From aeacfca5aea5df7c107cf0c12e72ab5d496c96e1 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 3 Jan 2023 19:48:17 +0100
Subject: More improvements to knowledge base

---
 source/know/concept/dielectric-function/index.md | 65 +++++++++++++++---------
 1 file changed, 41 insertions(+), 24 deletions(-)

(limited to 'source/know/concept/dielectric-function/index.md')

diff --git a/source/know/concept/dielectric-function/index.md b/source/know/concept/dielectric-function/index.md
index 529ce2a..d55cc91 100644
--- a/source/know/concept/dielectric-function/index.md
+++ b/source/know/concept/dielectric-function/index.md
@@ -13,7 +13,8 @@ The **dielectric function** or **relative permittivity** $$\varepsilon_r$$
 is a measure of how strongly a given medium counteracts
 [electric fields](/know/concept/electric-field/) compared to a vacuum.
 Let $$\vb{D}$$ be the applied external field,
-and $$\vb{E}$$ the effective field inside the material:
+and $$\vb{E}$$ the effective field inside the material,
+then $$\varepsilon_r$$ is defined such that:
 
 $$\begin{aligned}
     \boxed{
@@ -23,7 +24,7 @@ $$\begin{aligned}
 
 If $$\varepsilon_r$$ is large, then $$\vb{D}$$ is strongly suppressed,
 because the material's electrons and nuclei move to create an opposing field.
-In order for $$\varepsilon_r$$ to be well defined, we only consider linear media,
+In order for $$\varepsilon_r$$ to be well-defined, we only consider *linear* media,
 where the induced polarization $$\vb{P}$$ is proportional to $$\vb{E}$$.
 
 We would like to find an alternative definition of $$\varepsilon_r$$.
@@ -54,13 +55,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-
-## From induced charge density
-
-A common way to calculate $$\varepsilon_r$$ is from
+In practice, a common way to calculate $$\varepsilon_r$$ is from
 the induced charge density $$\rho_\mathrm{ind}$$,
 i.e. the offset caused by the material's particles responding to the field.
-We start from [Gauss' law](/know/concept/maxwells-equations/) for $$\vb{P}$$:
+Starting from [Gauss' law](/know/concept/maxwells-equations/) for $$\vb{P}$$:
 
 $$\begin{aligned}
     \nabla \cdot \vb{P}
@@ -68,27 +66,27 @@ $$\begin{aligned}
     = - \rho_\mathrm{ind}(\vb{r})
 \end{aligned}$$
 
-This is Poisson's equation, which has the following well-known
-[Fourier transform](/know/concept/fourier-transform/):
+This is Poisson's equation, which has a well-known solution
+via [Fourier transformation](/know/concept/fourier-transform/):
 
 $$\begin{aligned}
     \Phi_\mathrm{ind}(\vb{q})
     = \frac{\rho_\mathrm{ind}(\vb{q})}{\varepsilon_0 |\vb{q}|^2}
-    = V(\vb{q}) \: \rho_\mathrm{ind}(\vb{q})
+    \equiv V(\vb{q}) \: \rho_\mathrm{ind}(\vb{q})
 \end{aligned}$$
 
 Where $$V(\vb{q})$$ represents Coulomb interactions,
-and $$V(0) = 0$$ to ensure overall neutrality:
+and $$V(0) \equiv 0$$ to ensure overall neutrality:
 
 $$\begin{aligned}
     V(\vb{q})
-    = \frac{1}{\varepsilon_0 |\vb{q}|^2}
+    \equiv \frac{1}{\varepsilon_0 |\vb{q}|^2}
     \qquad \implies \qquad
     V(\vb{r} - \vb{r}')
     = \frac{1}{4 \pi \varepsilon_0 |\vb{r} - \vb{r}'|}
 \end{aligned}$$
 
-The [convolution theorem](/know/concept/convolution-theorem/)
+Note that the [convolution theorem](/know/concept/convolution-theorem/)
 then gives us the solution $$\Phi_\mathrm{ind}$$ in the $$\vb{r}$$-domain:
 
 $$\begin{aligned}
@@ -97,37 +95,56 @@ $$\begin{aligned}
     = \int_{-\infty}^\infty V(\vb{r} - \vb{r}') \: \rho_\mathrm{ind}(\vb{r}') \dd{\vb{r}'}
 \end{aligned}$$
 
-To proceed, we need to find an expression for $$\rho_\mathrm{ind}$$
-that is proportional to $$\Phi_\mathrm{tot}$$ or $$\Phi_\mathrm{ext}$$,
+To proceed to calculate $$\varepsilon_r$$ from $$\rho_\mathrm{ind}$$,
+one needs an expression for $$\rho_\mathrm{ind}$$
+that is proportional to $$\Phi_\mathrm{tot}$$ or $$\Phi_\mathrm{ext}$$
 or some linear combination thereof.
-Such an expression must exist for a linear material.
+Such an expression must exist for a linear medium,
+but the details depend on the physics being considered
+and are thus beyond our current scope;
+we will just show the general form of $$\varepsilon_r$$
+once such an expression has been found.
 
-Suppose we can show that $$\rho_\mathrm{ind} = C_\mathrm{ext} \Phi_\mathrm{ext}$$,
-for some $$C_\mathrm{ext}$$, which may depend on $$\vb{q}$$. Then:
+Suppose we know that $$\rho_\mathrm{ind} = c_\mathrm{ext} \Phi_\mathrm{ext}$$
+for some factor $$c_\mathrm{ext}$$, which may depend on $$\vb{q}$$.
+Then, since $$\Phi_\mathrm{tot} = \Phi_\mathrm{ext} \!+\! \Phi_\mathrm{ind}$$,
+we find in the $$\vb{q}$$-domain:
 
 $$\begin{aligned}
     \Phi_\mathrm{tot}
-    = (1 + C_\mathrm{ext} V) \Phi_\mathrm{ext}
+    = (1 + c_\mathrm{ext} V) \Phi_\mathrm{ext}
     \quad \implies \quad
     \boxed{
         \varepsilon_r(\vb{q})
-        = \frac{1}{1 + C_\mathrm{ext}(\vb{q}) V(\vb{q})}
+        = \frac{1}{1 + c_\mathrm{ext}(\vb{q}) V(\vb{q})}
     }
 \end{aligned}$$
 
-Similarly, suppose we can show that $$\rho_\mathrm{ind} = C_\mathrm{tot} \Phi_\mathrm{tot}$$,
-for some quantity $$C_\mathrm{tot}$$, then:
+Likewise, suppose we can instead show that
+$$\rho_\mathrm{ind} = c_\mathrm{tot} \Phi_\mathrm{tot}$$
+for some quantity $$c_\mathrm{tot}$$, then:
 
 $$\begin{aligned}
     \Phi_\mathrm{ext}
-    = (1 - C_\mathrm{tot} V) \Phi_\mathrm{tot}
+    = (1 - c_\mathrm{tot} V) \Phi_\mathrm{tot}
     \quad \implies \quad
     \boxed{
         \varepsilon_r(\vb{q})
-        = 1 - C_\mathrm{tot}(\vb{q}) V(\vb{q})
+        = 1 - c_\mathrm{tot}(\vb{q}) V(\vb{q})
     }
 \end{aligned}$$
 
+And in the unlikely event that an expression of the form
+$$\rho_\mathrm{ind} = c_\mathrm{ext} \Phi_\mathrm{ext} \!+\! c_\mathrm{tot} \Phi_\mathrm{tot}$$ is found:
+
+$$\begin{aligned}
+    (1 - c_\mathrm{tot} V) \Phi_\mathrm{tot}
+    = (1 + c_\mathrm{ext} V) \Phi_\mathrm{ext}
+    \quad \implies \quad
+    \varepsilon_r(\vb{q})
+    = \frac{1 - c_\mathrm{tot}(\vb{q}) V(\vb{q})}{1 + c_\mathrm{ext}(\vb{q}) V(\vb{q})}
+\end{aligned}$$
+
 
 
 ## References
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