application of statistical moment method to thermodynamic properties of metals at high pressures

Table III.. ment approximation of the atomic displacements. To our knowledge, this is the first time that the pressure dependencies of the thermodynamic quantities of metals have been st[r]

Vol 69, No 7, July, 2000, pp 2067-2075

Application of Statistical Moment Method to Thermodynamic Properties of Metals at High Pressures

Vu Van Hung∗ and Kinichi Masuda-Jindo

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

(Received September 16, 1999)

The moment method in statistical dynamics is used to study the thermodynamic properties of metals taking into account the anharmonicity effects of the lattice vibrations and hydrostatic pressures The explicit expressions of the lattice constant, thermal expansion coefficient, and the specific heats Cv and Cp of cubic (fcc) metals are derived within the fourth order moment

approximation The thermodynamic quantities of Al, Au, Ag, Cu, and Pt metals are calculated as a function of the pressure, and they are in good agreement with the corresponding exper-imental results The effective pair potentials work well for the calculations of transition and noble metals, compared to those of the sp-valence metals For obtaining better agreement of the thermodynamic quantities of metals like Al, it is required at least to use the more sophisticated electronic many body potentials In general, it has been shown that the anharmonicity effects of lattice vibration play a dominant role in determining the thermodynamic properties of metals under high pressures and at the finite (high) temperatures

KEYWORDS: anharmonic effect, lattice vibration, moment method, hydrostatic pressure, compressiblity, specific heat, Grăuneisen constant

Đ1 Introduction

The thermodynamic properties of materials at high pressures are of great interest not only from a funda-mental point of view but also for technological applica-tions.1, 2) One of the interesting problems is the

pres-sure induced structural transformations.3-9) The cohe-sive mechanisms and structural properties of a wide va-riety of materials have been studied for last two decades using the first principles density functional theories.4-7) In general, the first principles theory has been very suc-cessful in predicting the ground-state properties, such as crystal structure and atomic volume of the crystalline materials Calculations show that some materials un-dergo crystallographic phase transition upon compres-sion of the Mbar order and at low temperature region It is also known that the effects of pressure on the gap properties of the semiconducting alloys like AlxGa1−xAs

are of great importance and special attention has been given to the direct-to-indirect gap transitions.10)

In order to understand the pressure dependence of the thermodynamic properties of materials, it is highly de-sirable to establish an analytical method which enables us to evaluate the free energy of the system taking into account both the anharmonicity and quantum mechan-ical effect of the lattice vibration So far, the numeri-cal numeri-calculation methods, such as the molecular dynamics and Monte Carlo simulation techniques have been pre-sented However, it is generally difficult to get simple algebraic formula between the thermodynamic quanti-ties and physical insight of the phenomena, within the

∗Permanent address: Hanoi National Pedagogic University, km8

Hanoi-Sontay Highway, Hanoi, Vietnam

2067

non-analytical numerical simulation studies

The phase transformation as a function of pressure can be studied by calculating the Gibbs free energies of the competing phases, although we not go into details of the phase transitions in the present study In order to in-vestigate the thermodynamic properties of the materials, as a function of the pressure, we use the moment method in the statistical dynamics11-13) and derive the explicit expressions of the free energies of the system In par-ticular, the thermodynamic properties of face-centered cubic (fcc) metals are investigated within the fourth or-der moment approximation of the atomic displacements The nearest-neighbour distance, thermal expansion co-efficient, and specific heats Cv and Cp of cubic metals

are derived in terms of the hydrostatic pressure P and the temperature T The numerical calculations are per-formed for fcc Al, Au, Ag, Cu and Pt metals, using the effective pair potentials between the atoms For compari-son, we have also used the many body potentials derived from the microscopic electronic theory We will show that the theoretical calculations on the thermodynamic properties in good agreement with the corresponding experimental results Here, we note that the present the-ory can be applied straightforwardly to study the ther-modynamic properties of metals having lattice structures other than the fcc structure and also to materials having no crystallographic symmetry

In the present article, we study the pressure depen-dence of the thermodynamic properties of the fcc metals, going beyond the harmonic approximation of the lattice vibration We use the moment method in the statistical dynamics, and the essence of the scheme is outlined in the next§2 The detailed formulation of the pressure

de-are

by θ = kBT and x = ¯hω2θ, respectively In the above

eq (2), the summation is taken all over the neighbouring atoms i around 0-th atomic site The vibration frequency ω is related to the vibrational constant k by the well known relation

k =

X

i

à ∂2ϕ

io

∂u2 iβ

!

≡ maω2, (3)

where ma denotes the atomic mass, and β the x, y and

z components of the Cartesian coordinates

The pressure versus volume relation of the lattice is10, 11)

P νa=−r

·

∂U0

∂r + θ

2x coth x k

∂k ∂r ¸

, (4)

where P denotes the hydrostatic pressure and νa the

atomic volume νa = NV of the crystal, being νa = √

2 r

3

for the fcc lattice Using eq (4), one can find the near-est bour distance r1at pressure P and temperature

T However, for numerical calculations, it is convenient to determine firstly the nearest neighbour distance r1 at

pressure P and at absolute zero temperature T = For T = temperature, eq (4) is reduced to

P νa=−r

1

∂U0

∂r + ¯ hω

4k ∂k

∂r (5)

neigh

Table I Potential parameters D and r0of the fcc metals [3]

Metals m n r0(˚A) D/kB(K)

Al 5.5 12.5 2.8541 2995.6

Cu 5.0 8.0 2.5487 3401.1

Ag 5.5 11.5 2.8760 3325.6

Pt 5.5 10.5 2.7890 9914.2

Au 5.5 10.5 2.8751 4683.0

in§2.2 Some of the numerical examples will be given in §3 Final §4 is devoted to the conclusions

§2 Method of Calculation

So far, the anharmonicity of lattice vibration of the crystalline materials has been discussed on the basis of simplifying assumptions and simplified models The allowance for anharmonicity was made by Grăuneisen and by Mie14) in developing their equation of state These

authors assume a temperature-dependent cubic lattice constant a(T ) In the harmonic theory the oscillator or eigenfrequencies ωj of the lattice are independent of the

lattice constant a.15) If, however, one starts with the correct potential energy, then the coupling parameter is second order, and therefore also the ωj are functions of

a If, in the expansion, terms higher than quadratic ones are omitted, then part of the anharmonic effects is al-ready described by the dependence ωj(a) This

approxi-mation is called the quasi-harmonic approxiapproxi-mation.15, 16)

With these assumptions one can calculate the free en-ergy ψ(a, T ) as a function of the lattice constant a and temperature T By minimizing the free energy ψ with respect to the lattice parameter at constant temperature, one obtains the thermal expansion

(∂ψ/∂a)T = 0→ a(T )

The compressibility can also be calculated, being propor-tional to (∂2ψ/∂a2)

T at the equilibrium lattice constant

at the temperature T

Born and co-workers15) studied more thoroughly the

temperature dependence of the elastic constants of crys-talline solids at high temperatures, using also the quasi-harmonic approximation In this case the frequencies ωj

depend on the structure of the until cell, so that all elas-tic moduli can be calculated A method for the direct calculation of the adiabatic constants has been given by Stern.15)

In contrast to the previous studies on the anharmonic-ity of lattice vibration, we investigate the thermody-namic properties of materials fully taking into account the fourth order terms of the atomic displacements in the expansion of the free energy of the system We use the moment method in the statistical dynamics and cal-culate the thermodynamic quantities of the cubic metals 2.1 Pressure versus volume relation

In this subsection, we will derive the pressure versus volume relation of solids (cubic metals) retaining only the quadratic terms in the atomic displacements With the moment method,11-13)the Helmholtz free energy of the

system composed of N atoms can be given by (neglecting higher order terms than the third order term)

ψ0= 3N

½

6U0+ θ[x + ln(1− e

−2x)]¾, (1)

U0=

X

i

ϕ(roi), (2)

where U0represents the sum of effective pair interaction

energies ϕ(roi), on the 0-th atom and θ and x are defined

first

· ¸

For simplicity, we take the effective pair interaction energy in metal systems as the power law, similar to the Lennard-Jones potential

ϕ(r) = D

(n− m) h

m ³r0

r ´n

− n³r0

r ´mi

, (6)

where D and r0 are determined to fit to the

experimen-tal data (e.g., cohesive energy and elastic modulus and m≈ 10 and n ≈ are used as shown in Table I For fcc metals we take into account both the first nearest and second nearest-neighbour interactions One may ask here whether the phenomenological pair potentials are ade-quate to describe the change in the crystal energies due to the atomic displacements in the metals However, it is noted here that atomic displacements due to thermal vibration are relatively small and many body interac-tion effects not play a dominant role in determining the change in the total electronic energies of the system due to the atomic ts.17)In order to check this

point, we have also used the more sophisticated many body potentials derived from the electronic theory.18-21)

The details of using the many body potentials are given in the Appendix

Using the effective pair potentials of eq (6), it is straightforward to get the interaction energy U0 and the

U0=

D 2(n− m)

· mAn µ r0 r1 ¶n

− nAm

µ r0

r1

ảmá

, (7)

k = nmD

2r2

1(n− m)

½

[(n + 2) ¯An+4− An+2]

µ r0

r1

ản

[(m + 2) Am+4 Am+2]

à r0

r1

ảmắ

, (8)

where An, ¯Am,· · · are the structural sums for the given

crystal and defined by An=

n

X

i

Zi

νi

, and A¯m=

1 r2 m X i

Zili2

ν2 i

, (9)

where Zi is the coordination number of i-th nearest

neighbour atoms with radius ri (for fcc lattice ν1 =

and Z1 = 12, ν2 =

√

2 and Z2 = 6) and li denotes the

direction cosine The nearest neighbour distance r1 in

the lattice at absolute zero temperature is obtained by minimizing the total energy of the crystal as

r1= rn0−m

r An

Am

(10)

Then at the absolute zero temperature, one can de-termine the pressure dependence of the lattice constant from the pressure versus volume relation of eq (5) One may use eq (4) for the pressure versus volume relation of the crystal at finite temperatures The thermodynamic quantities at finite temperature T and pressure P can

be determined using the series expansion technique, and the details will be given in the next subsection§2.2 2.2 Thermodynamic quantities of metals under

pres-sure

In order to derive thermodynamic quantities like ther-mal expansion coefficient, specific heats, Grăuneisen con-stant and compressibility of the crystal, we firstly deter-mine the nearest neighbour distance r1(P, T ) at finite

temperature T in a following manner For the calcula-tion of the lattice spacing of the crystal at finite temper-ature, we now need fourth order vibrational constants γ at pressure P and T = K defined by

γ = 12 X i   Ã

∂4ϕ io ∂u4 iβ ! eq + Ã

∂4ϕ io ∂u2 iβ∂u iγ ! eq  

≡ 4(γ1+ γ2), (11)

∆r21(P, T ) =

2γ(P, 0)θ2A(P, T )

3[k3(P, 0)] , (12)

A(P, T ) = a1+

γ2(P, 0)θ2

k4(P, 0) a2+

γ3(P, 0)θ3

k6(P, 0) a3+

γ4(P, 0)θ4

k8(P, 0) a4, (13)

a1= +

x coth x

2 , (14)

a2=

13 +

47

6 x coth x + 23

6 x

2

coth2x +1 2x

3

coth3x, (15)

a3=−

µ 25

3 + 121

6 x coth x + 50

3 x

2

coth2x +16 x

3

coth3x +1 2x

4

coth4x ¶

, (16)

a4=

43 +

93

2 x coth x + 169

3 x

2

coth2x +83 x

3

coth3x +22 x

4

coth4x + 2x

5

coth5x, (17)

with

x = ¯hω(P, 0)

2θ , ω(P, 0) = r

k(P, 0)

m (18)

Then, one can find the nearest neighbour distance r1(P, T ) at pressure P and temperature T as

r1(P, T ) = r1(P, 0) + ∆r1(P, T ) (19)

From eq (12), it is straightforward to derive the thermal expansion coefficient

α(P, T ) = ∆r1(P, T ) r1(P, 0)T

µ +θ

2

A0(P, T ) A(P, T )

¶

, (20)

where A0(P, T ) = dA(P, T )dθ On the other hand, Grăuneisen constant G of the crystal is given as

γG=

1 ·

ln ω(P, T ) ω0(P, T )

Á

lnr1(P, T0) r1(P, T )

¸

(21)

For the numerical calculations we use the following simple scheme For the temperature region close to the temperature T0, one can expand the vibrational

frequency ω(P, T ) and the nearest neighbour distance a(P, T ) around the fixed temperature T0 as

ω(P, T )≈ ω(P, T0) +

∂ω(P, T ) ∂T

¯¯ ¯¯

T =T0

(T− T0), (22)

r1(P, T )≈ r1(P, T0) +

∂r1(P, T )

∂T ¯¯ ¯¯

T =T0

(T− T0) (23)

Furthermore, in the high-temperature region, one can use the approximation x¿ and ln(1+x) ≈ x, and find where β 6= γ = x, y, z It is noted here that for the calculation of the pressure versus volume relation as pre-sented in the previous subsection, the fourth order term gives minor contribution for low temperatures around T0,

but it plays an essential role for high temperature region than the Debye temperature

The thermally induced lattice expansion ∆r1(P, T ) at

the Grăuneisen constant as

γG(P, T ) =

1

r1(P, T0)

ω(P, T0)

∂ω(P, T ) ∂T

¯¯ ¯¯

T =T0

∂r1(P, T )

∂T ¯¯ ¯¯

T =T0

(24)

Using the above formula of γG, we show that the

Grăuneisen parameter G(P, T ) has the weak

tempera-ture dependence, in agreement with the tendency of the experimental results On the other hand, we find the change of the crystal volume at temperature T as

∆V

V =

r3

1(P, T )− r31(P, 0)

r3 1(P, 0)

(25)

Let us now consider the compressibility of the solid phase (fcc metals) According to the definition of the isothermal compressibility χT, it is given in terms of the

volume V and pressure P as11, 12)

χT =−

1 V0 V P ả T (26)

Specifically, for a cubic (fcc) crystal it is expressed as χT =(r1/r10)

3 r1 P r ả T (27)

Here, the pressure P is determined from the free energy ψ of the crystal by

P =

V

ả

T

=r1 3V r ả T (28)

Then, the isothermal compressibility can be given as χT =

3(r1/r10) 2P + √ r1 3N µ ∂2ψ

∂r2 ¶

T

(29)

Furthermore, from the definition of the linear thermal expansion coefficient, one obtains the following formula

α = kBχT P ả V =

2kBT

3r2

1 3N

∂2ψ

∂θ∂r (30) The specific heats of the crystal can be obtained by applying the Gibbs-Helmholtz relation We find the free energy of the crystal using the fourth order vibrational

§3 Results of Numerical Calculations

We now calculate the thermodynamic quantities of metallic systems, thermal expansion coefficient, Grăuneisen constant, specific heats and compressibility using the effective pair potentials between the metal atoms of eq (6).22, 23) The effective pair potentials

be-tween the atoms is chosen to be power law form (similar

Table II The change of volume of metals versus hydrostatic pressure P (GPa)

Metals P 0.49 0.98 1.47 1.96 2.45 2.94

Al ∆V /V0(%) 0.475 0.943 1.391 1.828 2.255 2.658

exp 0.668 1.312 1.932 2.520 3.009 3.642

Cu ∆V /V0(%) 0.268 0.542 0.803 1.051 1.311 1.558

exp 0.352 0.696 1.039 1.370 1.695 2.010

Ag ∆V /V0(%) 0.397 0.808 1.205 1.579 1.952 2.312

exp 0.938 1.385 1.820 2.236 2.619

Pt ∆V /V0(%) 0.170 0.338 0.507 0.675 0.843 0.999

exp 0.176 0.351 0.526 0.701 0.877 1.002

Au ∆V /V0(%) 0.289 0.574 0.859 1.131 1.404 1.675

exp 0.281 0.558 0.831 1.101 1.367 1.626

Experimental results are taken from refs 21 and 22

The third term in the above eq (31) gives the contri-bution from the anharmonicity of thermal lattice vibra-tions Then, the specific heat at constant volume Cv is

given by Cv = 3N kB

½ x2

sinh2x+ 2θ k2

·³ 2γ2+

γ1

3

´x3coth x

sinh2 +γ1

3 µ

1 + x

2

sinh2x ¶

−γ2

x4

sinh4x+

2x4coth2

x

sinh2x (32)

constants γ defined by eq (11) as ψ≈ U0+ ψ0+

3N θ2

k2

·

γ2x2coth2x

+γ1

à

2 + x

2

sinh2x ả

− 2γ2

x3coth x

sinh2x ¸

(31)

à ảáắ

The specific heat at constant pressure Cp and the

adi-abatic compressibility χs are determined from the well

known thermodynamic relations Cp= Cv+

9T V α2

χT

, and χs=

Cv

Cp

χT (33)

When the compressibilities χT and χs are known, one

can determine the inverses of them, i.e., the isothermal and adiabatic bulk moduli BT and Bs, as

BT =

1 χT

and Bs=

1 χs

(34)

to Lennard-Jones potentials) For the fcc metals Au, Ag, Al, Cu and Pt, the potential parameters D, r0, m and

n are taken from ref 22 These parameters are deter-mined so as to fit the experimental lattice constants and

cohesive properties Using these effective potentials, one can find the nearest neighbour distance r1(P, 0) at

pres-sure P and temperature T = K Then, we calculate the vibrational constants k and γ at the pressure P and

tem-Fig Changes in volume−∆V/V0(%) versus hydrostatic

perature T = K with the aid of eqs (3), (11) and (19) After determining the quantities at T = 0, the nearest neighbour distance r(P, T ), the thermal expansion coef-ficient α(P, T ) and the Grăuneisen parameter G(P, T )

at the pressure P and temperature T are calculated The changes in volume of metals under hydrostatic pressure P are calculated for Al, Cu, Ag, Pt and Au, using eq (12) as a function of the pressure P , at tem-perature T = 300 K The calculated results are presented in Table II and Figs 1(a), 1(b), 1(c) and 1(d), for Al, Cu, Ag, Pt and Au, metals respectively, together with the corresponding experimental results.24)In the figures,

dashed straight lines indicate the equilibrium bulk mod-uli, i.e., linear relationship between the pressure P and volume V of the metals, in the limit of the zero sure The calculated change of the volume under pres-sure are in good agreement with the experimental data (Table II) In general, the agreement of the calculated results of ∆V /V versus pressure P are better for noble and transition metals in which d-band cohesion is pre-dominant, compared to those of sp-valence metals like Al

We have also calculated the thermodynamic quanti-ties of the above metals using the many body poten-tials18-21, 25-28) derived from the microscopic electronic

theory For transition metals, the many body potentials are composed of two terms, i.e., contributions of band structure energy and the short-range repulsive energy The former band structure energy is due to the cohe-sion of the d-bands and the latter repulcohe-sion comes from the overlap between d-orbitals and the increase in the kinetic energy of sp-valence electrons upon the compres-sion The many body potentials for Ag, Au, Cu, and Pt metals are taken from Cleri and Rosato,21) and the

parameters are given in Table III The calculated values of ∆V /V versus P of Ag, Au, Cu and Pt metals using the many body potentials are presented in Figs 1(b), 1(c) and 1(d) by calc.2 curves In order to account the free electron nature of the valence electrons of Al metal,

Table III The parameter values of many body potentials (eV unit)

Metals A (eV) p ξ0 (eV) q

Al 0.1780 6.500 1.3831 2.070

Cu 0.07157 11.562 1.1485 2.02139

Ag 0.1028 10.928 1.1780 3.139

Pt 0.08435 14.8 1.2495 3.40

Au 0.2061 10.229 1.790 4.036

For Al, C1= 00947, S0 1= 00515 and C0 2= 01664 are used.0

Table IV Thermodynamic quantities of Al, Cu, Ag, Pt and Au metals at T = K and under pressure P

Metals P (GPa) 0.49 0.98 1.47 1.96 2.45 2.94

Al

a(0, P ) 2.8265 2.8224 2.8184 2.8146 2.8108 2.8072

a(P, T ) 2.8395 2.8350 2.8307 2.8266 2.8225 2.8186

χT· 10−3(GPa−1) 9.9298 9.5724 9.2394 8.9358 8.6454 8.3804

α· 10−5(K−1) 2.2019 2.1259 2.0548 1.9897 1.9274 1.8703

Cv(cal/mol·K) 5.3730 5.3655 5.3579 5.3503 5.3424 5.3346

Cp(cal/mol·K) 5.6804 5.6614 5.6430 5.6255 5.6081 5.5917

γG 3.0659 3.0543 3.0433 3.0331 3.0231 3.0138

Cu

a(0, P ) 2.5155 2.5133 2.5112 2.5092 2.5071 2.5051

a(P, T ) 2.5234 2.5211 2.5189 2.5168 2.5146 2.5125

χT· 10−3(GPa−1) 5.5360 5.4320 5.3348 5.2439 5.1511 5.0643

α· 10−5(K−1) 1.5973 1.5684 1.5412 1.5158 1.4899 1.4657

Cv(cal/mol·K) 5.5726 5.5700 5.5675 5.5650 5.5623 5.5597

Cp(cal/mol·K) 5.7763 5.7696 5.7632 5.7571 5.7508 5.7447

γG 2.6343 2.6284 2.6228 2.6175 2.6120 2.6068

Ag

a(0, P ) 2.8383 2.8346 2.8310 2.8276 2.8242 2.8209

a(P, T ) 2.8480 2.8441 2.8403 2.8367 2.8331 2.8297

χT· 10−3(GPa−1) 8.4882 8.2447 8.0161 7.8067 7.6046 7.4145

α· 10−5(K−1) 1.7962 1.7476 1.7019 1.6600 1.6195 1.5814

Cv(cal/mol·K) 5.7278 5.7273 5.7268 5.7261 5.7254 5.7247

Cp(cal/mol·K) 5.9693 5.9617 5.9545 5.9478 5.9411 5.9348

γG 2.7116 2.7023 2.6935 2.6852 2.6771 2.6994

Pt

a(0, P ) 2.7373 2.7358 2.7343 2.7328 2.7313 2.7277

a(P, T ) 2.7419 2.7404 2.7389 2.7373 2.7358 2.7343

χT· 10−3(GPa−1) 3.5102 3.4666 3.4238 3.3817 3.3402 3.3019

α· 10−5(K−1) 0.9008 0.8899 0.8791 0.8686 0.8582 0.8485

Cv(cal/mol·K) 5.7455 5.7445 5.7435 5.7425 5.7414 5.7404

Cp(cal/mol·K) 5.8766 5.8738 5.8710 5.8683 5.8656 5.8630

γG 2.8722 2.8679 2.8637 2.8595 2.8554 2.8515

Au a(P, 0) 2.8416 2.8390 2.8364 2.8339 2.8314 2.8289

a(P, 300 K) 2.8489 2.8461 2.8434 2.8408 2.8382 2.8356

χT· 10−3(GPa−1) 5.9845 5.8591 5.7372 5.6227 5.5112 5.4026

α· 10−5(K−1) 1.3646 1.3373 1.3107 1.2857 1.2614 1.2377

Cv(cal/mol·K) 5.7800 5.7814 5.7811 5.7808 5.7804 5.7800

Cp(cal/mol·K) 5.9800 5.9750 5.9705 5.9662 5.9620 5.9578

γG 2.8700 2.8664 2.8594 2.8527 2.8461 2.8395

ment approximation of the atomic displacements To our knowledge, this is the first time that the pressure dependencies of the thermodynamic quantities of metals have been studied using the analytic theoretical scheme, and numerically evaluated The analytic formulae of the present study allow us to calculate the thermodynamic quantities quite accurately without using the certain fit-ting and averaging procedures, like the least squares method The present formalism is not restricted to the applications of the effective pair potentials, but it is also incorporated with the energetics based on the ab ini-tio electronic theory In general, we have obtained good agreement in the thermodynamic quantities between the three oscillatory interactions are added21) as shown in

the Appendix

The more sophisticated cohesion theory of sp-valence metals has been developed recently by Hansen et al.,26) and their theory is also used in the present calculations of the thermodynamic quantities The results of ∆V /V versus P of Al metal by many body potentials of Cleri and Rosato21)and of Hansen et al.26) are reffered to as

calc.2 and calc.3 in Fig 1(a), respectively In general, better agreements of the ∆V /V versus P have been ob-tained by using the electronic many body potentials The better agreements are obtained because the bulk moduli of the metals at K temperature are well reproduced by the many body potentials rather than the pairwise Lennard-Jones potentials, and also metallic bondings are well described by the former electronic theory The more details are also given in the Appendix

The thermodynamic quantities at the pressure P and finite temperature T are obtained from those values of T = The numerical results of the specific heats Cvand

Cp, the thermal expansion coefficient , the isothermal

compressibility 1T , and Grăuneisen constant, are

pre-sented in Table IV One can see in Table IV that the Grăuneisen constant G for the fcc metals are almost

in-dependent of the external pressure This is one of the important theoretical findings of the present study and the tendency is in agreement with experimental observa-tions The calculation of the thermodynamic quantities of the crystals by the present statistical moment method is of great significance in the sense that the thermody-namic quantities are directly determined from the closed analytic expressions and it does not use the certain (ar-tificial) averaging procedures, as in the usual computer simulation studies based on the molecular dynamics and Monte Carlo methods

In the present study, we have used effective pair po-tentials for metal atoms to demonstrate the utility of the present theoretical scheme based on the moment method in the statistical dynamics However, as we see in the ∆V /V versus P calculations, it is straightforward to use more fundamental first principles potentials, like many body potentials18-20)describing the metallic bondings in

the lattice Our preliminary calculation shows that the thermodynamic quantities of the metals under hydro-static pressures can be calculated within the similar ac-curacy even when using the many body potentials We also note that the present theoretical scheme based on the statistical moment method is successfully applicable to the many important problems of the materials sci-ence, e.g., calculations of XAFS (X-ray absorption fine structure) and alloy phase diagrams taking into account thermal lattice vibration and the size-misfit among the constitute atoms.29-31)

§4 Conclusions

The present moment method in the statistical dynam-ics allows us to investigate the thermodynamic properties of metals under hydrostatic pressures and finite temper-atures The method is simple and physically transpar-ent, and thermodynamic quantities of the metals can be expressed in closed forms within the fourth order

mo-theoretical calculations and experimental results The calculated bulk moduli and the first pressure derivatives of the bulk moduli are in good agreement with experi-ments

Acknowledgments

One of the authors (V.V.H) thanks the Japan Soci-ety for the Promotion of Science for financial support and Department of Materials Science and Engineering, Tokyo Institute of Technology for support and hospital-ity during his stay from September to November 30, 1998

Appendix: Application of Electronic Many Body Potentials

The calculation of the thermodynamic quantities of materials can be done by using the statistical moment method and the energetics based on the electronic the-ory For this purpose, the energy term U0and the

deriva-tives of the atomic ts, k and γ, are aluated numerically for each atom when the analytic calculations are not possible

Transition metals are elements with partially filled narrow d band superimposed on a broad free electron-like s-p band Most of the properties of the transition metals are characterized by the filling of the d band The cohesive energy of a transition metal consists of two terms

Ecoh= Ebond+ Erep, (A.1)

where the second term Ereprepresents the repulsive

en-ergy arising from the overlap between d-orbitals and the increase of sp-valence electrons upon compression The functional form is given by the second moment approxi-mation as

which has five parameters ε0, ξ0, p, q, and r0, fitted to

empirical data such as the cohesive energies and elas-tic constants Highly reliable parameters are derived by fitting the first principles calculations within the general-ized gradient approximation (GGA) of density functional theory.28) Because of the summations under the square

Etot=

1 N

N

X

i=1

( A0

N

X

j6=i

exp µ

−p ·

rij

r0

− ¸¶

− ·

ξ02

X exp

µ −2q

· rij

r0

− ¸¶¸1

2

)

, (A.2)

root they are many-body potentials in the sense that they are not a sum of pairwise additive functions Cleri and Rosato21) fitted these parameters to experimental

data for 16 fcc and hexagonal-close-packed (hcp) tran-sition metals A different parametrization strategy was introduced by Sigalas and Papacostantopoulos in which the parameters were fitted to local density approxima-tion (LDA) calculaapproxima-tions of the total energy as a funcapproxima-tion of lattice constant

For sp-valance metals like Al, the additional oscilla-tory terms Eiosc are added to the many body potential

of (A.2) Eiosc=

X

j6=i

½

C1cos(2kFrij)

(rij/r0)3

+S1sin(2kFrij) (rij/r0)4

+C2cos(2kFrij) (rij/r0)5

¾

, (A.3) where kF denotes the Fermi wave vector of the metal

The C1, S1 and C2 values for Al metal are presented in

Table III

The many body potential scheme is similar to the so-called embedded atom method.19)In the embedded atom

method, each atom in a solid is viewed as an atom em-bedded in a host comprising all the other atoms A sim-ple approximation to embedding function F is the so-called local ximation, whereby the embedded atom experiences a locally uniform electron density This can be viewed as the lowest-order term of an expansion in-volving the successive gradients of the density The func-tional F is then approximated to yield

E = Fi(ρi(ri)) +

1

X

j

φij(rij), (A.4)

where φijis a pair potential representing the electrostatic

interaction, rij is the distance between atoms i and j,

and Fidenotes the embedding energy The total energies

of metals and alloys are given by a sum over all individual contributions:

Etot=

X

i

Fi(ρh,i) +

1

X

i,j i6=j

φij(rij), (A.5)

where the host density ρh,i at atom i is closely

approx-imated by a sum of the atomic densities ρj of the

con-stituent atoms

This conventional EAM has further refined by Erco-lessi and Adams and Hansen et al The ErcoErco-lessi-Adams interaction model for Al was constructed with so-called force matching method and it gives excellent structural and elastic properties for the bulk along with the correct surface interlayer relaxations Hansen et al refined the Ercolessi-Adams potential to introduce additional terms in order to account for (i) an exponential Born-Mayer-like repulsion at short Al–Al separation (for physical va-por deposition), and improving the embeding function F in the low-density region (for Al2dimer), and also

in-troducing the polynomial cut off function: The atomic density ρi is then given as

ρi=

X

j6=i

ρ(rij)× fc(rij, R0, D0) (A.6)

appro

The sum runs over all atoms that lie within the potential range R0+ D0 (5.56 ˚A), which is enforced by the cutoff

function fc(r, R, D) This function is zero for r

exceed-ing R + D and unity for r less than R− D For r within the interval (R− D, R + D) it is defined according to

fc(r, R, D) =−3

· r− R

D +

¸5

+15

· r− R

D +

¸4

−5 ·

r− R

D +

¸3

+ (A.7)

The pair potential term is written as ¯

φij = [φ(rij) + (A exp{−λrij} × fc(rij, Rφ, Dφ)− B)]

×fc(rij, R0, D0), (A.8)

where the first cutoff fc(rij, Rφ, Dφ) switches on the

ex-ponential repulsive term at small distances (r < 2.25 ˚A), while fc(rij, Rφ, Dφ) terminates the interaction range of

the potential The exponential term ensures that one gets a Born-Mayer repulsion at short separations for, e.g., diatomic molecules As mentioned in§3, the above mentioned many body potentials and EAM are very suc-cessful for the calculation of ∆V /V versus P of the fcc metals

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