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/debye-length/index.md | 36 +++++++++++++++----------------
 1 file changed, 18 insertions(+), 18 deletions(-)

(limited to 'source/know/concept/debye-length')

diff --git a/source/know/concept/debye-length/index.md b/source/know/concept/debye-length/index.md
index a42d137..e226ad9 100644
--- a/source/know/concept/debye-length/index.md
+++ b/source/know/concept/debye-length/index.md
@@ -16,10 +16,10 @@ This has the effect of **shielding** the object's presence
 from the rest of the plasma.
 
 We start from [Gauss' law](/know/concept/maxwells-equations/)
-for the [electric field](/know/concept/electric-field/) $\vb{E}$,
-expressing $\vb{E}$ as the gradient of a potential $\phi$,
-i.e. $\vb{E} = -\nabla \phi$,
-and splitting the charge density into ions $n_i$ and electrons $n_e$:
+for the [electric field](/know/concept/electric-field/) $$\vb{E}$$,
+expressing $$\vb{E}$$ as the gradient of a potential $$\phi$$,
+i.e. $$\vb{E} = -\nabla \phi$$,
+and splitting the charge density into ions $$n_i$$ and electrons $$n_e$$:
 
 $$\begin{aligned}
     \nabla^2 \phi(\vb{r})
@@ -28,9 +28,9 @@ $$\begin{aligned}
 
 The last term represents a *test particle*,
 which will be shielded.
-This particle is a point charge $q_t$,
-whose density is simply a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta(\vb{r})$,
-and is not included in $n_i$ or $n_e$.
+This particle is a point charge $$q_t$$,
+whose density is simply a [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(\vb{r})$$,
+and is not included in $$n_i$$ or $$n_e$$.
 
 For a plasma in thermal equilibrium,
 we have the [Boltzmann relations](/know/concept/boltzmann-relation/)
@@ -45,7 +45,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We assume that electrical interactions are weak compared to thermal effects,
-i.e. $k_B T \gg q \phi$ in both cases.
+i.e. $$k_B T \gg q \phi$$ in both cases.
 Then we Taylor-expand the Boltzmann relations to first order:
 
 $$\begin{aligned}
@@ -57,8 +57,8 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Inserting this back into Gauss' law,
-we arrive at the following equation for $\phi(\vb{r})$,
-where we have assumed quasi-neutrality such that $q_i n_{i0} = q_e n_{e0}$:
+we arrive at the following equation for $$\phi(\vb{r})$$,
+where we have assumed quasi-neutrality such that $$q_i n_{i0} = q_e n_{e0}$$:
 
 $$\begin{aligned}
     \nabla^2 \phi
@@ -70,7 +70,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We now define the **ion** and **electron Debye lengths**
-$\lambda_{Di}$ and $\lambda_{De}$ as follows:
+$$\lambda_{Di}$$ and $$\lambda_{De}$$ as follows:
 
 $$\begin{aligned}
     \boxed{
@@ -84,7 +84,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-And then the **total Debye length** $\lambda_D$ is defined as the sum of their inverses,
+And then the **total Debye length** $$\lambda_D$$ is defined as the sum of their inverses,
 and gives the rough thickness of the Debye sheath:
 
 $$\begin{aligned}
@@ -108,7 +108,7 @@ This has the following solution,
 known as the **Yukawa potential**,
 which decays exponentially,
 representing the plasma's **self-shielding**
-over a characteristic distance $\lambda_D$:
+over a characteristic distance $$\lambda_D$$:
 
 $$\begin{aligned}
     \boxed{
@@ -117,13 +117,13 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Note that $r$ is a scalar,
-i.e. the potential depends only on the radial distance to $q_t$.
+Note that $$r$$ is a scalar,
+i.e. the potential depends only on the radial distance to $$q_t$$.
 This treatment only makes sense
 if the plasma is sufficiently dense,
 such that there is a large number of particles
-in a sphere with radius $\lambda_D$.
-This corresponds to a large [Coulomb logarithm](/know/concept/coulomb-logarithm/) $\ln\!(\Lambda)$:
+in a sphere with radius $$\lambda_D$$.
+This corresponds to a large [Coulomb logarithm](/know/concept/coulomb-logarithm/) $$\ln\!(\Lambda)$$:
 
 $$\begin{aligned}
     1 \ll \frac{4 \pi}{3} n_0 \lambda_D^3 = \frac{2}{9} \Lambda
@@ -138,7 +138,7 @@ $$\begin{aligned}
     = \frac{A}{r} \exp(-B r)
 \end{aligned}$$
 
-Where $A$ and $B$ are scaling constants that depend on the problem at hand.
+Where $$A$$ and $$B$$ are scaling constants that depend on the problem at hand.
 
 
 
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