= Solution
For $\alpha=1/2$, the preceding <density of states> has
$$
b=3,
\qquad
B=\frac{2A}{\pi C^4}.
$$
When $T\gg T_D$, the <Bose-Einstein distribution> has the classical approximation
$$
\frac{\hbar\omega}{e^{\beta\hbar\omega}-1}
\sim k_BT.
$$
Each of the $2n$ modes therefore contributes $k_BT$ to the energy, so the <Dulong-Petit law> gives
$$
E_{\rm tot}\sim2nk_BT,
\qquad
\boxed{C_V\sim2nk_B}.
$$
When $T\ll T_D$, the upper integration limit tends to infinity. The <Bose integral>
$$
\int_0^\infty\frac{x^4}{e^x-1}\,dx
=\Gamma(5)\zeta(5)=24\zeta(5)
$$
then yields
$$
E_{\rm tot}
\sim\frac{24B\zeta(5)k_B^5}{\hbar^4}T^5.
$$
Hence the <heat capacity at constant volume> is
$$
\boxed{
C_V\sim
\frac{120B\zeta(5)k_B^5}{\hbar^4}T^4
=\frac{240A\zeta(5)k_B^5}{\pi C^4\hbar^4}T^4,}
$$
so $d=4$. The high-temperature constant is the classical <equipartition theorem> result, while the low-temperature power law makes $C_V\to0$ as required by the <Third law of thermodynamics>.
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