Update 2_debye_model.md - polish

src/2_debye_model.md

@@ -169,7 +169,7 @@ $$
 \begin{align}
 E &= \frac{L^3}{(2\pi)^3}\int\limits_0^{2π}d\varphi\int\limits_0^π \sin θ\;dθ\int\limits_0^∞ k^2 dk × 3 × \left(\frac{1}{2}\hbar\omega(k)+\frac{\hbar\omega(k)}{ {\rm e}^{\hbar\omega(k)/{k_B T}}-1}\right)\\
 &= \frac{L^3}{(2\pi)^3}\int_0^∞ 12 π k^2 dk \left(\frac{1}{2}\hbar\omega(k)+\frac{\hbar\omega(k)}{ {\rm e}^{\hbar\omega(k)/{k_{\rm B}T}}-1}\right)\\
-&= \frac{L^3}{(2\pi)^3}\int_0^∞ 12 π v^{-3} \omega^2 d\omega \left(\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{ {\rm e}^{\hbar\omega/{k_{\rm B}T}}-1}\right).
+&= \frac{L^3}{(2\pi)^3}\int_0^∞ 12 π v{\rm s}^{-3} \omega^2 d\omega \left(\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{ {\rm e}^{\hbar\omega/{k_{\rm B}T}}-1}\right).
 \end{align}
 $$
 
@@ -282,8 +282,8 @@ ax.legend(loc='lower right');
 
 1. Atoms in materials move in a collective fashion, forming sound waves with dispersion relation $ω = v|\mathbf{k}|$.
 2. The normal modes have a constant density of $(L/2π)^3$ in the reciprocal space.
-3. The total energy and the total heat capacity are given by integrating a contribution of individual modes over the $k$-space.
-4. The density of states $g(ω)$ counts the number of modes per unit frequency, and it is proportional to $ω^2$ for phonons in 3D.
+3. The total energy and heat capacity are given by integrating the contribution of the individual modes over $k$-space.
+4. The density of states $g(ω)$ counts the number of modes per unit frequency. $g(ω)$ is proportional to $ω^2$ for bosons in 3D with a dispersion relation $ω = v|\mathbf{k}|$.
 5. At low temperatures the phonon heat capacity is $∼T^3$.