Update figure comparing Debye and Einstein models. Added experimental data for silver.

src/2_debye_model.md

@@ -101,33 +101,37 @@ $$
 where $x=\frac{\hbar\omega}{k_{\rm B}T}$ and $\Theta_{\rm D}\equiv\frac{\hbar\omega_{\rm D}}{k_{\rm B}}$, the _Debye temperature_.
 
 ```python
-def integrand(y):
-    return y**4 * np.exp(y) / (np.exp(y) - 1)**2
 
-def c_einstein(T, T_E=1):
+T = [1.35,2.,3.,4.,5.,6.,7.,8.,10.,12.,14.,16.,20.,28.56,36.16,47.09,55.88,65.19,74.56,83.91,103.14,124.2,144.38,166.78,190.17,205.3]
+c = [0.,0.,0.,0.,0.,0.,0.0719648,0.1075288,0.2100368,0.364008,0.573208,0.866088,1.648496,4.242576,7.07096,10.8784,13.47248,15.60632,17.27992,18.6188,20.33424,21.63128,22.46808,23.05384,23.47224,23.68144]
+
+def c_einstein(T, T_E):
     x = T_E / T
-    return 3 * x**2 * np.exp(x) / (np.exp(x) - 1)**2
+    return 24.945 * x**2 * np.exp(x) / (np.exp(x) - 1)**2
+
+def integrand(y):
+    return y**4 * np.exp(y) / (np.exp(y) - 1)**2
 
 @np.vectorize
-def c_debye(T, T_D=1):
+def c_debye(T, T_D):
     x = T / T_D
-    return 9 * x**3 * quad(integrand, 0, 1/x)[0]
+    return 24.945 * 3 * x**3 * quad(integrand, 0, 1/x)[0]
 
-T = np.linspace(0.01, 1.5, 500)
 fig, ax = pyplot.subplots()
 
-ax.plot(T, c_einstein(T), label="Einstein model")
-ax.plot(T, c_debye(T), label="Debye model")
-
-ax.set_ylim(bottom=0, top=3.4)
-ax.set_xlabel('$T$')
-ax.set_ylabel(r'$\omega$')
-ax.set_xticks([1])
-ax.set_xticklabels([r'$\Theta_D$'])
-ax.set_yticks([3])
-ax.set_yticklabels(['$3Nk_B$'])
+ax.scatter(T, c)
+ax.title('Heat capacity of silver compared to the Debye and Einstein models')
+ax.plot(T, c_einstein(T, 151), label='Einstein model')
+ax.plot(T, c_debye(T, 215), label='Debye model')
+ax.set_ylim(bottom=0, top=26)
+ax.set_xlim(0, 215)
+ax.set_xlabel('T(K)')
+ax.set_ylabel(r'C(J/mol-K)')
+ax.set_xticks([0, 100, 200])
+ax.set_yticks([24.945])
+ax.set_yticklabels(['$3R$'])
 ax.legend(loc='lower right')
-pyplot.hlines([3], 0, 1.5, linestyles='dashed')
+pyplot.hlines([24.945], 0, T[-1], linestyles='dashed')
 draw_classic_axes(ax, xlabeloffset=0.3)
 ```