3 factor written twice
File in which the problem appears
https://gitlab.kwant-project.org/solidstate/lectures/-/blob/master/docs/2_debye_model_solutions.md
Problematic sentence
Going to spherical coordinates:
\begin{align}
E &= 3_p \frac{3_p L^3}{2\pi^2}\frac{1}{v_x v_y v_z}\int_{0}^{\kappa_D} \frac{\hbar\kappa^3}{e^{\beta\hbar\kappa} - 1} d\kappa + E_Z \\
& = \frac{3_p L^3}{2\pi^2\hbar^3\beta^4}\frac{1}{v_x v_y v_z}\int_0^{\beta\hbar\kappa_D} \frac{x^3}{e^x - 1}dx + E_Z,
\end{align}
Correct version
Going to spherical coordinates:
\begin{align}
E &= 3_p \frac{L^3}{2\pi^2}\frac{1}{v_x v_y v_z}\int_{0}^{\kappa_D} \frac{\hbar\kappa^3}{e^{\beta\hbar\kappa} - 1} d\kappa + E_Z \\
& = \frac{3_p L^3}{2\pi^2\hbar^3\beta^4}\frac{1}{v_x v_y v_z}\int_0^{\beta\hbar\kappa_D} \frac{x^3}{e^x - 1}dx + E_Z,
\end{align}