From f5ea076c60dd34ffef86d61ffef16c5071a4c648 Mon Sep 17 00:00:00 2001
From: Anton Akhmerov <anton.akhmerov@gmail.com>
Date: Wed, 24 Feb 2021 00:04:46 +0100
Subject: [PATCH] formatting and rephrasing
---
src/6_bonds_and_spectra.md | 30 ++++++++-----------
src/figures/interatomic_interaction.svg | 38 +++++++++----------------
2 files changed, 26 insertions(+), 42 deletions(-)
diff --git a/src/6_bonds_and_spectra.md b/src/6_bonds_and_spectra.md
index 6c438f1c..9edddba9 100644
--- a/src/6_bonds_and_spectra.md
+++ b/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).
+
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 $r = a$.
-<!--
-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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diff --git a/src/figures/interatomic_interaction.svg b/src/figures/interatomic_interaction.svg
index 3e17511b..319ccd1d 100644
--- a/src/figures/interatomic_interaction.svg
+++ b/src/figures/interatomic_interaction.svg
@@ -1,6 +1,4 @@
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