formatting and rephrasing

src/6_bonds_and_spectra.md

@@ -37,6 +37,7 @@ In the previous lecture, we observed that the diatomic molecule's bonding energy
 However, the trend is unphysical; the two atoms must start repelling eventually (at least when the nuclei get close, but really already much earlier).
 
 ![](figures/bonding_with_repulsion.svg)
+
 Hence from now on, we consider an interatomic repulsion between atoms that get sufficiently close.
 
 ## First steps towards phonons
@@ -45,13 +46,11 @@ The atoms are in equilibrium at the botom of the interatomic potential.
 Let the bottom of the potential be $U = U_0$ at the interatomic distance $r = a$.
 
 ![](figures/interatomic_interaction.svg)
-<!--
-Change delta_r to r in figrue and convert to python format
--->
+
 Near the minimum, the potential is approximately parabolic. We show this by Taylor expanding the interatomic potential around the minimum:
 
 $$
-U = U_0 + \frac{\kappa}{2!} (r - a)^2 - \frac{\kappa_3}{3!} (r - a)^3 + \ldots
+U = U_0 + \frac{\kappa}{2!} (r - a)^2 - \frac{\kappa_3}{3!} (r - a)³ + …
 $$
 
 Up to second order, the potential is quadratic and gives rise to a harmonic equations of motion. 
@@ -68,34 +67,29 @@ For this reason, we approximate the interatomic potential around the minimum to
 Given the quadratic interatomic potential, the material responds with a returning force:
 
 $$
-\begin{align}
-F &= - \frac{\mathrm{d} U}{\mathrm{d} r} \Bigr|_{r = a+\delta r}\\
-&= \kappa a \frac{\delta L}{L}.
-\end{align}
+F = - \frac{\mathrm{d} U}{\mathrm{d} r} \Bigr|_{r = a+\delta r} = \kappa a \frac{\delta L}{L}.
 $$
-We use this result to calculate the materials *compressibility*:
+This result directly leads us to a macroscopic material parameter, its *compressibility*:
 
 $$
-\begin{align}
-\beta &\equiv -\frac{1}{L} \frac{\partial L}{\partial F}\\
-&= \frac{1}{\kappa a}.
-\end{align}
+\beta \equiv -\frac{1}{L} \frac{\partial L}{\partial F}= \frac{1}{\kappa a}.
 $$
 
-The compressibility allows us to calculate the speed of sound by using the following relation:
+Furthermore, the compressibility allows us to calculate the speed of sound by using the relation:
 $$
 v = \sqrt{\frac{1}{\rho \beta}},
 $$
-where $\rho$ is the mass density of the material.
-By using mass density relation $\rho = m/a$ (in 1D), we obtain the speed of sound
+where $\rho$ is the density of the material.
+Because $\rho = m/a$ (in 1D), we finally arrive to the expression for the speed of sound, derived from the atomic properties:
 $$
 v = \sqrt{\frac{\kappa a^2}{m}}.
 $$
-The result might look familiar!
+This way we can already see how phonon heat capacity emerges from the microscopic material structure.
 
 ### Thermal expansion
+
 Let us consider the Taylor expansion of the interatomic potential around minimum to the third order. 
-On top of the harmonic term, we now also have the anharmonic term $\frac{\kappa_3}{3!}(r-a)^3$.
+On top of the harmonic term, we now also have the anharmonic term $\kappa_3(r-a)^3/6$.
 We compare the interatomic potential up to the second-order with the third-order expansion in the figure below. 
 Notice that the second-order approximation is symmetric around the minimum while the third-order term is not.
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