EQUATION OF STATE AND THERMODYNAMIC PROPERTIES OF BCC METALS

The moment method in statistical dynamics is used to study the equation of state and thermodynamic properties of the bcc 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 of the bcc metals are derived within the fourth order moment approximation. The thermodynamic quantities of W, Nb, Fe, and Ta metals are calculated as a function of the pressure, and they are in good agreement with the corresponding results obtained from the first principles calculations and experimental results. The effective pair potentials work well for the calculations of bcc metals. P V C C ,


INTRODUCTION
The study of high pressure behaviour of materials has become quite interesting in recent years since the discovery of new crystal structures and due to many geophysical and technological applications.A lot of theoretical models have been proposed in order to predict the P-V-T equation of state (EQS) at the high pressure domain.Using the input data as the volume , the bulk modulus ,etc., at the available low-pressure, these EQS models predict the highpressure behaviours of materials.However, the results obtained from these semi-empirical models depend on the input data and the kinds of model.0 V 0 T B So far, most path integral Monte Carlo (PIMC) [1,2] and path integral molecular dynamic (PIMD) [3,4] have been restricted to the calculation of structural and thermal properties of quantum solids or to the calculation of equations of state of condensed rare gases.Within the framework of the density-functional theory (DFT) [5], the thermodynamic properties of solids under a constant pressure can be calculated from the first-principles caculations .For ordered solids, the free energy at finite temperature has contributions from both the lattice vibrations and the thermal excitation of electrons.In the quasiharmonic approximation, the free energy is calculated by adding a dynamical contribution which is approximated by the free energy of a system of harmonic oscillators corresponding to the crystal vibrational modes (phonons)-to a static contribution-which is accessible to standard DFT calculations [6].Vibrational modes are treated quantum mechanically, but the full Hamiltonian is approximated by a harmonic expansion about the equilibrium atomic positions.Anharmonic effects are included through the explicit volume dependence of the vibrational frequencies.The static high pressure properties of the transition metals (for example tantalium with the body centred cubic (bcc) structure) obtained from the first principles by using the linearizing augmented plane wave (LAPW) method [7,8].Calculations based on various semiempirical models [9 -12] as well as on the first-principles methods [13 -16] demonstrate that the quasiharmonic approximation provides a reasonable description of the dynamic properties of many bulk materials below the melting point.
In the present study, we use the moment method in statistical dynamic [17 -20] to investigate the equation of state and thermodynamic properties of bcc metals.We will calculate the temperature and pressure dependence of the nearest neighbour distance and the thermodynamic properties of bcc metals.
The format of the present paper is as follows: In Sec. 2, the equation of state and the temperature and pressure dependence of thermodynamic properties of bcc metals are given.The calculation results of thermodynamic properties of W, Nb, Fe and Ta metals at various pressures are presented and discussed in Sec. 3 .

Pressure versus volume relation
The pressure versus volume relation of the lattice is [17] Pv = -a where , and P denotes the hydrostatic pressure and v is the atomic volume v = V/N of the crystal, being v = 3 3 3 4 a for the bcc lattice.Using eq.( 1), one can find the nearest neighbour distance at pressure P and temperature T.However, for numerical calculations, it is convenient to determine firstly the nearest neighbour distance at pressure P and at absolute zero temperature T = 0.For T = 0 temperature, eq. ( 1) is reduced to For simplicity, we take the effective pair interaction energy in metal systems as the power law, similar to the Lennard-Jones where are determined to fit to the experimental data (e.g., cohesive energy and elastic modulus).For bcc metals we take into account the first nearest, second, third, fourth and fifth nearest neighbour interactions.

, r D
Using the effective pair potentials of Eq.( 3), it is straighforward to get the interaction energy and the parameter k in the crystal as From eqs. (2), (4), and (6) we obtain equation of state of bcc crystal at zero temperature where y = a r o , In principle Eq. ( 9) permits to find the nearest neighbour distance at zero temperature and pressure P. Using the MAPLE V program and the values of parameters D and determined by the experimental data [21] (Table 1), Eq. ( 9) can be solved, we find the values of the nearest neighbour distance at temperature T = 0 and pressure P. Calculated results for the nearest neighbour distance of W, Nb, Ta and Fe metals at zero temperature and pressure P are presented in the Table 2.

Thermodynamic quantities of bcc metals at high pressure
For the calculation of the lattice spacing of the crystal at finite temperature and pressure P, we now need fourth order vibrational constants γ and k at pressure P and T = 0 K defined by = Using the effective pair potentials of Eq. ( 3), the parameter γ of the bcc crystal has the form where the structural sums equal to Using the obtained results of nearest neighbour distance ( Table 2) and Eqs. ( 5), ( 7), ( 13) and ( 14 The thermally induced lattice expansion y o (P,T) at pressure P and temperature T is given in a closed formula using the force balance criterion of the fourth order moment approximation as [17,18] ) , ( ) 0 , Then, one can find the nearest neighbour distance at pressure P and temperature T as Using the above formula of distance , we can find the change of the crystal volume at temperature T as Let us now consider the compressibility of the solid phase (bcc metals).The isothermal compressibility can be given as Furthermore, from the definition of the linear thermal expansion coefficient, one obtains the following formula We find the free energy of the crystal using the statistical moment method as [17,19] Then, the energy of the crystal equal to where represents the sum of effective pair interaction energies and the second term in the above Eq.( 23) given the contribution from the anharmonicity of thermal lattice vibrations and the fourth order vibrational constants 0 U 2 1 ,γ γ defined by Eq. (12).Then, the specific heat at constant volume is given by The specific heat at constant pressure , the adiabatic compressibility ( One can now apply the above formulae to study the thermodynamic properties of bcc metals under hydrostatic pressures.The pressure dependences of the crystal volume, isothermal compressibility, specific heats and the linear thermal expansion coefficient are calculated selfconsistently with the lattice spacing of the given bcc crystals.

RESULTS AND DISCUSSION
In order to check the validity of the present moment method for the study of the thermodynamic properties of the metallic systems described herein, we performed calculations for pure metals W, Ta , Fe and Nb.Using the experimental data of the parameters D and r 0 ( Table 1), and the MAPLE V program, Eq.( 9) can be solved, we find the values of the nearest neighbour distance a (P, 0) at temperature T = 0 and pressure P for W, Ta, Fe, and Nb metals.Using the obtained results of the nearest neighbor distance a(P, 0) (Tables 2) and Eqs. ( 5), ( 13), we find the values of parameters k(P, 0), and ) 0 , (P γ at pressure P and temperature T = 0 K. α for W, Ta, Fe and Nb metals.The calculated results are presented in Tables 3 -8 and Figs.1-4. Table 3 shows the lattice constants and bulk moduli for all of the bcc metal studied here, comparing them to first-principles LDA calculations, the tight-binding (TB) results [23], and to experiment [24,25].The lattice constant and bulk modulus at temperature T = 300 K and zero pressure calculated by the present theory are in good agreement with the first-principles results and experimental data.The lattice constant is within 2% of the SMM values for all of the bcc metals.Similarly, the bulk moduli are in excellent agreement with the experimental results, within < 1% for W, Fe, and Nb metals except tantalum, where the error is 9%.We not that for the bulk moduli of W, Fe and Nb metals, the present calculations give much better results compared to those by previous theoretical calculations.

Table 3:
Calculated results for the lattice constant, , and bulk modulus, , at T = 300 K and P = 0, comparing the results of tight-binding parametrization (TB), firstprinciples local density approximation (LDA) [23] results and experiment (Expt.)(Refs.24  In Table 4 we compare with the first-principles calculations and experiment the zero pressure volume,V , and the bulk moduli, for Ta and W metals.We show in Table 4 the results obtained by A. Strachan et al. [26] using the linearized augmented plane wave method with the GGA (denoted as LAPW-GGA) and the Embedded Atom Model force fields (named qEAM FF), and zero temperature calculations using full potential linear muffintin orbital method within the GGA approximation and with spin orbit interactions (denoted as FP LMTO GGA SC) by S derlind and Moriarty [27].The results obtained by Y. Wang et al. [29] using the densityfunctional theory (denoted as DFT), and room temperature experimental values by Cynn, Yoo [28] and A Dewaele et al. [30] are also presented in The Figs. 1 and 2 show the ratio , and bulk moduli for W, Nb and Ta metals as the functions of the pressure P. The present SMM calculations for the ratio V/ are in good agreement with experimental results which taken from McQueen et al [31] for Nb and Ta; and from McQueen and Marsh [32] for W. The lattice constants decrease due to the effect of increasing pressure, therefore the bulk modulus becomes larger.The Fig. 3 shows the bulk modulus of the W, Nb and Ta metals as a function of the temperature T at various pressures P. We have found that the bulk modulus, depends strongly both on the temperature and the pressure.The decrease of with increasing temperature arises from the thermal lattice expansion and the effects of the vibration entropy.24) and ( 25), respectiv .However, the evalutions by Eqs. ( 24) and ( 25) are the lattice contributions, a we do not include the contributions of lattice vacancies and electronic parts of the specific heats V C .The calculated values of the lattice specific heats V C and P C by the present SMM may not be directly compared with the corresponding experimental values for high temperature region ear the melting temperature, but the temperaturedependence (curvature) of P C for the W, Nb and Ta metals is in agreement with the experimental results.  .The present SMM calculations for the linear thermal expansion coefficient are in good agreement with the experimental results.The present formalism takes into account the quantum-mechanical zero-point vibrations as well as the higher-order anharmonic terms in the atomic displacements and it enables us to derive the thermodynamic quantities of the bcc metals for a wide temperature range.Fig. 4 shows that the thermal expansivity as a function of pressure.Theory predicts the thermal expansivity to drop rapidly with pressure, and the temperature dependence to decrease.These results are in good agreement with the first-principles calculations reported by R. E. Cohen and O. Gulsenren [16].

Table 8:
Temperature dependence of the linear thermal expansion coefficient for Nb and

CO LUSION
ulations are performed by using the ef NC M calc fective pair potential for the W, Fe, Nb and a metals.The use of the simple potentials is due to the fact that the purpose of the present study is to gain a general understanding of the effects of the anharmonicity of the lattice a n the thermodynamic properties for the bcc metals.
In The SM T vibr tion and pressure o utility of the present theoretical scheme based on the moment method in the statistical dynamics.The method is simple and physically transparent, and thermodynamic quantities of metals with bcc structures can be expressed in closed forms within the fourth order moment approximation of the atomic displacements.The present formalism is not restricted to the effective pair potentials, but it is also incorporated with the energetics based on the ab initio electronic theory.In general, we have obtained good agreement in the thermodynamic quantities between our theoretical calculations and first-principles results, and experimental values.

ω
is the frequency of lattice vibration, and ,... are the structural sums for the given crystal and defined by number of i-th nearest neighbour atoms with radius (for bcc lattice r k = υ k a o ), we find the values of parameters , at pressure P and T = 0K.

Table 1 :
Parameter D and determined by the experimental data [21]

Table 2 :
Calculated results for the nearest neighbour distance a(P, 0) at zero temperature and pressure P

Table 4 .
The present SMM calculations of the bulk mudulus and zero pressure volume at absolute zero and room temperatures agree well with the experimental values and previous theoretical calculations.The zero pressure volume,V , is in excellent agreement with the experimental results, within ~0.5% for W metal except tantalum, where the error is ~6%.

Table 5 :
Thermodynamic quantities of W. Nb, and Ta metals at pressure P = 0 and temperature T

Table 6 :
Temperature dependence of the present study, we have used effective pair potentials for metal atoms to demonstrate the applications of the .E., Berne, B.J., Marlyna, G.J., and Klein, M.L. (1993), J. Chem.Phys., vol.