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?
