Thermal behavior of the Debye-Waller factor and the specific heat of anharmonic crystals

We study, within the framework of the variational method in statistical mechanics, the influence of the cubic and quartic crystalline anharmonicity on the classical and quantum thermal behavior of the specific heat, Debye temperature $\Theta$, Debye-Waller factor $W$, crystalline expansion, and phonon spectrum. The systems we mainly focalize are the single oscillator, the monoatomic linear chain, and the simple cubic crystal. The trial Hamiltonian is a harmonic one, therefore the various anharmonic influences are mainly absorbed into the renormalization of $\Theta \mathrm{}\left(T\right)\mathrm{}$. Several differences between the classical and quantum results are exhibited. Satisfactory qualitative agreement with experience was obtained in the low-temperature regime, in particular on what concerns the existence of a minimum in $\Theta \mathrm{}\left(T\right)\mathrm{}$ which has been observed in Cu, Al, Ag, Au, and Pb. For the intermediate-temperature regime the customary linear behavior of $W\left(T\right)\mathrm{}$ [hence $\Theta \mathrm{}\left(T\right)\mathrm{}$ almost constant] is reobtained. Finally, in the high-temperature regime, the present treatment leads to a $\sqrt{T}$ dependence for the $W$ factor, which implies the wrong curvature with respect to experimental data. A possible explanation of this disagreement might be related to the melting phenomenon, which is not covered by the present theory.