Biassed Magnet

The negative bias put on the magnet does several jobs, it produces an potential energy well around the magnet to trap the ions, and it emits electrons that initiate the primary ionization channel of the neutral background gas. Under a quite wide parameter regime, from 400mTorr down to 10mTorr, the magnet supports a stable, annular plasma discharge that resembles a Saturnian ring. Like the previous case of electrode injection, the magnet exhibits arcing whenever outgassing is significant. Thus we conclude that there is nothing peculiar about the electrode electric fields that lead to arcing, since we obtain arcing without any electrode in the system.

Under a subset of voltages and magnet geometries, there appears to be a gap between the annular plasma ring and the surface of the magnet. It is not immediately apparent why there should be any space at all between the equatorially trapped plasma and the surface of the magnet. Certainly the neutral gas in the chamber makes the plasma diffusive, and should permit ion access to the magnet at all field strengths or aspect ratios. However we can understand this gap if we write a diffusion equation in radius, with local plasma density being the competition between transport, production and loss.

$$df/dt = D d^2 f/dr^2 - f/\Lambda + f/S_i + S_e$$ (3)

where $D$ is the diffusion coefficient, $r$ is the radius, $f$ is the phase space density, $\Lambda$ is the loss rate and $S_i$ the source rate due to ion collisions with neutrals, whereas $S_e$ is the source rate due to electron stripping of neutrals. Note that the diffusion coefficient arising from collisions with neutrals decreases both as the magnet field increases and as the ions lose energy because the ion gyroradii decrease, causing the average spatial step size in a collision to decrease.

Thus a gap at small radius occurs when the loss rate in this region exceeds the source rate. The electron source can only occur if the electrons, presumeably emitted from the surface of the magnet and streaming along the field lines, gain enough energy to strip an atom before they cross the field-line equator and climb back uphill. Since the electric potential is determined by the biassed magnet and the grounded chamber, the potential surfaces are roughly symmetric, and the largest field-aligned electric fields are at high magnetic latitudes that depart almost radially from the magnet. Thus we would expect that the innermost field-lines may have insufficient potential gradient to ionize atoms by electron impact, which may account for the lack of emission close to the magnet. Note, however, that if the space charge of the annulus generates a field-aligned potential, then the annulus will accelerate the electrons increasing $S_e$, and exhibiting the properties of a positive feedback mechanism.

In addition, the ion source term occurs as the ions gain (lose) energy adiabatically by diffusing toward (away from) the magnet. At these eV energies, the ionization cross section increases with energy, permitting the ions to multiply or at least emit more radiation. Simultaneously, the collisional transport rate decreases due to the stronger B-field. Therefore one would expect the plasma to become brighter as it moved inward until suddenly losses grew exponentially and produced a sharp inner edge. Thus the interplay between collisional losses and diffusive transport sources determines the position of the inner edge of the ion torus in the data and under specific conditions, generate the dark separation of the plasma torus from the main magnet.

That this model gives the correct qualitative response can be seen by observing that the plasma torus diameter shrinks as the magnet bias is made more negative. Increasing the negative voltage on the magnet should increase the diffusive energy gain of the ions, as well as the diffusive transport rate, the field-aligned potential gradients, and hence the source term $S_e$, so that the glowing plasma torus appears to approach the magnet and even "collide" with the surface, given enough potential.