- M.G.Haines, "Plasma Containment in Cusp-Shaped Magnetic Fields" (Review Paper), Nuclear Fusion 17 4(1977), pp.811-858
The theme of the sheath thickness appears throughout the paper. It is surprisingly complex. For our purposes (assuming the more recent literature does not significantly change the picture), I would say the standard answer, if you do everything right, is the hybrid Larmor radius. Bussard's assumption that the appropriate scale length is the electron gyroradius seems unsupported, but might be taken as a firm lower limit, that would require a small miracle to achieve. (More on problems with Bussard's model to follow.) In the words of Haines,Up to a couple years ago experiments were indicating a hole size comparable with an ion Larmor radius, a_i, in contrast with ideal sheath theory which predicts a sheath thickness of an electron Larmor radius, a_e. ... However, recently some differently initiated experiments have established a hole size much smaller than the ion Larmor radius and approximately (a_e*a_i)^1/2 in size. No theoretical model has been advanced in the literature, but in this review paper a fairly satisfactory model gives this result. (p.811)
[2 a_e] is the thinnest sheath that can be expected in a cusp experiment. (p.814)
However, because the radius r_0(z) of the beta = 1 plasma in a spindle cusp varies from zero at the point cusp radius to R at the ring cusp, and because the magnetic flux in the sheath is a constant, it follows that the sheath thickness delta(z) is approximately given by area conservationbecause for a beta = 1 plasma with a uniform pressure the magnetic field just outside the sheath is constant in magnitude. Here delta_p is the radius of the effective circular hole at the point cusp, and delta_L is the half-width of the effective sheath at the ring (or line) cusp.
- delta_p^2 = 2 delta(z) r_0(z) approx 2 delta_L R
This is far and away the most interesting thing I learned, although it is obvious once it has been pointed out. I have always been claiming that we can ignore the point cusps because the hole from the line cusps will be so much bigger. But actually we could ignore the line cusps and we would still have comparable losses from the point cusps.... the loss rates from the two point cusps equaled that from the line cusp. (p.829)
Thus Bussards's statement in the Valencia paper (p.9), "At this condition, the electrons inside the quasi-sphere “see“ small exit holes on the B cusp axes, whose size is 1.5-2 times their gyro radius at that energy and field strength.", is patently false, even if we assume that a miracle makes the sheath thickness a_e instead of (a_e*a_i)^1/2. Say we have a point cusp with a loss area of 1 mm^2. If we consider a circle around the cusp with 100 mm radius, then the sheath there can only be (1 mm^2)/(2*pi*100 mm) = 1.6 mu. There is no physics in hell that can keep the sheath thickness a thousandth of an electron gyroradius. I see this as confirmation - with a wholly different argument and much more robust - of my insistence on calculating losses based on line cusps.
Moving on to (in the language of polywells) recirculation, Haines states
I suspected something like this but wasn't quite sure. Haines leaves little doubt that at least one of the species (electrons or ions) will be lost at a rate comparable to the unplugged cusp loss rate.The use of electrostatic forces through grids or electrodes placed at the ring cusp and point cusp regions has, probably rightly, received little attention. ... At first sight the one-dimensional model in Fig. 34 in which electrostatic barriers of height Phi_i > kT_i/e followed by |-Phi_e| > kT_e/e to reflect ion and electrons, respectively, looks attractive.... Two consequences arise; first, the magnitude of E required to balance a pressure of n ~ 4 X 10^14 cm^-3 and T_e ~ T_i ~ 2 X 10^8 K is 6.8 MV/cm which is prohibitive; and second, ... the plasma must be able to support an electric field component parallel to the magnetic field of this magnitude. Clearly large currents and associated power losses and electrode loading would occur. (p.846)
He proceeds to consider scaling to a (D-T) power reactor and concludes,
Finally, he makes an intriguing comment on multiple point cusps:that a spindle cusp reactor is not practical, but a long [ > 250 m!] cusp or cusp-ended theta pinch is more feasible particularly if a sheath thickness of 2(a_e*a_i)^1/2 is confirmed.
These comments sound inconsistent to me, but maybe one of you would like to look more closely the the SM. The latest journal article referenced is Sadowsky, M., Rev.Sci.Instrum. 40 (1969) 1545.Sadowski and his co-workers have been studying plasma containment in a spherical multipole (SM) magnetic configuration for some years. It is hoped that since line cusps have been eliminated in favour of point cusps only (of large number) that plasma losses will be much reduced whilst still keeping the favourable minimum-B properties of the trap.
Some where, some time, some thread I estimated reactor size based on cusp losses, but I can't find it now. Can anybody help me? I thought I landed at something bloody big, but not out of this world. (Assuming D-T fuel cycle, of course.) I would like to check if these new insights change anything.