diff --git a/src/2_debye_model.md b/src/2_debye_model.md index be8175344ced9b51cbde1ea645f92bb2c4ae9f30..744e7ee230f75af41aa1aff4d9fcaa82b54e498e 100644 --- a/src/2_debye_model.md +++ b/src/2_debye_model.md @@ -131,18 +131,18 @@ draw_classic_axes(ax, xlabeloffset=0.3) ## Exercises -### Debye model: concepts -a) Describe the concepts of k-space and density of states. -b) Calculate the density of state $g(\omega)$ and $g(k)$ for a 3D, 2D and 1D systems with linear dispersion $\omega=vk$. -c) _draft: sketch the state of a standing wave. Here I'm not sure what you exactly want to see, Anton, and how to write it down in an easy way._ - -### Debye model in 2D -a) State the assumptions of the Debye theory. -b) Determine the energy of a two-dimensional solid as a function of $T$ using the Debye approximation (the integral can't be solved analytically). -c) Calculate the heat capacity in the limit of high $T$ (hint: it goes to a constant). -d) At low $T$, show that $C_V=KT^{n}$. Find $n$. Find $K$ in term of a definite integral. - -### Anisotropic velocities _this exercise looks a bit too hard in the end_ +### 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. _draft: sketch the state of a standing wave. Here I'm not sure what you exactly want to see, Anton, and how to write it down in an easy way._ + +### Exercise 2: Debye model in 2D +1. State the assumptions of the Debye theory. +2. Determine the energy of a two-dimensional solid as a function of $T$ using the Debye approximation (the integral can't be solved analytically). +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 During the lecture we derived the low-temperature heat capacity assuming that the longitudinal and transverse modes have the same sound velocity $v$. -a) Suppose that the longitudinal and transverse sound velocities are different ($v_L != v_T$). How does this change the Debye result? -b) Suppose now that the velocity is anisotropic ($v_x!=v_y!=v_z$), neglecting the difference between transverse and longitudinal modes. How does this change the Debye result? +1. Suppose that the longitudinal and transverse sound velocities are different ($v_L != v_T$). How does this change the Debye result? +2. Suppose now that the velocity is anisotropic ($v_x!=v_y!=v_z$), neglecting the difference between transverse and longitudinal modes. How does this change the Debye result?