Merge branch 'Fixes_w2' into 'master'

Fixes w2

See merge request solidstate/lectures!22

src/2_debye_model.md

@@ -240,10 +240,10 @@ $$
 Substitute in $E$, differentiate to $T$:
 
 $$
-\Rightarrow C=9Nk_{\rm B}\left(\frac{T}{\Theta_{\rm D}}\right)^3\int_0^{\Theta_{\rm D}/T}\frac{x^4{\rm e}^x}{({\rm e}^x-1)^2}{\rm d}x,
+\Rightarrow C=9Nk_{\rm B}\left(\frac{T}{T_{D}}\right)^3\int_0^{T_{D}/T}\frac{x^4{\rm e}^x}{({\rm e}^x-1)^2}{\rm d}x,
 $$
 
-where $x=\frac{\hbar\omega}{k_{\rm B}T}$ and $\Theta_{\rm D}\equiv\frac{\hbar\omega_{\rm D}}{k_{\rm B}}$, the _Debye temperature_.
+where $x=\frac{\hbar\omega}{k_{\rm B}T}$ and $T_{D}\equiv\frac{\hbar\omega_{D}}{k_{\rm B}}$, the _Debye temperature_.
 
 ```python
 def integrand(y):
@@ -281,8 +281,8 @@ ax.legend(loc='lower right');
 ## Exercises
 
 ### Exercise 1: Debye model: concepts
-1. Describe the concept of k-space. What momenta are allowed in a 2D system with dimensions $L\times L$?
-2. The probability to find an atom of a 1D solid that originally had a position $x$ at a displacement $\delta x$ is shown on this plot:
+
+Consider the probability to find an atom of a 1D solid that originally had a position $x$ at a displacement $\delta x$ shown below:
 
 ```python
 def psi_squared(delta_x, x):
@@ -301,13 +301,13 @@ cbar.set_ticks(())
 cbar.set_label(r'$|\psi^2|$')
 ```
 
-Describe how many phonons in which $k$-state this solid has.
-Explain your answer.
+1. Describe how many phonons in which $k$-state this solid has. Explain your answer.
 
-??? hint
+    ??? hint
 
-    There are $n=2$ phonons in the state with $k=4\pi/L$ and $n=2$ phonons in a state with $k=-4\pi/L$.
+        There are $n=2$ phonons in the state with $k=4\pi/L$ and $n=2$ phonons in a state with $k=-4\pi/L$.
 
+2. Describe the concept of k-space. What momenta are allowed in a 2D system with dimensions $L\times L$?
 3. Explain the concept of density of states.
 4. Calculate the density of states $g(\omega)$ for a 3D, 2D and 1D systems with linear dispersion $\omega=vk$.
 
@@ -316,10 +316,12 @@ Explain your answer.
 1. State the assumptions of the Debye theory.
 2. Determine the energy of a two-dimensional solid as a function of $T$ using the Debye approximation (the integral can't be solved analytically).
 3. Calculate the heat capacity in the limit of high $T$ (hint: it goes to a constant).
-4. At low $T$, show that $C_V=KT^{n}$. Find $n$. Express $K$ as a definite integral.
+4. At low $T$, show that $C_V=KT^{n}$. Find $n$. Express $K$ as an indefinite integral (similarly to what done during the lecture). The integral can be evaluated by using the Riemann zeta function (See Chapter 2.3 of the book for more details).
 
 
 ###  Exercise 3: Different phonon modes
+*(adapted from ex 2.6a of "The Oxford Solid State Basics" by S.Simon)*
+
 During the lecture we derived the low-temperature heat capacity assuming that all the phonons have the same sound velocity $v$.
 In reality the longitudinal and transverse modes have different sound velocities (see [Wikipedia](https://en.wikipedia.org/wiki/Sound#Longitudinal_and_transverse_waves) for an illustration of different sound wave types).
 
@@ -332,7 +334,9 @@ Assume that there are two types of excitations:
 2. Verify that at high $T$ you reproduce the Dulong-Petit law.
 3. Compute the behavior of heat capacity at low $T$.
 
-### Exarcise 4: Anisotropic sound velocities
+### Exercise 4: Anisotropic sound velocities
+*(adapted from ex 2.6b of "The Oxford Solid State Basics" by S.Simon)*
+
 Suppose now that the velocity is anisotropic ($v_x \neq v_y \neq v_z$) and $\omega = \sqrt{v_x^2 k_x^2 + v_y^2 k_y^2 + v_z^2 k_z^2}$.
 How does this change the Debye result for the heat capacity?