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.
1. Express the heat capacity in the low-$T$ limit in terms of $T_D$.
2. Make a sketch of the heat capacity in the low-$T$ limit 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?