thermodynamic quantities of metals investigated by an analytic statistical moment method

The free energy, thermal lattice expansion coefficients, mean-square atomic displacements, and specific heats at the constant volume and those at the constant pressure, C v and C p , are[r]

Thermodynamic quantities of metals investigated by an analytic statistical moment method K Masuda-Jindo,1Vu Van Hung,2and Pham Dinh Tam2

1

Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta 4259, Midori-ku, Yokohama 226-8503, Japan

2Department of Physics, Hanoi National Pedagogic University, km8 Hanoi-Sontay Highway, Hanoi, Vietnam

共Received 18 June 2002; published March 2003兲

The thermodynamic properties of metals are studied by including explicitly the anharmonic effects of the lattice vibrations going beyond the quasiharmonic approximations The free energy, thermal lattice expansion coefficients, mean-square atomic displacements, and specific heats at the constant volume and those at the constant pressure, Cv and Cp, are derived in closed analytic forms in terms of the power moments of the atomic displacements The analytical formulas give highly accurate values of the thermodynamic quantities, which are comparable to those of the molecular dynamics or Monte Carlo simulations for a wide temperature range The present formalism is well suited to calculate the thermodynamic quantities of metals and alloys by including the many body electronic effects and by combining it with the first-principles approaches DOI: 10.1103/PhysRevB.67.094301 PACS number共s兲: 64.10.⫹h, 65.40.⫺b

I INTRODUCTION

The first-principles determination of the thermodynamic quantities of metals and alloys are now of great importance for the understanding of structural phase transformations as well as for the phase diagrams computations.1–3 So far, the first-principles density functional theories4 – 8have been used extensively for the calculations of the ground state properties of various metal systems at the absolute zero temperature In the phase transformations occurring in metals and alloys at finite temperatures共under pressure P兲, the thermal lattice vi-brations共anharmonicity effects兲play an essentially important role.9,10However, most of the first-principles calculations for the structural phase transformations and alloy phase diagram computations have been done with the use of the lattice vi-bration theory in the quasiharmonic 共QH兲 approx-imation.11–15For the alloy phase diagram calculations, there have been difficulties in accounting for the anharmonicity of thermal lattice vibrations, especially for the higher tempera-ture region than the Debye temperatempera-ture because the thermal lattice expansion plays an important role and cannot be ne-glected The martensitic phase transformation in substitu-tional alloys such as the NixAl1⫺xsystem has also been

stud-ied with the QH approximation, and the temperature region treated by the QH theory is usually lower than the Debye temperature.16

The systems considered at high temperatures and high pressures require the allowance for anharmonic effects which are very essential in these regions The simplest way is to use the QH Debye-Gruăneisen theory.10 However, the results ob-tained in such a way are not always satisfactory It is noted that the Debye form of the harmonic approximation is rather crude theory The applicability of the QH method to the study of particular metals is often restricted by the isotropic Debye mode and the assumption of the mean sound velocity

v.17The temperature dependence of the lattice constant and the linear thermal expansion coefficient are calculated by minimizing the free energy with respect to the volume of the system Due to their simplicity, pair potentials are often used

for genetic studies of trends among a given class of metallic materials Therefore, they not account for mostly impor-tant many-body electronic effects in metallic systems, and they cannot be relied on for properties of real materials

A number of theoretical approaches have been proposed to overcome the limitations of the QH theories The first calculation of the lowest-order anharmonic contributions to the atomic mean-square displacement 具u2典 or the Debye-Waller factor was done by Maradudin and Flinn18 in the leading-term approximation for a nearest-neighbor central-force model Since then, many anharmonic calculations in-cluding the lowest-order anharmonic contributions have been performed for metal systems.19,20 The method requires ac-knowledge of a number of Brillouin-zone sums14 and the calculations are performed for the central-force model crys-tals Recently, some attempts have been made to take into account the bond length dependence of bond stiffness tensors in the calculations of the free energy of the substitutional alloys.21,22 The anharmonic effects of lattice vibrations on the thermodynamic properties of the materials have also been studied by employing the first-order quantum-statistical perturbation theory23–25 as well as by the first-order self-consistent共SC兲phonon theories.26 –31The theories have been used to analyze, e.g., the temperature dependence of ex-tended x-ray absorption fine-structure 共EXAFS兲spectra and the phonon frequencies However, the previous anharmonic-ity theories are still incomplete and have some inherent drawbacks and limitations

In the present study, we use the finite-temperature mo-ment expansion technique to derive the Helmholtz free ener-gies of metal systems, going beyond the QH approximations The thermodynamic quantities, mean-square atomic dis-placements, specific heats, and elastic moduli are determined from the explicit expressions of the Helmholtz free energies The Helmholtz free energy of the system at a given tempera-ture T will be determined self-consistently with the equilib-rium thermal lattice expansions of the crystal

evaluation of the internal energy of the system In metals the long-range Coulomb interaction and the partially filled va-lence bands lead to interatomic forces that are inherently many-body in nature For more than a decade, the embedded-atom method共EAM兲41– 45and the second-moment approximation 共SMA兲of the TB scheme have been the two most common approaches, able to overcome the major limi-tations of two-body pair potentials.18,46,47The physical basis of EAM models makes them valid, especially for normal or noble metals, whereas SMA is a priori well suited for tran-sition elements 共with narrow d-band bonding兲

In Sec II, we will make a general derivation of the ther-mal lattice expansion and Helmholtz free energy of the monoatomic cubic metals based on the fundamental prin-ciples of quantum-statistical mechanics The thermodynamic quantities of the metals are then derived in terms of the power moments of the atomic displacements from the Helm-holtz free energy of the system Section III includes our main calculation results of the thermodynamic quantities of some cubic metals Finally, Sec IV summarizes the present study

II THEORY

We derive the thermodynamic quantities of metals, taking into account the higher-共fourth-兲order anharmonic contribu-tions in the thermal lattice vibracontribu-tions going beyond the QH approximation The basic equations for obtaining thermody-namic quantities of the given crystals are derived in a fol-lowing manner: The equilibrium thermal lattice expansions are calculated by the force balance criterion and then the thermodynamic quantities are determined for the equilibrium lattice spacings The anharmonic contributions of the ther-modynamic quantities are given explicitly in terms of the power moments of the thermal atomic displacements

Let us first define the lattice displacements We denote uil

the vector defining the displacement of the ith atom, in the lth unit cell, from its equilibrium position The potential en-ergy of the whole crystal U(uil) is expressed in terms of the

positions of all the atoms from the sites of the equilibrium lattice We may assume that this function has a minimum when all the uilare zero, for the perfect lattice is presumably

a configuration of stable equilibrium We use the theory of small atomic vibrations, and expand the potential energy U as a power series in the Cartesian components, uilj , of the displacement vector uilaround this point

U⫽U0⫹兺

i,l, j 冋

⳵U ⳵uilj册

eq

uilj⫹ 兺

ii⬘,l,ll⬘, j j⬘冋 ⳵2U ⳵uilj⳵ui

⬘l⬘ j⬘ 册

eq

uiljuij⬘⬘l⬘

⫹¯, 共1兲

where U0 denotes the internal共cohesive兲energy of the sys-tem If we truncate the above expansion of Eq.共1兲up to the second-order terms, then the full interatomic potential is re-placed by its quadratic expansion about the equilibrium atomic positions The system is then equivalent to a collec-tion of harmonic oscillators, and diagonalizacollec-tion of the cor-responding dynamical matrix yields the squares of the

normal-mode frequencies共phonon spectrum兲.48This scheme is called as the QH approximation

In the present study the thermodynamic quantities are cal-culated with the use of the electronic many-body potentials or the potentials derived by EAM We note that the present analytic formulation is quite useful when we combine it with the ab initio theoretical scheme by numerically evaluating the harmonic k and anharmonic␥1and␥2parameters which will be defined in the subsequent derivations The SMA TB scheme is well suited to describe the cohesion of transition metals since they are elements with a partially filled narrow d band superimposed on a broad free-electron-like s-p band. The narrowness of the d band, especially in the 3d series, is a consequence of the relative constriction of the d orbitals compared with the outer s and p orbitals As one moves across the periodic table, the d band is gradually being filled. Most of the properties of the transition metals are character-ized by the filling of the d band and ignoring the sp electrons. This constitutes Friedel’s d-band model which further as-sumes a rectangular approximation for the density of states ␳i(E) such that the bonding energy of the solid is primarily

due to the filling of the d band and proportional to its width. In the SMA, the bonding energy is then proportional to the root of the second moments冑␮i(2) In metals, an important contribution to the structure comes from the repulsive term represented as a sum of pair potentials accounting for the short-range behavior of the interaction between ions There-fore, the cohesive energy of a transition metal consists of

Ecoh⫽Erep⫹Ebond 共2兲

The SMA has been used to suggest various functional form for interatomic potentials in transition metals such as the Finnis-Sinclair potential,34 the closely related embedded atom potential, and the TB SMA, also referred in the litera-ture as to Gupta potential.33The functional form we adopted here for elemental metals is that of the many-body SMA potential

Eci⫽1 Ni兺⫽1

N

冠A兺

j⫽i N

exp冋⫺p冉ri j r0

⫺1冊册

⫺再␰i j

2兺

j⫽i N

exp冋⫺2q冉ri j r0

⫺1冊册冎

1/2

冡, 共3兲 which has five parameters: ␧0, ␰i j 共for pure metals, ␰i j ⫽␰0), p, q, and r0 The total cohesive energy Ec of the

system is then written as the sum of the Eci The parameters

The SMA TB potentials have been further extended and revised not only for bulk metal systems but also for nanos-cale materials For Rh clusters, Chein, Blaston-Barojas, and Pederson38 proposed the size-dependent parameters of the SMA TB potentials, on the basis of their generalized gradient approximation 共GGA兲 calculations A different parametriza-tion strategy was introduced by Sigalas and Papaconstantopoulos39in which the parameters were fitted to local density approximation 共LDA兲 calculations of the total energy as a function of lattice constant Li, Barojas, and Papaconstantopoulos40 fitted the SMA TB potential param-eters to a LDA database that consists of the total energy as a function of the lattice constant for both bcc and fcc lattices, rather than the fitting procedure to experimental quantities To simulate the long-range nature of the metallic bonding by sp electrons in alkali metals, the interactions up to 12th-neighbor shells 共228 atoms in bcc crystal兲 are taken into account.40Their potentials fitted to the first-principles LDA results are available for various metals, and more refined nonorthogonal basis TB schemes39 are also proposed for the quantitative calculations The present thermodynamic formu-lation is well suited to couple with any kind of TB schemes mentioned above The SMA TB potential parameters used in the present calculations are given in Table I

We now consider a quantum system, which is influenced by supplemental forces ␣i in the space of the generalized

coordinates qi.49–51 For simplicity, we only discuss

mon-atomic metallic systems, and hereafter omit the indices l on the sublattices Then, the Hamiltonian of the crystalline sys-tem is given by

Hˆ⫽Hˆ0⫺兺

i ␣i

qˆi, 共4兲

where Hˆ0 denotes the crystalline Hamiltonian without the supplementary forces ␣i and the carets represent operators

The supplementary forces␣i act in the direction of the

gen-eralized coordinates qi The thermodynamic quantities of the

harmonic crystal 共harmonic Hamiltonian兲will be treated in the Einstein approximation In this respect, the present for-mulation is similar conceptually to the treatment of quantum Monte Carlo method by Frenkel.52,53

After the action of the supplementary forces␣ithe system

passes into a new equilibrium state For obtaining the statis-tical average of an thermodynamic quantity 具qk典a for the

new equilibrium state, we use the general formula for the correlation Specifically, we use a recurrence formula54based on the density matrix in the quantum statistical mechanics 共for more details see Appendix A兲

具Kˆn⫹1典a⫽具Kˆn典a具qˆn⫹1典a⫹␪ ⳵具Kˆn典a

⳵␣n⫹1

⫺␪兺

m⫽0

⬁ B

2m 共2m兲!冉

iប ␪冊

2m

冓⳵Kˆn共2m兲 ⳵␣n⫹1冔a

, 共5兲

where␪⫽kBT, m is the atomic mass, and Kˆn is the

correla-tion operator of the nth order:

Kˆn⫽

2n⫺1关 关qˆ1,qˆ2兴⫹qˆ3兴⫹ ]⫹qˆn]⫹ 共6兲 In Eq.共5兲above, the symbol具¯典expresses the thermal av-eraging over the equilibrium ensemble with the Hamiltonian Hˆ and B2n denotes the Bernoulli numbers.关qi,qj兴⫹

repre-sents the anticommutation relation The general decoupling formula of Eq.共5兲enables us to get all moments of the lattice system and to investigate the nonlinear thermodynamic prop-erties of the materials, taking into account the anharmonicity of the thermal lattice vibrations The Helmholtz free energy TABLE I Parameters of the second moment TB potentials for cubic metals

A共eV兲 ␰ 共eV兲 p q Ec 共eV/atom兲 a共Å兲

Al共1兲a 0.1221 1.316 8.612 2.516 ⫺3.339 4.050

Al共2兲b 0.0334 0.7981 14.6147 1.112 ⫺3.339 4.050

Ni 0.1368 1.7558 10.00 2.70 ⫺4.435 3.523

Cu 0.0993 1.3543 10.08 2.56 ⫺3.544 3.615

Rh 0.0629 1.660 18.450 1.867 ⫺5.752 3.803

Pd 0.1746 1.718 10.867 3.742 ⫺3.936 3.887

Ag共1兲a 0.1028 1.1780 10.928 3.139 ⫺2.960 4.085

Ag共2兲b 0.1231 1.2811 10.12 3.37 ⫺2.960 4.085

Au 0.2061 1.790 10.229 4.036 ⫺3.779 4.079

Pt 0.2975 2.695 10.612 4.004 ⫺5.853 3.924

Li 0.0333 0.3249 7.75 0.737 ⫺1.63 3.49

Na 0.0159 0.2910 10.13 1.30 ⫺1.13 4.29

K 0.0205 0.2625 10.58 1.34 ⫺0.93 5.24

Rb 0.0194 0.2464 10.48 1.40 ⫺0.85 5.61

Cs 0.0205 0.2421 9.62 1.45 ⫺0.80 6.04

aindicates parameters taken from Ref 36.

of the system can then be obtained by taking into account the higher-order moments 共up to fourth order兲

The atomic force acting on a given ith atom in the lattice can be evaluated by taking derivatives of the internal energy of the ith atomic site and evaluating the power moments of the atomic displacements If the ith atom in the lattice is affected by a supplementary force ␣␤, then the total force acting on it must be zero, and one gets the force balance relation as

兺␣ 冉 ⳵2Eci ⳵ui␣⳵ui␤冊eq具

ui␣典⫹

1 2兺

␣,␥ 冉 ⳵3E

ci

⳵ui␣⳵ui␤⳵ui␥冊eq具

ui␣ui␥典

⫹3!1 兺

␣,␥,␩ 冉

⳵4E

ci

⳵ui␣⳵ui␤⳵ui␥⳵ui␩冊eq具

ui␣ui␥ui␩典⫺␣␤⫽0

共7兲 Here, the subscript eq indicates evaluation at equilibrium The thermal averages of the atomic displacements 具ui␣ui␥典

and具ui␣ui␥ui␩典 共called second- and third-order moments兲at

given site Ri can be expressed in terms of the first moment 具ui␣典 with the aid of Eq.共5兲as

具ui␣ui␥典a⫽具ui␣典a具ui␥典a⫹␪⳵具ui␣典a ⳵␣␥

⫹ប␦a␥

2m␻coth冉 ប␻

2␪冊⫺ ␪␦a␥

m␻2, 共8兲

具ui␣ui␥ui␩典a⫽具ui␣典a具ui␥典a具ui␩典a⫹␪P␣␥␩具ui␣典a

⳵具ui␥典a ⳵␣␩

⫹␪2⳵

2具u

i␣典a

⳵␣␥⳵␣␩ ⫹

ប具ui␩典a␦␣␥

2m␻ coth冉 ប␻

2␪冊

⫺␪具ui␩典a␦␣␥

m␻2 共9兲

Here, P␣␥␩is (␣⫽␥⫽␩) or 0共otherwise兲depending on␣, ␥, and␩共Cartesian component兲and␻is the atomic vibration frequency similar to that defined in the Einstein model, which will be given by Eq.共11兲 Then Eq.共7兲is transformed into the new differential equation

␥i␪2

d2y

d␣2⫹3␥i␪y

d y d␣⫹␥iy

3⫹k

iy

⫹␥i

␪

k共X coth X⫺1兲y⫺␣␤⫽0, 共10兲

where X⬅ប␻/2␪ and y⬅具ui典 Here, ki and ␥i are

second-and fourth-order derivatives of Eci and defined by the

fol-lowing formulas:

ki⫽冉⳵ 2E

ci

⳵ui2␣冊 eq

⬅m␻2, 共11兲

␥i⫽1

6冋冉 ⳵4E

ci

⳵ui4␣冊 eq

⫹6冉 ⳵

4E

ci

⳵ui2␤⳵ui2␥冊 eq册

⬅1

6共␥1i⫹6␥2i兲, 共12兲

respectively In the SMA TB scheme, the parameters ki,␥1i, and ␥2i are composed of two contributions 共band structure and repulsive energies兲 and ki is given by the following

form:

ki⫽q r0冋␩i

共2兲⫺冉2 q r0冊␩i

共3兲册␮2i⫺1/2⫺

A共p/r0兲兺

j 冋

1⫺ᐉi j2 ri j ⫺ᐉi j

2冉 p r0冊册

exp关⫺p共ri j/r0⫺1兲兴, 共13兲

where␩i(2) and␩i(3) are defined, respectively, as ␩i共

2兲⫽兺

j 冋

1⫺li j

2

ri j 册␰i j

2

exp关⫺2q共ri j/r0⫺1兲兴, 共14兲

␩i共

3兲⫽兺

j

li j

2␰

i j

2

expb⫺2q共ri j/r0⫺1兲c, 共15兲

with

li j⫽冉⳵ri j

⳵x 冊⫽共xj⫺xi兲/ri j

After a bit of algebra, ␥1i defined by Eq.共12兲is given by

␥1i⫽冉rq

0冊冋 ⳵2␩

i

共2兲 ⳵x2 ⫺2

⳵2␩

i

共3兲 ⳵x2 冉

q r0冊册␮2i

⫺1/2

⫺冉rq

0冊

冋␩i共

2兲⫺2␩

i

共3兲冉 q r0冊册

2 ␮2i⫺3/2

⫹A冉p

r0冊兺j 冋

3共1⫺6li j2⫹5li j4兲 ri j3

⫹3共1⫺6li j

2⫹5l

i j

4兲 ri j

2 冉

p r0冊

⫺6li j

2

ri j 冉

p r0冊

2

⫹li j4冉 p r0冊

4

册exp兵⫺p共ri j/r0兲⫺1其 共16兲 The second derivatives of␩i(2)and␩i(3)appearing in the first term of the right-hand side of Eq.共16兲are also given explic-itly in terms of the TB potential parameters and the direction cosines li j and mi j between the central atom i and its

␥2i⫽冉 q r0冊冋

⳵2␩

i

共2兲 ⳵y2 ⫺2

⳵2␩

i

共3兲 ⳵y2 冉

q r0冊册␮2i

⫺1/2⫺ 2冋⳵␩i

共1兲 ⳵y 册

2

冉q r0冊

2

␮2i⫺3/2⫹冉q r0冊

2

冋␩i共

2兲⫺2␩

i

共3兲冉 q r0冊册

2 ␮2i⫺

5/2

⫹A冉p

r0冊兺j 冋

1⫺3li j2⫺3mi j2⫹15li j2mi j2

ri j3 ⫹

1⫺3li j2⫺3mi j2⫹15li j2mi j2

ri j2 冉

p r0冊

⫺li j

2⫹m

i j

2⫺6l

i j

2m

i j

2

ri j 冉

p r0冊

2

⫹li j2mi j2冉 p r0冊

3

册exp兵⫺p共ri j/r0⫺1兲其, 共17兲

where␩i(1) is defined by ␩i共

1兲⫽兺

j

li j␰i j

2

exp关⫺2q共ri j/r0⫺1兲兴 共18兲

Here, we note that ␥1i and ␥2i depend sensitively on the structure of crystals through factors including direction co-sines as can be seen in Eqs 共16兲 and 共17兲 The factors in-cluding direction cosines for cubic crystals are presented in Table II The derivatives of␩i(1),␩i(2), and␩i(3)with respect to the y variable are given in Appendix B.

In determining the atomic displacement具ui典, the

symme-try property appropriate for cubic crystals is used

具ui␣典⫽具ui␥典⫽具ui␩典⬅具ui典 共19兲

Then, the solutions of the nonlinear differential equation of Eq 共10兲can be expanded in the power series of the supple-mental force ␣as

y⫽⌬r⫹A1␣⫹A2␣2. 共20兲

Here, ⌬r is the atomic displacement in the limit of zero of supplemental force␣ Substituting the above solution of Eq TABLE II Lattice sums appearing in the harmonic k1 and anharmonic␥1and ␥2 parameters in cubic

metals.兺1⬅兺j⫽i1⫺6li j

⫹5li j

, 兺2⬅兺j⫽i1⫺3li j

⫺3mi j

⫹15li j

mi j

, 兺3⬅兺j⫽ili j

⫹mi j

⫺6li j

mi j

Crystal structure Neighbors

fcc Zi 12 24 12 24

Distance & ) 冑5

兺j⫽i

lij

2 4 2 8 4 8

兺j⫽i

lji

4 2 2 4 2 164/25

兺j⫽i

lij

mij

2 1 0 2 1 18/25

兺1 ⫺2 ⫺4 ⫺2 44/5

兺2

3 ⫺6 ⫺66/5

兺3

2 4 292/25

bcc Zi 12 24

Distance 2/) 2冑6/3 冑11/3

兺j⫽i

lij

2 8/3 2 4 8 8/3

兺j⫽i

lji

4 8/9 2 2 664/121 8/9

兺j⫽i

lij

mij

2 8/9 0 1 152/121 8/9

兺1 ⫺32/9 ⫺2 416/121 ⫺32/9

兺2

16/3 ⫺6 ⫺624/121 16/3

兺3

共20兲into the original differential equation Eq.共10兲, one can get the coupled equations on the coefficients A1 and A2, from which the solution of⌬r is given as

共⌬r兲2⬇关⫺C2⫹冑C2 2⫺4C

1C3兴/2C1, 共21兲 where

C1⫽3␥i,

C2⫽3ki冋1⫹

␥i␪

ki2 共X coth X⫹1兲册, 共22兲

C3⫽⫺

2␥i␪2 ki2 冉1⫹

X coth X 冊

Using Eqs 共8兲 and共21兲, it can be shown that mean square atomic displacement 共second moment兲 in cubic crystals is given by

具u2典⫽␪

kX coth X⫹

2␥␪2

k3 共1⫹X coth X/2兲

⫹2␥

2␪3

k5 共1⫹X coth X兲共1⫹X coth X/2兲 共23兲 Once the thermal expansion⌬r in the lattice is found, one can get the Helmholtz free energy of the system in the fol-lowing form:

⌿⫽U0⫹⌿0⫹⌿1, 共24兲 where⌿0 denotes the free energy in the harmonic approxi-mation and ⌿1 the anharmonicity contribution to the free energy.38 – 40 We calculate the anharmonicity contribution to the free energy⌿1 by applying the general formula

⌿⫽U0⫹⌿0⫹冕

␭

具Vˆ典␭d␭, 共25兲 where ␭Vˆ represents the Hamiltonian corresponding to the anharmonicity contribution It is straightforward to evaluate the following integrals analytically

I1⫽冕

␥1

具ui4典d␥1, I2⫽冕

␥2

具ui2典␥ 1⫽0

2 d␥2. 共26兲

Then the free energy of the system is given by ⌿⫽U0⫹3N␪关X⫹ln共1⫺e⫺2X兲兴

⫹3N再␪

2

k2冋␥2X

2coth2X⫺2 3␥1冉1⫹

X coth X 冊册,

⫹2␪

3

k4 冋 3␥2

2

X coth X冉1⫹X coth X

2 冊

⫺2␥1共␥1⫹2␥2兲冉1⫹X coth X

2 冊共1⫹X coth X兲册冎, 共27兲

where the second term denotes the harmonic contribution to the free energy

With the aid of the free energy formula⌿⫽E⫺TS, one can find the thermodynamic quantities of metal systems The specific heats and elastic moduli at temperature T are directly derived from the free energy⌿ of the system For instance, the isothermal compressibility␹Tis given by

␹T⫽3共a/a0兲3冒冋2 P⫹ 3N

&

a 冉 ⳵2⌿

⳵r2冊T册, 共28兲 where

⳵2⌿ ⳵r2 ⫽3N再

1

⳵2U ⳵r2 ⫹␪冋

X coth X 2k

⳵2k ⳵r2

⫺

4k2冉 ⳵k ⳵r冊

2

冉X coth X⫹ X

2

sinh2X冊册冎 共29兲 On the other hand, the specific heats at constant volume Cv is

Cv⫽3NkB再

X2 sinh2X⫹

2␪

k2 冋冉2␥2⫹ ␥1

3 冊

X3coth X sinh2X

⫹␥13 冉1⫹ X

2

sinh2X冊⫺␥2冉 X4 sinh4X⫹

2X4coth2X sinh2X 冊册冎

共30兲

The specific heat at constant pressure Cpis determined from

the thermodynamic relations

Cp⫽Cv⫹9TV␣T

␹T

, 共31兲

where ␣T denotes the linear thermal expansion coefficient

and␹Tthe isothermal compressibility In Eqs.共27兲,共29兲, and

共30兲above, the suffices i for the parameters k,␥1and␥2are omitted because each atomic site is equivalent in a mono-atomic cubic crystal with primitive structure The relation-ship between the isothermal and adiabatic compressibilities, ␹T and␹s, is simply given by

␹s⫽CCv p␹T

共32兲

␥G⫽V

C␷冋 ⳵S ⳵V册T⫽

␣TBSV

CP

, 共33兲

where BS⬅␹S⫺

1

denotes the adiabatic bulk modulus

III RESULTS AND DISCUSSIONS A Comparison with the quasiharmonic theory Firstly, we compare the thermodynamic quantities of met-als calculated by the present statistical moment method 共SMM兲with those by the QH theory.10The basic idea of the QH approximation is that the explicit dependence of the free

energy F(T,V) on the system volume V can be explored by homogeneous scaling of the atomic potentials 兵Ri0其 Then, for each temperature T the equilibrium volume V is obtained by minimizing Helmholtz energy F with respect to V In Fig. 1, we present the linear thermal expansion coefficients␣Tof

Cu, Pd, Ag, and Mo metals, calculated by the present theory, together with those of the QH theory by Moruzzi, Janak, and Schwarz共MJS model兲.10The linear thermal expansion coef-ficients␣Tby the present statistical moment theory and those

of the QH theory by Moruzzi et al are referred to as SMM and MJS, respectively In order to allow the direct compari-son between the two different schemes, the linear thermal expansion coefficients␣Tof the cubic metals are calculated

FIG Comparison of linear thermal expansion coefficients␣T of共a兲 Cu,共b兲Pd,共c兲Ag, and共d兲 Mo, calculated by using the Morse

potentials Solid and dot-dashed lines show the results of self-consistent 共SC兲 and non-self-consistent共NSC兲treatments of the statistical

with the use of the same Morse type of potentials, exactly identical forms as used in the QH calculations by MJS.10The four metals Cu, Pd, Ag, and Mo are chosen simply because the linear thermal expansion coefficients ␣T are well

repro-duced by the two-body Morse potentials as demonstrated by them.10

The solid lines in Fig show the linear thermal expan-sion coefficients ␣T calculated by the self-consistent 共SC兲 treatments of the present SMM scheme, while the dot-dashed ones are obtained by the non-self-consistent 共NSC兲 treat-ments In the SC treatments, the characteristic parameters k, ␥1, and ␥2 are determined self-consistently with the lattice constants aT at given temperature T However, in the NSC

treatments, the harmonic k, and anharmonic ␥1, and␥2 pa-rameters are fixed to those values evaluated at the appropri-ate reference temperature T0共e.g., absolute zero temperature or some reference temperature; here T0 is chosen to be K and taken to be constant for the whole temperature region兲 The calculated linear thermal expansion coefficients ␣T by

the present theory are in good agreement with those by QH theory for the lower temperature region below the Debye temperature and the agreement is better for the SC calcula-tions This indicates that the thermal lattice expansion gives rise to the significant reduction in the parameters k,␥1, and ␥2, and thereby changes the thermodynamic quantities ap-preciably even for the lower temperatures

B Thermodynamic quantities of metals by second moment TB potentials

With the use of the analytic expressions presented in Sec II, it is straightforward to calculate the thermodynamic quan-tities of metals and alloys at the thermal equilibrium Firstly, the equilibrium lattice spacings are determined, using Eqs 共20兲 and 共21兲, in the SC treatment including temperature 共bond length兲-dependent k,␥1, and ␥2 values The thermal lattice expansion can also be calculated by standard proce-dure of minimizing the Helmholtz energy of the system: We have checked that both calculations give almost identical re-sults on the thermal lattice expansions We calculate the ther-mal lattice expansion and mean-square atomic displacements of some fcc 共transition兲 metals and bcc alkali metals, for which the reliable many-body potentials are available, and compare them with those by the molecular dynamics共MD兲 and Monte Carlo共MC兲simulations So far, a number of the SMA base TB potentials have been proposed for fcc metals Specifically, we use the SMA TB potentials by Rosato et al.35 and by Cleri and Rosato36 for fcc metals, which are known to give good descriptions of cohesive properties of fcc elements For alkali metals Li, Na, K, Rb, and Cs, we use the potential parameters proposed recently by Li et al.40

In the TB scheme by Rosato et al.,35the interaction range is limited to the first nearest neighbors, while in the TB scheme by Cleri and Rosato,36 it is extended to the fifth neighbors In Fig 2, we present the linear thermal expansion coefficients ␣T and mean-square atomic displacements具u2典

of Cu crystal, together with the experimental values共by sym-bols 䊊兲.55–58For this calculation, the electronic many-body potentials are used for Cu crystal, but there are no large

differences in the calculated quantities when we use the Lennard-Jones 共LJ兲type of pair potentials The bold line in Fig 2共a兲represents the calculated ␣T by the present SMM, while the dashed line␣Tvalues by the Lennard-Jones type of

potential; ␸(r)⫽D0兵(r0/r)n⫺(n/m)(r0/r)m其, with n

⫽9.0, m⫽5.5, r0⫽2.5487 Å, and D0⫽4125.7 K 共0.35553 eV兲, respectively The overall agreement between the calcu-lated and experimental ␣T values is better for the calcula-tions by the SMA TB potential, although LJ potential param-eters are not best fitted to reproduce the experimental ␣T

values We note that the classical MD simulation,59shown by the dot-dashed curve in Fig 2共a兲, not reproduce the cor-rect curvature of the linear thermal expansion coefficient␣T,

and is qualitatively incorrect due to the absence of the

quan-FIG 共a兲The linear thermal expansion coefficient␣T 共a兲and

共b兲mean-square atomic displacements具u2典of Cu crystal calculated

by the present method The corresponding experimental values are

tum mechanical vibration effects One also sees in Fig 2共b兲 that the agreements between the calculated and experimental results of the mean square atomic displacements具u2典 in Cu crystal are quite excellent for the SMA TB calculations, com-pared to those by two-body potentials This implies that the present SMM scheme with SMA TB potentials provides us fully quantitative estimates for the thermodynamic quantities of elemental metals

We show in Fig 3共a兲 the mean-square atomic displace-ments 具u2典 of Al crystal as a function of temperature T, together with those values by the MD simulation60 and ex-perimental data.61 The present calculations by using SMM

differ significantly from those results by MD simulations, especially for the lower temperature region, i.e., below the Debye temperature This is due to the fact that in the classi-cal MD simulations the quantum mechaniclassi-cal vibration ef-fects are not taken into account One sees that the quantum mechanical zero point vibrations give main contributions at lower temperature region T⭐100 K The agreement between the present calculation and the experimental results is fairly good for the whole temperature region, from zero to ⬃800 K, much higher than the Debye temperature In Fig 3共b兲, we show the mean-square atomic displacements 具u2典 of Ag crystal calculated by the present SMM using the SMA TB potentials of Refs 35 and 36, together with the experimental results.62One sees in Fig 3共b兲that TB parameters by Rosato, Guillope, and Legrand35 共first-neighbor TB potential兲 leads to larger mean-square atomic displacement 具u2典 in Ag crys-tal compared to those results by using the TB parameters by Cleri and Rosato36 共5th neighbor TB potential兲 The similar tendency is also found for the thermal expansion coefficients ␣Tof Ag crystal, larger␣Tvalues by TB potential by Rosato,

Guillope, and Legrand.35 In the present formalism, the ther-mal lattice expansion and mean-square atomic displacements are characterized by the harmonic k and anharmonic ␥ pa-rameters In particular, the thermal lattice expansion 共 mate-rial dependence兲 is predicted by a ratio of ␥/k2 and the mean-square displacement具u2典by␥/k2 共and also by␥2/k5) parameter as well The ratios ␥/k2 of Cu crystal calculated by using the TB potential by Rosato, Guillope, and Legrand35 are in fact larger than those results by Cleri and Rosato36 for whole temperature region The mean square atomic displacement具u2典 in Ag crystal by the fifth-neighbor TB potential36are in fairly good agreement with the experi-mental results for the whole temperature region, and they are in good agreement with the MD simulation results for high temperature region

The calculated mean-square atomic displacements具u2典of Ag crystal by the present method is also compared with those by the cluster variation method 共CVM兲 As is well known, CVM63– 65is an analytical statistical method that di-rectly gives us the free energy of a system The CVM was originally designed for the statistical mechanics of the Ising model on a fixed lattice, and extended recently to treat sys-tems with continuous degrees of freedom, such as the lattice site distortion, due to thermal vibrations, thermal dilatation, and mixture of atoms of different sizes In general, in CVM treatments the correlations in the atomic displacements are taken into account within the small atomic clusters 共e.g., small clusters such as pair, tetrahedron, or octahedron clus-ters兲 Finel and Te´tot gave the first application of the Gauss-ian CVM65for the thermodynamic quantities of some transi-tion metals It has been demonstrated that Gaussian CVM gives the excellent results of the thermodynamic quantities of metals 共the CPU time is several orders of magnitude smaller than the one needed for numerical MD or MC simu-lations兲 The thin dot-dashed and thin dashed curves in Fig 3共b兲represent the mean-square atomic displacement具u2典 of Ag crystal obtained by the Gaussian CVM65 using the SMA TB potentials of Refs 35 and 36, respectively Both CVM

FIG Mean-square atomic displacements 具u2典 of 共a兲Al and

共b兲Ag crystals as a function of temperature In共a兲, the dashed line

shows the results of MD simulations by Papanicolaou et al.共Ref

calculations of ␣T are generally in agreement with the

ex-perimental results We note that for 具u2典 calculations of Ag crystal, however, the present analytic SMM gives much effi-cient analytic calculations and much better results compared to those by CVM calculations

The calculated thermodynamic quantities of cubic metals, fcc共in addition to Cu, Ag, and Al presented above兲and alkali 共bcc兲metals, by the present method are summarized in Table III In the present calculations, we use the TB potential pa-rameters by Li, Barojas, and Papaconstantopoulos40 for al-kali metals Li, Na, K, Rb, and Cs This TB model takes into account the interatomic interactions up to 12th neighbors, i.e., 228 atoms in bcc lattice The relative magnitudes of linear thermal expansion coefficients of fcc共transition兲 met-als are in good agreement with the experimental results However, the thermal lattice expansion coefficients␣ of al-kali metals are systematically larger 共⬃10%兲 than those of experimental results, although their relative magnitudes are in good agreement with the experimental results The calcu-lated Gruăneisen constants and elastic moduli are also pre-sented in Table III The anharmonicity of the lattice vibra-tions is well described by the Gruăneisen constant G The

material of larger value of␥Gmay be regarded as the

mate-rial with higher lattice anharmonicity So, the evaluation of the Gruăneisen constant is of great significance for the assess-ment of anharmonic thermodynamic properties of metals and alloys The experimental Gruăneisen constantsGof fcc

met-als are larger than except for Ni, while those of alkali metals are less than and take values around 1.5 The calculated Gruăneisen constants G of fcc metals are also

larger than 2, while those values of alkali metals are less than 2, in agreement with the experimental results The calculated ␥Gvalues by the present method have the weak temperature

dependence, i.e., show the slight increase with increasing temperature as in the calculations by QH theory.10The

tabu-lated Gruăneisen constants G for low temperatures are well

compared with the experimental values which are deduced from the low共room-兲temperature specific heats

The lattice specific heats Cv and Cp at constant volume and at constant pressure are calculated using Eqs 共30兲 and 共31兲, respectively However, the evaluations by Eqs.共30兲and 共31兲are the lattice contributions, and their values may not be directly compared with the corresponding experimental val-ues We not include the contributions of lattice vacancies and electronic parts of the specific heats Cv, which are known to give significant contributions in metals for higher temperature region near the melting temperature In particu-lar, it has been demonstrated that lattice vacancies make a large contribution to the specific heats for the high-temperature region.66The electronic contribution to the spe-cific heat at constant volume Cveleis proportional to the tem-perature T and given by Cvele⫽␥eT, ␥e being the electronic

specific heat constant.56,66The electronic specific heats Cvele values are estimated to be 0.8 –13.4% of Cvlatfor metals con-sidered here by the free-electron model.56 Therefore, the present formulas of the lattice contribution to the specific heats, both Cv and Cp, for the cubic metals tend to

under-estimate the specific heats for higher temperature region, when compared with the experimental results The lattice contribution of specific heats Cp calculated for Cu crystal is

shown in Fig 4, together with the experimental results58and those of MD simulation results As expected from above mentioned reasonings, the calculated Cp values of solid Cu

are smaller than the experimental values at high tempera-tures However, the temperature dependence 共curvature兲 of Cpof Cu crystal by the present method is in good agreement with the experimental results, in contrast to the MD simula-tion results In the MD simulasimula-tions, the heat capacities per atom at constant pressure Cp can be obtained for metals by TABLE III Bulk modulus, linear thermal expansion, and Gruăneisen constant calculated with the use of

the SMA TB potentials Experimental values of Na*共RT兲are those values for 250 K

Element

BT(GPa) ␣(10⫺6K⫺1) ␥G

Calc

Expt Calc Expt Calc Expt

T⫽0 RT

Al 87 75 72 24.5 23.6 2.09 2.19

Cu 153 137 137 15.9 16.7 2.21 2.00

Ni 190 182 184 14.7 12.7 2.01 1.88

Ag 114 96 101 23.5 19.7 2.78 2.36

Rh 306 280 271 10.9 8.2 2.19 2.43

Pd 204 171 181 14.3 11.6 2.22 2.18

Au 185 164 173 17.2 14.2 3.21 3.04

Pt 301 259 278 11.2 8.9 3.06 2.56

Li 16.8 12.4 11.6 65.4 56.0 1.18 1.18

Na* 6.5 4.3 6.8 83.9 71.0 1.53 1.31

K 5.3 3.6 3.2 98.7 83.0 1.54 1.37

Rb 4.0 2.8 3.1 104.6 90.0 1.65 1.67

taking the numerical derivative of the internal energy with respect to temperature.59 The MD simulations by Mei, Dav-enport, and Fernando59 give reasonable values of Cp for

higher temperature region when compared with the experi-mental data However, it should be noted that the MD simu-lations are only adequate above the room temperature, and the calculated Cp value deviates from the experimental data

at low temperatures because quantum effects are not taken into account in the classical MD simulations

The bulk moduli BT of cubic metals are evaluated at

ab-solute zero temperature and at room temperature 共RT兲 and presented also in Table III The ratios of bulk moduli, BT/B0, with respect to those of the absolute zero tempera-ture are calculated to be 0.85–0.90 共fcc metals兲 and ⬃0.7 共alkali metals兲 which are favorably compared with the ex-perimental results In general, the calculated bulk moduli BT(RT) and B0are in good agreement with the experimental results as well as QH calculations

As a final remark of this section, we note that the present statistical moment method can be incorporated in a straight-forward manner with the first-principles density functional theory, by simply evaluating three kinds of derivatives 共one for harmonic and two for anharmonic contributions兲 of the atomic total energy with respect to the Cartesian coordinates The density functional TB67 and TBTE68共tight-binding total energy兲methods with Slater-Koster parameters derived from the first-principles theories can be readily applied to evaluate k,␥1, and␥2values, on the basis of Hellman-Feynman theo-rem The full density functional theories such as the linear-response approach by Giannozzi et al.69 and real-space finite-element density matrix method70 can also be used for

the evaluations of k, ␥1, and ␥2, and thus for the calcula-tions of thermodynamic quantities of the present study

IV CONCLUSIONS

We have presented an analytic formulation for obtaining the thermodynamic quantities of metals and alloys based on the finite temperature moment expansion technique in the statistical physics The thermal lattice expansion of mon-atomic crystals共fourth-order anharmonic contribution兲is de-rived explicitly in terms of the three characteristic param-eters, k1, ␥1, and ␥2 The present formalism takes into account the quantum-mechanical zero-point vibrations as well as the higher-order anharmonic terms in the atomic dis-placements and it enables us to derive the various thermody-namic quantities of metals and alloys for a wide temperature range We are able to calculate the thermodynamic quantities quite efficiently and accurately by using the analytic formu-las and taking into account the many-body electronic effects in metallic systems The calculated thermodynamic quanti-ties of metals are in good agreement with the experimental results as well as with those by MD and MC simulations共in some cases, better results by the present method兲

Although in this paper we only used many-body elec-tronic potentials, the extension to coupling the present SMM scheme with the ab initio density functional theories is straightforward This can be done by evaluating three char-acteristic parameters k,␥1, and␥2 for cubic systems It can also be applied directly for the composition-temperature phase diagrams calculations of alloys for the full temperature range from absolute zero to the melting temperatures Tm

ACKNOWLEDGMENTS

The authors would like to thank Professor S Tsuneyuki of Institute for Solid State Physics of the University of Tokyo for valuable discussions The support of the supercomputing facility of Institute for Solid State Physics, the University of Tokyo is also acknowledged

APPENDIX A: MOMENT DEVELOPMENT BY DENSITY MATRIX FORMALISM

To derive the mean-square atomic displacement 共second moment兲 and higher-order power moments of the thermal lattice vibrations, we use the formalism based on the density matrix ␳ˆ , which is defined by

␳ˆ⫽exp冉⌿⫺H ˆ

␪ 冊, 共A1兲

where⌿and Hˆ denote the Helmholz free energy and Hamil-tonian of the system, respectively In the presence of the constant supplemental forces ␣1, ␣2, ,␣N in the system,

the Hamiltonian Hˆ is given by Hˆ⫽Hˆ0⫺兺i␣iqˆi The density

matrix ␳ˆ is normalized so as to satisfy the condition Tr␳ˆ

⫽1 and given by the solution of the Liouville equation

FIG Specific heats per atom at constant pressure Cpplotted

against temperature T for Cu crystal, in unit of Boltzmann’s

con-stant kB The experimental data共Ref 58兲 are shown as the open

iប⳵␳ˆ

⳵t⫽关Hˆ ,␳ˆ兴⫺ 共A2兲

We use the following identities on the derivatives of an op-erator function Eˆ␭ composed of two different operators Aˆ and Bˆ :

Eˆ␭共␶,Aˆ ,Bˆ兲⬅exp关␶共Aˆ⫹␭Bˆ兲兴 共A3兲

⳵Eˆ␭共␶,Aˆ ,Bˆ兲

⳵␭ ⫽再␶Bˆ⫹n兺⫽1

⬁ ␶n⫹1共iប兲n

共n⫹1兲! 关Aˆ⫹␭Bˆ关Aˆ⫹␭Bˆ 关Aˆ

⫹␭Bˆ 兴兴兴冎Eˆ␭共␶,Aˆ ,Bˆ兲

⫽Eˆ␭共␶,Aˆ ,Bˆ兲再␶Bˆ⫺兺

n⫽1

⬁ 共⫺␶兲n⫹1共iប兲n

共n⫹1兲!

⫻关Aˆ⫹␭Bˆ关Aˆ⫹␭Bˆ 关Aˆ⫹␭Bˆ 兴兴兴冎, 共A4兲

where 关Aˆ⫹␭Bˆ ,Bˆ兴⫽

iប兵共Aˆ⫹␭Bˆ兲Bˆ⫺Bˆ共Aˆ⫹␭Bˆ兲其 By differentiation of the density matrix␳ˆ with respect to the constant force␣i, one can get the relation

1 ␪

⳵⌿ ⳵␣i

⫹1␪冋⫺具qˆi典a⫺兺

n⫽1

⬁ 1

共n⫹1兲!冉 iប

␪冊

n

具qˆi共n兲典a册⫽0, 共A5兲 where

具qˆi共n兲典⫽Tr关关 关qˆi,Hˆ兴⫺Hˆ 兴␳ˆ 共A6兲

For equilibrium state, one can show that具qˆi(n)典⫽0 because ⳵␳ˆ /⳵t⫽关Hˆ ,␳ˆ兴⫺⫽0 is satisfied One can then derive from 共A5兲the relation

具qˆk典a⫽

⳵⌿ ⳵␣k

共A7兲

Using the relation 具Fˆ典a⫽Tr关Fˆ␳ˆ兴 and Eqs 共A3兲 and 共A4兲, one can get the following identities:

具共Fˆ⫺具Fˆ典兲共qˆ⫺具qˆ典兲典⫽⫺␪冉⳵具F ˆ典

⳵␣ ⫺冓

⳵Fˆ ⳵␣冔冊

⫹␪兺

m⫽1 ⬁

共⫺1兲mBm m! 冉

iប ␪冊

m

冓⳵Fˆ共m兲 ⳵␣ 冔,

共A8兲

具共qˆ⫺具qˆ典兲共Fˆ⫺具Fˆ典兲典

⫽⫺␪冉⳵⳵␣具Fˆ典⫺冓⳵⳵␣Fˆ冔冊

⫹␪兺

m⫽1 ⬁

共⫺1兲mBm m!冉⫺

iប ␪ 冊

m

冓⳵Fˆ共m兲

⳵␣ 冔 共A9兲

Then, one gets the decoupling formula

1

2具关Fˆ ,qˆk兴⫹典a⫺具Fˆ典a具qˆk典a

⫽␪⳵具Fˆ典a

⳵␣k

⫺␪兺

n⫽0

⬁ B

2n 共2n兲!冉

iប ␪冊

2n 冓⳵Fˆ共2n兲

⳵␣k 冔a

, 共A10兲 In the above Eqs 共A8兲–共A10兲, B2n denotes the Bernoulli number and Fˆ(k) is defined by

共A11兲 Substituting Fˆ⫽qˆk into Eq 共A10兲, one can get the

mean-square atomic displacement from the thermal equilibrium po-sition, as

具共qˆi⫺具qˆi典a兲2典a⫽␪

⳵具qˆi典a

⳵␣i

⫺␪兺

n⫽0 ⬁

B2n

共2n兲!冉

iប

␪ 冊

2n 冓⳵qˆi共2n兲

⳵␣i 冔a

共A12兲 Equation 共A12兲 is used to derive Eq 共8兲 in the text The similar formulas can be given for higher order moments as well

APPENDIX B: DERIVATIVES OF COUPLING

PARAMETERS k AND␥

The second derivatives such as ⳵2␩i(2)/⳵x2 and

⳵2␩

i

(3)

/⳵x2, appearing in Eq.共16兲in the text are given by the following forms, respectively:

⳵2␩

i

共2兲

⳵x2 ⫽兺j 冋⫺

3共1⫺6li j2⫹5li j4兲ri j⫺3

⫺2共1⫺8li j2⫹7li j4兲ri j-2冉q r0冊

⫹4li j2共1⫺li j2兲ri j⫺1冉q r0冊

2 册␰i j

2 exp关⫺2q共r

i j/r0⫺1兲兴,

⳵2␩

i

共3兲

⳵x2 ⫽兺

j 冋

2共1⫺5li j2⫹4li j4兲ri j⫺2⫹10li j2共1⫺li j2兲ri j⫺1冉q r0冊

⫹4li j

4冉 q r0冊

2

册␰i j

2

exp关⫺2q共ri j/r0⫺1兲兴 共B2兲

On the other hand, the first and second derivatives such as ⳵␩i

(1)/⳵y , ⳵2␩

i

(2)/⳵y2, and⳵2␩

i

(3)/⳵y2 with respect to the y variable are given by

⳵2␩

i

共1兲

⳵y ⫽兺j 冋

li jmi j•ri j⫺

1⫹ 2li jmi j冉

q r0冊册␰i j

2

⫻exp关⫺2q共ri j/r0⫺1兲兴, 共B3兲

⳵2␩

i

共2兲

⳵y2 ⫽⫺兺

j 冋共

1⫺3li j2⫺3mi j2⫹15li j2mi j2兲ri j⫺3

⫹2共1⫺3mi j

2⫺ li j

2⫹ 7li j

2 mi j

2兲 ri j⫺

1冉q r0冊

⫺4共m2⫺l2m2兲ri j⫺1冉q r0冊

2

册␰i j

2

⫻exp关⫺2q共ri j/r0⫺1兲兴, 共B4兲

and ⳵2␩

i

共3兲

⳵y2 ⫽兺j 冋⫺

2共li j

2⫺ 4li j

2 mi j

2兲 ri j⫺

2⫺ 2li j

2共

⫺5m2

i j 兲

ri j⫺1冉 q r0冊⫹4li j

2 mi j2冉 q

r0冊

册␰i j

2

⫻exp关⫺2q共ri j/r0⫺1兲兴, 共B5兲

respectively

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