Anharmonicity in metals from the universal energy equation
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Anharmonicity in metals from the universal energy equation L. A. Girifalco and K. Kniaz Department of Materials Science and Engineering and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6272 (Received 2 February 1996; accepted 13 September 1996)
A theoretical computation of vibrational anharmonicity is presented which is a generalization of the simple Gruneisen approach. The calculation was based on a model that defines a simple relationship between the binding energy of a solid and the variation of vibration frequencies with volume. The agreement between calculated and experimental Gruneisen constants is good.
A simple description of anharmonicity in solids can be of great value in understanding a variety of crystal properties. While the Gruneisen approach provides such a description, it suffers from the fact that it is almost completely empirical since it defines an anharmonicity parameter, the Gruneisen constant, which can only be obtained from the values of the measurable quantities it seeks to describe. A method of computing this parameter that is independent of the thermodynamic anharmonic properties would be useful. Recently, it was pointed out that the Gruneisen parameter can be calculated if the crystal energy is known as a function of lattice parameter.1 This idea was successfully applied to computation of the anharmonic properties of the fcc phase of C60 1 and C70 2 using a crystal energy obtained from a spherically averaged pair potential for the intermolecular interaction. All that is needed to compute the Gruneisen parameter for metals is an expression for the crystal energy as a function of lattice parameter. Such an expression is available from the work of Rose et al.3 who showed that the energy obtained from first principles calculations could all be represented by a simple equation in terms of reduced energy and reduced Wigner–Seitz radius units. In this paper, we show how this reduced energy equation yields a Gruneisen parameter that is a function of volume that is useful for calculating the anharmonic properties of crystals. The reduced equation for the static energy of metal crystals given by Rose et al. is u0 gsad ; 0 2s1 1 a 1 0.05a3 de2a (1) u0 u00 is the potential energy per atom, exclusive of all vibrational contributions, when the Wigner–Seitz radius rs is the equilibrium value rs0 for the static crystal at zero pressure. u0 is the potential energy per atom of the static crystal as a function of a, which is defined by a
rs 2 rs0 , L
(2)
J. Mater. Res., Vol. 12, No. 2, Feb 1997
http://journals.cambridge.org
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where the distance parameter L is defined by v Ä Ä u u u00 Lt . 12prs0 B00
(3)
B00 is the static bulk modulus at zero pressure defined by µ 2 ∂ d u0 0 , (4) B0 y dy 2 0 with y being the atomic volume. In terms of the Wigner–Seitz radius, the Gruneisen parameter is defined by g2
1 d ln nj , 3 d ln rs
(5)
where nj are the normal mode f
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