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/plancks-law/index.md | 38 ++++++++++++++++----------------
 1 file changed, 19 insertions(+), 19 deletions(-)

(limited to 'source/know/concept/plancks-law')

diff --git a/source/know/concept/plancks-law/index.md b/source/know/concept/plancks-law/index.md
index 2db783d..6c2cd2e 100644
--- a/source/know/concept/plancks-law/index.md
+++ b/source/know/concept/plancks-law/index.md
@@ -18,14 +18,14 @@ and photons are bosons
 (see [Pauli exclusion principle](/know/concept/pauli-exclusion-principle/)),
 this system must obey the
 [Bose-Einstein distribution](/know/concept/bose-einstein-distribution/),
-with a chemical potential $\mu = 0$ (due to the freely varying population):
+with a chemical potential $$\mu = 0$$ (due to the freely varying population):
 
 $$\begin{aligned}
     f_B(E)
     = \frac{1}{\exp(\beta E) - 1}
 \end{aligned}$$
 
-Each photon has an energy $E = \hbar \omega = \hbar c k$,
+Each photon has an energy $$E = \hbar \omega = \hbar c k$$,
 so the [density of states](/know/concept/density-of-states/)
 is as follows in 3D:
 
@@ -37,9 +37,9 @@ $$\begin{aligned}
     = \frac{8 \pi V E^2}{h^3 c^3}
 \end{aligned}$$
 
-Where the factor of $2$ accounts for the photon's polarization degeneracy.
-We thus expect that the number of photons $N(E)$
-with an energy between $E$ and $E + \dd{E}$ is given by:
+Where the factor of $$2$$ accounts for the photon's polarization degeneracy.
+We thus expect that the number of photons $$N(E)$$
+with an energy between $$E$$ and $$E + \dd{E}$$ is given by:
 
 $$\begin{aligned}
     N(E) \dd{E}
@@ -47,16 +47,16 @@ $$\begin{aligned}
     = \frac{8 \pi V}{h^3 c^3} \frac{E^2}{\exp(\beta E) - 1} \dd{E}
 \end{aligned}$$
 
-By substituting $E = h \nu$, we find that the number of photons $N(\nu)$
-with a frequency between $\nu$ and $\nu + \dd{\nu}$ must be as follows:
+By substituting $$E = h \nu$$, we find that the number of photons $$N(\nu)$$
+with a frequency between $$\nu$$ and $$\nu + \dd{\nu}$$ must be as follows:
 
 $$\begin{aligned}
     N(\nu) \dd{\nu}
     = \frac{8 \pi V}{c^3} \frac{\nu^2}{\exp(\beta h \nu) - 1} \dd{\nu}
 \end{aligned}$$
 
-Multiplying by the energy $h \nu$ yields the distribution of the radiated energy,
-which we divide by the volume $V$ to get Planck's law,
+Multiplying by the energy $$h \nu$$ yields the distribution of the radiated energy,
+which we divide by the volume $$V$$ to get Planck's law,
 also called the **Plank distribution**,
 describing a black body's radiated spectral energy density per unit volume:
 
@@ -70,8 +70,8 @@ $$\begin{aligned}
 
 ## Wien's displacement law
 
-The Planck distribution peaks at a particular frequency $\nu_{\mathrm{max}}$,
-which can be found by solving the following equation for $\nu$:
+The Planck distribution peaks at a particular frequency $$\nu_{\mathrm{max}}$$,
+which can be found by solving the following equation for $$\nu$$:
 
 $$\begin{aligned}
     0
@@ -81,7 +81,7 @@ $$\begin{aligned}
     = 3 \nu^2 (\exp(\beta h \nu) - 1) - \nu^3 \beta h \exp(\beta h \nu)
 \end{aligned}$$
 
-By defining $x \equiv \beta h \nu_{\mathrm{max}}$,
+By defining $$x \equiv \beta h \nu_{\mathrm{max}}$$,
 this turns into the following transcendental equation:
 
 $$\begin{aligned}
@@ -98,14 +98,14 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Which states that the peak frequency $\nu_{\mathrm{max}}$
-is proportional to the temperature $T$.
+Which states that the peak frequency $$\nu_{\mathrm{max}}$$
+is proportional to the temperature $$T$$.
 
 
 ## Stefan-Boltzmann law
 
-Because $u(\nu)$ represents the radiated spectral energy density,
-we can find the total radiated energy $U$ per unit volume by integrating over $\nu$:
+Because $$u(\nu)$$ represents the radiated spectral energy density,
+we can find the total radiated energy $$U$$ per unit volume by integrating over $$\nu$$:
 
 $$\begin{aligned}
     U
@@ -116,9 +116,9 @@ $$\begin{aligned}
     = \frac{8 \pi}{\beta^4 h^3 c^3} \int_0^\infty \frac{x^3}{\exp(x) - 1} \dd{x}
 \end{aligned}$$
 
-This definite integral turns out to be $\pi^4/15$,
+This definite integral turns out to be $$\pi^4/15$$,
 leading us to the **Stefan-Boltzmann law**,
-which states that the radiated energy is proportional to $T^4$:
+which states that the radiated energy is proportional to $$T^4$$:
 
 $$\begin{aligned}
     \boxed{
@@ -126,7 +126,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $\sigma$ is the **Stefan-Boltzmann constant**, which is defined as follows:
+Where $$\sigma$$ is the **Stefan-Boltzmann constant**, which is defined as follows:
 
 $$\begin{aligned}
     \sigma
-- 
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