eos - PHZ 5156 Final project Finite-temperature equation of...

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PHZ 5156 Final project Finite-temperature equation of state For low enough temperatures, the energy of a crystalline lattice can be described within the harmonic approximation. In other words, the energy is given by temperature- independent constant U 0 plus the phonon energy, U = U 0 + X λ ( n λ + 1 / 2) ~ ω λ where ω λ is a phonon frequency for mode λ , and n λ is the number of phonons in mode λ . In thermal equilibrium, we are interested in the average energy U , and hence we need the average occupation h n λ i , which is given by, h n λ i = ( e ~ ω λ k B T - 1) - 1 Convince yourself analytically that the free energy F(V,T) is given by, F ( V,T ) = U 0 + k B T X λ log ± 2 sinh ² ~ ω λ 2 k B T ³´ The volume dependence enters because the phonon frequencies ω λ depend on volume V , usually characterized by the Gruneisen parameter, γ λ = V ω λ ∂ω λ ∂V By determining the volume V where F is minimized, the temperature dependence of the lattice parameters can be found. For higher temperatures (e.g. about 1/2
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This note was uploaded on 08/08/2011 for the course PHZ 5156 taught by Professor Johnson,m during the Fall '08 term at University of Central Florida.

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eos - PHZ 5156 Final project Finite-temperature equation of...

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