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docs/lecture_1.md

@@ -5,7 +5,7 @@ _Based on chapter 2 of the book_
 In this lecture we will:
 
 - discuss specific heat of a solid based on atomic vibrations (_phonons_)
-- disregard periodic lattice $\rightarrow$ consider homogeneous medium
+- disregard periodic lattice $\Rightarrow$ consider homogeneous medium
     - _(chapter 9: discuss phonons in terms of atomic masses and springs)_
 - discuss the Einstein model
 - discuss the Debye model
@@ -19,10 +19,10 @@ This can be explained by considering a _quantum_ harmonic oscillator:
 
 $$\varepsilon_n=\left(n+\frac{1}{2}\right)\hbar\omega$$
 
-Phonons are bosons $\rightarrow$ they follow Bose-Einstein statistics.
+Phonons are bosons $\Rightarrow$ they follow Bose-Einstein statistics.
 
 $$
-n(\omega,T)=\frac{1}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}\rightarrow\bar{\varepsilon}=\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}
+n(\omega,T)=\frac{1}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}\Rightarrow\bar{\varepsilon}=\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}
 $$
 
 ![](figures/bose_einstein.svg)
@@ -58,8 +58,6 @@ $$E=\int\limits_0^\infty\left(\frac{1}{2}\hbar\omega+\frac{\hbar\omega}{ {\rm e}
 $g(\omega)$ is the _density of states_: the number of normal modes found at each position along the $\omega$-axis. How do we calculate $g(\omega)$?
 
 
-
-
 #### Reciprocal space, periodic boundary conditions
 Each normal mode can be described by a _wave vector_ ${\bf k}$. A wave vector represents a point in _reciprocal space_ or _k-space_. We can find $g(\omega)$ by counting the number of normal modes in k-space and then converting those to $\omega$.
 
@@ -96,7 +94,7 @@ Substitute $x\equiv\frac{\hbar\omega}{k_{\rm B}T}$:
 
 $$\Rightarrow E=E_{\rm Z}+\frac{3V}{2\pi^2 v_{\rm s}^3}\frac{\left(k_{\rm B}T\right)^4}{\hbar^3}\int\limits_0^\infty\frac{x^3}{ {\rm e}^x-1}{\rm d}x$$
 
-The integral on the right is a constant, $\left(\frac{\pi^4}{15}\right)$ $\rightarrow$ $C=\frac{ {\rm d}E}{ {\rm d}T}\propto T^3$.
+The integral on the right is a constant, $\left(\frac{\pi^4}{15}\right)$ $\Rightarrow$ $C=\frac{ {\rm d}E}{ {\rm d}T}\propto T^3$.
 
 #### Debye's interpolation for medium T
 The above approximation works very well at low temperature. But at high temperature, $C$ should of course settle at $3k_{\rm B}$ (the Dulong-Petit value). The reason why the model breaks down, is that it assumes that there is an infinite number of harmonic oscillators up to infinite frequency.