Update 2_debye_model.md

src/2_debye_model.md

@@ -131,9 +131,9 @@ draw_classic_axes(ax, xlabeloffset=0.3)
 
 ## Exercises
 
-### Exercise 1: What is a phonon
-
-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:
+### Exercise 1: Debye model: concepts
+1. Describe the concepts of k-space. What momenta are allowed in a 2D system of dimensions LxL?
+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:
 
 ```python
 def psi_squared(delta_x, x):
@@ -159,9 +159,8 @@ Explain your answer.
 
     There are $n=2$ phonons in the state with $k=4\pi/L$ and $n=2$ phonons in a state with $k=-4\pi/L$.
 
-### Exercise 1: Debye model: concepts
-1. Describe the concepts of k-space and density of states.
-2. Calculate the density of state $g(\omega)$ and $g(k)$  for a 3D, 2D and 1D systems with linear dispersion $\omega=vk$.
+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 $\omega=vk$.
 
 ###  Exercise 2: Debye model in 2D
 1. State the assumptions of the Debye theory.
@@ -169,8 +168,11 @@ Explain your answer.
 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 $n$. Find $K$ in term of a definite integral.
 
-###  Exercise 3: Anisotropic velocities
+###  Exercise 3: Different phonon modes
 During the lecture we derived the low-temperature heat capacity assuming that the longitudinal and transverse modes have the same sound velocity $v$.
+Materials usually have different velocities of the longitudinal and transverse sound waves ($v_L=\omega_{\parallel}/k$;$v_T=\omega_{\bot}/k$ with $v_L \neq v_T$).
+How does this change the Debye result? (hint: remember the exercise about lithium from last week).
 
-1. Materials usually have different velocities of the longitudinal and transverse sound waves are different ($v_L \neq v_T$). How does this change the Debye result?
-2. 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}, neglecting the difference between transverse and longitudinal modes. How does this change the Debye result?
+### Exarcise $: Anisotropic sound velocities
+Suppose now that the velocity is anisotropic ($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}, neglecting the difference between transverse and longitudinal modes.
+How does this change the Debye result for the heat capacity?