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

29 2008 5155–5166c World Scientific Publishing Company ANHARMONIC EFFECTIVE POTENTIAL, LOCAL FORCE CONSTANT AND EXAFS OF HCP CRYSTALS: THEORY AND COMPARISON TO EXPERIMENT NGUYEN VAN HUNG

Vol 22, No 29 (2008) 5155–5166

c World Scientific Publishing Company

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

∗ hungnv@vnu.edu.vn

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 con-stant, three first cumulants, a novel anharmonic factor, thermal expansion coefficient and anhamonic contributions to EXAFS amplitude and phase have been derived This anharmonic theory is applied to analyze the EXAFS of Zn and Cd at 77 K and 300 K, measured at HASYLAB (DESY, Germany) Numerical results are found to be in good agreement with experiment, where unnegligible anharmonic effects have been shown in the considered theoretical and experimental quantities.

Keywords: Effective potentials; anharmonicity; cumulant; EXAFS.

1 Introduction

EXAFS and its parameters are often measured at low temperatures and well-analyzed by the harmonic procedure1 because the anharmonic contributions to atomic thermal vibrations can be neglected However, EXAFS may provide appar-ently different information on structure and on other parameters of the substances

at different high temperatures due to anharmonicity.2–25 Analysis of these effects

is necessary to evaluate thermal atomic vibration of the substances The formalism for including anharmonic effects in EXAFS is often based on the cumulant expan-sion approach.3,4Using this procedure, the anharmonic effects in EXAFS are often

5155

valuated by the ratio method,3–10 which is based on the comparison of EXAFS data measured at two different temperatures Another way is direct calculation and analysis of EXAFS and its parameters, including anharmonic effects at any temperature For this purpose, an anharmonic factor has been introduced14–16 to take into account the anharmonic contributions to the mean square relative dis-placement (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 in-cluding 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 develop-ment presented in Sec 2 is the derivation of analytical expressions for the anhar-monic interatomic effective potential, effective local force constant, correlated Ein-stein 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 inter-actions 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,27,28to the investigation of local force constants of transition metal dopants in a nickel host,22 compared to Mossbauer studies, and to theoretical ap-proaches to the EXAFS.19Anharmonic 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 gen-eralized 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 cumu-lants 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

2 Formalism

Based on cumulant expansion approach, the EXAFS oscillation function is given

by21

χ(k) = F (k)e

−2R/λ(k)

kR2 Im

(

"

2ikR +X

n

(2ik)n n! σ (n)

#)

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, re-spectively; R = hri with r as the instantaneous bond length between absorber and backscattering atoms and σ(n)(n = 1, 2, 3, ) are the cumulants

The total mean square relative displacement (MSRD) σ2

tot(T ) or anharmonic DWF at a temperature T is given as the sum of the harmonic σ2(T ) and anharmonic

σ2

A(T ) contributions

σtot2 (T ) = σ2(T ) + σ2A(T ) , σA2 = β0(T )[σ2(T ) − σ2] , β0(T ) = 2γG

∆V

V , (2) where γG is Gr¨uneisen parameter, ∆V /V is the relative volume change due to thermal expansion, and σ2 is zero-point contribution to σ2(T )

The anharmonic effective potential in our approach is expressed as a function of the displacement x = r−r0along the ˆR0direction, r and r0being the instantaneous and equilibrium distances between absorber and backscattering atoms, respectively

Veff(x) ≈ 1

2k0x

where k0is effective local force constant, and k3is cubic parameter giving the asym-metry due to anharmonicity (Here and in the following, the constant contributions are neglected)

A Morse potential is assumed to describe the interatomic interaction, and ex-panded to the third order around its minimum

V (x) = D(e−2αx− 2e−αx) ∼= D(−1 + α2x2− α3x3+ · · ·) , (4) where α describes the width of the potential and D is the dissociation energy

In the case of relative vibrations of absorber and backscatterer atoms, including the effect of correlation and taking into account only the nearest or first shell neighbor interactions, the effective interatomic potential is given by

Veff(x) = V (x) + X

i=a,b X j6=a,b

V 1

2x ˆR

0

· ˆRij



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

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

For calculation of thermodynamic parameters, we use the further definition

y = x − a, a = hxi,12,17 to write Eq (5) as

Veff(y) ∼= (k0+ 3k3a)ay +1

2keffy

where keff is an effective local force constant, which is in principle different from

k0

In accordance with Eq (7), the anharmonic effective potential can be expressed

as the sum of the harmonic contribution and a perturbation δV due to the weak anharmonicity

Veff(y) ∼= 1

2keffy

where now the force constant is different from the one of Eq (6) for Veff(x), i.e.,

keff = 5Dα2



1 − 109αa



= µωE2 , θE=~ωE

kB

, µ = MAMS

MA+ MS

(9) from which we obtain the correlated Einstein frequency ωEand temperature θE; kB

is Boltzmann constant; µ is reduced mass of absorber and backscatterer with masses

MAand MS, respectively; and the perturbation δV due to the weak anharmonicity

is given by

δV (y) ∼= 5Dα2



ay −203 αy3



Making use of quantum statistical methods,29 the physical quantity is deter-mined by an averaging procedure using canonical partition function Z and statis-tical density matrix ρ, e.g.,

hymi = Z1Tr(ρ ym) , m = 1, 2, 3, (11) Atomic vibrations are quantized in terms of phonons, and anharmonicity is the result of phonon-phonon interaction, that is why we express y in terms of annihilation and creation operators, ˆa and ˆa+, respectively

y ≡ a0(ˆa + ˆa+) , a20= ~ωE

which have the following properties

[ˆa, ˆa+] = 1 , ˆ+|ni =√n + 1|n + 1i , a|ni =ˆ √n − 1|n − 1i ,

ˆ+ˆa|ni = n|ni ,

(13)

and use the harmonic oscillator state |ni 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 de-picted 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

σ2= hy2i =Z1

0 X n

where the canonical partition function is given by

Z ∼= Z0=X

n

∞ X n=0

zn= 1

1 − z, z = e

Applying Eqs (13) to calculate the matrix element in Eq (14), we obtain the second cumulant

σ2(T ) = σ201 + z

1 − z, σ

2

10Dα2, z = e−θE /T (16) Using Eq (11) to evaluate the traces, the remaining odd moments are given by

hymi = 5Dα

2

Z0 X

e−βE n− e−βEn0

En− En 0 hn|ay − 3

20αy

3

|n0ihn0|ym|ni , m = 1, 3 (17)

Calculating the matrix elements in Eq (17), we obtain the first (m = 1) and the third (m = 3) cumulants as

σ(1)(T ) = a(T ) = σ0(1)1 + z

1 − z =

9α

20σ

20σ

2

σ(3)(T ) = σ(3)0 1 + 10z + z

2 (1 − z)2 = σ(3)0 [3(σ2/σ02)2− 2] , σ0(3)= 3α

10(σ

2

and the thermal expansion coefficient

αT = 1 R

da

dT = α

0 T

z| ln(z)|2 (1 − z)2 = α0

T

 5Dα

kBT

2 [(σ2)2− (σ2)2] ,

α0

100DαR,

(20)

where σ0(1), σ2, σ0(3) are zero-point contributions to σ(1), σ2, σ(3), respectively, and

α0

T is the constant value which αT approaches at high temperatures

From the above results, we obtain the following cumulant relations

αTrT σ2

σ(3) = 3z(1 + z) ln(1/z)

(1 − z)(1 + 10z + z2),

σ(1)σ2

6 − 4(σ2/σ2)2 , (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

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

β0(T ) = 9α

2

8 σ 2



1 + 3α 4Rσ 2



1 + 3α 4Rσ 2



Note that all the above derived thermodynamic quantities are expressed by the sec-ond cumulant σ2, hence leading our work mainly to the calculation of this quantity The anharmonic contribution to the EXAFS phase at a given temperature is the difference between the total phase and the one of the harmonic EXAFS, and it

is given by

ΦA(T, k) = 2k



σ(1)(T ) − 2σA2(T ) 1

R +

1 λ(k)



−23σ(3)(T )k2



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 ) =X

j

S2

kR2 j

(24) where S2is 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

It is obvious that in Eq (24), σ2

A(T ) determines the anharmonic contributions

to the amplitude characterizing attenuation, and ΦA(k, T ) the anharmonic contri-butions 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, but at high temperatures, it approaches the classical limit results including anhar-monic effects, so that the present anharanhar-monic procedure contains the haranhar-monic and classical theories as its special cases

3 Discussion of Experimental and Numerical Results

EXAFS spectra at 77 K and 300 K for Zn and Cd have been measured at HASY-LAB (DESY, Germany) Some results are presented in Fig 1 The EXAFS for Zn

Table 1 Calculated and experimental values of Morse parameters D, α, r 0 , of effective local force constant k eff , correlated Einstein frequency ωEand temperature θE for Zn and Cd.

Bond D (eV) α (˚ A −1 ) r0(˚ keff(N/m) ω (× 10 13 Hz) θE(K) Zn-Zn, Calc 0.1698 1.7054 2.7931 39.5616 2.6917 205.6101 Zn-Zn, Expt 0.1685 1.7000 2.7650 39.0105 2.6729 204.1730 Cd-Cd, Calc 0.1675 1.9069 3.0419 48.7927 2.2798 174.1425 Cd-Cd, Expt 0.1653 1.9053 3.0550 48.0711 2.2628 172.8499

k(Å -1 )

-30

-20

-10

0

10

20

30

77K 300K

(a)

Zn

3Å -1 < k < 13Å -1

R(Å)

0.0 0.1 0.2 0.3 0.4

77K x2, 300K

(b) Fig 1 (a) Experimental EXAFS of Zn and (b) its Fourier transform magnitudes at 77 K and

300 K.

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 com-pare 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 tem-peratures have been calculated using the obtained Morse parameters The results are written in Table 1 They are used for calculation of the anharmonic EXAFS

and its parameters Figure 2 shows the calculated Morse potential (a) and anhar-monic effective potentials (b) for Zn and Cd compared to experiment and to their harmonic terms The anharmonic effective interatomic potentials are quite different from the Morse pair potentials due to inclusion of the interaction of nearest neigh-bors of the absorber and backscattering atoms Figure 3 illustrates the temperature dependence of the calculated total MSRD σ2

tot(T ) or anharmonic DWF for Zn and

Cd, compared to their harmonic ones σ2(T ) of the anharmonic contribution σ2

to the MSRD (b) According to Eq (24), σ2

A(T ) is also the anharmonic contribu-tion to the EXAFS amplitude, and it increases as the temperature increases The calculated values of σ2

tot(T ) and σ2

A(T ) agree well with experiment at 77 K and

300 K We see that both σ2

tot(T ) and σ2(T ) contain zero-point contribution at low temperatures and σ2(T ) is linearly proportional to the temperature at high tem-perature, satisfying all standard properties of these quantities.21 Based on Eq (22) and the results of Fig 3, the value of σ2

A is about 3.50% at 300 K and 8.30% at

700 K of σ2 for Zn, but negligible at 77 K

Due to this anharmonic contribution, σ2

tot(T ) is slightly not linear at high tem-peratures The cumulant relations Eqs (21) have been calculated for Zn and Cd, and the results are shown in Fig 4 They agree with experiment at 77 K and 300 K The first relation equals zero at 0 K and approaches the constant value of 0.5 at high temperatures, the second one equals 1.5 at 0 K and also approaches 0.5 at high temperatures, as these properties were shown in theory12,17 and in experi-ment7for the other crystals They are slightly different for different materials only below the Einstein temperature Hence, we can conclude that these cumulant re-lations 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 contri-bution is negligible at 77 K, but valuable at 300 K and 500 K, especially at high

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

contributions to EXAFS phase for Zn at

Fig 4 Temperature dependence of cumulant relations Eq (17) for Zn and Cd compared to experiment at 77 K and 300 K.

Based on Eq

(22) and the results of Fig 3, the value of

ΦA

-5 -4 -3 -2 -1 0 1

77K 300K 500K

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 FEFF1by adding our anharmonic contributions For XANES, the multiple scatter-ing is important, but for EXAFS, the sscatter-ingle scatterscatter-ing is dominant,31and 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 scatter-ing 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

Zn, Present theory, single scattering, 1st shell

k(Å -1 )

-15

-10

-5

0

5

10

15

77K 300K

(a)

for the other crystals They are

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

R(Å)

0.0 0.1 0.2 0.3 0.4

77K 300K

(b) Fig 6 (a) Calculated EXAFS and (b) their Fourier transform magnitudes for the first shell for single scattering of Zn at 77 K, 300 K and 500 K.

Zn, 300K

3Å -1 < k < 13.5Å -1

R(Å)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Expt.

Present theory FEFF

(a)

3Å -1 < k < 12.85Å -1

R(Å)

0.00 0.05 0.10 0.15 0.20 0.25 0.30

FEFF Present theory exp

(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 contribu-tions 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 occurs at high temperatures because the anharmonicity is not included