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

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.

Solved by gpt-5.6-sol high.