Add warm-up exercises.

src/2_debye_model.md

@@ -150,7 +150,7 @@ There is therefore exactly one allowed ${\bf k}$ per volume $\left(\frac{2\pi}{L
 
 When we consider larger and larger box sizes $L→∞$, the volume per allowed mode becomes smaller and smaller, and eventually we obtain an integral:
 $$
-\sum_\mathbf{k}  \rightarrow \frac{L^3}{(2\pi)^3}\iiint\limits_{-∞}^{∞}dk_x dk_y dk_z )
+\sum_\mathbf{k}  \rightarrow \frac{L^3}{(2\pi)^3}\iiint\limits_{-∞}^{∞}dk_x dk_y dk_z
 $$
 
 
@@ -291,6 +291,15 @@ ax.legend(loc='lower right');
 
 ## Exercises
 
+### 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. Einstein model has the free fitting parammeter $\omega$, but Debye model doesn't require any fitting parameters to properly describe the low temperature limit? There is, however, a material dependent parameter in the Debye model. Which one is it?
+
 ### Exercise 1: Debye model: concepts
 
 Consider the probability to find an atom of a 1D solid that originally had a position $x$ at a displacement $\delta x$ shown below: