fixes typo's

src/2_debye_model.md

@@ -150,7 +150,7 @@ Therefore, we consider a material with a simple shape to make the calculation fo
 The easiest option people have invented so far is a box of size $V = L^3$ with **periodic boundary conditions**[^2].
 
 Periodic boundary conditions require that the atomic displacement $\mathbf{\delta r}$ is periodic inside the material. 
-Let us consider a translation by $L$ in the $x$-directionL
+Let us consider a translation by $L$ in the $x$-direction
 
 $$
 \mathbf{\delta r}(\mathbf{r} + L\mathbf{\hat{x}}) = \mathbf{\delta r}(\mathbf{r}).
@@ -328,7 +328,7 @@ g(\omega) = \left\{
 \right.
 $$
 
-Let us now compute $\omega_D$?
+Let us now compute $\omega_D$.
 We know that for a 3D system with $N$ atoms has to have exactly $3N$ phonon modes.
 $$
 \begin{align}
@@ -340,7 +340,7 @@ which gives us
 $$
 \omega_D = v_s (6\pi^2 N)^{1/3} / L.
 $$
-Both $N$ and $L$ are arbitrary, however we are considering an $L×L×L$ cube with $N$ atoms, so $L / N^{1/3}$ is the distance between neighboring atoms, and therefore $\omega_D$ does not depend on the box size.
+Both $N$ and $L$ are arbitrary, however we are considering an $L×L×L$ box with $N$ atoms, so $L / N^{1/3}$ is the distance between neighboring atoms, and therefore $\omega_D$ does not depend on the box size.
 
 Using the corrected expression for the total energy that includes the high frequency cut-off, the total energy without the zero-point motion part is
 $$