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authorPrefetch2023-01-03 19:48:17 +0100
committerPrefetch2023-01-03 19:48:27 +0100
commitaeacfca5aea5df7c107cf0c12e72ab5d496c96e1 (patch)
tree6d89742cdf29fe0ad46590586858396a4c560fca /source/know/concept/dielectric-function
parentb1a9b1b9b2f04efd6dc39bd2a02c544d34d1259c (diff)
More improvements to knowledge base
Diffstat (limited to 'source/know/concept/dielectric-function')
-rw-r--r--source/know/concept/dielectric-function/index.md65
1 files changed, 41 insertions, 24 deletions
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