Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 26 insertions, 26 deletions

diff --git a/source/know/concept/spitzer-resistivity/index.md b/source/know/concept/spitzer-resistivity/index.md
index f9fd969..283b7fd 100644
--- a/source/know/concept/spitzer-resistivity/index.md
+++ b/source/know/concept/spitzer-resistivity/index.md
@@ -9,17 +9,17 @@ layout: "concept"
 ---
 
 If an [electric field](/know/concept/electric-field/)
-with magnitude $E$ is applied to the plasma, the electrons experience
-a [Lorentz force](/know/concept/lorentz-force/) $q_e E$
+with magnitude $$E$$ is applied to the plasma, the electrons experience
+a [Lorentz force](/know/concept/lorentz-force/) $$q_e E$$
 (we neglect the ions due to their mass),
-where $q_e$ is the electron charge.
+where $$q_e$$ is the electron charge.
 
 However, collisions slow them down while they travel through the plasma.,
-This can be modelled as a drag force $f_{ei} m_e v_e$,
-where $f_{ei}$ is the electron-ion collision frequency
-(we neglect $f_{ee}$ since all electrons are moving together),
-$m_e$ is their mass,
-and $v_e$ their typical velocity relative to the ions in the background.
+This can be modelled as a drag force $$f_{ei} m_e v_e$$,
+where $$f_{ei}$$ is the electron-ion collision frequency
+(we neglect $$f_{ee}$$ since all electrons are moving together),
+$$m_e$$ is their mass,
+and $$v_e$$ their typical velocity relative to the ions in the background.
 Balancing the two forces yields the following relation:
 
 $$\begin{aligned}
@@ -27,7 +27,7 @@ $$\begin{aligned}
     = f_{ei} m_e v_e
 \end{aligned}$$
 
-Using that the current density $J = q_e n_e v_e$,
+Using that the current density $$J = q_e n_e v_e$$,
 we can rearrange this like so:
 
 $$\begin{aligned}
@@ -37,12 +37,12 @@ $$\begin{aligned}
     = \eta J
 \end{aligned}$$
 
-This is Ohm's law, where $\eta$ is the resistivity.
-From our derivation of the [Coulomb logarithm](/know/concept/coulomb-logarithm/) $\ln(\Lambda)$,
-we estimate $f_{ei}$ to be as follows,
-where $n_i$ is the ion density,
-$\sigma$ is the collision cross-section,
-and $\mu$ is the [reduced mass](/know/concept/reduced-mass/)
+This is Ohm's law, where $$\eta$$ is the resistivity.
+From our derivation of the [Coulomb logarithm](/know/concept/coulomb-logarithm/) $$\ln(\Lambda)$$,
+we estimate $$f_{ei}$$ to be as follows,
+where $$n_i$$ is the ion density,
+$$\sigma$$ is the collision cross-section,
+and $$\mu$$ is the [reduced mass](/know/concept/reduced-mass/)
 of the electron-ion system:
 
 $$\begin{aligned}
@@ -52,15 +52,15 @@ $$\begin{aligned}
     \approx \frac{1}{2 \pi} \frac{Z q_e^4}{\varepsilon_0^2 m_e^2} \frac{n_e}{v_e^3} \ln(\Lambda)
 \end{aligned}$$
 
-Where we used that $\mu \approx m_e$,
-and $q_i = -Z q_e$ for some ionization $Z$,
-and as a result $n_e \approx Z n_i$ due to the plasma's quasi-neutrality.
-Beware: authors disagree about the constant factors in $f_{ei}$;
+Where we used that $$\mu \approx m_e$$,
+and $$q_i = -Z q_e$$ for some ionization $$Z$$,
+and as a result $$n_e \approx Z n_i$$ due to the plasma's quasi-neutrality.
+Beware: authors disagree about the constant factors in $$f_{ei}$$;
 recall that it was derived from fairly rough estimates.
 This article follows Bellan.
 
-Inserting this expression for $f_{ei}$ into
-the so-called **Spitzer resistivity** $\eta$ then yields:
+Inserting this expression for $$f_{ei}$$ into
+the so-called **Spitzer resistivity** $$\eta$$ then yields:
 
 $$\begin{aligned}
     \boxed{
@@ -70,10 +70,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-A reasonable estimate for the typical velocity $v_e$
+A reasonable estimate for the typical velocity $$v_e$$
 at thermal equilibrium is as follows,
-where $k_B$ is Boltzmann's constant,
-and $T_e$ is the electron temperature:
+where $$k_B$$ is Boltzmann's constant,
+and $$T_e$$ is the electron temperature:
 
 $$\begin{aligned}
     \frac{1}{2} m_e v_e^2
@@ -85,8 +85,8 @@ $$\begin{aligned}
 
 Other choices exist,
 see e.g. the [Maxwell-Boltzmann distribution](/know/concept/maxwell-boltzmann-distribution/),
-but always $v_e \propto \sqrt{T_e/m_e}$.
-Inserting this $v_e$ into $\eta$ then gives:
+but always $$v_e \propto \sqrt{T_e/m_e}$$.
+Inserting this $$v_e$$ into $$\eta$$ then gives:
 
 $$\begin{aligned}
     \eta