---
title: "Dielectric function"
sort_title: "Dielectric function"
date: 2022-01-24
categories:
- Physics
- Electromagnetism
- Quantum mechanics
layout: "concept"
---

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,
then $$\varepsilon_r$$ is defined such that:

$$\begin{aligned}
    \boxed{
        \vb{D} = \varepsilon_0 \varepsilon_r \vb{E}
    }
\end{aligned}$$

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:

$$\begin{aligned}
    \boxed{
        \varepsilon_r
        \equiv \frac{\Phi_\mathrm{ext}}{\Phi_\mathrm{tot}}
    }
\end{aligned}$$

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.
Starting 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 a well-known solution
via [Fourier transformation](/know/concept/fourier-transform/):

$$\begin{aligned}
    \Phi_\mathrm{ind}(\vb{q})
    = \frac{\rho_\mathrm{ind}(\vb{q})}{\varepsilon_0 |\vb{q}|^2}
    \equiv V(\vb{q}) \: \rho_\mathrm{ind}(\vb{q})
\end{aligned}$$

Where $$V(\vb{q})$$ represents Coulomb interactions,
and $$V(0) \equiv 0$$ to ensure overall neutrality:

$$\begin{aligned}
    V(\vb{q})
    \equiv \frac{1}{\varepsilon_0 |\vb{q}|^2}
    \qquad \implies \qquad
    V(\vb{r} - \vb{r}')
    = \frac{1}{4 \pi \varepsilon_0 |\vb{r} - \vb{r}'|}
\end{aligned}$$

Note that 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 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 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 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:

$$\begin{aligned}
    \Phi_\mathrm{tot}
    = (1 + c_\mathrm{ext} V) \Phi_\mathrm{ext}
    \quad \implies \quad
    \boxed{
        \varepsilon_r(\vb{q})
        = \frac{1}{1 + c_\mathrm{ext}(\vb{q}) V(\vb{q})}
    }
\end{aligned}$$

Likewise, suppose we can instead show that
$$\rho_\mathrm{ind} = c_\mathrm{tot} \Phi_\mathrm{tot}$$
for some quantity $$c_\mathrm{tot}$$, then:

$$\begin{aligned}
    \Phi_\mathrm{ext}
    = (1 - c_\mathrm{tot} V) \Phi_\mathrm{tot}
    \quad \implies \quad
    \boxed{
        \varepsilon_r(\vb{q})
        = 1 - c_\mathrm{tot}(\vb{q}) V(\vb{q})
    }
\end{aligned}$$

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
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.