Some of the profusion of new and almost new grid shapes being offered seems to be deviating from the limits expressed by Bussard. Some seem variations on the baseball seam theme and Bussard pointed out the inadequacy of that. See page 4 of:
http://prometheusfusionperfection.files ... tent08.pdf
To evangelize my understanding ( and thereby offering a chance to correct any of my misunderstanding) I will try to describe the cusp geometries from a volume perspective.
With an opposing two ring magnet Penning type trap there are two types of cusps. The point cusps that are relatively small cusp escape areas, and line or equatorial cusps that consist of much more area.
To keep things simple I will use squares to compare dimensions, as opposed to spheres and complex shapes.
In two dimensions an opposing magnet penning trap and a Polywell look similar. It is the third dimension that changes the picture.
Take a penning trap with two magnets 100 cm in diameter and separated by 1 cm.
Assume the equatorial or line cusp is 1 cm wide. Assume that the confining magnetic field extends 1/2 the distance to the center axis. So the line cusp would have a loss area of 1 cm * the distance around the square, or 400 cm^2. The confined volume is 1 cm * 50 cm per side * 1 cm wide or 2500 cm^3. Thus the loss area/ containment volume is 400 cm^2 / 1500 cm^3 or ~ 1cm^2 of loss area per 6 cm^3 of confined volume. This compares to the lower end of what the 2008 patent application referred to as mirror reflection confinement. If you separate the two magnets more the internal volume increases, but the opposing magnetic field intersection weakens, so the loss area of the line cusps increases, in what I will assume is a proportional manor so this ration does not change.
In a Polywell with it's magnets arranged evenly around a sphere (cube in my examples) the internal volume is increased, but the line cusp widths stay about the same, because the cusp formed between the magnets is dependent on the strength, but more importantly, on the separation of the magnets that produce the cusps and these are unchanged in the Polywell, irregardless of the overall size of the machine. The important point here is not the shape of the magnets, but that that magnets on opposite sides of a line cusp are kept as close as possible.
Take the penning trap example above. Add the 4 extra magnets. Now you have 6 magnets forming a cube, 100 cm on a side, that would have a containment volume of 50 cm * 50 cm * 50cm, or 125,000 cm^3. Because of the extra magnets the length of the total line cusps would increase by a factor of ~3 ( I think). But the width of these line cusps would remain ~ 1 cm. So the total line cusp loss areas would increase to ~ 400 cm^2 * 3 or 1200 cm^2.
The resulting loss area/ containment volume is 1200 cm^2 / 125,000 cm^3 or 1/104. This is ~ twice what the has been referred to as cusp confinement. I think the corner areas of these line cusps would account for the increased losses, because the magnet edges are separated further from each other so the magnetic fields are weaker and the corresponding line cusp widths are wider. I dislike referring to these areas as 'point cusps' or virtual point cusps, as they are actually the intersection of three (or more) widening line cusps. Having sharper corners would lessen this separation but there are limits imposed by arcing concerns, and how acutely you can bend copper or superconducting wires, etc. How this perspective would affect ideas of using squares rotated 90 degrees so that there are much larger virtual point cusps is uncertain, but it seems they would significantly compromise the containment based solely on the above arguments. though still much less than trying to contain the same volume with only two opposing magnets.
So, even without a Wiffleball, the Polywell has an ~ 10 fold or greater containment advantage over opposing magnet penning type mirror machines.
The Wiffleball is claimed to pinch the cusps so that they are much narrower. Or, perhaps it would be more accurate to say that the collection areas that feed into the cusps are pinched. The actual cusp width itself would be unchanged. For my example I will assume this effect decreases the above line cusps to ~ 0.1 cm widths. That would decrease the total line cusp loss area to ~ 120 cm^2. So the loss to containment volume ratio would be ~ 1/1000. This approaches the claimed performance of Wiffleball containment.
The important consideration is that the line cusps are between adjacent magnets and the space separating the magnets in a Polywell is minimized irregardless of the overall size of the machine. This allows the volume (or density) to be rapidly increased with only modest increases in loss area. The limits to this scaling is either the mechanical strength necessary to hold the increasing strong magnets together, or the background density that leads to arcing and loss of the potential well. I believe this second limit is reached before the other becomes significant. Thus the top end density is the background arc limiting density (assume ~ 10^-6 atm) times the Wiffleball trapping effect (assume ~ 1000X), giving the claimed obtainable plasma densities of ~ 10^19 particles / M^3 (~ 10^-6 atm) * 1000, or 10^22 particles / M^3.
True point cusps have been ignored because I understand their contribution to losses are much smaller. Though a Polywell has more point cusps compared to an opposing magnet penning trap, the effect on the overall losses described above would be tiny.
Dan Tibbets
To error is human... and I'm very human.
