Recirculation redux

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bsmythe
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Recirculation redux

Post by bsmythe »

I was reading Art Carlson's all that can go wrong with recirculation thread and had a few questions / observations. The biggest one is:
is the current understanding of recirculation now that the electrons do not go out one cusp and back into a different one, ie, they go back in the same one they go out?
While the thread definitely went in that direction, there did not seem to be any definitive statement to that effect (say by Dr Nebel). Also, if this is the current thinking, is there still a need for circular cross section coils and coil casings? Wouldn't square ones be fine as long as the corners didn't protrude into the loss cones of the cusps?

I also had two observations that I did not see mentioned in the thread (possibly because they are either wrong or not significant effects).

For one, wouldn't the cusp loss cones be much bigger from the outside looking in than from the inside looking out when operating at high Beta. Under those conditions, you have diamagnetic effects of the plasma pushing the field out from the inside and closing up the cusps but there is no such effect on the outside. So wouldn't the holes an electron sees be smaller going out than those it sees going back in.

The second observation is that there is also an electron beam being injected into each cusp which will create a magnetic field perpendicular to the field created by the coils. The field would have the effect of pushing an electron going towards the core into the cusp where the beam is aimed (i.e. something like the pinch effect). Of course it would have exactly the opposite effect on electrons as they leave the cusps before they are turned around by the grid field. I still think, under certain conditions, it could have a net effect. My reasoning is as follows. At the exact center of the beam, its field would cancel itself out and there would be no effect on an electron traveling along this line. Since the beam has a finite width, this area of effectively zero field is also finite. An electron traveling straight out of a cusp would thus go a finite distance before E x B drift, space charge effects and other drift terms would sweep it out of the area of effective zero field and where the beam's field would cause it to diverge. If the electron goes a distance X before being turned around by the the coil's electric field then it would go some distance A before leaving the effective zero field region and being acted on by the force from the e beam. From there on, it would be acted on by beam's field for a distance of X -A. On it's return path when the electron reached a distance of X-A it would still be outside of the effective zero field area because of the various drift terms and thus would continue to be acted on by the by the e-beam's magnetic field, thus canceling out some of the drift. For this to be a significant effect, A would have to be a significant fraction of X. Also, the net current of the e-beam would need to be significant. If all of the electrons leave the cusps and then hit the chamber wall then the net current would be zero and the effect would be zero so the losses to the wall vs other loss channels would have to be below some level in order to have a significant effect. Can anyone tell me if this sounds reasonable or am I missing something?

Bill

drmike
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Re: Recirculation redux

Post by drmike »

bsmythe wrote:I was reading Art Carlson's all that can go wrong with recirculation thread and had a few questions / observations. The biggest one is:
is the current understanding of recirculation now that the electrons do not go out one cusp and back into a different one, ie, they go back in the same one they go out?
I don't think we have good enough models yet to say either way. It doesn't really matter either - you can't tell on electron from another.
While the thread definitely went in that direction, there did not seem to be any definitive statement to that effect (say by Dr Nebel). Also, if this is the current thinking, is there still a need for circular cross section coils and coil casings? Wouldn't square ones be fine as long as the corners didn't protrude into the loss cones of the cusps?
You want round surfaces to make the electric field smooth. If you have sharp corners on the magnets, and that surface is charged, you will enhance arcing.

I also had two observations that I did not see mentioned in the thread (possibly because they are either wrong or not significant effects).

For one, wouldn't the cusp loss cones be much bigger from the outside looking in than from the inside looking out when operating at high Beta. Under those conditions, you have diamagnetic effects of the plasma pushing the field out from the inside and closing up the cusps but there is no such effect on the outside. So wouldn't the holes an electron sees be smaller going out than those it sees going back in.
Exactly right. That is the whole point of the Wiffle-ball effect. By making the loss cone tiny from the inside, you enhance overall confinement. You also make it easier for lost electrons to get back in, which is another positive effect.
The second observation is that there is also an electron beam being injected into each cusp which will create a magnetic field perpendicular to the field created by the coils. The field would have the effect of pushing an electron going towards the core into the cusp where the beam is aimed (i.e. something like the pinch effect). Of course it would have exactly the opposite effect on electrons as they leave the cusps before they are turned around by the grid field. I still think, under certain conditions, it could have a net effect. My reasoning is as follows. At the exact center of the beam, its field would cancel itself out and there would be no effect on an electron traveling along this line. Since the beam has a finite width, this area of effectively zero field is also finite. An electron traveling straight out of a cusp would thus go a finite distance before E x B drift, space charge effects and other drift terms would sweep it out of the area of effective zero field and where the beam's field would cause it to diverge. If the electron goes a distance X before being turned around by the the coil's electric field then it would go some distance A before leaving the effective zero field region and being acted on by the force from the e beam. From there on, it would be acted on by beam's field for a distance of X -A. On it's return path when the electron reached a distance of X-A it would still be outside of the effective zero field area because of the various drift terms and thus would continue to be acted on by the by the e-beam's magnetic field, thus canceling out some of the drift. For this to be a significant effect, A would have to be a significant fraction of X. Also, the net current of the e-beam would need to be significant. If all of the electrons leave the cusps and then hit the chamber wall then the net current would be zero and the effect would be zero so the losses to the wall vs other loss channels would have to be below some level in order to have a significant effect. Can anyone tell me if this sounds reasonable or am I missing something?

Bill
It would be really interesting to try to model that. The collective effects get complicated and it is hard to predict what will happen. If you get two beams running into each other in a plasma you get the "two stream instability" which can destroy both beams. With a magnetic field present it will be even more complicated. Certainly an interesting physics problem!

D Tibbets
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Post by D Tibbets »

Some speculation from a layman's point of view.

The advantage of WB6 over WB4 is mostly the round geometry (and the spacing between the coils). And this is what Dr Bussard claimed made all the difference. I suppose a square cross section magnet would be easier to build, and would have a sharper turn(?) of the magnetic field lines entering the cusps, but once beyond this point the field lines from opposing magnets would be more parallel and thus have less decelerating effects on the electrons- ie square coils might have narrower throats but have smaller stopping power. Apparently (or not) this is less desirable. I'm guessing the 'Wiffleball' effect is similar- tighter, narrower cusps / targets for the electrons to hit before they can escape. Also, the Beta 1(?) conditions must be very close to the coils at these field strengths so the critical lines would intercept the metal case in a square coil before the maximal stoping power was achieved (assuming this was the whole story and there was no recirculation). So, both conditions (Wiffleball effect and coil geometry have synergistic effects that add up to the claimed critical confinement. Would a square, or oval or trapezoidal, etc cross section have any benefits over a round cross section coil provided the magnetic field strength was high enough to keep the electrons away from the innermost field lines that intercept the coil casing? And, even in round crossection magnetic coils how much stronger does the field have to be to allow for the larger diameter of over lying layers needed for cooling and insulating the magrids of a power producing machine?

I guess that recirculation is real, based on Bussard et el's claims that the round geometry with appropriate spacing makes such a difference. In my (very) modest opinion I do not believe electrons can recirculate through the same cusp. If the electron can fight it's way past the area where the magnetic field lines are most curved- area just before or at the nearest approach to two adjacent round magnetic fields (?)- they will not reverse their direction unless they reach another area of greater magnetic field curvature. Based on my visual understanding of the Polywell, most of the electrons that escape, do so though the edge- linear cusps. Because of the milder field line curvature they would then have a fair chance of 'orbiting ' back through the nearest center cusp (center of one of the round donuts). I would not expect an electron escaping through the center cusp to recirculate through a corner cusp( it would either have enough kinetic energy to leave the field line and fly to the vacuum vessel wall (assuming the field line does't reach out that far) or it would reverse and travel back through the center cusp (perhaps the same result in the end)). The eliptical orbit could reach the vacuum vessel wall if it is not far enough away, but the positive charge on the magrid would help to keep the electron in a tighter orbit (lower field line), thus increasing the chances of it recirculating. Once back inside the magrid the electron could possibly recirculate again (orbit) if it was in an isolated system, but I suspect this would be rare because the electron will probably interact with other electrons or ions and change its path, in a sense resetting the confinement status. Without knowing weather any of this makes sense, I speculate that some might say that if the electron has enough initial speed to escape through the cusp, it has enough speed to 'sling shot' away from the magnetic field line, thereby escaping to the vessel wall. But, the positively charged magrid might dampen this(?). And, I understand that once a charged particle is trapped on a magnetic field line it will stay on that line- orbit if a monopole or a circular field like in the Polywell, or oscillate back and forth, until an external force displaces it.

I don't follow your last paragraph well (original poster), but if a tightly focused/collimated electron gun shot it's electrons towards an opposing cusp then I expect that the vast majority of electrons would exit through that cusp. But if the electron gun's electron output was not tightly focused then only a corresponding percentage of the electrons would exit on the first pass. Also, if more than one electron gun is used then the beams would intersect in the center. The increased electrons in this area would repel each other and further defocus the beams. And, once ions are added the conditions would become even more chaotic.

The location of electrons inside the Pollywell once they have completed one pass confuses me (every thing else is of course perfectly clear :) ). The high energy electron is injected and immediately decelerates as it approaches the center due to mutual electron repulsion and the effects of the positively charged magrid behind it. Once past the center it start accelerating towards the now closer magrid on the opposite side. If it is aimed precisely at the cusp it exits, otherwise it interacts with the magnetic field and it's forward motion is translated into lateral motion. If it is thereby contained, I would not expect it to move back towards the center as it is attracted to the positively charged grid. As it moves back and forth along a magnetic field line, it decelerates as it approaches the cusps (greater curvature of the field lines) and speeds up as it leaves these areas. Because of this I would expect it to spend more time near the cusps, so there would be greater numbers of electrons near the cusps (would this effect electron or ion containment?). This would end up with a single shot of electrons passing through the center of the 'well', but with time the electrons would pile up in a shell near the magnets, till they managed to drain through the cusps through up scattering, etc. I'm guessing that the injection of new electrons have to be great enough to dominate this effect (if real). The presence of positively charged ions would complicate the situation, especially if a virtual anode forms in the center.




Dan Tibbets
To error is human... and I'm very human.

bsmythe
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Post by bsmythe »

D Tibbets wrote: I don't follow your last paragraph well (original poster), but if a tightly focused/collimated electron gun shot it's electrons towards an opposing cusp then I expect that the vast majority of electrons would exit through that cusp. But if the electron gun's electron output was not tightly focused then only a corresponding percentage of the electrons would exit on the first pass. Also, if more than one electron gun is used then the beams would intersect in the center. The increased electrons in this area would repel each other and further defocus the beams. And, once ions are added the conditions would become even more chaotic.
Dan Tibbets
Dan
The part about the electron counter current is a bit confusing. I am assuming that the e beams injected into each cusp go into the core and do not exit through the opposite cusp. Dr Krall's paper on forming and maintaining a potential well in the HEPS experiment concluded that this is what was happening. What I was talking about in the last paragrapg is that in steady state operation for a constant potential well depth, the e-beam current in has to equal the losses of electrons in the device. This is just a restatement of conservation of charge where:

charge in - charge out = storage of charge in the wiffle ball

In steady state storage is constant and is porportional to well depth so e-beam injection has to be balanced out by electron losses. Some losses are cross field diffusion of electrons to the grid and neutral charge exchange. Another loss may be envisioned as an electron flying out of a cusp and going straight to the vacuum chamber wall. The losses of this sort can be envisioned as an e-beam traveling in the opposite direction as the injection beam and partially canceling out the magnetic field the injection beam creates. The net current going into the cusp is therefore the injection current minus the losses to the wall through that cusp.

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Post by TallDave »

While the thread definitely went in that direction, there did not seem to be any definitive statement to that effect (say by Dr Nebel).
Reconstructing from memory, Nebel said something like "Behavior on the outside is adiabatic, thus what goes out comes back in." Don't think he said where, but seems to imply it could come back in anywhere.
Also, if this is the current thinking, is there still a need for circular cross section coils and coil casings?
It's best that they be conformal to the field. That's the optimal shape to reduce cross-field transport to the Magrid (i.e. it puts the surface of the casings the farthest possible distance from the electrons trying to get to them through the magnetic field).
So wouldn't the holes an electron sees be smaller going out than those it sees going back in.
Yes, the geometry at beta=1 probably looks something like a funnel facing inward: easy to get in, hard to get out.
The second observation is that there is also an electron beam being injected into each cusp
Not sure this is true. I think there are many more cusps than electron guns.
Last edited by TallDave on Mon Dec 01, 2008 4:38 pm, edited 2 times in total.

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Post by TallDave »

The part about the electron counter current is a bit confusing. I am assuming that the e beams injected into each cusp go into the core and do not exit through the opposite cusp.
Heh, they better not. That would directly contradict Bussard's "1e5 electron transits" statement.

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Post by drmike »

To get a picture on the electron sources, imagine two fire hoses pointed at each other. What happens when the water hits in the middle between the hoses is not exactly what happens when electron beams collide, but it's not far off. Not many electrons are going to get back up the other stream.

bsmythe
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Post by bsmythe »

I went back an looked at the recirculation thread again looking for this.
Art Carlson wrote:
I wrote:Along the same lines, for the line cusps I expect an electric field in the plane of the electron fan. That should push them aside so that they possibly will miss the sweet spot coming back in. Even if the net push per excursion is less than a gyroradius, this effect may limit the effective recycling factor. Grad-B and curvature drifts could still be an issue, too.
It is not too hard to quantitatively estimate the size of these effects. I am considering a plane passing through opposing edges of the cube, and consequently also through the diagonals of opposing faces. This is the plane of the line cusps. By symmetry, for all points in this plane, both the electric and magnetic fields will lie in the plane. The fields will be purely radial for the lines passing through the corners of the cube, the midpoints of the sides, and the centers of the faces. I am considering what happens to electrons making an excursion through a cusp and being reflected back toward the cusp, somewhere between the purely radial field lines.

If the electric and magnetic fields are not perfectly aligned, there will be an EXB drift pushing the electrons out of the plane. This drift will be in the same direction on the way out as on the way back in, so it accumulates. The excursion time is about R/v, and the drift velocity is EXB/B^2, so the displacement is delta = (R*EXB)/(v*B^2). If the displacement is greater than an electron gyroradius, then the electrons will miss the hole when they bounce back and not be recirculated, so we should compare the displacement to rho = mv/eB: delta/rho = (R*EXB)/(v*B^2)/(mv/eB) = (R*e*EXB)/(m*v^2*B). R*e*E is the potential difference, which must be comparable to the electron kinetic energy m*v^2/2, so we find that delta/rho is on the order of the cosine of the angle between E and B. How big will that be? We have the symmetry that forces them both to be in the plane we are considering, and also the symmetry that makes them perfectly aligned in 8 directions within this plane, so the misalignment won't be large, but there is no reason to expect it to vanish. Electric fields and magnetic field ultimately have a completely different topology. My guess is a misalignment in the range of 10^-2. This would lead to a maximum recirculation coefficient of 100, far less than the 1000 or 10,000 hypothesized and required.

The calculation for the curvature drift is similar. The drift velocity is m*v^2/(e*B*R_c). Thus delta/rho, which was the cosine of the angle between E and B above, is here the ratio of the device size R to the radius of curvature of the magnetic field R_c. R_c will certainly be much larger than R, but there is no reason to expect it to be infinite. My guess is something like R_c ~ 100*R, again leading to a maximum recirculation around 100. (Note that these effects are independent of each other, so the larger one wins.)

Afterthought: Since I am insisting that this sheet must be quasineutral or at least have a significant ion density nearby, and the curvature drift is in opposite directions for opposite polarities of charge, one must consider the interaction of the electrons and ions. This will be essentially a charge separation resulting in a (large) electric field perpendicular to the sheet, which will push the plasma perpendicular to B within the sheet. I am not sure what the consequences of this would be. It would possibly negate or at least mitigate the curvature effect, leaving the EXB effect to dominate.
I wasn't sure about how the E x B / Hole Size ration translated into the recirculation ratio but now I think I get it. He estimates that the E x B and curvature drifts are in the range of 1/100th of the hole width of the line cusps. So if an electron exits a cusp at 1/100 of the cusp width from the edge then it will drift out of the cusp opening and bounce off the cusp from the outside. If the distribution of electrons escaping the cusps is spatially uniform, then one percent of them will be within 1/100th of the edge. The upshot of this is if the cusp width is 1/100th (of its inside width) wider on the outside then E x B drift will not be enough to push them out of the outside cusp opening and cause them to be reflected. He put the curvature drift in the same ballpark so Hole size outside >= 1.01 hole size inside should serve as a back of the envelope metric for evaluating weather the bigger holes on the outside can mitigate E x B and curvature drift.

So this leads to the question of how much bigger are the holes on the outside than on the inside. M Simon's blog has a link to Dolan's Fusion Research, Principals, Experiments and Technology at:
http://www.sunist.org/shared%20document ... s%20Dolan/
Chapter 11 is on magnetic mirrors and cusps. For magnetic mirrors, the mirror ratio is altered by the plasma beta according to one of the following relationships: (Page 277)
R = Rv/(1-beta)^1/2 for long thin plasmas or:
R = Rv/(1-1/2*Beta) for short thick plasmas.
Where R is the mirror ration with plasma and Rv is the mirror ration without plasma, i.e in a vacuum. So the mirror ratio obviously changes by greater than 1% for Beta = 1. Further on in the chapter (starting at page 303), he addresses cusps. On page 305 he gets to the issue of the line cusp width which is roughly the hybrid lamor radius (Lambdaions*Lamdaelectrons) ^1/2. The derivation has no dependence on Beta. So perhaps the line cusp width is roughly constant regardless of Beta. However this paper http://pdf.aiaa.org/preview/CDReadyMJPC ... 8_4639.pdf pins the ring cusp half width for low beta plasmas at twice the hybrid lamor radius making the total width 4 x the hybrid lamor radius. The following google books link long url (you might have to scroll down to page 159) also puts the line cusp width at 4x the hybrid lamor radius for "very low pressure plasmas". I think this covers us on the cusps being sufficiently larger on the outside to mitigate for various drift terms.

Moving on to the next question, is the scenario Art lays out what really happens at the line cusps and if not does this effect the drift calculations enough to land us back with the electrons not being recirculated? I tend to buy Art's argument that the field lines at the line cusp openings are straight enough to hit the wall before curving around to another opening. Beyond his arguments for the electron path being straight, there is the fact that the numbers can be made to work for this model. If the electrons go out one hole and back in another (I'll call this loop recirculation) they will traverse substantial regions where the E and B fields are highly divergent if not even at right angles and the E X B drift would be substantially greater. The small E x B drift term Art got was due to the E and B fields being well aligned, throw this out and it's a whole different ball game. Since Dr B claimed that there was recirculation and Dr Nebel is not throwing up his hands, I will go with the assumption that recirculation is real and the out and back in the same hole model (I'll call this linear recirculation) can be shown to be a feasible recirculation path.

However, it could still work out the other way; does anyone want to take a crack at estimating the drift magnitudes for a loop recirculation path? I suppose that one thing this route has going for it is that for line cusp exit to point cusp reentry, you are potentially going out back in through a mirror loss cone at low Beta rather than the small hole of a line cusp and this might be more forgiving in terms of various types drift pushing the electrons out of the reentry envelope. You would be trading shooting for a small hole in real space to shooting for a potentially much bigger hole in velocity space. Does anyone want to try to model / calculate this?

Another question is how applicable is Art's BOE calculation to the actual field configuration at the line cusps. One thing that worries me is that his calcs use the ratio of the lamor radius to the E x B drift terms so that the magnitudes of E and B drop out and you just need the angle between them. This seems fine for spatially invariant B and E but both of these vary over the electron's path while the cusp width (i.e. the lamor radius) stays the same. Art's calculation is conservative though since E and B both get smaller as you move away from the coils. Another worry is that the E X B drift has another term for cases with spatially varying E fields. This extra term has the lapalcian of E x B/B^2 though so I think it should change sign when the electron turns around and largely cancel itself out. So I would conclude that Art's calculations are good enough to rule out E x B and curvature drift as limiting processes for linear electron recirculation (assuming the holes really are 4x bigger on the outside). Other candidates for the rate limiting process are scattering (outside the ma grid), charge exchange collisions (inside and out) and drift from space charge effects pushing the electrons out of the reentry envelope.

So, where does this leave us?
-Linear recirculation from line cusps seems to have cleared some theoretical challenges. Examination of other loss terms for this mode need to be done to show it can achieve the needed recirculation rates. Issues such as debey screening and the experimental evidence such as the apparent closeness of WB 7 to the chamber wall may still rule this out as an actual recirculation mechanism. As an aside, if Debey screening becomes a problem at larger scales, it can be overcome with electrostatic repellers on the cusp axes and the Debey screening should keep these from opening up an ion loss channel.
-Loop recirculation from line cusps faces greater challenges from electron drift but has not been ruled out by detailed calculations. This may require numeric modeling to accomplish.
-Recirculation from the point cusps has not been touched on in this or Art Carlson's tread in any great depth.

So at this point some of you may be wondering when is he going to stop? Why does he just keep writing and writing? Well, Art Carlson's discussions on this forum brought into sharp focus something that has bothered me for some time. That something is that we really don't have any detailed theoretical understanding of how this device works. I don't like having to rely on arguments like "Dr Nebel seems happy so there must be electron recirculation". Now there are two ways out of this. We can wait until the WB-8 work is done and hope it gets published or we can start to develop a theoretical framework for the device ourselves. The latter solution sounds much more fun and goes much better with my sensibilities. So this post is my first contribution to that effort; hopefully more will follow.

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Post by Art Carlson »

Good analysis. I admit I don't understand enough about cusp confinement to even pretend that I know what is really going on. I took a stab at it based on what I do understand about magnetized plasmas and raised some issues I thought could be important, including BOE but quantitative estimates of the size the effects could have. I think Rick Nebel pointed out in response to my post that there may be symmetry principles that are more general and could be used to show that the (linear) recirculation rates could be high. I think the answers are within our grasp, but it will take some serious work by the best physicists among us.
bsmythe wrote:So, where does this leave us?
-Linear recirculation from line cusps seems to have cleared some theoretical challenges. Examination of other loss terms for this mode need to be done to show it can achieve the needed recirculation rates. Issues such as debey screening and the experimental evidence such as the apparent closeness of WB 7 to the chamber wall may still rule this out as an actual recirculation mechanism. As an aside, if Debey screening becomes a problem at larger scales, it can be overcome with electrostatic repellers on the cusp axes and the Debey screening should keep these from opening up an ion loss channel.
What I would most like to understand about linear recirculation is how it is affected by phase-space conservation laws. They may allow us to leapfrog over a bunch of detailed arguments and calculations. I am afraid we may be making some mistakes by taking the idea of a hole too seriously. It is an _effective_ hole but in reality a combination of real space and velocity space effects. This could be important when we ask the question whether the hole looks bigger from the low beta side (which I strongly doubt).
The second thing nagging on my mind is whether there is any combination of electric fields that can plug both electron and ion losses at the same time. I am sure this issue has been dealt with in the magnetic mirror program, but I understand almost as little about mirror confinement as I do about cusp confinement.
-Loop recirculation from line cusps faces greater challenges from electron drift but has not been ruled out by detailed calculations. This may require numeric modeling to accomplish.
I'd say the biggest problem with the idea of loop recirculation is still the simple geometrical argument that the loops will hit the wall. Second, the symmetry arguments that might make line recirculation work will not generally apply to loop recirculation.
-Recirculation from the point cusps has not been touched on in this or Art Carlson's tread in any great depth.
I assume that the losses from the true point cusps (on the faces of the cubes) will be much smaller than the line- or quasi-line-cusp losses, so I'm not too interested in them (although it is also an interesting physics question).

bsmythe
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Post by bsmythe »

OK
This is mildly scary but I think that I might have some foggy idea of whats going on with the line cusps. Dolan doesn't go into a whole lot of detail or theory but from numerous re-readings, filling in the blanks, reading between the lines and other platitudinous acts that don't come to mind right now, this is what I think is going on:

Firstly, cusp confinement is not confinement at all; it's a loss channel. Dolan says that there are three classes of particles in a cusp type system, unconfined particles, particles confined by the magnetic forces and particles confined by electrostatic forces. He says that magneticly confined particles have confinement lifetimes dictated by mirror confinement. The unconfined particles are ones that behave non adiabatically. Somehow, this non adiabatic behavior is either caused or made more probable by magnetic field nulls. I'm pretty fuzzy on why and how and there is no real explanation in the chapter. A spindle or multipole magnetic cusp has a field null at the exact center where the central field lines of opposing coils meet and cancel each other out. So far all of this is in spelled out in the book.

From here on, I have to start reading between the lines and guessing. I think at low Beta operation, there is only this one tiny field null and the mirror confinement stinks anyway so mirror losses dominate. At higher Beta, the loss cones start to shrink and the losses from non adiabatic particles become more significant. At Beta=1, the mirror loss cones are choked down even further (or closed off entirely if you use his approximation for Beta modified mirror ratio for long thin plasmas) but you now have a much bigger field null from the area where diamagnetism in the plasma has completely expelled the field. Now the non adiabatic losses dominate. Now, I don't quite get why the non adiabatic particles come out the point or line cusps. I'm guessing it either because that behavior only or preferentially occurs in certain regions of velocity space or that the magnetic field configuration inside the device funnels them out of these cusp areas. The important thing is that they come out here and come out in a very narrow region of both real and velocity space. That is the end of the story inside of the device, now we switch our attention to the outside.

So the particles get spat out of the cusps get turned about and head back for the "holes" they came out of. Do they make it back through the "Holes"? No, they don't because there are no holes from the outside looking in. Now that they are outside the device, there are no field nulls to be found; no built in ones from the devices magnetic topology and no created ones from diamagnetic effects in the plasma. As Dr Nebel said, everything on the outside is adiabatic and so all the particles see is a magnetic mirror with no nice hole in it to get back through. But from the outside in, its a wimpy low to no Beta mirror and the exit from the "holes" has left them highly collimated in real and velocity space so they easily pass through the magnetic mirror on the way back.

Now of course this still leaves us with drift to contend with and to and see if this will push the electrons out of the loss cone of the mirror. However, I think the reentry requirements are greatly reduced.

Now, a lot of this is supposition and guess work but somewhere in chapter 11, Dolan recommends the following reference for more information on the theory behind cusp confinement:
M. G. Haines, "Plasma confinement in cusp shaped magnetic fields", Nuclear Fusion 17, 811-858 (1977)
I don't have access to e-journals but I will be near the state university this Weds and will swing by the library and try to lay my hands on a copy then.

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Post by drmike »

In the field null region the phase space is uniform - all particle velocities are distributed in all directions. So under a line cusp, particles that shoot out along the cusp have no magnetic coupling - they go from zero field in the null region to high field with their velocity vector along the field line. If there is no E field to buck them (as in most mirror or multi-coil systems) then they are lost to the walls.

With the present grid system I don't see how you can stop electrons from leaving the cusp regions (hole or line). I think this is why minimizing these losses was such a high priority, and why electron guns are used to put them back into the center.

I so wish I had time to model this fully - it would be a blast. But I'm having a hard time finding time to build hardware as it is! Way too much fun here. :D

TallDave
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Post by TallDave »

Firstly, cusp confinement is not confinement at all; it's a loss channel. Dolan says that there are three classes of particles in a cusp type system, unconfined particles, particles confined by the magnetic forces and particles confined by electrostatic forces.
Well, I think in a classic mirror machine the electrons that escape just go flying off. In a Polywell, though, an electron outside sees the positively-charged Magrid so it curves around on the field line, trying to get to the Magrid.
Cross-field transport constitutes an
unavoidable loss to coil structure, while cusp axis flow need
not be a “loss“ if the device is open and the electrons can
recirculate along the cusp axes to the outside of the machine,
thence to return along cusp axes field lines. This type of
recirculating machine with magnetically protected coil
surfaces is called a MagneticGrid (or MagGrid; MG)
machine. It requires that the machine, itself, be centered
inside of a containing wall or shell, that is held at a potential
below that of the machine proper, by the voltage used to
drive the electron injectors.
It's reasonable to infer from the WB-6/7 (open recirculating) vs. WB-5 (closed-box) results that recirculation does happen. If it wasn't happening, WB-6/7 would have performed no better than WB-5, which did not work as hoped because instead of recirculating electrons would hit the enclosing box.
WB-5 was an
attempt to revisit to the first large scale closed-box
experimental work (Ref. 6), to see how well electron
confinement had been improved by the understanding of
MaGrid insulation reached in the tests of WB-2,3,4 and
MPG. It was expected that greatly increased electron
trapping would result in higher electron densities at higher
system starting pressures, at the same currents of e- drive. It
was found that electron trapping was 1000x better than in the
earlier large machine (called HEPS), with comparable
electron densities at pressures over 1000x those attained in
the earlier work. .However, when increased drive currents
were employed to try to drive the internal densities to still
higher values, the machine was unable to go significantly
beyond this 1000-fold increased level, except with extreme
higher currents (30 kA and up).
Extensive detailed experimental studies showed that this was
due to e- losses along B-field intersect lines into the corners
and seams (where the B fields run directly into the tank
metal) of the containing tank. WB-5 was a closed box
machine, like HEPS, with its coils outside - so that it could
not allow e- recirculation out and back through its magnetic
cusps. These losses were extensive, and attempts to reduce
them by use of floating ceramic repellers placed along about
1/2 of the seam lines reduced e-losses by 2.5x but only at the
price of opening up huge loss areas for trapped ions. This
did show exactly how bad the unshielded metal problem
was; very bad in HEPS, less so in WB-5, but actually totally
intolerable in ANY machine. No matter the SHAPE of the
coil/coil joint (whether sharp-corner touching or line cusplike)
what matters is that (almost) NO metal must be there at
all. The coils MUST not touch and MUST be spaced apart.
This is the e-loss analogue of the effect of line cusp flow
paths at the spaced corners on overall trapping factors,
discussed above.
So the particles get spat out of the cusps get turned about and head back for the "holes" they came out of. Do they make it back through the "Holes"? No, they don't because there are no holes from the outside looking in. Now that they are outside the device, there are no field nulls to be found; no built in ones from the devices magnetic topology and no created ones from diamagnetic effects in the plasma. As Dr Nebel said, everything on the outside is adiabatic and so all the particles see is a magnetic mirror with no nice hole in it to get back through. But from the outside in, its a wimpy low to no Beta mirror and the exit from the "holes" has left them highly collimated in real and velocity space so they easily pass through the magnetic mirror on the way back.
Yes..ish. My understanding is the mirror holes are the same as the cusp holes, they've just been made smaller by the deformation of the magnetic field by the electron "pushback," to the point the system is best described as "cusp confined" (as opposed to mirror confined); this is the Wiffle-Ball effect. There's a nice picture of this in Valencia, next to the following text:
Initially, when the electron density is small, internal B field
trapping is by simple “mirror reflection“ and interior
electron lifetimes are increased by a factor Gmr, proportional
linearly to the maximum value of the cusp axial B field.
This trapping factor is generally found to be in the range of
10-60 for most practical configurations. However, if the
magnetic field can be “inflated“ by increasing the electron
density (by further injection current), then the thus-inflated
magnetic “bubble“ will trap electrons by “cusp confinement“
in which the cusp axis flow area is set by the electron gyroradius in the maximum central axis B field. Thus, cusp
confinement scales as B2. The degree of inflation is
measured by the electron “beta“ which is the ratio of the
electron kinetic energy density to the local magnetic energy
density, thus beta = 8(pi)nE/B2. Figure 16 shows two
means of reaching WB beta = one conditions.
Since the deformation only happens on the interior of the device (where electron density is ~1E3 times greater), the holes are much easier to get through from the outside.
I'd say the biggest problem with the idea of loop recirculation is still the simple geometrical argument that the loops will hit the wall.
As I recall, there is a picture here somewhere that shows how the interactions between the magnets bring the field lines in much closer. It's somewhat counterintuitive (well, it was to me anyway).

I remember too that a few of us were surprised at how close to the wall WB-7 appeared to be situated in the picture from EMC2.

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