# Dielectrics¶

In dielectric materials (i.e. insulators), unlike conductive materials, charges in the molecules or atoms cannot freely move but will shift slightly upon application of an electric field, thereby causing the molecules to become polarized. The cumulative effect of the dipole moments of these polarized molecules or atoms alters the electric field.

## Theory¶

The following is a brief summary of the equations and definitions pertaining to electric fields in the presence of dielectrics. For more details and general background, see standard texts like Griffiths and Wikipedia:Dielectric.

*Gauss’s Law* in differential form, which is one of the
*Maxwell Equations* and readily derives the
*Poisson Equation*, is .
In the presence of insulative materials (dielectrics), the charge density
is a sum of bound charge
() from the polarization of the dielectric
and free charge () from other sources (e.g. electrode surface
charges and space-charge). Since
(where **P** is the *polarization*, i.e. the dipole moment per unit volume, of
the dielectric), Gauss’
law may alternately be written in terms of free charge as
, where we define
as the *electric displacement*.
The advantage is that is more likely known than ,
so we can use the *Poisson Solver in SIMION* to solve for given
. In some cases
we can then derive **E** from **D**.

In a *linear dielectric*, ,
where is the *electric susceptibility* (and is dimensionless). We may
alternately express
as
,
where is *permittivity*,
and is the *relative permittivity*,
also called the (relative) *dielectric constant*. In a *non-linear dielectric* (but still
isotropic), is a function of the
magnitude of the local electric field. In an *anisotropic dielectric*,
**P** and **E** are not necessarily parallel, and
are tensors rather than scalars.

Putting this together and using , we have
for a linear dielectric, ,
which is a form of the *Poisson Equation*, which SIMION can solve via the
*Poisson Solver in SIMION*. This may be solved iteratively, in a self-consistent manner,
in the case of a non-linear, isotropic dielectric.

## SIMION Specific - Scope¶

SIMION 8.1.1.0 can calculate electric fields in the presence of dielectrics
(via the *Poisson Solver in SIMION* Refine function).
It supports linear anisotropic dielectrics, where the relative dielectric constant varies
as a function of position. An example is also provided of the more challenging
problem of handling non-linear isotropic dielectrics via an iterative (self-consistent)
approach. Anistropic dielectrics are not currently implemented. The field solving
can incorporate both dielectrics and space-charge effects at the same positions
(e.g. a charged dielectric), and *anistropically scaled grid cells*
(not to be confused with anisotropic dielectrics) are supported with dielectrics.

The related problem of solving current densities in the presence of materials of varying conductivity
is also supported (see *Current Density*).

In some cases, the dielectrics are far enough removed from your electrode surfaces that they have little effect on electric fields in the region where particles fly, so you can ignore the dielctrics in the simulation, such as by treating them as non-electrode points. For example, SIMION Example: einzel, there would actually be insulative spacers between the cylinder electrodes, but we ignore those in the model.

Note

This page is abridged from the full SIMION 8.1.1 “Supplemental Documentation” (Help file). The following additional sections can be found in the full version of this page accessible via the “Help > Supplemental Documentation” menu in SIMION 8.1.1:- Modeling
- Fast Adjust PA# files
- Types of dielectrics: Non-linear dielectrics
- Types of dielectrics: Isotropic v.s. anisotropic dielectrics
- With space-charge
- Dielectric charging by a beam (and avoidance of)
- Obtaining highest field accuracy on dielectric boundaries

## See Also¶

*Poisson Solver in SIMION**Floating Conductor*- Engineering with non-linear dielectrics X. Qi, Z. Zheng, S. Boggs. DEIS. Nov/Dec 2004 - Vol. 20, No. 6. 27-34.