The linear theory has been used by several authors to analyze the
electrostatic electron cyclotron instability. In [3], Chu
considered a neutral, uniform and infinite plasma approximation.
Assuming an uniform external magnetic field 
and a
cold electron beam, represented by the distribution
, he found an approximated linear growth rate for the
cyclotron harmonic, 
, given by
where 
, 
are the non-relativistic
electron cyclotron and plasma frequency respectively,
is the Larmor radius,
is the initial perpendicular velocity normalized
to the velocity of light c,
is the initial
relativistic factor,
is the perpendicular component of the
wave vector and 
is the order of the Bessel function.
The more elaborate model of a perfectly aligned gyrocenter beam has been considered in [10, 11]. The electron density across the magnetic field of such a beam, assuming that it is cold and centered at x=0, has the following dependence
where 
is the beam density averaged over the beam
diameter 
. The growth rate at the cyclotron harmonic
is then
where 
is the averaged non-relativistic
plasma frequency. Notice that both growth rates as given in
Eqs. (6) and (8) exhibit the same dependence in
the perpendicular velocity and the density, together
with a weak decrease for increasing harmonic numbers n.
A one dimensional electrostatic simulation can readily be done for
the model used in the analytical results presented above, by considering
a slab with the Cartesian phase space coordinates 
of
Eq. (1).
The matrix 
and the charge array 
of the discretized Poisson equation
(4a) now simplify to
where 
is the area of the simulated beam slab. The simulation
starts by loading
a cold and perfectly-aligned guiding-center electron distribution.
A small perturbation
is then imposed on the uniformly distributed
gyro-angles 
, to excite the
cyclotron harmonic mode. The growth rates calculated from the 1D slab
simulation are compared to the analytical predictions in
Fig. 1, showing good agreement, especially in the low
density region: the small discrepancy can be attributed to the low
density assumption used to obtain Eq. (8).
For high harmonic numbers, the simulation growth rates are slightly
smaller than the theoretical estimates, as can be seen in Fig. 2.
