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

1 files changed, 49 insertions, 49 deletions

diff --git a/source/know/concept/drude-model/index.md b/source/know/concept/drude-model/index.md
index 28e6dc2..b175e64 100644
--- a/source/know/concept/drude-model/index.md
+++ b/source/know/concept/drude-model/index.md
@@ -18,19 +18,19 @@ as found in metals and doped semiconductors.
 
 An [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
 has an oscillating [electric field](/know/concept/electric-field/)
-$E(t) = E_0 \exp(- i \omega t)$
+$$E(t) = E_0 \exp(- i \omega t)$$
 that exerts a force on the charge carriers,
-which have mass $m$ and charge $q$.
+which have mass $$m$$ and charge $$q$$.
 They thus obey the following equation of motion,
-where $\gamma$ is a frictional damping coefficient:
+where $$\gamma$$ is a frictional damping coefficient:
 
 $$\begin{aligned}
     m \dvn{2}{x}{t} + m \gamma \dv{x}{t}
     = q E_0 \exp(- i \omega t)
 \end{aligned}$$
 
-Inserting the ansatz $x(t) = x_0 \exp(- i \omega t)$
-and isolating for the displacement $x_0$ yields:
+Inserting the ansatz $$x(t) = x_0 \exp(- i \omega t)$$
+and isolating for the displacement $$x_0$$ yields:
 
 $$\begin{aligned}
     - x_0 m \omega^2 - i x_0 m \gamma \omega
@@ -40,10 +40,10 @@ $$\begin{aligned}
     = - \frac{q E_0}{m (\omega^2 + i \gamma \omega)}
 \end{aligned}$$
 
-The polarization density $P(t)$ is therefore as shown below.
-Note that the dipole moment $p$ goes from negative to positive,
-and the electric field $E$ from positive to negative.
-Let $N$ be the density of carriers in the gas, then:
+The polarization density $$P(t)$$ is therefore as shown below.
+Note that the dipole moment $$p$$ goes from negative to positive,
+and the electric field $$E$$ from positive to negative.
+Let $$N$$ be the density of carriers in the gas, then:
 
 $$\begin{aligned}
     P(t)
@@ -52,8 +52,8 @@ $$\begin{aligned}
     = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} E(t)
 \end{aligned}$$
 
-The electric displacement field $D$ is thus as follows,
-where $\varepsilon_r$ is the unknown relative permittivity of the gas,
+The electric displacement field $$D$$ is thus as follows,
+where $$\varepsilon_r$$ is the unknown relative permittivity of the gas,
 which we will find shortly:
 
 $$\begin{aligned}
@@ -63,8 +63,8 @@ $$\begin{aligned}
     = \varepsilon_0 \bigg( 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} \bigg) E
 \end{aligned}$$
 
-The parenthesized expression is the desired dielectric function $\varepsilon_r$,
-which depends on $\omega$:
+The parenthesized expression is the desired dielectric function $$\varepsilon_r$$,
+which depends on $$\omega$$:
 
 $$\begin{aligned}
     \boxed{
@@ -82,26 +82,26 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-If $\gamma = 0$, then $\varepsilon_r$ is
-negative $\omega < \omega_p$,
-positive for $\omega > \omega_p$,
-and zero for $\omega = \omega_p$.
+If $$\gamma = 0$$, then $$\varepsilon_r$$ is
+negative $$\omega < \omega_p$$,
+positive for $$\omega > \omega_p$$,
+and zero for $$\omega = \omega_p$$.
 Respectively, this leads to
-an imaginary index $\sqrt{\varepsilon_r}$ (high absorption),
-a real index tending to $1$ (transparency),
+an imaginary index $$\sqrt{\varepsilon_r}$$ (high absorption),
+a real index tending to $$1$$ (transparency),
 and the possibility of self-sustained plasma oscillations.
-For metals, $\omega_p$ lies in the UV.
+For metals, $$\omega_p$$ lies in the UV.
 
-We can refine this result for $\varepsilon_r$,
-by recognizing the (mean) velocity $v = \idv{x}{t}$,
+We can refine this result for $$\varepsilon_r$$,
+by recognizing the (mean) velocity $$v = \idv{x}{t}$$,
 and rewriting the equation of motion accordingly:
 
 $$\begin{aligned}
     m \dv{v}{t} + m \gamma v = q E(t)
 \end{aligned}$$
 
-Note that $m v$ is simply the momentum $p$.
-We define the **momentum scattering time** $\tau \equiv 1 / \gamma$,
+Note that $$m v$$ is simply the momentum $$p$$.
+We define the **momentum scattering time** $$\tau \equiv 1 / \gamma$$,
 which represents the average time between collisions,
 where each collision resets the involved particles' momentums to zero.
 Or, more formally:
@@ -111,9 +111,9 @@ $$\begin{aligned}
     = - \frac{p}{\tau} + q E
 \end{aligned}$$
 
-Returning to the equation for the mean velocity $v$,
-we insert the ansatz $v(t) = v_0 \exp(- i \omega t)$,
-for the same electric field $E(t) = E_0 \exp(-i \omega t)$ as before:
+Returning to the equation for the mean velocity $$v$$,
+we insert the ansatz $$v(t) = v_0 \exp(- i \omega t)$$,
+for the same electric field $$E(t) = E_0 \exp(-i \omega t)$$ as before:
 
 $$\begin{aligned}
     - i m \omega v_0 + \frac{m}{\tau} v_0 = q E_0
@@ -121,7 +121,7 @@ $$\begin{aligned}
     v_0 = \frac{q \tau}{m (1 - i \omega \tau)} E_0
 \end{aligned}$$
 
-From $v(t)$, we find the resulting average current density $J(t)$ to be as follows:
+From $$v(t)$$, we find the resulting average current density $$J(t)$$ to be as follows:
 
 $$\begin{aligned}
     J(t)
@@ -129,8 +129,8 @@ $$\begin{aligned}
     = \sigma E(t)
 \end{aligned}$$
 
-Where $\sigma(\omega)$ is the **AC conductivity**,
-which depends on the **DC conductivity** $\sigma_0$:
+Where $$\sigma(\omega)$$ is the **AC conductivity**,
+which depends on the **DC conductivity** $$\sigma_0$$:
 
 $$\begin{aligned}
     \boxed{
@@ -145,7 +145,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We can use these quantities to rewrite
-the dielectric function $\varepsilon_r$ from earlier:
+the dielectric function $$\varepsilon_r$$ from earlier:
 
 $$\begin{aligned}
     \boxed{
@@ -164,12 +164,12 @@ which can be treated as free particles
 moving in the bands of the material.
 
 The Drude model can also be used in this case,
-by replacing the actual carrier mass $m$
-by the effective mass $m^*$.
+by replacing the actual carrier mass $$m$$
+by the effective mass $$m^*$$.
 Furthermore, semiconductors already have
-a high intrinsic permittivity $\varepsilon_{\mathrm{int}}$
+a high intrinsic permittivity $$\varepsilon_{\mathrm{int}}$$
 before the dopant is added,
-so the diplacement field $D$ is:
+so the diplacement field $$D$$ is:
 
 $$\begin{aligned}
     D
@@ -177,9 +177,9 @@ $$\begin{aligned}
     = \varepsilon_{\mathrm{int}} \varepsilon_0 E - \frac{N q^2}{m^* (\omega^2 + i \gamma \omega)} E
 \end{aligned}$$
 
-Where $P_{\mathrm{int}}$ is the intrinsic undoped polarization,
-and $P_{\mathrm{free}}$ is the contribution of the free carriers.
-The dielectric function $\varepsilon_r(\omega)$ is therefore given by:
+Where $$P_{\mathrm{int}}$$ is the intrinsic undoped polarization,
+and $$P_{\mathrm{free}}$$ is the contribution of the free carriers.
+The dielectric function $$\varepsilon_r(\omega)$$ is therefore given by:
 
 $$\begin{aligned}
     \boxed{
@@ -188,8 +188,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where the plasma frequency $\omega_p$ has been redefined as follows
-to include $\varepsilon_\mathrm{int}$:
+Where the plasma frequency $$\omega_p$$ has been redefined as follows
+to include $$\varepsilon_\mathrm{int}$$:
 
 $$\begin{aligned}
     \boxed{
@@ -198,14 +198,14 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The meaning of $\omega_p$ is the same as for metals,
-with high absorption for $\omega < \omega_p$.
-However, due to the lower carrier density $N$ in a semiconductor,
-$\omega_p$ lies in the IR rather than UV.
+The meaning of $$\omega_p$$ is the same as for metals,
+with high absorption for $$\omega < \omega_p$$.
+However, due to the lower carrier density $$N$$ in a semiconductor,
+$$\omega_p$$ lies in the IR rather than UV.
 
-However, instead of asymptotically going to $1$ for $\omega > \omega_p$ like a metal,
-$\varepsilon_r$ tends to $\varepsilon_\mathrm{int}$ instead,
-and crosses $1$ along the way,
+However, instead of asymptotically going to $$1$$ for $$\omega > \omega_p$$ like a metal,
+$$\varepsilon_r$$ tends to $$\varepsilon_\mathrm{int}$$ instead,
+and crosses $$1$$ along the way,
 at which point the reflectivity is zero.
 This occurs at:
 
@@ -214,9 +214,9 @@ $$\begin{aligned}
     = \frac{\varepsilon_{\mathrm{int}}}{\varepsilon_{\mathrm{int}} - 1} \omega_p^2
 \end{aligned}$$
 
-This is used to experimentally determine the effective mass $m^*$
+This is used to experimentally determine the effective mass $$m^*$$
 of the doped semiconductor,
-by finding which value of $m^*$ gives the measured $\omega$.
+by finding which value of $$m^*$$ gives the measured $$\omega$$.