### Exercise 1*: Deriving the density of states for the linear dispersion relation of the Debye model.
### Exercise 1: Deriving the density of states for the linear dispersion relation of the Debye model.
1. $\omega = v_s|\mathbf{k}|$
2. The distance between nearest-neighbour points in $\mathbf{k}$-space is $2\pi/L$. The density of $\mathbf{k}$-points in 1, 2, and 3 dimensions is $L/(2\pi)$, $L^2/(2\pi)^2$, and $L^3/(2\pi)^3$ respectively.
3. Express the number of states between frequencies $0<\omega<\omega_0$ as an integral over k-space. Do so for 1D, 2D and 3D. Do not forget the possible polarizations. We assume that in $d$ dimensions there are $d_p$ polarizations.
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2. The high-$T$ limit implies $\beta \rightarrow 0$. Therefore, $n_B \approx k_B T/\hbar\omega$, and the integral becomes particularly illuminating:
$$
E = \int_0^\omega_D \hbar\omega n_B(\omega) g(\omega) d\omega \approx \int_0^{\omega_D} k_B T g(\omega) d\omega = N_\text{modes} k_B T
E = \int_{0}^{\omega_D}\hbar\omega n_B(\omega) g(\omega) d\omega \approx \int_{0}^{\omega_D} k_B T g(\omega) d\omega = N_\text{modes} k_B T
$$
where we neglected the zero-point energy. In 3D, we have $N_\text{modes} = 3_p N_\text{atoms}$, so that we recover the Dulong Petit $C_v = dE/dT = 3 k_B$ per atom
