@@ -261,14 +261,19 @@ Also a sanity check: when the energy is close to the bottom of the band, $E = E_

1. What is a normal mode? What is a phonon? Why do phonons obey Bose statistics?

2. From the dispersion relation of a 1D monatomic chain given in the lecture notes, calculate the group velocity $v_g$ and density of states $g(\omega)$.

??? hint

To deal easily with the absolute value, you can split the dispersion for positive and negative $k$.

3. Sketch them.

4. From the 1D dispersion relation $\omega(k)$ in the picture below, sketch the group velocity $v_g(k)$ and the density of states $g(\omega)$.

4. From the 1D dispersion relation $\omega(k)$ in the picture below, sketch the group velocity $v_g(k)$ and the density of states $g(\omega)$ as a function of $k$.

![](figures/NNNdispersion.svg)

### Exercise 2: Vibrational heat capacity of a 1D monatomic chain

1. Give an integral expression for the total energy $U$ of a 1D monatomic chain (similarly to what was done within the Debye theory). To do so, first derive the density of states from the appropriate dispersion relation given in the lecture notes.

1. Give an integral expression for the total energy $U$ of a 1D monatomic chain (similarly to what was done within the Debye theory). Use the density of state calculated in exercise 1.2.

2. Give an integral expression for the heat capacity $C$.

3. Compute the heat capacity numerically, using e.g. MatLab or Python.

4. Do the same for $C$ in the Debye model and compare the two. What differences do you see?