Next: Normalization scheme
Up: Particle-in-cell codes
Previous: Particle-in-cell codes
Consider an unmagnetized, uniform, 1-dimensional plasma consisting of
electrons and
unit-charged
ions.
Now, ions are much more massive than electrons.
Hence, on short time-scales, we can treat the ions as a static
neutralizing background, and only consider the motion of the electrons. Let
be the
-coordinate of the
th electron. The equations of motion of the
th electron are written:
where
is the magnitude of the electron charge,
the electron mass, and
the
-component
of the electric field-strength at position
. Now, the electric field-strength can be
expressed in terms of an electric
potential:
![\begin{displaymath}
E(x) = - \frac{d\phi(x)}{dx}.
\end{displaymath}](img1088.png) |
(291) |
Furthermore, from the Poisson-Maxwell equation, we have
![\begin{displaymath}
\frac{d^2\phi(x)}{dx^2} = -\frac{e}{\epsilon_0}\,\{n_0- n(x)\},
\end{displaymath}](img1089.png) |
(292) |
where
is the permittivity of free-space,
the electron number
density (i.e.,
is the number of electrons in the interval
to
),
and
the uniform ion number density. Of course, the average value of
is
equal to
, since there are equal numbers of ions and electrons.
Let us consider an initial electron distribution function consisting of two counter-propagating
Maxwellian beams of mean speed
and thermal spread
: i.e.,
![\begin{displaymath}
f(x,v) = \frac{n_0}{2}\left\{\frac{1}{\sqrt{2\,\pi}\,v_{th}}...
...,\pi}\,v_{th}}\,{\rm e}^{-(v+v_b)^2/2\,v_{th}^{\,2}}
\right\}.
\end{displaymath}](img1096.png) |
(293) |
Here,
is the number of electrons between
and
with velocities
in the range
to
. Of course,
.
The beam temperature
is related to the thermal velocity via
,
where
is the Boltzmann constant. It is well-known that if
is significantly
larger than
then the above distribution is unstable to a plasma instability
called the two-stream instability.38 Let us investigate this instability numerically.
Next: Normalization scheme
Up: Particle-in-cell codes
Previous: Particle-in-cell codes
Richard Fitzpatrick
2006-03-29