From 8a9fb5fef2a97af3274290e512816e1a4cac0c02 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Mon, 24 Jan 2022 19:29:00 +0100
Subject: Rewrite "Lindhard function", split off "dielectric function"

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+---
+title: "Dielectric function"
+firstLetter: "D"
+publishDate: 2022-01-24
+categories:
+- Physics
+- Electromagnetism
+- Quantum mechanics
+
+date: 2022-01-20T22:04:13+01:00
+draft: false
+markup: pandoc
+---
+
+# Dielectric function
+
+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:
+
+$$\begin{aligned}
+    \boxed{
+        \vb{D} = \varepsilon_0 \varepsilon_r \vb{E}
+    }
+\end{aligned}$$
+
+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,
+where the induced polarization $\vb{P}$ is proportional to $\vb{E}$.
+
+We would like to find an alternative definition of $\varepsilon_r$.
+Consider that the usual electric fields $\vb{E}$, $\vb{D}$, and $\vb{P}$
+can each be written as the gradient of an electrostatic potential like so,
+where $\Phi_\mathrm{tot}$, $\Phi_\mathrm{ext}$ and $\Phi_\mathrm{ind}$
+are the total, external and induced potentials, respectively:
+
+$$\begin{aligned}
+    \vb{E}
+    = -\nabla \Phi_\mathrm{tot}
+    \qquad \qquad
+    \vb{D}
+    = - \varepsilon_0 \nabla \Phi_\mathrm{ext}
+    \qquad \qquad
+    \vb{P}
+    = \varepsilon_0 \nabla \Phi_\mathrm{ind}
+\end{aligned}$$
+
+Such that $\Phi_\mathrm{tot} = \Phi_\mathrm{ext} + \Phi_\mathrm{ind}$.
+Inserting this into $\vb{D} = \varepsilon_0 \varepsilon_r \vb{E}$
+then suggests defining:
+
+$$\begin{aligned}
+    \boxed{
+        \varepsilon_r
+        \equiv \frac{\Phi_\mathrm{ext}}{\Phi_\mathrm{tot}}
+    }
+\end{aligned}$$
+
+
+## From induced charge density
+
+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 $\vb{P}$:
+
+$$\begin{aligned}
+    \nabla \cdot \vb{P}
+    = \varepsilon_0 \nabla^2 \Phi_\mathrm{ind}(\vb{r})
+    = - \rho_\mathrm{ind}(\vb{r})
+\end{aligned}$$
+
+This is Poisson's equation, which has the following well-known
+[Fourier transform](/know/concept/fourier-transform/):
+
+$$\begin{aligned}
+    \Phi_\mathrm{ind}(\vb{q})
+    = \frac{\rho_\mathrm{ind}(\vb{q})}{\varepsilon_0 |\vb{q}|^2}
+    = V(\vb{q}) \: \rho_\mathrm{ind}(\vb{q})
+\end{aligned}$$
+
+Where $V(\vb{q})$ represents Coulomb interactions,
+and $V(0) = 0$ to ensure overall neutrality:
+
+$$\begin{aligned}
+    V(\vb{q})
+    = \frac{1}{\varepsilon_0 |\vb{q}|^2}
+    \qquad \implies \qquad
+    V(\vb{r} - \vb{r}')
+    = \frac{1}{4 \pi \varepsilon_0 |\vb{r} - \vb{r}'|}
+\end{aligned}$$
+
+The [convolution theorem](/know/concept/convolution-theorem/)
+then gives us the solution $\Phi_\mathrm{ind}$ in the $\vb{r}$-domain:
+
+$$\begin{aligned}
+    \Phi_\mathrm{ind}(\vb{r})
+    = (V * \rho_\mathrm{ind})(\vb{r})
+    = \int_{-\infty}^\infty V(\vb{r} - \vb{r}') \: \rho_\mathrm{ind}(\vb{r}') \dd{\vb{r}'}
+\end{aligned}$$
+
+To proceed, we need to find an expression for $\rho_\mathrm{ind}$
+that is proportional to $\Phi_\mathrm{tot}$ or $\Phi_\mathrm{ext}$,
+or some linear combination thereof.
+Such an expression must exist for a linear material.
+
+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:
+
+$$\begin{aligned}
+    \Phi_\mathrm{tot}
+    = (1 + C_\mathrm{ext} V) \Phi_\mathrm{ext}
+    \quad \implies \quad
+    \boxed{
+        \varepsilon_r(\vb{q})
+        = \frac{1}{1 + C_\mathrm{ext}(\vb{q}) V(\vb{q})}
+    }
+\end{aligned}$$
+
+Similarly, suppose we can show that $\rho_\mathrm{ind} = C_\mathrm{tot} \Phi_\mathrm{tot}$,
+for some quantity $C_\mathrm{tot}$, then:
+
+$$\begin{aligned}
+    \Phi_\mathrm{ext}
+    = (1 - C_\mathrm{tot} V) \Phi_\mathrm{tot}
+    \quad \implies \quad
+    \boxed{
+        \varepsilon_r(\vb{q})
+        = 1 - C_\mathrm{tot}(\vb{q}) V(\vb{q})
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  H. Bruus, K. Flensberg,
+    *Many-body quantum theory in condensed matter physics*,
+    2016, Oxford.
+2.  M. Fox,
+    *Optical properties of solids*, 2nd edition,
+    Oxford.
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