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 5 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, C v and C p, 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

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-Gru¨neisen 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 do 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 –31

The 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

We will use the electronic many-body potentials, i.e., second-moment tight-binding 共TB兲 potentials,32– 40 for the

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(u il) 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, u il j , of the

displacement vector uilaround this point

U ⫽U0⫹兺i,l, j 冋⳵U

⳵u il j册eq

uil j⫹ii⬘,l,ll兺⬘, j j⬘冋 ⳵2U

⳵uil j⳵ui

⬘l⬘

j⬘ 册eq

uil jui j⬘⬘l⬘

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

E ci⫽1

N i兺⫽1N 冠A兺j ⫽i N exp冋⫺p冉r i j

r0⫺1冊册

⫺再␰i j

2兺j ⫽i N exp冋⫺2q冉r i 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 E c of the

system is then written as the sum of the E ci The parameters

A, ␰0, p, and q are fitted to reproduce some experimental quantities at zero temperature 共cohesive energy E c, lattice

parameter a, bulk modulus and elastic constants兲 In the

sum-mations over the index j in Eq 共3兲 are either limited to the Z1 first neighbors only, and in that case we use the parameters

A, ␨0, p, and q determined by Rosato, Guillope, and

Legrand,35or extended up to the fifth neighbors, and in that case we use the parameters of Cleri and Rosato.36Cleri and Rosato36 fitted these parameters to experimental data for 16 fcc and hexagonal-close-packed 共hcp兲 transition metals

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 q i.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

H ˆ ⫽Hˆ0⫺兺i ␣i qˆ i, 共4兲

where H ˆ0 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 q i 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q k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K ˆ n⫹1典a⫽具K ˆ n典a具qˆ n⫹1典a⫹␪⳵具K ˆ n典a

⫺␪m兺⬁⫽0 B 2m

共2m兲!冉iប

冓⳵K ˆ n 共2m兲

where␪⫽k B T, m is the atomic mass, and K ˆ n is the

correla-tion operator of the nth order:

K ˆ n⫽ 1

2n⫺1关 关qˆ1,qˆ2兴⫹qˆ3兴⫹ ]⫹qˆ n]⫹. 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

H ˆ and B 2n denotes the Bernoulli numbers.关q i ,q j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

Al共1兲a

aindicates parameters taken from Ref 36

bindicates parameters taken from other sources: Al共2兲 from Ref 60 and Ag共2兲 from Ref 35

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 ⳵2E ci

⳵u i␣⳵u i␤冊eq

具u i␣典⫹1

2兺␣,␥ 冉 ⳵3E ci

⳵u i␣⳵u i␤⳵u i␥冊eq

具u i␣u i␥典

⫹3!1 ␣,␥,兺␩ 冉 ⳵4E ci

⳵u i␣⳵u i␤⳵u i␥⳵u i␩冊eq

具u i␣u i␥u i␩典⫺␣␤⫽0

共7兲 Here, the subscript eq indicates evaluation at equilibrium

The thermal averages of the atomic displacements 具u i␣u i␥典

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

given site Ri can be expressed in terms of the first moment

具u i␣典 with the aid of Eq.共5兲 as

具u i␣u i␥典a⫽具u i␣典a具u i␥典a⫹␪⳵具u i␣典a

⫹ប␦a␥

2m␻coth冉ប␻

2␪冊⫺␪␦a␥

m␻2, 共8兲

具u i␣u i␥u i␩典a⫽具u i␣典a具u i␥典a具u i␩典a⫹␪P␣␥ ␩具u i␣典a

⳵具u i␥典a

⫹␪2⳵2具u i␣典a

ប具u i␩典a␦␣␥

2m␻ coth冉ប␻

2␪冊

⫺␪具u i␩典a␦␣␥

Here, P␣␥ ␩is 1 (␣⫽␥⫽␩) 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␪2d2y

d␣2⫹3␥i␪y d y

d␣⫹␥i y3⫹k i y

⫹␥i

␪

k 共X coth X⫺1兲y⫺␣␤⫽0, 共10兲

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

second-and fourth-order derivatives of E ci and defined by the

fol-lowing formulas:

k i⫽冉⳵2E ci

⳵u i2␣冊eq

6冋 冉⳵4E ci

⳵u i4␣冊eq

⫹6冉 ⳵4E ci

⳵u i2␤⳵u i2␥冊eq册⬅1共␥1i⫹6␥2i兲,

共12兲

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

k i⫽r q

0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 j

2

r i j

⫺ᐉi j

2冉 p

r0冊 册exp关⫺p共r i j /r0⫺1兲兴, 共13兲 where␩i

(2) and␩i

(3) are defined, respectively, as

2

r i j 册␰i j

2 exp关⫺2q共r i j /r0⫺1兲兴, 共14兲

␩i共3兲⫽兺j l i j2␰i j

2 expb⫺2q共r i j /r0⫺1兲c, 共15兲

with

l i j⫽冉⳵r i j

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

r0冊 冋⳵2␩i共2兲

r0冊 册␮2i⫺1/2

⫺冉q

r0冊2

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⫺6l i j

2⫹5l i j

4兲

r i j3

⫹3共1⫺6l i j

2⫹5l i j

4兲

r i j2 冉 p

r0冊⫺6l i j

2

r i j 冉p

r0冊2

⫹l i j

4冉 p

r0冊4

册exp兵⫺p共r i 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 l i j and m i j between the central atom i and its neigh-boring atoms j共see Appendix B兲.␥2i is expressed explicitly as

r0冊 冋⳵2␩i共2兲

r0冊 册␮2i⫺1/2⫺2冋⳵␩i共1兲

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⫺3l i j

2⫺3m i j

2⫹15l i j

2m i j2

r i j3 ⫹1⫺3l i j

2⫺3m i j

2⫹15l i j

2m i j2

r i j2 冉 p

r0冊

⫺l i j

2⫹m i j

2⫺6l i j

2m i j2

r i j 冉p

r0冊2

⫹l i j

2m i j2冉 p

r0冊3

where␩i

(1)

is defined by

␩i共1兲⫽兺j

l i j␰i j

2 exp关⫺2q共r i 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u i典, the symme-try property appropriate for cubic crystals is used

具u i␣典⫽具u i␥典⫽具u i␩典⬅具u i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⫺6l i j

2

⫹5l i j

4

, 兺2⬅兺j ⫽i1⫺3l i j

2

⫺3m i j

2

⫹15l i j

2

m i j2, 兺3⬅兺j ⫽i l i j

2

⫹m i j

2

⫺6l i j

2

m i j2

兺j ⫽i l ij

兺j ⫽i l ij

2

兺2

兺3

兺j ⫽i l ij

兺j ⫽i l ij

2

兺2

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C22⫺4C1C3兴/2C1, 共21兲

where

C1⫽3␥i,

C2⫽3k i冋1⫹␥i␪

k i2 共X coth X⫹1兲册, 共22兲

C3⫽⫺2␥i␪2

k i2 冉1⫹X coth X2 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⫽␪

k X coth X⫹23 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0

␭

where ␭Vˆ represents the Hamiltonian corresponding to the

anharmonicity contribution It is straightforward to evaluate

the following integrals analytically

I1⫽冕0

␥ 1

具u i4典d␥1, I2⫽冕0

␥ 2

具u i2典␥21⫽0d␥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␥2X2coth2X⫺23␥1冉1⫹X coth X2 冊册,

⫹2␪3

k4 冋4

3␥2

2

X coth X冉1⫹X coth X2 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⫹ 1

3N

&

a 冉⳵2⌿

where

6

⳵r2 ⫹␪冋X coth X

2k

⳵2k

⳵r2

⫺ 1

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 C v

is

C v ⫽3Nk B再 X2

sinh2X⫹2␪

k2 冋冉2␥2⫹␥1

3 冊X3coth X

sinh2X

⫹␥1

3 冉1⫹ X

2 sinh2X冊⫺␥2冉 X4

sinh4X⫹2X

4coth2X

sinh2X 冊册冎

共30兲

The specific heat at constant pressure C pis determined from the thermodynamic relations

C p ⫽C v⫹9TV␣T

2

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⫽C C v

One can also find ‘‘thermodynamic’’ Gru¨neisen constant as

C␷冋⳵S

⫽␣T B S V

where B S⬅␹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R i0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 1 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

moment method, respectively, while the dashed ones are the results of the QH theory by Moruzzi, Janak, and Schwarz共MJS兲

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 1 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 a T 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 0 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, do 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 2 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 presented by symbols䊊

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 3 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

60兲, while the solid circles are the experimental values

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 Gru¨neisen constants and elastic moduli are also

pre-sented in Table III The anharmonicity of the lattice

vibra-tions is well described by the Gru¨neisen constant ␥G The

material of larger value of␥Gmay be regarded as the

mate-rial with higher lattice anharmonicity So, the evaluation of

the Gru¨neisen constant is of great significance for the

assess-ment of anharmonic thermodynamic properties of metals and

alloys The experimental Gru¨neisen constants␥Gof fcc

met-als are larger than 2 except for Ni, while those of alkali

metals are less than 2 and take values around ⬃1.5 The

calculated Gru¨neisen 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 Gru¨neisen 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 C v and C p 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 do not include the contributions of lattice vacancies

and electronic parts of the specific heats C v, 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 C veleis proportional to the

tem-perature T and given by C vele⫽␥e T, ␥e being the electronic specific heat constant.56,66The electronic specific heats C vele values are estimated to be 0.8 –13.4% of C vlatfor metals con-sidered here by the free-electron model.56 Therefore, the present formulas of the lattice contribution to the specific

heats, both C v and C p, 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 C p 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 C p values of solid Cu are smaller than the experimental values at high tempera-tures However, the temperature dependence 共curvature兲 of

C pof 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 C p can be obtained for metals by

TABLE III Bulk modulus, linear thermal expansion, and Gru¨neisen constant calculated with the use of the SMA TB potentials Experimental values of Na*共RT兲 are those values for 250 K

Element

Calc