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Diffstat (limited to 'source/know/concept/dielectric-function')
| -rw-r--r-- | source/know/concept/dielectric-function/index.md | 65 |
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 |