diff --git a/src/2_debye_model.md b/src/2_debye_model.md index 84d7c9133d88f3884b61c528b45d2c6c98bb7cc7..79ee30d29fe46d6ced60111be2f36533b9ffb82e 100644 --- a/src/2_debye_model.md +++ b/src/2_debye_model.md @@ -289,6 +289,15 @@ 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: