Update 1_einstein_model.md - polish

src/1_einstein_model.md

@@ -235,7 +235,7 @@ $$
 \bar{\varepsilon}=\frac{1}{2}\hbar\omega_0+\frac{\hbar\omega_0}{ {\rm e}^{\hbar\omega_0/k_{\rm B}T}-1}
 $$
 
-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.
+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 $T=0$ because of the uncertainty principle. Moreover, we see that the energy in the oscillator only starts to increase significantly when $kT>\hbar \omega$. I.e., the heat capacity only becomes significant for $kT>\hbar \omega$ and goes to zero when $T\rightarrow0$ 
 
 ```python
 xline = [1, 1];
@@ -251,6 +251,7 @@ ax.set_xticklabels(['$0$', '$k_B T$'])
 ax.set_ylabel('$n$')
 ax.set_yticks([1, 2])
 ax.set_yticklabels(['$1$', '$2$'])
+ax.text(1, 0.5, r'$\hbar \omega = kT$', ha='left', color='r');
 draw_classic_axes(ax, xlabeloffset=.2)
 temps = np.linspace(0.01, 2)
 ax2.plot(temps, 1/2 + 1/(np.exp(1/temps)-1))
@@ -264,8 +265,7 @@ ax2.set_yticklabels([r'$\hbar\omega/2$'])
 draw_classic_axes(ax2, xlabeloffset=.15)
 ```
 
-
-To calculate the heat capacity per atom $C$, we need to differentiate $\bar{\varepsilon}$ to $T$.
+We now calculate the heat capacity per atom $C$ explicitly. To do so, we need to differentiate $\bar{\varepsilon}$ with respect to $T$.
 $$
 \begin{multline}
 C = \frac{\partial\bar{\varepsilon}}{\partial T}