Merge branch 'master' into local

src/1_einstein_model.md

@@ -235,7 +235,7 @@ $$
 \langle E \rangle=\frac{1}{2}\hbar\omega_0+\frac{\hbar\omega_0}{ {\rm e}^{\hbar\omega_0/k_{\rm B}T}-1}
 $$
 
-The left plot below shows the Bose-Einstein distribution vs energy. We see that low-energy states are more likely to be occupied than high-energy states. The right plot shows the increasing thermal energy in the oscillator for increasing temperature and highlights the zero-point energy $\hbar\omega_0/2$ that remains in the oscillator at $T=0$ (a consequence of the uncertainty principle). Moreover, we see that the energy in the oscillator is approximately constant when $k_{\rm B}T<\hbar \omega_0$. I.e., the heat capacity becomes small for $k_{\rm B}T<\hbar \omega_0$ and goes to zero when $T\rightarrow0$. 
+The left plot below shows the Bose-Einstein distribution vs energy. We see that low-energy states are more likely to be occupied than high-energy states. The right plot shows the increasing thermal energy in the oscillator for increasing temperature and highlights the zero-point energy $\hbar\omega_0/2$ that remains in the oscillator at $T=0$ (a consequence of the uncertainty principle). Moreover, we see that the energy in the oscillator becomes approximately constant when $k_{\rm B}T\ll\hbar \omega_0$. This implies that the heat capacity becomes small when $k_{\rm B}T<\hbar \omega_0$ and goes to zero when $T\rightarrow0$. 
 
 ```python
 xline = [1, 1];
@@ -263,7 +263,7 @@ ax2.set_ylabel(r"$\langle E \rangle$")
 ax2.set_yticks([1/2])
 ax2.set_yticklabels([r'$\hbar\omega_0/2$'])
 draw_classic_axes(ax2, xlabeloffset=.15)
-ax2.text(1.05, 0.95, r'$k_{\rm B}T=\hbar \omega_0$', ha='left', color='r');
+ax2.text(1.05, 0.75, r'$k_{\rm B}T=\hbar \omega_0$', ha='left', color='r');
 ```
 
 We now calculate the heat capacity per atom $C$ explicitly. To do so, we need to differentiate $\langle E \rangle$ with respect to $T$.
@@ -277,7 +277,7 @@ C = \frac{\partial\langle E \rangle}{\partial T}
 $$
 There is still one step to do, rewriting this equation more elegantly:
 $$
-C = k_b \left(\frac{T_E}{T}\right)^2\frac{e^{T_E/T}}{(e^{T_E/T} - 1)^2)},
+C = k_b \left(\frac{T_E}{T}\right)^2\frac{e^{T_E/T}}{(e^{T_E/T} - 1)^2},
 $$
 where we introduce the *Einstein temperature* $T_E \equiv \hbar \omega_0 / k_B$.
 This is a characteristic temperature when the thermal excitations "freeze out" in the harmonic oscillator.
@@ -289,7 +289,7 @@ def c_einstein(T, T_E=1):
     return 3 * x**2 * np.exp(x) / (np.exp(x) - 1)**2
 
 xline = [1, 1];
-yline = [0, 1.5];
+yline = [0, 1.1];
 temps = np.linspace(0.01, 1.5, 500)
 fig, ax = pyplot.subplots()
 
@@ -305,7 +305,7 @@ ax.set_yticks([1])
 ax.set_yticklabels(['$k_B$'])
 pyplot.hlines([1], 0, 1.5, linestyles='dashed')
 draw_classic_axes(ax)
-ax.text(1.05, 0.95, r'$k_{\rm B}T = \hbar \omega$', ha='left', color='r');
+ax.text(1.01, 0.5, r'$k_{\rm B}T = \hbar \omega$', ha='left', color='r');
 ```
 
 The dashed line is the classical value, $k_{\rm B}$.
@@ -332,8 +332,10 @@ ax.set_ylim((0, 3));
 ## Conclusions
 
 1. The law of Dulong–Petit is an observation that all materials have $C≈3k_B$ per atom.
-2. Oscillations of atoms are *quantum*, and they *freeze* at a low temperature, leaving only zero-point motion
-3. Using Bose–Einstein distribution we explain how $C$ drops with $T$.
+2. Oscillations of atoms are *quantum*, and they *freeze out* when $k_BT \\ \hbar \omega_0$, leaving only zero-point motion.
+3. The Einstein model describes each atom in a solid as a quantum harmonic oscillator.
+4. Using the Bose–Einstein distribution, we derived the thermal energy and heat capacity per atom in the Einstein model.
+5. The Eisntein model correctly predicts that the heat capacity drops to 0 as $T\rightarrow 0$.
 
 
 ## Exercises