<!--- Still need to make the expression for the temperature correct in the figure above -->
Having found an expression for $\langle E \rangle$ as a function of $T$, we can now calculate the heat capacity per atom $C$ explicitly. To do so, we need to differentiate $\langle E \rangle$ with respect to $T$.
Having found an expression for $\langle E \rangle$ as a function of $T$, we can now calculate the heat capacity per atom $C$ explicitly. To do so, we need to differentiate $\langle E \rangle$ with respect to $T$.
The horizontal dashed line is the classical value, $k_{\rm B}$. The shaded area is the difference between the classical value $k_B$ and the value predicted by the Einstein model. Integrating over the shaded area yields $\frac{1}{2}\hbar\omega_0$, which is the zero-point energy of the oscillator, which cannot be extracted from the system. The vertical dashed line depicts the Einstein temperature $T_E$, at which the heat capacity $C \approx 0.92 k_B$.
The horizontal dashed line is the classical value, $k_{\rm B}$. The shaded area is the difference between the classical value $k_B$ and the value predicted by the Einstein model. Integrating over the shaded area yields $\frac{1}{2}\hbar\omega_0$, which is the zero-point energy of the oscillator, which cannot be extracted from the system. The vertical dashed line depicts the Einstein temperature $T_E$, at which the heat capacity $C \approx 0.92 k_B$.
ax.set_title(r'Emperical and predicted heat capacity of diamond as a function of $T$')
ax.set_xlabel('$T[K]$')
ax.set_xlabel('$T[K]$')
ax.set_ylabel('$C/k_B$')
ax.set_ylabel('$C/k_B$')
ax.set_ylim((0,3));
ax.set_ylim((0,3));
...
@@ -362,11 +365,11 @@ Although the Einstein model fits the experimental data quite well, it still devi
...
@@ -362,11 +365,11 @@ Although the Einstein model fits the experimental data quite well, it still devi
### Quick warm-up exercises
### Quick warm-up exercises
1.Sketch the Bose Einstein distribution as a function of $\omega$ for two different values of $T$
1.Why is the heat capacity per atom of an ideal gas typically $3k_B/2$ and not $3 k_B$?
2.Sketch the heat capacity of an Einstein solid for two different values of $T_E$
2.What is the high-temperature heat capacity of an atom in a solid with two momentum and two spatial coordinate degrees of freedom?
3.What is the high-temperature heat capacity of an atom in a solid with two momentum and two spatial coordinate degrees of freedom?
3.Sketch the Bose Einstein distribution as a function of $\omega$ for two different values of $T$
4.Why is the heat capacity per atom of an ideal gas typically $3k_B/2$ and not $3 k_B$?
4.Explain which behaviour of the function $1/(e^{-\hbar\omega/k_BT}-1)$ tells you it is not the Bose Einstein distribution.
5.Explain which behaviour of the function $1/(e^{-\hbar\omega/k_BT}-1)$ tells you it is not the Bose Einstein distribution.
5.Sketch the heat capacity of an Einstein solid for two different values of $T_E$
### Exercise 1: Heat capacity of a classical oscillator.
### Exercise 1: Heat capacity of a classical oscillator.
...
@@ -383,36 +386,34 @@ $$
...
@@ -383,36 +386,34 @@ $$
$$
$$
Z = \int_{-\infty}^{\infty}dp \int_{-\infty}^{\infty} dx e^{-\beta H(p,x)}.
Z = \int_{-\infty}^{\infty}dp \int_{-\infty}^{\infty} dx e^{-\beta H(p,x)}.
$$
$$
where $\beta = 1/k_B T$
2. Using the solution of 1., compute the expectation value of the energy.
2. Using the solution of 1., compute the expectation value of the energy.
3. Compute the heat capacity. Check that you get the law of Dulong-Petit but with a different prefactor.
3. Calculate the heat capacity. Does it depend on the temperature?
4. Explain the difference in the prefactor by considering the number of degrees of freedom.
### Exercise 2: Quantum harmonic oscillator
### Exercise 2: Quantum harmonic oscillator
Consider a 1D quantum harmonic oscillator. Its eigenstates are:
Consider a 1D quantum harmonic oscillator. Its energy eigenvalues are:
$$
$$
E_n = \hbar\omega(n+\frac{1}{2}),
E_n = \hbar\omega(n+\frac{1}{2}),
$$
$$
1. Sketch the wave function of this harmonic oscillator for $n=3$.
1. Compute the partition function using the following expression:
2. Compute the quantum partition function using the following expression:
$$
$$
Z = \sum_j e^{-\beta E_j}.
Z = \sum_j e^{-\beta E_j}.
$$
$$
3. Using the partition function, compute the expectation value of the energy.
2. Using the partition function found in 2.1, compute the expected value of the energy.
4. Compute the heat capacity. Check that in the high temperature limit you get the same result as in Exercise 1.3.
3. Compute the heat capacity. Check that in the high temperature limit you get the same result as in Exercise 1.3.
- What temperature can be considered high?
4. Plot the found heat capacity and roughly indicate in the plot where the Einstein temperature is.
- What is the expectation value of $n$?
5. What is the expected value of $n$?
### Exercise 3: Total heat capacity of a diatomic material
### Exercise 3: Total heat capacity of a diatomic material
One of the assumptions of the Einstein model states that every atom in a solid oscillates with the same frequency $\omega_0$. However, if the solid contains different types of atoms, it is unreasonable to assume that the atoms oscillate with the same frequency. One example of such a solid is a lithium crystal, which consists of the [two stable isotopes](https://en.wikipedia.org/wiki/Isotopes_of_lithium) $^6$Li (7.5%) and $^7$Li (92.5%) in their natural abundance. Let us extend the Einstein model to take into account the different masses of these different isotopes.
We consider a crystal of lithium, which consists of the [two stable isotopes](https://en.wikipedia.org/wiki/Isotopes_of_lithium) $^6$Li (7.5%) and $^7$Li (92.5%) in their natural abundance. Let us extend the Einstein model to take into account the different masses of these different isotopes.
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 the two different isotopes in the lithium crystal?
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 the two different isotopes in the lithium crystal?
2. Write down the total energy stored in the vibrations of the atoms of the lithium crystal, assuming that all $^6$Li atoms are in $n=2$ vibrational state and all $^7$Li atoms are in $n=4$ vibrational state.
2. Write down the total energy stored in the vibrations of the atoms of the lithium crystal, assuming that all $^6$Li atoms are in $n=2$ vibrational mode and all $^7$Li atoms are in $n=4$ vibrational mode.
3.Write down the total energy stored in the vibrations of the atoms in the lithium crystal at a temperature $T$ by modifying the Einstein model.
3.In the case where the osccilators can occupy any vibrational mode, write down the total energy stored in the vibrations of the atoms in the lithium crystal at a temperature $T$ by modifying the Einstein model.
4. Compute the heat capacity of the lithium crystal as a function of $T$.
4. Compute the heat capacity of the lithium crystal as a function of $T$.
