Update 2_debye_model.md - polish

src/2_debye_model.md

@@ -131,10 +131,7 @@ Periodic boundary conditions imply that the atomic displacement $\mathbf{\delta
 $$
 \mathbf{\delta r}(\mathbf{r} + L\mathbf{\hat{x}}) = \mathbf{\delta r}(\mathbf{r})
 $$
-To satisfy this equation, we arrive at the condition $k_x=p 2 \pi/L$, with $p= ..., -2, -1, 0, 1, 2$ in $\mathbb{Z}$.
-
-The same condition holds for the $x$- and $y$-direction. We see that periodicity implies that not all the points in $k$-space are allowed.
-Instead only waves for which each component $k_x, k_y, k_z$ of the $\mathbf{k}$-vector belongs to the set
+To satisfy this equation, we arrive at the condition $k_x=p 2 \pi/L$, with $p= ..., -2, -1, 0, 1, 2, ...$ in $\mathbb{Z}$. The same condition holds for the $x$- and $y$-direction. We see that periodicity implies that not all the points in $k$-space are allowed. Instead only waves for which each component $k_x, k_y, k_z$ of the $\mathbf{k}$-vector belongs to the set
 $$k_{x,y,z}=…, \frac{-4\pi}{L}, \frac{-2\pi}{L}, 0, \frac{2\pi}{L}, \frac{4\pi}{L}, …$$
 satisfy the periodic boundary conditions.
 
@@ -142,7 +139,7 @@ The allowed $k$-vectors form a regular grid in $k$-space (also referred to as *r
 
 ![](figures/DOS_periodic.svg)
 
-In three dimensions, there is exactly one allowed ${\bf k}$-value per volume $\left(\frac{2\pi}{L}\right)^3$ in reciprocal space.
+We see that in three dimensions, there is exactly one allowed ${\bf k}$-value per volume $\left(\frac{2\pi}{L}\right)^3$ in reciprocal space.
 
 When we consider larger and larger box sizes $L→∞$, the volume per allowed mode becomes smaller and smaller, and eventually we obtain an integral:
 $$
@@ -152,7 +149,7 @@ $$
 
 ## Density of states
 
-Let's use this knowledge to continue our calculation of the total energy:
+Let's use this knowledge to continue our calculation of the total energy stored in the normal modes of our solid at temperature $T$:
 $$
 \begin{align}
 E &= \frac{L^3}{(2\pi)^3}\iiint\limits_{-∞}^{∞}dk_x dk_y dk_z × 3×\left(\frac{1}{2}\hbar\omega(\mathbf{k})+\frac{\hbar\omega(\mathbf{k})}{ {\rm e}^{\hbar\omega(\mathbf{k})/{k_{\rm B}T}}-1}\right),\\
@@ -161,7 +158,7 @@ E &= \frac{L^3}{(2\pi)^3}\iiint\limits_{-∞}^{∞}dk_x dk_y dk_z × 3×\left(\f
 $$
 The factor $3$ accounts for three possible directions of displacement (wave polarizations).
 
-To compute this integral, we observe that the integrand depends only on $|\mathbf{k}|$, and therefore switching to spherical coordinates is the way to go:
+To compute the integral, we observe that the integrand depends only on $|\mathbf{k}|$, and therefore switching to spherical coordinates is the way to go:
 
 $$
 \begin{align}
@@ -171,9 +168,9 @@ E &= \frac{L^3}{(2\pi)^3}\int\limits_0^{2π}d\varphi\int\limits_0^π \sin θ\;d
 \end{align}
 $$
 
-In the last expression everything inside the brackets is about Bose-Einstein statistics, while all the prefactors together are specific to the problem we are studying.
+In the last expression everything inside the brackets is about Bose-Einstein statistics, while the prefactors are specific to the problem we are studying.
 
-We can emphasize this further by introducing a new concept, _density of states_, $g(\omega)$, which is a central concept in this course.
+We can emphasize this further by introducing a new concept: the _density of states_ $g(\omega)$
 
 > The density of states $g(ω)$ is the number of available normal modes per infinitesimal interval $δω$.