
and agree well with experiment at 77 K and 300 K. They show unnegligible anharmonic effects in the theoretical and experimental considered quantities, which denotes the importance of including anharmonic contributions in
the temperature dependent EXAFS and the efficiency of the effective potential
procedure.

Acknowledgments
The authors thank J. J. Rehr, P. Fornasini and A. I. Frenkel for useful comments.
One of the authors (N. V. H.) thanks the BUGH Wuppertal for hospitality and
appreciates the partial supports of the basic science research project No. 405806
and the special research project of VNU Hanoi QG.08.02.


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