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, Gr¨ uneisen constant

§1 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 do 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

pendence of the thermodynamic quantities is presented

by θ = kBT and x = ¯hω, 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 = 1 2

X

i

Ã

∂2ϕio

∂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

· 1 6

∂U0

∂r +

θ

2x coth x

1 k

∂k

∂r

¸

where P denotes the hydrostatic pressure and νa the atomic volume νa = NV of the crystal, being νa = √22r3

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 = 0 For

T = 0 temperature, eq (4) is reduced to

P νa=−r 1

6

∂U0

∂r +

¯ hω 4k

∂k

neigh

Table I Potential parameters D and r 0 of the fcc metals [3].

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 Gr¨uneisen 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 1

6U0+ θ[x + ln(1− e−2x)]¾

U0=X

i

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

(n− m)

h m

³r0 r

´n

− n³r0

r

´mi

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 ≈ 5 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 do 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 vibrational constant k in the crystal as

diplacemen

U0= D

2(n− m)

·

mAn

µ

r0

r1

¶n

− nAm

µ

r0

r1

¶m¸ , (7)

2r2(n− m)

½ [(n + 2) ¯An+4− An+2]

µ

r0

r1

¶n

−[(m + 2) ¯Am+4− Am+2]

µ

r0

r1

¶m¾

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 1

m

X

i

Zil2 i

ν2 i

where Zi is the coordination number of i-th nearest

neighbour atoms with radius ri (for fcc lattice ν1 = 1

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

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, Gr¨uneisen 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 = 0 K defined by

γ = 1 12

X

i





Ã

∂4ϕio

∂u4 iβ

!

eq

+ 6

Ã

∂4ϕio

∂u2

iβ∂u2 iγ

!

eq





∆r21(P, T ) = 2γ(P, 0)θ

2A(P, T )

A(P, T ) = a1+γ

2(P, 0)θ2

k4(P, 0) a2+

γ3(P, 0)θ3

k6(P, 0) a3+

γ4(P, 0)θ4

a1= 1 + x coth x

a2= 13

3 +

47

6 x coth x +

23

6 x

2

coth2x +1

2x

3

a3=−

µ 25

3 +

121

6 x coth x +

50

3 x

2

coth2x +16

3 x

3

coth3x +1

2x

4

coth4x

¶

a4= 43

3 +

93

2 x coth x +

169

3 x

2

coth2x +83

3 x

3

coth3x +22

3 x

4

coth4x + 1

2x

5

coth5x, (17)

with

x = ¯hω(P, 0)

2θ , ω(P, 0) =

r k(P, 0)

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

1 +θ 2

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

¶

where A0(P, T ) = dA(P, T )dθ On the other hand,

Gr¨uneisen constant γG of the crystal is given as

γG= 1 3

·

ln ω(P, T )

ω0(P, T )

Á

lnr1(P, T0)

r1(P, T )

¸

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¿ 1 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 pressure P and temperature T is given in a closed for-mula using the force balance criterion of the fourth order moment approximation as11, 12)

the Gr¨uneisen constant as

γG(P, T ) = 1

3

r1(P, T0) ω(P, T0)

∂ω(P, T )

∂T

¯¯

¯¯

T =T 0

∂r1(P, T )

∂T

¯¯

¯¯

T =T 0

Using the above formula of γG, we show that the

Gr¨uneisen parameter γG(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

r3(P, T )− r3(P, 0)

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

µ

∂V

∂P

¶

T

Specifically, for a cubic (fcc) crystal it is expressed as

χT =−(r1/r10)3 3

r1

µ

∂P

∂r

¶

T

Here, the pressure P is determined from the free energy

ψ of the crystal by

P =−

µ

∂ψ

∂V

¶

T

=−r1

3V

µ

∂ψ

∂r

¶

T

Then, the isothermal compressibility can be given as

χT = 3(r1/r

0

1)3 2P +

√ 2

r1

1 3N

µ

∂2ψ

∂r2

¶

T

Furthermore, from the definition of the linear thermal

expansion coefficient, one obtains the following formula

α = kBχT

3

µ

∂P

∂θ

¶

V

=−

√ 2kBχT

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, Gr¨uneisen 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).

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

2

sinh2x

¶

−γ2

x4

sinh4x+

2x4coth2x

constants γ defined by eq (11) as

ψ≈ U0+ ψ0+3N θ

2

k2

·

γ2x2coth2x

+γ1 3

µ

2

sinh2x

¶

− 2γ2

x3coth x sinh2x

¸ (31)

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 α

2

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

One can now apply the above formulae to study the thermodynamic properties of materials under hydro-static pressures The pressure dependences of the crystal volume, isothermal compressibility, Gr¨uneisen constant, and specific heats are calculated self-consistently with the lattice spacing of the given crystal

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 = 0 K Then, we calculate the vibrational constants k and γ at the pressure P and

tem-Fig 1 Changes in volume −∆V/V 0 (%) versus hydrostatic pres-sure P (GPa) for Al, Cu, Ag, Pt and Au metals.

perature T = 0 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 Gr¨uneisen parameter γG(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).

For Al, C 1 = 00947, S 0 1 = 00515 and C 0 2 = 01664 are used 0

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

Al

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

Cu

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

Ag

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

Pt

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

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

300

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 0 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 = 0 The numerical results of the specific heats Cvand

Cp, the thermal expansion coefficient α, the isothermal

compressibility χ−1T , and Gr¨uneisen constant, are

pre-sented in Table IV One can see in Table IV that the

Gr¨uneisen constant γG 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 1 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

j 6=i

exp

µ

−p

·

rij

r0

− 1

¸¶

−

·

ξ02X exp

µ

−2q

·

rij

r0

− 1

¸¶¸1 )

, (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

j 6=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

2

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

2

X

i,j

i 6=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

j 6=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

¸5

+15 2

·

r− R

¸4

−5

·

r− R

¸3

The pair potential term is written as

¯

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

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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