Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 35A / b

Use periodic boundary conditions on the square crystal. Each allowed wavevector occupies area $(2\pi {)}^2/A$ in two-dimensional $k$-space. Including the two phonon polarizations, the number of modes in the disk $|k|\le K$ isThe phonon dispersion relation $\omega =C|k{|}^{\alpha }$ gives $K=(\omega /C{)}^{1/\alpha }$, soDifferentiating the cumulative mode count produces the density of states for a two-dimensional power-law phonon dispersion:

The Debye model fixes its cutoff by requiring the continuum to contain the $2n$ vibrational degrees of freedom:Therefore the Debye frequency and Debye temperature are

A phonon has zero chemical potential and energy $\hslash \omega$. The total internal energy is consequentlyWith the change of variable $x=\beta \hslash \omega$ and $\beta \hslash {\omega }_D=T_D/T$, this becomes

For $\alpha =1/2$, the preceding density of states hasWhen $T\gg T_D$, the Bose-Einstein distribution has the classical approximationEach of the $2n$ modes therefore contributes $k_{B}T$ to the energy, so the Dulong-Petit law gives

When $T\ll T_D$, the upper integration limit tends to infinity. The Bose integralthen yieldsHence the heat capacity at constant volume isso $d=4$. The high-temperature constant is the classical equipartition theorem result, while the low-temperature power law makes $C_V\to 0$ as required by the Third law of thermodynamics.