DSpace at VNU: Anharmonic effective potential, correlation effects, and EXAFS cumulants calculated from a morse interaction potential for fcc metals

Anharmonic Effective Potential, Correlation Effects, and EXAFS Cumulants Calculated from a Morse Interaction Potential for fcc Metals Nguyen VANHUNGand Paolo FORNASINI1 Department of Physi

Anharmonic Effective Potential, Correlation Effects, and EXAFS Cumulants Calculated from a Morse Interaction Potential for fcc Metals

Nguyen VANHUNGand Paolo FORNASINI1 Department of Physics, University of Science, VNU Hanoi, 334 Nguyen Trai, Hanoi, Vietnam

1 Dipartimento di Fisica, Universita` di Trento, Via Sommarive 14, I-38050 Povo (Trento), Italy (Received March 2, 2007; accepted June 13, 2007; published August 10, 2007)

Anharmonic effective pair potentials and effective local force constants have been studied for fcc metals, assuming an interaction Morse potential and taking into account the influence of nearest

neighbours of absorber and backscatterer atoms Analytical expressions for the first three extended X-ray

absorption fine structure (EXAFS) cumulants, as well as for the atomic mean square displacements, have

been derived as a function of the Morse parameters Numerical results for copper and nickel are

compared with experimental data A good agreement is found for the second cumulant Non-negligible

discrepancies are instead found for the first and third cumulants, which are tentatively attributed to the

central nature of the Morse potential, which neglects many-body effects

KEYWORDS: anharmonicity, EXAFS, Debye–Waller factor

DOI: 10.1143/JPSJ.76.084601

1 Introduction

Extended X-ray absorption fine structure (EXAFS) has

developed into a powerful probe of local atomic structure

and thermal effects of substances.1–12)To take into account

thermal vibrations, the EXAFS function for a given

scatter-ing path is expressed in terms of a canonical average of all

distance-dependent factors:4)

ðkÞ / = f ðk; ; RÞe2iðkÞ e

2ikre2r=ðkÞ

r2

where r is the instantaneous distance and R an average

distance The canonical average of eq (1) is equivalent to a

real space average:13)

e2ikre2r= ðkÞ

r2

¼

Z 1 0 Pðr; Þe2ikrdr ð2Þ

where

is an effective distribution of distances, ðrÞ being the real

distribution.14)In case of moderate disorder, the integral on

the right-hand side of eq (2) can be conveniently

para-metrized in terms of a few leading cumulants14)of the real

distribution, typically the first three: ð1Þ, ð2Þ, and ð3Þ The

cumulants of the real distribution have been elsewhere11,15)

indicated by Ci (i ¼ 1; 2; 3; ), with a different meaning of

the first cumulant C1is the average value of the distribution

of distances,11,15)ð1Þis the thermal expansion with respect

to the equilibrium distance.6,8) In principle, C

1 and ð1Þ should have the same temperature dependence, but differ in

absolute values The third cumulant ð3ÞC

3 measures the asymmetry of the distribution The second cumulant 2

C is the variance of the distribution, and corresponds, to a

good accuracy, to the parallel mean square relative

displace-ment (MSRD)

2¼ hu2ki ¼Dð ^R0 ðubuaÞ2E

ð4Þ

where ub and ua are the instantaneous displacements of backscatterer and absorber atoms, respectively, and ^R0 is the unit vector along the equilibrium distance.2) Since EXAFS is sensitive to the relative motion, the MSRD is the sum of the uncorrelated mean square displacements (MSDs)

u2 of the two atoms minus the displacement correlation function (DCF) CR

For a two-atomic molecule, the EXAFS cumulants can be expressed as a function of the force constant of the one-dimensional bare interaction potential.6,16)For many-atomic systems, the EXAFS cumulants can be connected by the same analytical expressions to the force constants of a one-dimensional effective pair potential The relationship be-tween EXAFS cumulants and effective potential on the one hand, and physical properties of many-atomic systems on the other, is still a matter of debate, in particular with reference

to the very meaning of the effective potential13,17) and its possible dependence on temperature.18,19) The interest towards this subject has been enhanced by recent temper-ature dependent EXAFS studies, which have questioned the equivalence of the first and third cumulants for measuring the thermal expansion of interatomic bonds.11,20)

A method for calculating the EXAFS cumulants from the force constants of the crystal potential, based on first principles finite temperature many-body perturbation theory, has been proposed by Fujikawa and Miyanaga.21) The method has been thoroughly applied mainly to one-dimen-sional systems,22)and some attempts have been done also for fcc crystals.23) Promising results on three-dimensional systems have been obtained by path-integral techniques, based on the use of effective potentials24)or on Monte Carlo sampling.25)

A simpler phenomenological approach was developed by Hung and Rehr8)(henceforth cited as HR), who derived an anharmonic correlated model for the effective potential taking into account the interaction of absorber and back-scatterer atoms with their nearest neighbours via a Morse potential HR expressed the second and third EXAFS cumulants, as well as the coefficient of thermal expansion,

as a function of the Morse potential parameters for the case



E-mail: hungnv@vnu.edu.vn

Vol 76, No 8, August, 2007, 084601

#2007 The Physical Society of Japan

084601-1

of an fcc metal, and compared the theoretical results with the

then available experimental data for copper The HR

approach, focussing on a local cluster, is suited for treating

amorphous and crystalline systems on the same ground

New refined experimental data have recently been

published on EXAFS cumulants11) and on diffraction

thermal factors26) of copper, stimulating a more accurate

check of theoretical results This paper is a natural extension

of HR Analytical expressions of the anharmonic effective

pair potentials and local force constants have been obtained

for the first coordination shell of copper and nickel, whence

the espressions of EXAFS cumulants have been calculated

Besides, a calculation has been attempted also for the

single-atom effective potential and the uncorrelated MSD

Theoretical results have been compared with recent

experimental data from both EXAFS and diffraction

experi-ments, allowing for a careful evaluation of strengths and

limitations of the HR procedure.8) The HR approach

represents some sort of a ‘‘theoretical laboratory’’, where

the connection between the three-dimensional properties of

solids and the one-dimensional effective pair potentials can

be attempted on the basis of a simple starting model and

subsequently ameliorated by a progressive refinement of

approximations The comparison with experimental results

performed in this work represents a check of the first step,

and gives suggestions for further steps

In §2 the theoretical approach of HR8) is presented with

some detail for the case of fcc metals, to make it more

accessible to non specialists, and extended to the

uncorre-lated MSDs In §3 the EXAFS cumulants and the MSDs

calculated as a function of temperature from the Morse

potential parameters are presented and compared with

experimental data The possible origin of discrepancies is

discussed Section 4 is dedicated to conclusions

For perfect monatomic crystals, the parallel MSRD is

given by

Here the MSD has been defined as

u2¼ ðu a ^R0Þ2

¼ ðu b ^R0Þ2

where ua¼ub are the instantaneous displacements of

absorber and backscatterer atoms, respectively, and ^R0 is

the unit vector connecting the two atoms, so that the DCF is

given by

CR¼2 ðu a ^R0Þðub ^R0Þ

¼2u22: ð7Þ The anharmonic effective potential can be expressed as a

function of the displacement x ¼ r  r0 along the R^0

direction, r and r0 being the instantaneous and equilibrium

distances between absorber and backscatterer atoms,

respec-tively:

VeffðxÞ ’1

2k0x

2

where k0is an effective local force constant and k3is a cubic

parameter giving the asymmetry due to anharmonicity

(Here and in the following, the constant contributions are

neglected.)

For the calculation of thermodynamical parameters, we use the further definitions a ¼ hr  r0i and y ¼ x  a,6) to rewrite eq (8) as

VeffðyÞ ’ ðk0þ3k3aÞay þ1

2keffy

2þk3y3; ð9Þ

where keff is in principle different from k0 Making use of quantum statistical methods,6,27) the physical quantities are determined by an averaging proce-dure using the statistical density matrix  and the canonical partition function Z, e.g.,

hymi ¼ 1

ZTrðy

mÞ; m ¼ 1; 2; 3; ð10Þ

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, ^aa and ^ay, respectively

y  a0ðaa þa^yÞ; a20¼h!E=2keff; ð11Þ and use the harmonic oscillator state jni as the eigenstate with the eigenvalue En¼nh!E, ignoring the zero-point energy for convenience (here !E is the Einstein frequency)

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

VðxÞ ¼ D e 2x2ex

’D 1 þ  2x23x3

; ð12Þ where  describes the width of the potential and D is the dissociation energy

The above expressions will be used in the next sub-sections, where the presence or absence of an apex s will label the quantities concerning the MSD and the MSRD, respectively

2.1 EXAFS cumulants and mean square relative displacement

The case of relative vibrations of absorber and back-scatterer atoms, including the effect of correlation, has been treated in HR.8) According to eq (2) of HR, taking into account only the nearest neighbour interactions, the effective pair potential is given by

VeffðxÞ ¼ VðxÞ þX

i¼a;b

X j6¼a;b

2x ^R

0 ^Rij

¼VðxÞ þ 2V x

2

þ8V x

4

þ8V x 4

; ð13Þ

where the first term on the right concerns only absorber and backscatterer atoms, the remaining sums extend over the remaining neighbours, and the second equality is for monatomic fcc crystals

Applying the Morse potential (12) to eq (13) and comparing with eq (8), for the case of correlated atomic vibrations the force constants are now expressed in terms of the Morse parameters as

k0 ¼5D2; k3¼ 3

4D

3

In accordance with eq (9), the potential (13) can be expressed as

084601-2

VeffðyÞ ’ 5D2a 1  9

20a

y

þ5

2D

2 1  9

10a

y23

4D

3y3; ð15Þ

where the force constant is different from that of the

potential VeffðxÞ:

keff¼5D2 1  9

10a

¼ !2E; k3 ¼ 3

4D

3 : ð16Þ

Here !E is the correlated Einstein frequency, to which it

E¼h!E=kB, and is the reduced mass Note that the third order

coefficient has here the correct value 3/4 instead of 5/4 as

reported in HR

Using the above results for the correlated atomic

vibrations and the procedure depicted by eqs (10) and

(11), as well as the first order thermodynamic perturbation

theory considering the anharmonic component in the

potential (15), we derived the EXAFS cumulants The

second cumulant (parallel MSRD) is expressed as

2ðTÞ ¼ 021 þ z

1  z; z ¼ expðh!E=kBTÞ; ð17Þ where 2

0 ¼h!E=10D2 is the zero-point contribution to the

MSRD The first cumulant is

ð1ÞðTÞ ¼9

20

directly proportional to the second cumulant The third

cumulants is

ð3ÞðTÞ ¼3

10 3ð

2Þ22ð02Þ2

ð19Þ

and the thermal expansion coefficient

T¼ 9D

3 4RkBT2 ð2Þ2 ð20Þ2

where R is the bond length

To calculate the total MSRD including anharmonic

contributions10)

2tot¼2þð220Þ ð21Þ

an anharmonic factor has been derived

ðTÞ ¼9

2

2 1 þ3

4R

2 1 þ3

4R 2

2.2 Mean square displacement and correlation function

Taking into account only the influence of the N nearest

atomic neighbors, a single-atom effective potential28)can be

obtained

Veffs ðxÞ ¼XN

j¼1

Vðx ^R0 ^RjÞ

¼VðxÞ þ VðxÞ þ 4V x

2

þ4V x

2

where ^Rj are the unit vectors of neighbouring atoms

with respect to the equilibrium position of the central

atom, and the second equality is valid for monatomic

fcc crystals

Applying the Morse potential (12) to eq (23) and comparing with eq (8), for the case of single atomic vibrations we obtain the effective force constant and the cubic parameter

k0s¼mð!sEÞ2¼8D2; ks3¼0: ð24Þ The fact that k3¼0 shows that taking into account the influence of all nearest neighbours compensates the anhar-monic component present in each separate term of the potential (23)

The Einstein frequency !s

E, and the corresponding temperature, can the be expressed as

!sE¼2 ffiffiffiffiffiffiffiffiffiffiffiffi

2D=m

p

; Es ¼h!sE=kB; ð25Þ

m and kBbeing the atomic mass and the Boltzmann constant, respectively

In accordance with eq (9), the potential (23) can be expressed as

Veffs ðyÞ ’ 8D2ay þ 4D2y2; ð26Þ with the same force constants (24) as the potential Vs

effðxÞ:

keff¼k0 Based on the unperturbed statistical density matrix 0and the unperturbed canonical partition function Z0

Z0¼Trð0Þ ¼X

n expðnh!sEÞ

¼X1 n¼0

zns ¼ 1

1  zs;

ð27Þ

where ¼ 1=kBT and zs¼expðh!s

E=kBTÞ, we determined the MSD

u2¼ hy2i ’ 1

ZTrð0y

2

Þ ¼1 Z

X n

enh!sEhnjy2jni

¼2a20ð1  zsÞX

n ð1 þ nÞzns ¼ h!s

E 2ks eff

1 þ zs

1  zs

¼ h!s E 16D2

1 þ zs

1  zs¼u

2 0

1 þ zs

where u2 ¼h!s

E=16D2 is the zero-point contribution to the MSD

2.3 Low and high temperature limits

It is useful to consider the high-temperature (HT) limit, where the classical approach is applicable, and the low-temperature (LT) limit, where the quantum theory must be used.8)In the HT limit we use the approximation

zðzsÞ ’1  h!Eð!sEÞ=kBT ð29Þ

to simplify the expressions for the thermodynamic param-eters In the LT limit, zðzsÞ !0, so that we can neglect zðz2

sÞ and higher power terms These results are written in Table I

Table I Expressions of u 2 ,  2 , C R for fcc crystals in the LT and HT limits.

C R 2u 2 ð1 þ 2z s Þ   2 ð1 þ 2zÞ k B T=20D 2

084601-3

Note that from Table I the functions u2, 2, CRare linearly

proportional to the temperature at high-temperatures and

contain the zero-point contributions at low-temperatures,

satisfying all standard properties of these quantities.29)

2.4 Comparison with single-bond model

For comparison, we apply the Morse potential eq (12) to

the anharmonic correlated single bond model of ref.6

VSBðxÞ ¼1

2k

SB

to obtain the anharmonic correlated single bond potential

VSBðyÞ ¼ D2ð2  3aÞay þ1

2kSBy

2

þkSB3 y3 ð31Þ

and the local force constants

kSB¼2D2ð1  3aÞ; kSB3 ¼ D3: ð32Þ

Since a  1, the comparison of eq (32) with eq (16)

shows that the effective force constant is non-negligibly

strenghtened if the interaction with nearest neighbours is

included, while the cubic constant is slightly reduced The

effective pair potential cannot be confused with the bare

interaction potential By the same procedure used for the

effective potential, we derive the first three cumulants: The

second cumulant (parallel MSRD) is

SB2 ðTÞ ¼ ð02ÞSB1 þ zSB

1  zSB

; zSB¼expðh!SBE =kBTÞ; ð33Þ

where ð02ÞSB¼h!SBE =4D2 is the zero-point contribution

First and third cumulants are

ð1ÞSBðTÞ ¼3

2 

2

ð3ÞSBðTÞ ¼  3ð2Þ2SB2ð02Þ2SB

ð35Þ

and the thermal expansion coefficient is

T¼ 3D

3

RkBT2 ð2Þ2SB ð20Þ2SB

3 Results and Discussion

The expressions derived in §2 have been used to evaluate

the force constants of the effective pair and single-particle

potentials of copper and nickel The first three EXAFS

cumulants and the diffraction MSDs have then been

calculated We used the D and  parameters of the Morse

potential evaluated by Girifalco and Weizer30) from the

experimental values of vaporisation energy, lattice constants

and compressibility of Cu and Ni (Table II)

The calculated values of the force constants and Einstein angular frequencies are listed in Table III for the single atom and pair effective potentials [eqs (24) and (16), respective-ly] The values calculated for the single-bond potential [eq (32)] are also shown for comparison In Fig 1, the single-bond, effective single atom and effective pair poten-tials of copper are shown The single atom potential is, to first order, harmonic Let us compare the pair potentials: the force constant keff is much larger and the modulus of the cubic parameter k3 smaller for the effective potential with respect to the single-bond potential; the effect of the interaction with nearest neighbours on the absorber-back-scatterer pair is a strenghtening of the spring constant and reduction of its anharmonicity

3.1 Mean square relative displacement The second cumulants (MSRDs) of copper and nickel calculated from the Morse potential parameters are shown in

Table II Parameters of the Morse potentials for copper and nickel, from Lincoln et al., 31Þ Girifalco and Weizer, 30Þ Pirog et al 32Þ

(eV)

 (A˚1 )

D 2

(eV/A˚2 )

D 3

(eV/A˚3 )

Table III Calculated and experimental values of force constants and Einstein angular frequencies for the single atom effective, pair effective and single bond potentials, for copper and nickel.

k s eff

(N/m)

! s E

(10 13 rad/s)

k eff

(N/m)

! E

(10 13 rad/s)

k 3

(eV/A ˚ 3 )

k SB eff

(N/m)

! SB E

(10 13 rad/s)

k SB 3

(eV/A ˚ 3 )

0 0.4 0.8 1.2

x (Å)

Cu

Fig 1 Comparison of calculated effective single atom potential (dashed line), effective pair potential (continuous line), and single bond pair potential (dotted line) for Cu.

084601-4

Fig 2 The dashed lines represent the harmonic expression

eq (17), while the continuous lines include the anharmonic

contribution [eq (21)] The difference is of the order of 3%

at 600 K, and can be considered negligible with respect to

the typical accuracy of experimental data The MSRDs

calculated from the single-bond potentials (dash-dotted

lines) are instead much larger, due to the smaller value

of the force constant keff

The calculated MSRDs are compared with recently

published data from EXAFS experiments: Fornasini et al.11)

for copper from 4 to 500 K (full circles), Pirog et al.32)for

both copper and nickel from 300 to 630 K (crossed squares),

Yokoyama24)for nickel from 40 to 300 K (open circles) The

agreement is particularly good for copper, where the

discrepancy between theory and experiments is comparable

with the discrepancy between the experiments In the case of

nickel, a discrepancy is observed not only between

theoret-ical calculations, but also between experimental values It is

anyway worth noting that the calculations by Katsumata

et al.23)for nickel are based on significantly different values

of the Morse parameters than the present calculations

These results, besides stressing the difference between

single-bond and effective potential, show that a relatively

simple model based on the interaction with

nearest-neigh-bours via a Morse potential can in principle reproduce the

amplitude of relative displacements of absorber and

back-scatterer atoms as a function of temperature in fcc metals

A reverse procedure was used by Pirog et al.,32) to

evaluate the parameters of the Morse potentials from the

experimental values of the first three EXAFS cumulants,

based on the relations of HR in the high-temperature

classical approximation The values of Morse parameters

obtained by Pirog et al are shown in Table II One can see

that the values of the product D2 of Pirog et al and of

Girifalco and Weizer agree for copper, but are slightly different for nickel, and this explains the slight discrepancy between theory and experiment for the MSRD of nickel The difference of D2 values for nickel is however of the same order of magnitude of the difference between the values of Girifalco and Weizer30)and of Lincoln et al.31)for copper It

is thus difficult to assess wether the slight discrepancy between theory and experiment found for nickel is due to the approximation of the HR model or to inadequacy of the starting Morse parameters

3.2 Third EXAFS cumulant and thermal expansion The third cumulants evaluated from the Morse potential parameters for copper and nickel are shown in Fig 3 and compared with experimental results The inadequacy of the single-bond model (dash-dotted line) with respect to the effective potential model in reproducing experimental data is evident: the single-bond third cumulant is much higher than both the experimental and the effective potential third cumulants The influence of neighbouring atoms reduces the asymmetry of the distance distribution, as a joint effect of reduction of the cubic parameter k3and of the MSRD 2

A non negligible residual discrepancy can however be noticed also between the experimental values and the values calculated from the effective pair potential (continuous line) For copper, theoretical values underestimate the experimen-tal ones by about 30% For nickel, a much better agreement

is found, in particular for the low-temperature values One possible cause of the discrepancy between theory and experiment could be the inadequacy of a purely central potential in accounting for the asymmetry of the effective potential Actually, in a recent paper25) dedicated to the calculation of the EXAFS cumulants of the first shell of copper by the path-integral Monte Carlo technique, a good

0

0.01

0.02

Cu

0

0.01

0.02

T (K)

2 )

Ni

Fig 2 Temperature dependence of the MSRD  2

tot ðTÞ (continuos lines) for Cu (upper panel) and Ni (lower panel) calculated from eq (21); the

dashed line is the harmonic part, while the dash-dotted line is calculated

from the single-bond potential The experimental data are from ref 32

(crossed squares) for Cu and Ni, from ref 11 (full circles) for Cu and

from ref 24 (open circles) for Ni The dotted line is the theoretical

calculation for Ni, from ref 23.

0 10 0

3 10-4

6 10-4

0 Cu

0 100

3 10-4

6 10-4

T (K)

Ni

Fig 3 Temperature dependence of the third cumulant  ð3Þ ðTÞ (continuos lines) for Cu (upper panel) and Ni (lower panel) calculated from eq (19); the dash-dotted line is calculated from the single-bond potential The experimental data are from ref 32 (crossed squares) for Cu and Ni, from ref 11 (full circles) for Cu and from ref 24 (open circles) for Ni The dotted line is the theoretical calculation for Ni, from ref 23.

084601-5

agreement with the experimental values of the third

cumulant could be obtained only when a many-body

interaction potential was used

Single-bond and effective pair potentials also differ in the

evaluation of the first cumulant ð1Þ, corresponding to the

thermal expansion with respect to the equilibrium distance

(Fig 4, top panel) The thermal expansion of the distance

between a pair of isolated atoms interacting via a Morse

potential is strongly reduced when the atomic pair is

embedded within a crystal Different are also the zero-point

thermal expansions

The comparison with experimental results for copper is

done in the lower panel of Fig 4 An EXAFS study of

copper in the temperature range from 4 to 500 K has been

recently published by Fornasini et al.;11) the data analysis,

performed by the ratio method, led to relative values of the

first cumulant CðTÞ ¼ CðTÞ  Cð4 KÞ, where C is the

average value of the distribution of distances (circles in

Fig 4) In the same paper, the thermal expansion was also

evaluated according to the Frenkel and Rehr6) formula,

a ¼ 3k3C

2=k0, using the quadratic and cubic constants k0

and k3 calculated from the second and third experimental

cumulants Since no zero-point thermal expansion could be

measured by the relative values C

1ðTÞ of the first cumulant obtained through the ratio method, relative values ð1Þ

(continuous line) and a (diamonds) have been plotted in

Fig 4, lower panel, in order to facilitate comparisons Pirog

et al.32) have plotted the interatomic distances measured

by EXAFS in the temperature interval from 290 to 570 K;

the values have been vertically shifted so as to match, on

the average, the C data of Fornasini et al Also shown

in Fig 4 is the first-shell thermal expansion Rc measured

by Bragg diffraction (dashed line)

It is well known11,33,34)that the average distance measured

by the first EXAFS cumulant C ¼ hjrbrajiis larger than the distance between average atomic positions measured by Bragg diffraction, Rc¼ jhrbi  hraij, owing to the effect of thermal vibrations perpendicular to the bond direction Correspondingly, the thermal expansion C (circles) is larger than the thermal expansion Rc (dashed line) It has also been experimentally verified11,20) that the thermal expansion due solely to the shape of the effective potential, and measured by6) a ¼ 3k3C=k0 (diamonds) does not correspond to the first cumulant thermal expansion (circles) The discrepancy has been tentatively explained in terms

of temperature dependence of the minimum position of the effective potential, but a thorough investigation on the subject is still lacking

The first cumulant ð1Þ, which has been calculated from the Morse potential parameters through eq (18), only depends on the shape of the effective potential and should then directly compared with the parameter a of ref.6 One should then expect a correspondence between the theoretical values ð1Þ (continuous line) and the experimental values

a (diamonds) The non negligible discrepancy between

ð1Þand a confirms the limitations of the model based on the central Morse potential

3.3 Mean square displacement The uncorrelated single-atom MSD calculated for copper from the Morse potential parameters [eq (28)] is compared

in Fig 5 with recent experimental values obtained by diffraction of Mo¨ssbauer gamma rays.26,35) The theoretical model underestimates the MSD u2by about 30%, due to an overestimate of the force constant ks

0 According to eq (5), since the MSRD 2 is well reproduced, the model under-estimates also the DCF CR

The inefficiency in reproducing the MSDs (Fig 5), in spite of the ability in reproducing the MSRDs (Fig 2), can

be explained as follows The model here used, based on the simple projection of the nearest-neighbour interaction potentials along an interatomic direction, is quite ineffective

in accounting for the long-wavelengths acoustic phonons, which can strongly affect the uncorrelated MSD but have negligible influence on the MSRDs

0

0.02

0.04

0.06

(a)

0

0.02

0.04

(b)

T (K)

Fig 4 (a) Temperature dependence of the calculated first cumulant

 ð1Þ ðTÞ of the first shell of copper, calculated from the effective potential

(continuos line) and from the single-bond potential (dash-dotted line).

(b) Variation  ð1Þ ðTÞ of the calculated first cumulants with respect to

the zero kelvin value, compared with the experimental EXAFS data:

from ref 32 (first cumulant, crossed squares) and from ref 11 (first

cumulant, circles, and thermal expansion from third cumulant, diamonds),

as well as with the vlues of thermal expansion from Bragg diffraction

(dashed line).

0 0.01 0.02

2 )

T (K) Cu

Fig 5 Temperature dependence of the MSD u 2 for copper calculated from Morse potential parameters through eq (28) (continuous line) and measured by Day et al 26) (diamonds) and by Martin and O’Connor 35) (open squares) The dashed line is the MSD reconstructed from the MSRD on the hypothesis of validity of the Debye model (see text).

084601-6

An approximate method for obtaining the MSD values

of a metal from the experimental MSRD values can be

introduced, in the hypothesis that the temperature

depend-ence of both quantities can be described by suitable Debye

models, uncorrelated and correlated, respectively, having

the same Debye temperature.2,3) The ratio u2=2 can be

calculated from the Debye models and used to recover the

values of u2 from the values of 2 The ratio is in principle

temperature dependent, but, as a first approximation, one

can use its high-temperature asymptotic value

As an example, for the case of copper we used the Debye

asymptotic value, u2=2’5=6, to evaluate the MSD u2

(dashed line in Fig 5) from the MSRD 2 calculated from

the Morse potential parameters (continuous line in Fig 2)

This procedure based on the Debye ratio can be

interest-ing for the followinterest-ing reasons: a) It allows a better evaluation

of the MSD from the Morse potential parameters; b) It

allows the approximate evaluation of the MSD from the

experimental values of MSRD (it is generally easier to get

accurate experimental values of the MSRD from EXAFS

than of the MSD from diffraction)

4 Conclusions

In this work we explored the effectiveness of anharmonic

effective pair and single-atom potentials, evaluated from the

parameters of a Morse interaction potential and taking into

account nearest-neighbour interactions, in reproducing the

first three EXAFS cumulants and the uncorrelated MSD of

fcc metals copper and nickel The main conclusions can be

summarized as follows

a) The effective pair potential is significantly stronger and

less asymmetric (keff larger and k3 smaller) than the

single-bond potential, and much better reproduces the

experimental EXAFS cumulants

b) A very good agreement is found between theory and

experiment for the second EXAFS cumulant (parallel

MSRD) of copper; less good is the agreement for

nickel, where however a non negligible discrepancy is

present also between the available experimental data

c) The theoretical values underestimate the experimental

values of the third EXAFS cumulant; the discrepancy,

larger for copper than for nickel, has been tentatively

attributed to the radial nature of the Morse potential,

which neglects many-body effects

d) The theoretical values underestimate also the

exper-imental first cumulant; the discrepancy is due to: d1)

the inadequacy of the third cumulant (ð1Þ and ð3Þ in

HR model are strictly connected); d2) the intrinsic

difference between the experimental values C1 and

the theoretical values ð1Þ, due to vibrations

perpen-dicular to the bond, which are neglected by the present

formulation of the HR model

e) The theoretical values underestimate the uncorrelated

MSD, probably due to the difficulty of the model in

properly reproducing the effects of long-wavelendth

acoustic modes

The good agreement found for the second EXAFS

cumulant suggests that further improvements of the HR model could lead to a better reproduction also of the third cumulant and of thermal expansion

Acknowledgments The authors thanks J J Rehr and A I Frenkel for helpful comments One of the authors (N.V.H.) appreciates the partial supports by projects Nos 4 058 06 and QG.05.04 The authors are grateful to the Institute of Pure and Applied Physics for financial support in publication

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084601-7

effective pair and single-atom potentials, evaluated from the

parameters of a Morse interaction potential and taking into

account nearest-neighbour interactions,... panel) and Ni (lower panel) calculated from eq (19); the dash-dotted line is calculated from the single-bond potential The experimental data are from ref 32 (crossed squares) for Cu and Ni, from. .. inadequacy of a purely central potential in accounting for the asymmetry of the effective potential Actually, in a recent paper25) dedicated to the calculation of the EXAFS cumulants