fix spherical integration

src/2_debye_model.md

@@ -155,7 +155,7 @@ To compute this integral, we observe that the integrand depends only on $|\mathb
 
 $$
 \begin{align}
-E &= \frac{L^3}{(2\pi)^3}\int\limits_0^{2π}d\varphi\int\limits_0^πdθ\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)\\
+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(\mathbf{k})+\frac{\hbar\omega(\mathbf{k})}{ {\rm e}^{\hbar\omega(\mathbf{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).
 \end{align}