Pressure-volume relationships of some solids based on double potential energy functions

We present an analysis for the pressure-volume relationship of substances viz. Neon, Argon, Aluminum, Copper, Lithium Hydride, and Magnesium Oxide using interatomic potential functions due to Morse, Rydberg, and Davydov. The formulations for P-V relationship have been obtained using these potential functions. The results for pressure as a function of volume are determined up to a range of V/(V_0=0.5) for each solid. The results are mapped with the values obtained from the Shanker equation and the Hama-SuitoEOS. The relationship between P and V through different equations of state(EOS) for different solids depends on the values of K_(0 )as well as K_0^’. A material would be more incompressible if both K_(0)and 〖K〗_0^’are high. For example in case of Cu, K_(0 )is somewhat less than that for MgO but 〖K〗_0^’ is larger for Cu than that for MgO. This makes Cu to be more incompressible than MgO. This is evident from the results which reveal P = 620GPa for Cu, and P = 430GPa for MgO both at V/V_0 = 0.5, the maximum compressions.


INTRODUCTION
The physical and the chemical properties of a solid found to be relatedwith its interatomic forces and therefore any change of temperature valuechanges the inter-atomic distance of the solids which in turn changes its different properties. EOS is the technique whichprovides us the usefulinformation about the association between various thermodynamic variables like pressure, volume, and temperature [1][2][3]. All thermodynamic system has it's particular EOS which is independent of others EOS's. EOS reveals Behavior unlike a single divisive system it from others. In the present endeavor, we have calculated the pressure-volume association for monatomic and diatomic solids with different nature of chemical bonds using different EOS. For this purpose, we will discuss here four types of mono-atomic solids (a) FCCsp metal-Aluminum [4,5],(b) BCC-rare-gas metal-Copper [6,7],(c) FCC-difficult to metalize Substance-Neon [8,9] and (d) FCC-large gap but small bulk modulus-Argon [8] . For diatomic solids we will discuss Lithium Hydride (LiH)which have small bulk modulus [9,10] and one another Magnesium Oxide(MgO) which is identified as the substance having large value of bulk modulus [5] .For diatomic substances, the Rydberg-Vinet equations and others deviatesat theoretical pressure at large compression

METHOD OF FORMULATIONS
.Equation of state for calculating the pressure induced compression of solids predates the work of Morse [14]. MorseEOS has been obtained using the double exponential potential energy functions and can be expressed as follows and f = K 0 ′ − 1 .
r and ro being the intermolecular separation for the repulsive and the attractive forces. Vinet et al [15] have derived the EOSusing the potential energy function given by Rydberg [9]. This EOS is written as follows Davydov obtained another alternative form of EOS which has been mentioned by Zharkov and Kalinin [16].This EOS is based on a different potential energy function which yields Shanker et.al [17] have derived an EOSfrom the derivatives of potential energy addressing the volume-interatomic force constant relationship of the solids. This EOS is written as follows Hama and Suito [18] have derivedan EOS by the augmented plane wave (APW) methods and the quantum statistical model. The Hama-Suito EOS has been found to be in good agreement with the data for different types of substances and various ranges of pressure from low to extremely high values. The Hama-Suito EOS is given below

RESULTS AND ANALYSIS.
From the above formulation, the values of P for the substances viz.Neon, Argon, Aluminum, Copper, Lithium Hydride, and Magnesium Oxide are calculated. The input values (Table 1) for 0 , 0 ′ and 0 ′′ are taken from ab initio results for different solids due to Hama and Suito [18].In order to make the results significant, we have used the valuesof 0 , 0 ′ and 0 ′′ in all the EOSwith zeromodifications. The P values are given in Table 2 for the purpose of compression up to a range of 0 = 0.5 ⁄ for each solid.It is found from the results given in Table 2 that the EOS based on potential functions yield good agreement with each other and also with the Hama-Suito EOS derived from the first principles based on the APW method and the quantum statistical model. The pressure required for different solids to produce the maximum compression 0 ⁄ = 0.5 are quite different from each other. Thus neon (Ne) and argon (Ar) are more compressible (less amount of pressure is required at 0 = ⁄ 0.5) as compared to the other solids e.g. copper(Cu) and magnesium oxide (MgO). The bulk modulus represents incompressibility of a material. For Cu and MgO, the bulk moduli have largest values (Table1),and therefore these solids are 0 is somewhat less than that for MgO but 0 ′ is larger for Cu than that for MgO. This makes Cu to be more incompressible than MgO. This is evident from the results which reveal P=620GPa for Cu, and P=430GPa for MgO both at 0 ⁄ = 0.5, the maximum compressions. The results obtained in the present study are useful for investigating high-pressure thermo-elastic properties of materials.

Conclusions.
From the above study, it is concluded that the relationship based on double potential energy functions between P and V through different equations of state (EOS) for different solids depends on the values of 0 as well as 0 ′ . A material would be more incompressible if both 0 and 0 ′ are high.

5.ACKNOWLEDGEMENTS.
The author is sincerely thankful to Dr. Jai Shanker, Agra, India for his valuable guidance and useful discussions.