DSpace at VNU: Pressure effects in Debye-Waller factors and in EXAFS

Simulated EXAFS of Cu and its Fourier transform magnitude using our calculated pressure-induced change in the 1st shell are found to be in a reasonable agreement with those using X-ray d

Pressure effects in Debye–Waller factors and in EXAFS

, Vu Van Hungb, Ho Khac Hieua,c, Ronald R Frahmd

a

University of Science,VNU Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam.

b

Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam.

c National University of Civil Engineering, 55 Giai Phong, Hai Ba Trung, Hanoi, Vietnam.

d

Bergische Universit¨ at-Gesamthochschule Wuppertal, FB: 8-Physik, Gauß Straße 20, 42097 Wuppertal, Germany

a r t i c l e i n f o

Article history:

Received 15 July 2010

Received in revised form

1 October 2010

Accepted 5 November 2010

Keywords:

EXAFS

Pressure dependence effects

Debye–Waller factors and cumulants

a b s t r a c t

Anharmonic correlated Einstein model (ACEM) and statistical moment method (SMM) have been developed to derive analytical expressions for pressure dependence of the lattice bond length, effective spring constant, correlated Einstein frequency and temperature, Debye–Waller factors (DWF) or second cumulant, first and third cumulants in Extended X-ray Absorption Fine Structure (EXAFS) at a given temperature Numerical results for pressure-dependent DWF of Kr and Cu agree well with experiment and other theoretical values Simulated EXAFS of Cu and its Fourier transform magnitude using our calculated pressure-induced change in the 1st shell are found to be in a reasonable agreement with those using X-ray diffraction (XRD) experimental results

&2010 Elsevier B.V All rights reserved

1 Introduction

The anharmonic EXAFS providing information on structure and

thermodynamic parameters of substances has been analyzed by

means of cumulant expansion approach[1,2] In this formulation,

an EXAFS oscillation functionw(k) is given by[3]

wðkÞ ¼FðkÞ

kR2e2R= l ðkÞIm ei f ðkÞexp 2ikR þX

n

ð2ikÞn n! sðNÞ

, ð1Þ

where k andlare the wave number and mean free path of emitted

photoelectrons, respectively, F(k) is the real atomic backscattering

amplitude,fð Þk the net phase shift andsðnÞðn ¼ 1,2,3, .Þ are the

cumulants

EXAFS is sensitive to pressure[4,5], which can cause certain

changes of cumulants including DWF leading to uncertainties in

physical information taken from EXAFS

The pressure dependence of the DWF or EXAFS second cumulant

has been measured at the wiggler beamlines of Stanford

Synchro-tron Radiation Laboratory (SSRL, USA) for Cu [6], and at the

Laboratoire Pour I’Utisation du Rayonnement Electromagne´tique

(LURE) (Orsay, France) for Kr[7,8] Such pressure effects have been

calculated by correlated Debye model[6], as well as by

Monte-Carlo (MC) simulation[7]and by Loubeyre’s model[8]to interpret

experimental results

Recently the ACEM and SMM have been used for calculation of

temperature dependence of EXAFS cumulants of crystals at zero

pressure[9] The purpose of this work is to develop the ACEM and SMM for calculation and analysis of pressure effects in DWF and in EXAFS of crystals at a given temperature Our new development presented in Section 2 is establishing and solving the equation of state to obtain the pressure-dependent lattice bond length, and then is the derivation of the analytical expressions for pressure dependence of the effective spring constant, correlated Einstein frequency and temperature, DWF or second cumulant, first and third cumulants in EXAFS Morse and Lennard-Jones (L-J) potentials have been used to characterize the interaction between each pair of atoms Numerical calculations (Section 3) using the developed ACEM and SMM show similar results for the pressure-dependent DWF of Cu (fcc), crystal Kr (fcc) and semiconductor Si (dia), where the calculated values for Kr and Cu agree well with experiment and other theoretical results [6–8] Since the experimental data for pressure dependence of EXAFS of Cu is not yet available, we compare the pressure-dependent EXAFS of Cu and its Fourier transform magnitude simulated using the FEFF code [10] and our calculated pressure-induced change in the 1st shell to those using XRD results[11] They show a reasonable agreement

2 Formalism

Using the expressions for free energy and pressure [12]we obtain the equation of state describing the pressure versus volume relation of crystal lattice in the form

Pv ¼ a 1 6

@U0

@a þyzcoth z

1 2K

@K

@a

, z ¼_o

2 , y¼kBT, ð2Þ

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n

Corresponding author.

E-mail address: hungnv@vnu.edu.vn (N Van Hung).

where P denotes the hydrostatic pressure and v the atomic volume

v ¼V/N of a crystal having volume V and N atoms, kBis Boltzmann

constant The frequencyoand the nearest neighbor distance a(P,T)

depend on pressure P and temperature T

Hence, solving this equation we can obtain the pressure- and

temperature-dependent nearest neighbor distance a(P,T)

How-ever, for our numerical calculations it is convenient to determine

first the value a(P,0) at pressure P and temperature T¼0 K In this

case Eq (2) is reduced to

Pv ¼ a 1

6

@U0

@a þ

_o0

4K

@K

@a

whereo0is the value ofoat zero temperature

We can take the pair interaction energy in crystals as the power

law using the Morse potential

jðrÞ ¼ D e2aðrr0 Þ

2e a ðrr0Þ

whereadescribes the width of potential, D the dissociation energy

and r the instantaneous bond length between the two immediate

neighboring atoms with r0being its equilibrium value; or using the

(L-J) potential

jðrÞ ¼ e

mn n

s

r

 m

m s

r

 n

where e describes the dissociation energy, and r has the

same definition as for Morse potential in Eq (4), but withsbeing

its equilibrium value The parameterse,s, n and m are determined

by fitting experimental data (e.g., cohesive energy and elastic

modulus), and the results for some crystals[13,14]are shown in

Table 1

Using (L-J) potential Eq (5), it is straightforward to get the

interaction energy U0, the quantity K and the parameterg in a

crystal[15]as

U0¼ e

nm mAn

s

a

 n

nAm s

a

 m

ð6Þ

KðaÞ ¼1

2

X

i

@2j

@u2

i b

¼Cn s

a

 n

Cm s

a

 m

¼Mo2

ð7Þ

containing the coefficients

Cn¼ enm

2a2ðnmÞ ðn þ 2ÞA

a 2 ix

n þ 4An þ 2

;

Cm¼ enm

2a2ðnmÞ ðmþ 2ÞA

a 2 ix

m þ 4Am þ 2

and

gðaÞ ¼ enm

12a4ðnmÞ Dn

s

a

 n

Dm

s

a

 m

containing the coefficients

Dn¼ ðn þ2Þðn þ 4Þðn þ6Þ Aa4ix

n þ 8þ6Aa

2

ix a 2 iy

n þ 8

18ðn þ2Þðn þ 4ÞAa2ix þ9ðn þ 2ÞAn þ 4;

Dm¼ ðm þ2Þðm þ 4Þðm þ 6Þ Aa4ix

2

ix a 2 iy

m þ 8

18ðm þ 2Þðm þ 4ÞAa2ix

Here M is the atomic mass, An, Am, Aa2ix

n , Aa2ix

m are the structural sums

of the given crystal

From Eqs (3), (5)–(7) we obtain the equation of state for crystals

at zero temperature as

Pv ¼ c1

s

r

 m

c2

s

r

 n

þ c3ðs=aÞnc4ðs=aÞm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c5ðs=aÞnc6ðs=aÞm

containing the coefficients

c1¼An enm 6ðnmÞ ; c2¼Am

enm 6ðnmÞ;

c3¼1 a

_

4 ffiffiffiffiffi M p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie nm 2ðnmÞ

r

ðn þ2Þ ðn þ2ÞAa2ix

n þ 4An þ 2

;

c4¼1 a

_

4 ffiffiffiffiffi M p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffienm 2ðnmÞ

r

ðm þ2ÞAa2ix

m þ 4Am þ 2

;

c5¼ ðn þ 2ÞAa2ix

n þ 4An þ 2; c6¼ ðm þ 2ÞAa2ix

Solving Eq (11) using the given parameters n, m, e and s

(Table 1) we get the nearest neighbor distance a(P,0) Substituting the obtained a(P,0) in Eqs (7), (9), we find the quantities K(P,0),

g(P,0) at pressure P and temperature T¼0 K

In SMM the nearest neighbor distance a(P,T) between the two immediate atoms at pressure P and temperature T can be defined as

a P,Tð Þ ¼a P,0ð Þ þy0ðP,TÞ: ð13Þ Hence, in order to calculate a(P,T) of Eq (13) using the obtained a(P,0) we substitute the obtained K(P,0), g(P,0) in the following expression for the thermally induced lattice expansion[12,16]

y0ðP,TÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2gðP,0Þy2

3K3ðP,0ÞAðP,TÞ

s

where

AðP,TÞ ¼ a1þg2ðP,0Þy2

K4ðP,0Þ a2þ

g3ðP,0Þy3

K6ðP,0Þ a3þ

g4ðP,0Þy4

K8ðP,0Þ a4,

a1¼1 þZ

2, a2¼

13

3 þ

47

6Z þ

23

6 Z

2

þ1 2

3,

a3¼  25

3 þ

121

6 Z þ

50

3 Z

2

þ16

3 Z

3

þ1 2

4

a4¼43

3 þ

93

2 Z þ

169

3 Z

2

þ83

3 Z

3

þ22

3 Z

4

þ1 2

5

z ¼_oðP,0Þ

2 , Z ¼ zcothðzÞ, oðP,0Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi KðP,0Þ M

r :

To a good approximation the parallel mean square relative displacement (MSRD) corresponds to the second cumulant

s2¼h!R:ð!ui!u0Þi2

¼ u2

, uiu0

 ui hu0i, u2

ð16Þ Here, u0and uiare the atomic displacements of the zeroth and the ith sites from their equilibrium positions, R!is the unit vector at the zeroth site pointing towards the ith site, and the brackets /S denote the thermal average The first two terms on the right-hand side are the uncorrelated mean square displacement (MSD), while the third term is the parallel displacement correlation function

Table 1

Lennard-Jones potential parameters m, n,eandsfor Cu, Kr and Si [13,14]

In SMM using the expression of the second-order moment[12]

we obtain the MSD

u2

2

þyA1þy

A1¼1

K 1 þ

2g2

K4 1 þZ

2



ðZ þ 1Þ

From Eqs (14), (16)–(18) we derive the second cumulant or

DWF of crystals at pressure P and temperature T in SMM as

s2ðP,TÞ 4g2ðP,0Þy3

K5ðP,0Þ 1 þ

Z 2



ðZ þ 1Þ þ 2 KðP,0ÞZ: ð19Þ The ACEM [17] has been used in analysis of

temperature-dependent EXAFS data[3,17,19–23] In this model the cumulants

have been calculated using the effective atomic interaction potential

jeffðxÞ 1

2keffx þ k3x3þ ¼jðxÞ þX

i a j

j m

Mi

^

R12U ^Rij

, ð20Þ

where x equals either rr0 for Morse potential or r–s for (L-J)

potential, keff is effective spring constant and k3 is the cubic

anharmonicity parameter This model has been defined as an

oscillation of a single bond pair of atoms with masses M1and M2

(e.g., absorber and backscatterer) in a given system Their oscillation is

influenced by their neighbors In the center-of-mass frame of this

bond, the model is defined further by the second equation, here

m¼M1M2=ðM1þM2Þ is reduced mass, ^R is the bond unit vector,

the sum i is over absorber (i¼ 1) and backscattering atom (i¼ 2), and

the sum j is over all their nearest neighbors, excluding absorber

and backscatterer themselves whose contributions are described by

the termj(x)

Our development in this work is the derivation of pressure

dependence of the effective spring constant keff, the correlated

Einstein frequency oE and temperature yE and the cumulants

starting from the definition of Gr ¨uneisen parameter

gG¼ @lnoE

This parametergGdepends on volume which, in turn, depends

on pressure With a slight modification of the formula found in

Refs.[18,24], the simplest parameterization of the volume

depen-dence for crystals[6]can be written as

gGðVðPÞÞ

VðPÞ ¼

gGðV0Þ

V0

where the subscript 0 indicates zero pressure

From Eqs (21), (22) we derive pressure dependence of the

correlated Einstein frequencyoEand temperatureyEas

oEðVðPÞÞ ¼oEðV0Þexp gGðV0Þ 1VðPÞ

V0

, yEðVðPÞÞ ¼_oEðVðPÞÞ

kB

, ð23Þ

VðPÞ

V0

¼a3ðP,TÞ

where a(P,T) is calculated by Eq (13)

Using these results, we obtain the pressure dependence of the

effective spring constant as

keffðVðPÞÞ ¼mo2ðVðPÞÞ: ð25Þ

From Eqs (23)–(25) we derive the EXAFS DWF of crystals as a

function of pressure P at a given temperature T

s2ðP,TÞ ¼s2ðVðPÞÞ1 þ zðVðPÞ,TÞ

1zðVðPÞ,TÞ,

s2ðVðPÞÞ ¼ _oEðVðPÞÞ

2keffðVðPÞÞ,zðVðPÞ,TÞ ¼ exp yEðVðPÞÞ=T

: ð26Þ

Making use of quantum statistical methods[25]the expressions for the 1st cumulantsð1Þ and the 3rd cumulant sð3Þ have been derived as the functions of pressure P at a given temperature T

sð1ÞðP,TÞ ¼sð1Þ0 ðVðPÞÞ1 þ zðVðPÞ,TÞ

1zðVðPÞ,TÞ, sð1Þ0 ðVðPÞÞ ¼ f1ðs2ðVðPÞÞÞ, ð27Þ

sð3ÞðP,TÞ ¼sð3Þ

0 ðVðPÞÞ3ðs2ðVðPÞ,TÞÞ22ðs2ðVðPÞÞÞ2

ðs2ðVðPÞÞÞ2 ,

sð3Þ

0 ðVðPÞÞ ¼ f3ðs2ðVðPÞÞÞ, ð28Þ

wheresð1Þ

0 ðVðPÞÞ,s2ðVðPÞÞandsð3Þ

0 ðVðPÞÞare pressure-dependent zero-point contributions to the 1st, 2nd and 3rd cumulants, respectively They are described by the functions of the zero-point contribution to the 2nd cumulants2ðVðPÞÞwith explicit form depending on crystal structure and on our using Morse or (L-J) potential

The pressure-dependent EXAFS can be calculated using Eq (1) with the above obtained pressure-dependent DWF and cumulants

3 Numerical results and discussions

Now we apply the expressions derived in previous section to numerical calculations for metal Cu, crystal Kr and semiconductor

Si The (L-J) potential parameters of Cu, Kr and Si are given in

Table 1 Morse potential parameters have the values D ¼0.3429 eV,

a¼1.3588 ˚A 1 for Cu taken from Ref [26], and D ¼0.9956 eV,

a¼1.3621 ˚A 1for Si calculated by our generalizing the method for the cubic crystals[26]to diamond structure These Morse potential parameters were obtained using experimental values of the energy

of sublimation, the compressibility and the lattice constant The pressure dependence of DWF, i.e., the MSRD or the second cumulants2(Fig 1) and the MSD u2(Fig 2) for crystal Kr at 300 K has been calculated by the ACEM and by the SMM using (L-J) potential Both methods provide similar results where the values fors2 (Fig 1) calculated by the present theory agree well with experiment[7,8]and with the other theoretical values calculated

by the MC simulation[7]and by the Loubeyre’s model (private communication)[8] Our calculated results for MSD (Fig 2) agree well with those calculated by the MC simulation [7] Fig 3

illustrates the pressure dependences2ðPÞs2ð0Þ for Cu at 300 K calculated by the ACEM using Morse potential and by the SMM using (L-J) potential The calculated results of both methods for

Fig 1 Pressure dependence of MSRD ors2 of Kr calculated by ACEM and SMM

DWF are also similar and found to be in a good agreement with

those calculated by the correlated Debye model[6] and by the

model 2 for calculation of the moments of the nearest neighbor

distance distribution from an expansion to third order of the

potential energy [6] Fig 4 shows the pressure dependence

s2ðPÞs2ð0Þ for Si at 300 K calculated by the ACEM using Morse

potential and by the SMM using (L-J) potential They have the form

similar to those for Cu but with lower values.Fig 5demonstrates

the pressure dependence of our calculated pressure induced

change in the 1st shellDr for Cu.Fig 6illustrates our calculated

pressure dependencesð3Þð ÞP sð3Þð Þ0 for the 3rd cumulant of Cu at

300 K compared to the results calculated by the methods P3 and P4

[6] Since the experimental pressure-dependent EXAFS data of Cu is

not yet available, we compared our by FEFF[10]code simulated

EXAFS spectrum of Cu (Fig 7) for the 1st shell in single scattering

and its Fourier transform magnitude (Fig 8) at 10 GPa using our

calculated pressure-induced change in the 1st shellDr¼0.041 ˚A

(Fig 4) to those usingDr ¼0.05 ˚A taken from XRD[11]and to the

Fig 2 Pressure dependence of MSD u 2 of Kr calculated by ACEM and by SMM using

(L-J) potential compared to MC simulation [7]

Fig 3 Pressure dependences2 ðPÞs2 ð0Þ of Cu calculated by ACEM using Morse

potential and by SMM using (L-J) potential compared to Debye [6] and model 2 [6]

results.

Fig 4 Pressure dependences2 ðPÞs2 ð0Þ of Si calculated by ACEM using Morse potential and by SMM using (L-J) potential.

-0.045 -0.04 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0

P (GPa)

Cu

Fig 5 Pressure dependence of calculated pressure-induced change in the 1st shell

Dr ¼ rðPÞrð0Þof Cu.

Fig 6 Pressure dependencesð3Þ ðPÞsð3Þ ð0Þ calculated by ACEM using Morse and

results at P¼0 The EXAFS amplitude and the peak height of its

Fourier transform magnitude increase, the phase of EXAFS is shifted

to the right, and the position of its Fourier transform magnitude is

shifted to the left as the pressure increases.Figs 7 and 8show a

reasonable agreement of our calculated results with those obtained

from the XRD A small discrepancy in Dr of 0.009 ˚A may be

tentatively attributed to neglecting many-body effects in the Morse

and (L-J) potentials used in our theory while XRD is

three-dimensional technique

4 Conclusions

In this work a formalism based on our further development of the

ACEM and SMM using Morse and (L-J) potentials for investigation of

the pressure effects in cumulants including DWF and in EXAFS has been developed Both methods provide similar results for the pressure dependence of DWF

Our development is establishing and solving equation of state to get the pressure dependence of the lattice bond length, and then is the derivation of the analytical expressions of pressure dependence for the 2nd cumulant or DWF, as well as for the 1st and 3rd cumulants, the effective spring constant, the correlated Einstein frequency and temperature

The good agreement of our calculated results of the pressure-dependent DWF for Kr and Cu with experiment and other theore-tical values; the reasonable agreement of our simulated EXAFS spectrum of Cu and its Fourier transform magnitude with those using experimental XRD data; as well as the significant increase in the height and the shifting of simulated EXAFS amplitude of Cu and its Fourier transform magnitude under pressure increase denote a further step of applying the ACEM and SMM to reproduction of the pressure-dependent experimental DWF and to description of the pressure effects in EXAFS

Acknowledgements

The authors thank J.J Rehr, P Fornasini and M Vaccari for useful comments This work is supported by the research project QGTD.10.02 of VNU Hanoi and in part by the research project

no 103.01.09.09 of NAFOSTED; one of the authors (V.V.H.) acknowledges the partial support of the research project no 103.01.2609 of NAFOSTED

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Fig 7 Pressure dependence of EXAFS of Cu simulated by FEFF [10] using our

calculated 2nd, 3rd cumulant,Dr ( Fig 5 ) andDr of XRD [11]

Fig 8 Fourier transform magnitudes of pressure-dependent EXAFS of Cu of Fig 7

EXAFS spectrum of Cu (Fig 7) for the 1st shell in single scattering

and its Fourier transform magnitude (Fig 8) at 10 GPa using our

calculated pressure- induced... (L-J) potentials for investigation of

the pressure effects in cumulants including DWF and in EXAFS has been developed Both methods provide similar results for the pressure dependence of... significant increase in the height and the shifting of simulated EXAFS amplitude of Cu and its Fourier transform magnitude under pressure increase denote a further step of applying the ACEM and SMM