Describe how many phonons in which $k$-state this solid has.
Explain your answer.
??? hint
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@@ -161,7 +162,6 @@ Describe how many phonons in which $k$-state this solid has.
### 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. Discuss what it means to have $n=3$ phonons occupying a state with $k=(0, 0, 2\pi/L)$. Draw the amplitudes of the atomic displacements in a state with $
### Exercise 2: Debye model in 2D
1. State the assumptions of the Debye theory.
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@@ -171,5 +171,6 @@ Describe how many phonons in which $k$-state this solid has.
### 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 $v$.
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?
1. Materials usually have different velocities of the longitudinal and transverse sound waves are different ($v_L \neq v_T$). How does this change the Debye result?
2. Suppose now that the velocity is anisotropic ($v_x \neq v_y \neq v_z$) and $\omega = \sqrt{v_x^2 k_x^2 + v_y^2 k_y^2 + v_z^2 k_z^2}, neglecting the difference between transverse and longitudinal modes. How does this change the Debye result?
