1. Assume that the strength of the returning force $k$ experienced by each atom is the same. What is the difference in the oscillation frequencies of different isotopes of lithium in the lithium crystal?
2. Write down the total energy of lithium assuming that all $^6$Li atoms are in $n=2$ vibrational state, and all $^7$Li atoms are in $n=4$ vibrational state.
3. Write down the total energy of lithium at a temperature $T$ by modifying the Einstein model.
4. Compute the heat capacity of lithium.
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4. Compute the heat capacity of lithium as a function of $T$.
1. Describe the concept of k-space. What momenta are allowed in a 2D system of dimensions $L^2$?
1. Describe the concept of k-space. What momenta are allowed in a 2D system with dimensions $L\times L$?
2. The probability to find an atom of a 1D solid that originally had a position $x$ at a displacement $\delta x$ is shown on this plot:
```python
...
...
@@ -163,16 +163,29 @@ Explain your answer.
4. Calculate the density of states $g(\omega)$ for a 3D, 2D and 1D systems with linear dispersion $\omega=vk$.
### 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.
4. At low $T$, show that $C_V=KT^{n}$. Find $n$. Express $K$ as a definite integral.
### Exercise 3: Different phonon modes
During the lecture we derived the low-temperature heat capacity assuming that the longitudinal and transverse modes have the same sound velocity $v$.
Materials usually have different velocities of the longitudinal and transverse sound waves ($v_L=\omega_{\parallel}/k$;$v_T=\omega_{\bot}/k$ with $v_L \neq v_T$).
How does this change the Debye result? (hint: remember the exercise about lithium from last week).
During the lecture we derived the low-temperature heat capacity assuming that all the phonons have the same sound velocity $v$.
In reality the longitudinal and transverse modes have different sound velocities (see [Wikipedia](https://en.wikipedia.org/wiki/Sound#Longitudinal_and_transverse_waves) for an illustration of different sound wave types).
Assume that there are two types of excitations:
* One longitudinal mode with $\omega = v_\parallel |k|$
* Two transverse modes with $\omega = v_\bot |k|$
1. Write down the total energy of phonons in this material *(hint: use the same reasoning as in the [Lithium exercise](1_einstein_model.md#exercise-4-total-heat-capacity-of-a-diatomic-material))*.
2. Verify that at high $T$ you reproduce the Dulong-Petit law.
3. Compute the behavior of heat capacity at low $T$.
### Exarcise $: Anisotropic sound velocities
Suppose now that the velocity is anisotropic ($v_x \neq v_y \neq v_z$) and $\omega = \sqrt{v_x^2 k_x^2 + v_y^2 k_y^2 + v_z^2 k_z^2}, neglecting the difference between transverse and longitudinal modes.
### Exarcise 4: Anisotropic sound velocities
Suppose now that the velocity is anisotropic ($v_x \neq v_y \neq v_z$) and $\omega = \sqrt{v_x^2 k_x^2 + v_y^2 k_y^2 + v_z^2 k_z^2}$.
How does this change the Debye result for the heat capacity?
??? hint
Write down the total energy as an integral over $k$, then change the integration variables so that the spherical symmetry of the integrand is restored.
