A beam of ions penetrating a plasma perpendicular to a homogeneous magnetic field is investigated. The particle density of the beam may be modulated by varying the intensity of the ion source with the frequency ω. For simplicity, the ions are assumed to move with equal velocity w. The modulation of the beam produces oscillations of the plasma and the ions of the beam will loose energy; therefore it should be possible to trap the injected particles in the plasma. Furthermore the kinetic energy of the trapped particles will be transformed into thermal energy of the plasma 2.
The ion source is assumed to deliver a linear beam (of sufficiently small diameter). Upon being shot into a plasma with a magnetic field the ions will travel along a curved line. In order to treat the problem exactly one would have to solve simultaneously the equations of motion for the plasma coupled with those for the individual ions of the beam. We simplify the problem in that we do not solve the equation of motion of the ions in the beam. Instead we imagine that they are forced to travel in a straight beam.
The problem becomes especially simple if we assume the ion source to be a slit instead of a point source. Then the ions do not all travel along the same straight line, but in parallel straight lines in the plasma. The direction of motion is always perpendicular to the magnetic field. Then we may distinguish two cases of the relation of the field direction to the direction of the plane:
1) the field lines are parallel to the plane of the beam (parallel case, Fig. 1),
2) the field lines are normal to the plane (normal case, Fig. 3).
If the modulating frequency of the beam is small compared to the gyrofrequency of the electrons and compared to the plasma frequency, if the conductivity is infinite, and if the gas pressure in the plasma may be neglected, one obtains for the mean relative energy loss of an ion per cm of path, in the parallel case
(N1 number of particles per cm2 of beam surface of the unmodulated part of the beam, N2 the number of ions per cm2 of the modulated part, ri the classical ion radius, P the square of the ratio of ion velocity to ALFVÉN velocity, Q the square of the ratio of the modulating frequency to the ion gyro-frequency.) In case of P equal to one the energy loss of the beam is infinite. For P < 1 the energy loss will vanish identically. In the normal case the energy loss of each particle is in general of the same order of magnitude, and vanishes identically when P < 1 and ω is greater than the gyrofrequency of the ions. If the ions move with the ALFVÉN velocity, one has a resonance with an infinitely great loss of energy by radiation. In reality of course there will be damping because of the finite conductivity. Assuming N1=N2=107 particles (per cm2 of the beam), the formula mentioned above in case of Q+P—1\P(P—l)½ ≈ 1 will give a relative loss of energy per cm path of about 10-8, which means that a particle has to travel about 6 miles in order to suffer an energy loss of about 1%. This disappointingly small loss leads us to te conclusion that only in cases of resonance measurable effects can be expected. It is certain that the energy loss in the case of the resonance P=1 will exceed the value estimated above by several powers of ten. A detailed discussion of this resonance has to take into account finite conductivity of the plasma.