1. For low T, $\beta \rightarrow \infty$. The heat capacity is then given as:
$$
C \overset{\mathrm{low \: T}}{\approx} 9Nk_{\mathrm{B}}\left(\frac{T}{T_{D}}\right)^3\int_0^{T_{D}/T}\frac{x^4{\mathrm{e}}^x}{({\mathrm{e}}^x-1)^2}{\mathrm{d}}x.
C \overset{\mathrm{low \: T}}{\approx} 9Nk_{\mathrm{B}}\left(\frac{T}{T_{D}}\right)^3\int_0^{T/T_D}\frac{x^4{\mathrm{e}}^x}{({\mathrm{e}}^x-1)^2}{\mathrm{d}}x.
$$
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.
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@@ -43,7 +43,7 @@ $$
5. The Debye frequency $\omega_D$.
6. The wavelength is of the order of the interatomic spacing: