The dielectric function or relative permittivity
is a measure of how strongly a given medium counteracts
electric fields compared to a vacuum.
Let be the applied external field,
and the effective field inside the material,
then is defined such that:
If is large, then is strongly suppressed,
because the material’s electrons and nuclei move to create an opposing field.
In order for to be well-defined, we only consider linear media,
where the induced polarization is proportional to .
We would like to find an alternative definition of .
Consider that the usual electric fields , , and
can each be written as the gradient of an electrostatic potential like so,
where , and
are the total, external and induced potentials, respectively:
Such that .
Inserting this into
then suggests defining:
In practice, a common way to calculate is from
the induced charge density ,
i.e. the offset caused by the material’s particles responding to the field.
Starting from Gauss’ law for :
This is Poisson’s equation, which has a well-known solution
via Fourier transformation:
Where represents Coulomb interactions,
and to ensure overall neutrality:
Note that the convolution theorem
then gives us the solution in the -domain:
To proceed to calculate from ,
one needs an expression for
that is proportional to or
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
once such an expression has been found.
Suppose we know that
for some factor , which may depend on .
Then, since ,
we find in the -domain:
Likewise, suppose we can instead show that
for some quantity , then:
And in the unlikely event that an expression of the form
- H. Bruus, K. Flensberg,
Many-body quantum theory in condensed matter physics,
- M. Fox,
Optical properties of solids, 2nd edition,