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Consider the solution of Poisson's equation:

(303) 
where
. Note that in normalized
units. Let
and
.
We can write
which automatically satisfies the periodic boundary conditions and .
Note that
, since
. The other are obtained
from

(306) 
for .
The Fourier transformed version of Poisson's equation yields

(307) 
and

(308) 
for , where
. Finally,

(309) 
for to , which ensures that the remain real.
The discretized version of Eq. (297) is

(310) 
Of course, and are special cases which can be resolved using the periodic
boundary conditions.
Finally, suppose that the coordinate of the th electron lies between the th and th
gridpoints: i.e.,
. We can then use linear interpolation to
evaluate the electric field seen by the th electron:

(311) 
Next: An example 1D PIC
Up: Particleincell codes
Previous: Evaluation of electron number
Richard Fitzpatrick
20060329