diff --git a/src/1_einstein_model.md b/src/1_einstein_model.md
index 29b8b94d4e4f0da1841dd34fa44940d397fb5cc0..d7171d9e5ae8e9c9da41de1c63c8c97057a76cc4 100644
--- a/src/1_einstein_model.md
+++ b/src/1_einstein_model.md
@@ -78,7 +78,7 @@ fig = go.Figure(
title='Heat capacity of chemical elements',
autosize=True,
yaxis=go.layout.YAxis(
- title='$C/k_B$',
+ title='$C/k_B [-]$',
tick0=1,
dtick=2,
),
@@ -108,9 +108,10 @@ c = [0.384, 0.578, 0.683, 0.798, 0.928, 1.069, 1.343, 1.656, 1.833, 2.671, 2.720
fig, ax = pyplot.subplots()
ax.scatter(T, c)
-ax.set_xlabel('$T[K]$')
-ax.set_ylabel('$C/k_B$')
-ax.set_ylim((0, 3));
+ax.set_xlabel('$T [K]$')
+ax.set_ylabel('$C/k_B [-]$')
+ax.set_ylim((0, 3))
+ax.set_title('Heat capacity of diamond as a function of the temperature');
```
We observe that:
@@ -254,7 +255,7 @@ xline = [1, 1];
yline = [0, 1.5];
fig, (ax, ax2) = pyplot.subplots(ncols=2, figsize=(10, 5))
omega = np.linspace(0.1, 2)
-ax.plot(omega, 1/(np.exp(omega) - 1), '-', xline, yline, 'r--')
+ax.plot(omega, 1/(np.exp(omega) - 1), '-')
ax.set_ylim(0, top=3)
ax.set_xlim(left=0)
ax.set_xlabel('$\hbar \omega$')
@@ -263,10 +264,11 @@ ax.set_xticklabels(['$0$'])
ax.set_ylabel('$n_{BE}(\omega,T)$')
ax.set_yticks([0,1, 2])
ax.set_yticklabels(['$0$','$1$', '$2$'])
+ax.set_title(r'$n_{BE}$ as a function of $\hbar \omega$')
# draw_classic_axes(ax, xlabeloffset=.2)
# ax.text(1.05, 0.95, r'$\hbar \omega = k_{\rm B}T$', ha='left', color='r');
temps = np.linspace(0.01, 2)
-ax2.plot(temps, 1/2 + 1/(np.exp(1/temps)-1), '-', [1,1], [0, 1.1], 'r--')
+ax2.plot(temps, 1/2 + 1/(np.exp(1/temps)-1), '-', [0.55,0.55], [0, 0.7], 'r--')
ax2.set_ylim(bottom=0)
ax2.set_xlabel('$k_B T$')
ax2.set_xticks([0])
@@ -274,12 +276,11 @@ ax2.set_xticklabels(['$0$'])
ax2.set_ylabel(r"$\langle E \rangle$")
ax2.set_yticks([1/2])
ax2.set_yticklabels([r'$\hbar\omega_0/2$'])
+ax2.set_title(r'$\langle E \rangle$ as a function of $T$')
draw_classic_axes(ax2, xlabeloffset=.15)
-ax2.text(1.05, 0.65, r'$k_{\rm B}T=\hbar \omega_0$', ha='left', color='r');
+ax2.text(0.65, 0.35, r'$k_{\rm B}T=\hbar \omega_0$', ha='left', color='r');
```
-<!--- 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 $T$.
$$
\begin{align}
@@ -306,10 +307,11 @@ xline = [1, 1];
yline = [0, 1.1];
temps = np.linspace(0.01, 1.5, 500)
fig, ax = pyplot.subplots()
-
-ax.plot(temps, c_einstein(temps)/3, '-', xline, yline, 'r--')
+pyplot.hlines([1], 0, 1.5, linestyles='dashed', label = r'Classical')
+ax.plot(temps, c_einstein(temps)/3, '-', label = r'Einstein model')
+ax.plot(xline, yline, 'r--')
ax.fill_between(temps, c_einstein(temps)/3, 1, alpha=0.5)
-
+ax.set_title('Heat capacity for the Einstein model and the equipartition theorem')
ax.set_ylim(bottom=0, top=1.2)
ax.set_xlabel('$T$')
ax.set_ylabel(r'$C$')
@@ -317,9 +319,10 @@ ax.set_xticks([0])
ax.set_xticklabels(['$0$'])
ax.set_yticks([1])
ax.set_yticklabels(['$k_B$'])
-pyplot.hlines([1], 0, 1.5, linestyles='dashed')
draw_classic_axes(ax)
-ax.text(1.01, 0.5, r'$T= T_E= \hbar \omega_0/k_{\rm B}$', ha='left', color='r');
+ax.text(1.01, 0.5, r'$T= T_E= \hbar \omega_0/k_{\rm B}$', ha='left', color='r')
+ax.legend(loc = 'lower right')
+fig.show();
```
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$.
@@ -337,11 +340,11 @@ T_E = fit[0][0]
delta_T_E = np.sqrt(fit[1][0, 0])
fig, ax = pyplot.subplots()
-ax.scatter(T, c)
+ax.scatter(T, c, label = r'Experimental value')
temps = np.linspace(10, T[-1], 100)
-ax.plot(temps, c_einstein(temps, T_E));
-
+ax.plot(temps, c_einstein(temps, T_E), label = r'Einstein model');
+ax.set_title(r'Emperical and predicted heat capacity of diamond as a function of $T$')
ax.set_xlabel('$T[K]$')
ax.set_ylabel('$C/k_B$')
ax.set_ylim((0, 3));
@@ -362,11 +365,11 @@ Although the Einstein model fits the experimental data quite well, it still devi
### Quick warm-up exercises
-1. Sketch the Bose Einstein distribution as a function of $\omega$ for two different values of $T$
-2. Sketch the heat capacity of an Einstein solid for two different values of $T_E$
-3. What is the high-temperature heat capacity of an atom in a solid with two momentum and two spatial coordinate degrees of freedom?
-4. Why is the heat capacity per atom of an ideal gas typically $3k_B/2$ and not $3 k_B$?
-5. Explain which behaviour of the function $1/(e^{-\hbar\omega/k_BT}-1)$ tells you it is not the Bose Einstein distribution.
+1. Why is the heat capacity per atom of an ideal gas typically $3k_B/2$ and not $3 k_B$?
+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. Sketch the Bose Einstein distribution as a function of $\omega$ for two different values of $T$
+4. 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.
@@ -383,36 +386,34 @@ $$
$$
Z = \int_{-\infty}^{\infty}dp \int_{-\infty}^{\infty} dx e^{-\beta H(p,x)}.
$$
+where $\beta = 1/k_B T$
2. Using the solution of 1., compute the expectation value of the energy.
-3. Compute the heat capacity. Check that you get the law of Dulong-Petit but with a different prefactor.
-4. Explain the difference in the prefactor by considering the number of degrees of freedom.
+3. Calculate the heat capacity. Does it depend on the temperature?
### Exercise 2: Quantum harmonic oscillator
-Consider a 1D quantum harmonic oscillator. Its eigenstates are:
+Consider a 1D quantum harmonic oscillator. Its energy eigenvalues are:
$$
E_n = \hbar\omega(n+\frac{1}{2}),
$$
-1. Sketch the wave function of this harmonic oscillator for $n=3$.
-2. Compute the quantum partition function using the following expression:
+1. Compute the partition function using the following expression:
$$
Z = \sum_j e^{-\beta E_j}.
$$
-3. Using the partition function, compute the expectation 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.
- - What temperature can be considered high?
- - What is the expectation value of $n$?
+2. Using the partition function found in 2.1, compute the expected value of the energy.
+3. Compute the heat capacity. Check that in the high temperature limit you get the same result as in Exercise 1.3.
+4. Plot the found heat capacity and roughly indicate in the plot where the Einstein temperature is.
+5. What is the expected value of $n$?
### Exercise 3: Total heat capacity of a diatomic material
-
-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.
+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.
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.
-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.
+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. 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 $T$.