1. For low T, $\beta\rightarrow \infty$. The heat capacity is then given as:
1. For low T, $1/T\rightarrow \infty$. The heat capacity is then given as:
$$
C \overset{\mathrm{low \: T}}{\approx} 9Nk_{\mathrm{B}}\left(\frac{T}{T_{D}}\right)^3\int_0^{T/T_D}\frac{x^4{\mathrm{e}}^x}{({\mathrm{e}}^x-1)^2}{\mathrm{d}}x.
C \overset{\mathrm{low \: T}}{\approx} 9Nk_{\mathrm{B}}\left(\frac{T}{T_{D}}\right)^3\int_0^{\infty}\frac{x^4{\mathrm{e}}^x}{({\mathrm{e}}^x-1)^2}{\mathrm{d}}x.
$$
2. See plot below (shown for $T_{D,1} < T_{D,2}$)
3. The polarization is related to the direction of the amplitudes of the waves with respect to the direction of the wave.
