From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/density-of-states/index.md | 74 +++++++++++++-------------
 1 file changed, 37 insertions(+), 37 deletions(-)

(limited to 'source/know/concept/density-of-states')

diff --git a/source/know/concept/density-of-states/index.md b/source/know/concept/density-of-states/index.md
index 60b81d5..1c3b511 100644
--- a/source/know/concept/density-of-states/index.md
+++ b/source/know/concept/density-of-states/index.md
@@ -8,15 +8,15 @@ categories:
 layout: "concept"
 ---
 
-The **density of states** $g(E)$ of a physical system is defined such that
-$g(E) \dd{E}$ is the number of states which could be occupied
-with an energy in the interval $[E, E + \dd{E}]$.
-In fact, $E$ need not be an energy;
+The **density of states** $$g(E)$$ of a physical system is defined such that
+$$g(E) \dd{E}$$ is the number of states which could be occupied
+with an energy in the interval $$[E, E + \dd{E}]$$.
+In fact, $$E$$ need not be an energy;
 it should just be something that effectively identifies the state.
 
 In its simplest form, the density of states is as follows,
-where $\Gamma(E)$ is the number of states with energy
-less than or equal to the argument $E$:
+where $$\Gamma(E)$$ is the number of states with energy
+less than or equal to the argument $$E$$:
 
 $$\begin{aligned}
     g(E)
@@ -25,9 +25,9 @@ $$\begin{aligned}
 
 If the states can be treated as waves,
 which is often the case,
-then we can calculate the density of states $g(k)$ in
-$k$-space, i.e. as a function of the wavenumber $k = |\vb{k}|$.
-Once we have $g(k)$, we use the dispersion relation $E(k)$ to find $g(E)$,
+then we can calculate the density of states $$g(k)$$ in
+$$k$$-space, i.e. as a function of the wavenumber $$k = |\vb{k}|$$.
+Once we have $$g(k)$$, we use the dispersion relation $$E(k)$$ to find $$g(E)$$,
 by demanding that:
 
 $$\begin{aligned}
@@ -37,14 +37,14 @@ $$\begin{aligned}
     = g(k) \dv{k}{E}
 \end{aligned}$$
 
-Inverting the dispersion relation $E(k)$ to get $k(E)$ might be difficult,
+Inverting the dispersion relation $$E(k)$$ to get $$k(E)$$ might be difficult,
 in which case the left-hand equation can be satisfied numerically.
 
 
-Define $\Omega_n(k)$ as the number of states with
-a $k$-value less than or equal to the argument,
-or in other words, the volume of a hypersphere with radius $k$.
-Then the $n$-dimensional density of states $g_n(k)$
+Define $$\Omega_n(k)$$ as the number of states with
+a $$k$$-value less than or equal to the argument,
+or in other words, the volume of a hypersphere with radius $$k$$.
+Then the $$n$$-dimensional density of states $$g_n(k)$$
 has the following general form:
 
 $$\begin{aligned}
@@ -54,14 +54,14 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $D$ is each state's degeneracy (e.g. due to spin),
-and $k_{\mathrm{min}}$ is the smallest allowed $k$-value,
-according to the characteristic length $L$ of the system.
-We divide by $2^n$ to limit ourselves to the sector where all axes are positive,
-because we are only considering the magnitude of $k$.
+Where $$D$$ is each state's degeneracy (e.g. due to spin),
+and $$k_{\mathrm{min}}$$ is the smallest allowed $$k$$-value,
+according to the characteristic length $$L$$ of the system.
+We divide by $$2^n$$ to limit ourselves to the sector where all axes are positive,
+because we are only considering the magnitude of $$k$$.
 
-In one dimension $n = 1$, the number of states within a distance $k$ from the
-origin is the distance from $k$ to $-k$
+In one dimension $$n = 1$$, the number of states within a distance $$k$$ from the
+origin is the distance from $$k$$ to $$-k$$
 (we let it run negative, since its meaning does not matter here), given by:
 
 $$\begin{aligned}
@@ -69,9 +69,9 @@ $$\begin{aligned}
     = 2 k
 \end{aligned}$$
 
-To get $k_{\mathrm{min}}$, we choose to look at a rod of length $L$,
+To get $$k_{\mathrm{min}}$$, we choose to look at a rod of length $$L$$,
 across which the function is a standing wave, meaning that
-the allowed values of $k$ must be as follows, where $m \in \mathbb{N}$:
+the allowed values of $$k$$ must be as follows, where $$m \in \mathbb{N}$$:
 
 $$\begin{aligned}
     \lambda = \frac{2 L}{m}
@@ -79,9 +79,9 @@ $$\begin{aligned}
     k = \frac{2 \pi}{\lambda} = \frac{m \pi}{L}
 \end{aligned}$$
 
-Take the smallest option $m = 1$,
-such that $k_{\mathrm{min}} = \pi / L$,
-the 1D density of states $g_1(k)$ is:
+Take the smallest option $$m = 1$$,
+such that $$k_{\mathrm{min}} = \pi / L$$,
+the 1D density of states $$g_1(k)$$ is:
 
 $$\begin{aligned}
     \boxed{
@@ -91,8 +91,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-In 2D, the number of states within a range $k$ of the
-origin is the area of a circle with radius $k$:
+In 2D, the number of states within a range $$k$$ of the
+origin is the area of a circle with radius $$k$$:
 
 $$\begin{aligned}
     \Omega_2(k)
@@ -100,8 +100,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Analogously to the 1D case,
-we take the system to be a square of side $L$,
-so $k_{\mathrm{min}} = \pi / L$ again.
+we take the system to be a square of side $$L$$,
+so $$k_{\mathrm{min}} = \pi / L$$ again.
 The density of states then becomes:
 
 $$\begin{aligned}
@@ -112,14 +112,14 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-In 3D, the number of states is the volume of a sphere with radius $k$:
+In 3D, the number of states is the volume of a sphere with radius $$k$$:
 
 $$\begin{aligned}
     \Omega_3(k)
     = \frac{4 \pi}{3} k^3
 \end{aligned}$$
 
-For a cube with side $L$, we once again find $k_{\mathrm{min}} = \pi / L$.
+For a cube with side $$L$$, we once again find $$k_{\mathrm{min}} = \pi / L$$.
 We thus get:
 
 $$\begin{aligned}
@@ -130,17 +130,17 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-All these expressions contain the characteristic length/area/volume $L^n$,
+All these expressions contain the characteristic length/area/volume $$L^n$$,
 and therefore give the number of states in that region only.
-Keep in mind that $L$ is free to choose;
+Keep in mind that $$L$$ is free to choose;
 it need not be the physical size of the system.
 In fact, we typically want the density of states
 per unit length/area/volume,
-so we can just set $L = 1$ in our preferred unit of distance.
+so we can just set $$L = 1$$ in our preferred unit of distance.
 
 If the system is infinitely large, or if it has periodic boundaries,
-then $k$ becomes a continuous variable and $k_\mathrm{min} \to 0$.
-But again, $L$ is arbitrary,
+then $$k$$ becomes a continuous variable and $$k_\mathrm{min} \to 0$$.
+But again, $$L$$ is arbitrary,
 so a finite value can be chosen.
 
 
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