typo Debye

docs/2_debye_model.md

@@ -123,11 +123,11 @@ As discussed in the previous lecture, the quantum mechanical excitations of a ha
 The expected value of the total energy stored in the oscillators (which, from now on, we will simply denote as the total energy $E$) is given by the sum of the energy stored in the individual oscillators. These oscillators are characterized by their wavevector $\mathbf{k}$:
 
 \begin{align}
-E &= 3 \sum_\mathbf{k} \left(\frac{1}{2}\hbar\omega(\mathbf{k})+\hbar \omega(\mathbf{k}) n_{B}(\hbar \omega(\mathbf{k}))\right)\\
+E &= 3 \sum_\mathbf{k} \left(\frac{1}{2}\hbar\omega(\mathbf{k})+\hbar \omega(\mathbf{k}) n_{B}(\beta \hbar \omega(\mathbf{k}))\right)\\
 &= 3 \sum_\mathbf{k} \left(\frac{1}{2}\hbar\omega(\mathbf{k})+\frac{\hbar\omega(\mathbf{k})}{ e^{\hbar\omega(\mathbf{k})/{k_BT}}-1}\right).
 \end{align}
 
-Here we used that the expected occupation number is $n_B(\hbar \omega(\mathbf{k}))$.
+Here we used that the expected occupation number is $n_B(\beta \hbar \omega(\mathbf{k}))$.
 
 ??? question "Where does the factor 3 come from?"