Update 2_debye_model.md - polish

src/2_debye_model.md

@@ -90,7 +90,7 @@ Peter Debye (1884 – 1966) suggested to instead consider _normal modes_: sound
 
 > A (running) sound wave is a collective motion of atoms through this solid, where the displacement of each atom $\mathbf{\delta r}$ depends on its position $\mathbf{r}$ and time $t$ through a relation
 > $$\mathbf{\delta r} = \mathbf{\delta r}_0 e^{i(\mathbf{kr}-\omega t)},$$
-> with $\mathbf{\delta r}_0$ the wave amplitude, and $\mathbf{k}$ the _wave vector_ (the wave length $\lambda = 2\pi/|\mathbf{k}|$).
+> with $\mathbf{\delta r}_0$ the wave amplitude, and $\mathbf{k}$ the _wave vector_ (the wavelength $\lambda = 2\pi/|\mathbf{k}|$).
 
 > Because the shape of the wave depends on time only through the factor $\exp(i\omega t)$, these waves are _normal modes_—stable oscillations of the medium.
 Each normal mode has a _wave vector_ $\mathbf{k}$.
@@ -110,7 +110,7 @@ $$
 
 We still have several open questions:
 
-* Normal modes depend on the material's shape. What impact does this have on the heat capacitance?
+* Normal modes depend on the material's shape. What impact does this have on the heat capacity?
 * What $\mathbf{k}$ are possible and what aren't?
 * If all $\mathbf{k}$ are possible, shouldn't $E$ be infinite?
 
@@ -124,18 +124,14 @@ Therefore the simpler shape we consider, the easier is the calculation.
 
 The easiest option people have invented so far is a box $L×L×L$ with **periodic boundary conditions**[^2].
 
-This means that the displacement $\mathbf{\delta r}(\mathbf{r}$ as well as the velocity of the solid is periodic:
+Periodic boundary conditions imply that the atomic displacement $\mathbf{\delta r}$ is periodic. Let's consider a displacement by $L$ in the $x$-direction
 
 $$
-\begin{align}
-\mathbf{\delta r}(\mathbf{r} + (L, 0, 0)^T) &= \mathbf{\delta r}(\mathbf{r})\\
-\frac{d\mathbf{\delta r}(\mathbf{r} + (L, 0, 0)^T)}{dt} &= \frac{d\mathbf{\delta r}(\mathbf{r})}{dt}\\
-\end{align}
+\mathbf{\delta r}(\mathbf{r} + L\mathbf{\hat{x}}) &= \mathbf{\delta r}(\mathbf{r})
 $$
+To satisfy this equation, we arrive at the condition $k_x=p 2 \pi/L$]$, with $p= ..., -2, -1, 0, 1, 2$ in $\mathbb{Z}$.
 
-(of course similar periodicity applies to $y$ and $z$ coordinates)
-
-Periodicity means that not all the points in $k$-space are allowed.
+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.
@@ -291,12 +287,11 @@ ax.legend(loc='lower right');
 
 ### Quick warm-up exercises
 
-1. Express the three-dimensional density of states in terms of $\omega_D$.
-2. Express the heat capacity for low $T$ in terms of $T_D$. 
-3. Make a sketch of the heat capacity in the low $T$ for two different Debye temperatures. 
-4. Why are there only 3 polarizations when there are 6 degrees of freedom in three-dimensions for an oscillator?
-5. Convert the two-dimensional integral $\int\mathrm{d}k_x\mathrm{d}k_y$ to a one-dimensional integral.
-6. The Einstein model has the eigenfrequency $\omega_0 = k_\mathrm{B} T_E/\hbar$ of the quantum harmonic oscillators modeling the atoms as a material-dependent free fitting parameter. What is the material-dependent parameter that plays a similar role in the Debye model?
+1. Express the heat capacity for low $T$ in terms of $T_D$. 
+2. Make a sketch of the heat capacity in the low $T$ for two different Debye temperatures. 
+3. Why are there only 3 polarizations when there are 6 degrees of freedom in three-dimensions for an oscillator?
+4. Convert the two-dimensional integral $\int\mathrm{d}k_x\mathrm{d}k_y$ to a one-dimensional integral.
+5. The Einstein model has the eigenfrequency $\omega_0 = k_\mathrm{B} T_E/\hbar$ of the quantum harmonic oscillators modeling the atoms as a material-dependent free fitting parameter. What is the material-dependent parameter that plays a similar role in the Debye model?
 
 ### Exercise 1: Debye model: concepts