This is about some nice geometrical properties of spheroids which may, or may not, have application to offshoots of the baseline Polywell concept. I don't know enough plasma physics to do more than speculate, but I can always hope that some physicist might get inspired...
First, some background for readers not familiar with spheroids. A spheroid is a rotationally-symmetric, stretched (prolate) or squashed (oblate), sphere-like surface whose intersection with a plane containing its symmetry axis is an ellipse. This ellipse generates the spheroid when rotated about a major or minor axis. Prolate spheroids have two distinct focal points, namely the foci of the generating ellipse. Oblate spheroids have a focal circle, traced out by the foci of the generating ellipse. A perfect sphere, with the foci merged into one point at its center, is the intermediate surface. A nice visual demonstration is found here (click on "watch web preview"):
http://demonstrations.wolfram.com/SpheroidalFoci/
This paper (available electronically at many college libraries via www.jstor.org) points out an interesting property of ellipses and ellipsoids (spheroids being a special case of ellipsoids):
M. Frantz, "A Focusing Property of the Ellipse", American Mathematical Monthly, 101 (1994), p 250-258
In a nutshell, rays (particles) emitted from or passing through one focus of an ellipse, which then get reflected (bounced) off of the ellipse interior to pass through the other focus, tend to become focused along the line connecting the foci. For prolate spheroids, the focusing is along the symmetry axis. For oblate spheroids, the focal region is a disk containing the focal circle, perpendicular to the symmetry axis. Assuming perfect reflection, the "intensity" increases exponentially when considering reflections in a plane. An example given in the paper shows that a circular wavefront of uniform intensity departing one focus of an ellipse (eccentricity 0.96) will return to that same focus, after two reflections, with intensity increased by a factor of about 6 million. [added edit] See Figure 7 in the paper to make this clearer (elements of the initially uniform wavefront get pushed around the wavefront circle until they are all moving along a straight line containing the foci).
Now the wild speculation. Some necessary but almost certainly not sufficient starting assumptions:
1) The quasi-sphericity of Polywell fields is not so "quasi" near the center (those cool pictures of plasma blobs sure look spherical).
2) Current in some or all of the B field coils is independently controllable.
3) Coil currents can be manipulated to slightly stretch or squash the total field into reasonable approximations of spheroids.
Forcing a static, non-spherical field would probably be a bad idea (as Dr. Bussard said, naturally occurring fusion reactors are all spherical), but might there be some advantage in modifying the Periodically Oscillating Plasma Sphere (POPS) concept to cause field shape to oscillate between spherical and prolate spheroidal, spherical and oblate spheroidal, or oblate and prolate spheroidal? I only have a cursory understanding of POPS, so I'm also assuming here that it normally tries to keep things as spherical as possible. Would there be any tendency for particles in a spheroidal field to concentrate near the focal line or disk? In Slough's concept, FRCs are linearly translated into collision. IF Polywell particles did tend to concentrate near spheroid focal regions, could they be slammed together by driving the prolate foci, or the oblate focal circle, into a central point? Would a prolate field be of benefit for getting fusion products to exit in one direction for QED-ARC propulsion? Would an oblate field be of benefit for getting alphas to exit in a plane for direct conversion purposes?
OK, I'm done.
Spheroidal Foci and POPS?
Not all stars are strickly spherical. Rapidly spinning stars can have considerable equatorial bulges. Some spin so fast that they come close to tearing themselfs apart.
Concerning quasispherical Polywell geometry. I'm guessing this would be the best if extream confluence/ focusing of the ions are required for success, but Dr. Nebel has stated that a high degree of confluence is not needed. I've speculated if a more cylindrical geometry would work. The intrenal volume could be increased without as great a diameter increase (eg: X axis= Y axis= 1/2 Z axis.) I think the need for symetrical opposing magnetic fields would still be maintained. Any confluence would be more towards a central line instead of a point. How this would affect the cusps, annealing, electron thermalizing time, fusion product escape,etc would be interesting.
I could see the the shape of the Wiffleball being varied with geometry of the magnets, realitive time dependant magnetic field strengths, variation in drive energies, slight variations in cusp sizes (eg, allow cusps to open slightly more on one end to preferentially allow fusion product ions to exit there, at least on a temperary basis). plasma rotation, radient layering effects from POPS type mechanisms., etc, etc.
A good theory would allow playing with these parameters without having to build hardware. But, unfortionatly, the models are apparently limited. Thermalalized neutral plasma models might answer some questions, but according to Dr Bussard, these can be misleading or outright wrong when concidering the quasineutral, non thermalized, magnetically, and electrodynamically confined plasmas in a Polywell.
Dan Tibbets
Concerning quasispherical Polywell geometry. I'm guessing this would be the best if extream confluence/ focusing of the ions are required for success, but Dr. Nebel has stated that a high degree of confluence is not needed. I've speculated if a more cylindrical geometry would work. The intrenal volume could be increased without as great a diameter increase (eg: X axis= Y axis= 1/2 Z axis.) I think the need for symetrical opposing magnetic fields would still be maintained. Any confluence would be more towards a central line instead of a point. How this would affect the cusps, annealing, electron thermalizing time, fusion product escape,etc would be interesting.
I could see the the shape of the Wiffleball being varied with geometry of the magnets, realitive time dependant magnetic field strengths, variation in drive energies, slight variations in cusp sizes (eg, allow cusps to open slightly more on one end to preferentially allow fusion product ions to exit there, at least on a temperary basis). plasma rotation, radient layering effects from POPS type mechanisms., etc, etc.
A good theory would allow playing with these parameters without having to build hardware. But, unfortionatly, the models are apparently limited. Thermalalized neutral plasma models might answer some questions, but according to Dr Bussard, these can be misleading or outright wrong when concidering the quasineutral, non thermalized, magnetically, and electrodynamically confined plasmas in a Polywell.
Dan Tibbets
To error is human... and I'm very human.
Correct, stars tend to be oblate due to rotation. They also have very complex dynamics... maybe solar oscillations (ringing) were an inspiration for POPS, I don't know. In any event, I have to treat plasma physics as a black (or at least, very dark gray) box due to ignorance on my part. Even very simple lumped-parameter nonlinear systems are capable of infinitely complex behavior under the right circumstances. How much more so for a distributed-parameter, high-dimensional, highly-coupled nonlinear system where fields tell particles how to move and particles tell fields how to bend.
What I was hoping for above is that the intrinsic focusing properties of spheroids might somehow survive all the complicated particle/field dynamics and produce effects that could be exploited. Maybe the effects would be small, but still statistically significant enough to be useful. I'm guessing it would be most important to have established maximum sphericality (spherocity?) via the wiffeball effect before attempting to slightly deform the field. The time-averaged field would preferably need to be as close to spherical as practical. Careful experimentation, or a really expensive supercomputer simulation (assuming there is enough theory to describe it and the equations are numerically tractable), will be needed.
In my post above I should have written "Polywell propulsion" (unspecified architecture/Isp/thrust) instead of "QED-ARC propulsion", since I think Dr B. planned on driving the QED-ARC relativistic electron beam (to heat reaction mass) from direct-converted power, not fusion products streaming from the core. I have an SSTO fixation since that's where an advance is most needed. There are lots of pending options once you get out of the gravity well.
I also did not want to imply that high energy alphas would be channelled by the slightly distorted overall field. Maybe there would be a statistical shift in alpha emission directions due to the warped field? If you think of electrons as bouncing around, billiards-like, inside a wiffleball spheroid, without interference (BIG assumptions), then they would tend towards the prolate axis or the oblate disk, and this would alter ion convergence. What's the relation, if any, between p-B11 impact direction and alpha direction? Are any laser-like coherence effects possible?
What I was hoping for above is that the intrinsic focusing properties of spheroids might somehow survive all the complicated particle/field dynamics and produce effects that could be exploited. Maybe the effects would be small, but still statistically significant enough to be useful. I'm guessing it would be most important to have established maximum sphericality (spherocity?) via the wiffeball effect before attempting to slightly deform the field. The time-averaged field would preferably need to be as close to spherical as practical. Careful experimentation, or a really expensive supercomputer simulation (assuming there is enough theory to describe it and the equations are numerically tractable), will be needed.
In my post above I should have written "Polywell propulsion" (unspecified architecture/Isp/thrust) instead of "QED-ARC propulsion", since I think Dr B. planned on driving the QED-ARC relativistic electron beam (to heat reaction mass) from direct-converted power, not fusion products streaming from the core. I have an SSTO fixation since that's where an advance is most needed. There are lots of pending options once you get out of the gravity well.
I also did not want to imply that high energy alphas would be channelled by the slightly distorted overall field. Maybe there would be a statistical shift in alpha emission directions due to the warped field? If you think of electrons as bouncing around, billiards-like, inside a wiffleball spheroid, without interference (BIG assumptions), then they would tend towards the prolate axis or the oblate disk, and this would alter ion convergence. What's the relation, if any, between p-B11 impact direction and alpha direction? Are any laser-like coherence effects possible?
For clarity, let's remember that the Whiffleball is in fact a reference to the Magnetic Field produced by the MAGRID. The Plamsmoid is actually in general terms the geometric opposite of the field. Think of the magnetic field as your hand, and the plasmoid as a ball of playdough you are squeezing. Our whole goal here is to minimize the finger gaps that let the playdough squirt out.
POPS would be like squeezing and releasing your hand on a rubbery playdough ball. The plasma is not neccessarily sphere, more like a spikeyball as my six year old might say. Mathematically you could "average" it out, but the variance introduced by POPS would also cause a variance in the ball's "spikes" and corresponding possible loss mechanism. That would be hard to model without some more data. The photos in another old post of a permanent magnetic Magrid construct in with a magrid actually showed the "spikes" nicely. I took a spin, but did not find it yet. If I do I will link it.
POPS would be like squeezing and releasing your hand on a rubbery playdough ball. The plasma is not neccessarily sphere, more like a spikeyball as my six year old might say. Mathematically you could "average" it out, but the variance introduced by POPS would also cause a variance in the ball's "spikes" and corresponding possible loss mechanism. That would be hard to model without some more data. The photos in another old post of a permanent magnetic Magrid construct in with a magrid actually showed the "spikes" nicely. I took a spin, but did not find it yet. If I do I will link it.
Re: Spheroidal Foci and POPS?
yeah, but there are so many people with 'Dr' and 'Prof' in front of their names that believe man-made fusion reactors of the future will be toroidal, then (according to the apparent rules of deference this board likes to observe) it must surely be toroidal!DeltaV wrote:as Dr. Bussard said, naturally occurring fusion reactors are all spherical
Personally, I think your thinking is sound, but seeing as you'd end up with/need two centres of attraction and that the coils do not confine the shape of the reaction volume but hold the centres, I'm not sure how that geometry would hold together. Also, it would mean that reciprocating particles, were they to be heading towards one, would have a non-radial velocity towards the other and no-one else here, apart from Art, have ever expressed any agreement with me that non-radial velocities would occur in a Polywell, so not sure how that'd all work out.
I guess there are fixes possible. Maybe the coils would follow a peanut shape so as to generate the two centrums for electrons. It could be called a 'peawell' instead, though that conjours up images that might not be appreciated here. Or how about 'nutball'?
Be worth a try...
Sure be worth a try once a round' one can be got working...
Um, as I see it, there may be uses to extending the 'hot-spot', one being a gradual extension to linear, to rocket engine or plasma source. These notions hanker back to the original fusion experiments with toroidal or figure-8 configuration. Not tokomak, mind, but before them...
FWIW, here's an interesting page...
http://www.ornl.gov/sci/fed/mhd/mhd.html
{Ooh ! Look at the pretty predicted plasma piccies ;-}
Um, as I see it, there may be uses to extending the 'hot-spot', one being a gradual extension to linear, to rocket engine or plasma source. These notions hanker back to the original fusion experiments with toroidal or figure-8 configuration. Not tokomak, mind, but before them...
FWIW, here's an interesting page...
http://www.ornl.gov/sci/fed/mhd/mhd.html
{Ooh ! Look at the pretty predicted plasma piccies ;-}
ladajo
Maybe you're referring to the test plasma picture at www.emc2fusion.org . Regarding the wiffleball, as I understand it, not only do the inwardly-convex magnetic fields compress the electrons, but the confined electrons also push back and tend to sphericize and seal the surrounding magnetic fields via a diamagnetic effect (field line expulsion). Without the diamagnetic effect, the wiffleball holes would never get small enough. There's been a lot of debate on this forum about how leaky the wiffleball really is... I have nothing to add to that, but I hope it works as advertised because the entire Polywell concept depends on it. A related topic of debate was whether or not electrons lost through wiffleball holes would recirculate and reenter the fray. Dr. B. said a certain amount of leakage was expected and that efficient recirculation was the solution, which was vastly improved by rounding and separating the coils for WB-6 (at least enough to get more Navy funding).
chrismb
I think they were arguing that non-radial velocities would not occur with the baseline spherical geometry, not spheroids. I don't know enough to say either way. Regarding centers of attraction, I was thinking more of a fuzzy region, or a statistical tendency, rather than two distinct centers for the prolate case. The ellipse focusing property (ignoring repulsion by like charges, electron recombinations with ions, etc.) requires the particle to pass through a focus before it gets "trapped" on the road to focal line/disk "confinement". Most particles would initially not pass through or near a focus, but if a spheroidal field was held long enough, eventually more and more would tend towards that behavior (ignoring all the BIG assumptions, of course). It would depend on how many electron passes occur during a spheroidal cycle, among other things. Regarding coil shapes, I'd rather keep the baseline coil shapes and do all the field tweaking via software/electronics. If for some strange reason it turned out that a two-lobed construction was appropriate, I'd prefer the names "duopoly" or "dumbball".
Nik
I agree, spherical field is the first priority. The pictures of toroidal plasma instability got me thinking... What if oscillating the Polywell field between prolate and oblate spheroids caused (gakk!) toroidal magnetohydrodynamic vortices to form, maybe just one, or maybe two or more, stacked up... All of a sudden, DOE fusion research money might start pouring in by the billions!
Maybe you're referring to the test plasma picture at www.emc2fusion.org . Regarding the wiffleball, as I understand it, not only do the inwardly-convex magnetic fields compress the electrons, but the confined electrons also push back and tend to sphericize and seal the surrounding magnetic fields via a diamagnetic effect (field line expulsion). Without the diamagnetic effect, the wiffleball holes would never get small enough. There's been a lot of debate on this forum about how leaky the wiffleball really is... I have nothing to add to that, but I hope it works as advertised because the entire Polywell concept depends on it. A related topic of debate was whether or not electrons lost through wiffleball holes would recirculate and reenter the fray. Dr. B. said a certain amount of leakage was expected and that efficient recirculation was the solution, which was vastly improved by rounding and separating the coils for WB-6 (at least enough to get more Navy funding).
chrismb
I think they were arguing that non-radial velocities would not occur with the baseline spherical geometry, not spheroids. I don't know enough to say either way. Regarding centers of attraction, I was thinking more of a fuzzy region, or a statistical tendency, rather than two distinct centers for the prolate case. The ellipse focusing property (ignoring repulsion by like charges, electron recombinations with ions, etc.) requires the particle to pass through a focus before it gets "trapped" on the road to focal line/disk "confinement". Most particles would initially not pass through or near a focus, but if a spheroidal field was held long enough, eventually more and more would tend towards that behavior (ignoring all the BIG assumptions, of course). It would depend on how many electron passes occur during a spheroidal cycle, among other things. Regarding coil shapes, I'd rather keep the baseline coil shapes and do all the field tweaking via software/electronics. If for some strange reason it turned out that a two-lobed construction was appropriate, I'd prefer the names "duopoly" or "dumbball".
Nik
I agree, spherical field is the first priority. The pictures of toroidal plasma instability got me thinking... What if oscillating the Polywell field between prolate and oblate spheroids caused (gakk!) toroidal magnetohydrodynamic vortices to form, maybe just one, or maybe two or more, stacked up... All of a sudden, DOE fusion research money might start pouring in by the billions!
Ah, but you see the problem with non-radial velocities is that you'll get particles diffusing into cirumferential velocity space, and as soon as that happens the whole thing thermalises. Creating two focii simply accelerates that thermalisation of particle energy into circumferential velocity space.DeltaV wrote: chrismb
I think they were arguing that non-radial velocities would not occur with the baseline spherical geometry, not spheroids. I don't know enough to say either way.
Sorry, fella, but there are ideas that it might work as a thermalised system, so your idea is not without its own elements of merit.
I think you might be thinking of this thread.ladajo wrote:... The photos in another old post of a permanent magnetic Magrid construct in with a magrid actually showed the "spikes" nicely. I took a spin, but did not find it yet. If I do I will link it.
viewtopic.php?t=1081&highlight=polywell+wannabe
Not as clean as the EMC2 picture of WB7, but within the limits of the perminate magnets, it does show the 'spikes', and also demonstrapes the convex shape of the 'sorta Wiffleball' border regions.
Concerning the relaxation of the electrons into a thermalized shell constrained by the Wiffleball border, I understand it is inevitable, but the dynamic nature of the electron population- the 'old thermalizing and periferalizing lazy electrons' leak out preferentially through the cusps ( or are otherwise lost through diffusion to surfaces) and are reborn as high energy radial electrons due to recirculation, and / or new replacement electron gun sourced electrons.
I'm not certain the dynamics of this would be changed in retangular shaped magrid, as opposed to a cubic magrid. So long as the referenced center is concidered to be a line along the long axis instead of a point in the center of a cube.
Concerning high energy charged fusion products, in a Polywell they are created, and fly off in random directions. They exit at cusps because the magnetic fields everywhere else bounce them back towards the interior (once a certain magrid size/ magnetic field strength condition is reached. Thus the mega electron volt kinetic energy charged particles are contained magnetically, untill they can find a hole (cusp). Dr Nebel said they may bounce ~ a thousand times before escaping). I assume this represents the upper end of the confinement times achievable in a purely magnetic bottle (like a Penning trap) for charged ions of this energy. With the lower energy fuel ions (eg:100,000 eV) the electrostatic (electrodynamic?) field created by the excess electrons hang onto the lower energy ions more efficiently than the leaky magnetic bottle, so the fuel ion confinement times are at least several orders of magnitude longer.
The electrons do not thermalize or accumulate too much at the Wiffleball border within thier lifetimes because they are lost too fast. The key to the Polywell is not that the electrons are prevented from escaping indefinatly, just long enough that they do not quite thermaalize (within some limit). I understand that Dr Bussard struggled with these competing effects for years. It would be easy to have electrons leak out profusely before they can thermalize, but the energy cost of replacing them with high energy radially directed electrons would always cost too much energy for breakeven to occur. The benifit of recirculation is that it restores the electrons to like new condition with very little input energy penalties.
Dan Tibbets
To error is human... and I'm very human.
Dan - that was it. Should have searched "pressure cooker" instead of "microwave"!
The linked pics you have at the archive show the plasma forms better.
It would be cool to do that experiment again, but with (for argument's sake) x8
stronger magnets and compare the pics.
I remember when you first posted it and I had thought what a great idea to visualize lots of math arguments.
DeltaV - the main debate around recirc is whether they go in out on the same cusp or loop around as it seemed Dr Bussard had suggested.
As I understand, the shaping of the magnets was focused to control arcing. The spacing of the magnets accounts for electron gyroradius and arcing.
If Dan can figure a way to recycle his experiment with a magnitude stonger magnetic field, the pics would be very telling I think on scaling effects re: the plasma pressure. I am guessing we would see his pics long before we see WB8 or WB7.1 for that matter...
Speaking of pics: has everyone seen that Famulus got his pump back and tried to blow hisself up again? Nice pics of Fusor "Blue" Deuterium Plasma (although theory to practice holds it should maybe be red/pink.)
http://prometheusfusionperfection.com/
I try to limit my use of big words
The linked pics you have at the archive show the plasma forms better.
It would be cool to do that experiment again, but with (for argument's sake) x8
stronger magnets and compare the pics.I remember when you first posted it and I had thought what a great idea to visualize lots of math arguments.
DeltaV - the main debate around recirc is whether they go in out on the same cusp or loop around as it seemed Dr Bussard had suggested.
As I understand, the shaping of the magnets was focused to control arcing. The spacing of the magnets accounts for electron gyroradius and arcing.
If Dan can figure a way to recycle his experiment with a magnitude stonger magnetic field, the pics would be very telling I think on scaling effects re: the plasma pressure. I am guessing we would see his pics long before we see WB8 or WB7.1 for that matter...
Speaking of pics: has everyone seen that Famulus got his pump back and tried to blow hisself up again? Nice pics of Fusor "Blue" Deuterium Plasma (although theory to practice holds it should maybe be red/pink.)
http://prometheusfusionperfection.com/
I try to limit my use of big words
Deuterium lamps...
Sure is blue-ish when pressure and voltage suit, but I hope he's taking precautions against the UV...
Um, not in 20A league, but I changed and aligned a lot of spectrophotometer deuterium lamps...
Slightly OT: For a while, the *very best* deuterium lamps were hand-made by an elderly craftsman in Kobe, Japan. Well, IIRC, he had to go into hospital for a prostate op, and his son lacked the knack, could produce quantity *and* quality. Supplies of several essential lamps dropped to a trickle, which is how we learned that our 'independent' suppliers were merely re-badging the Sensei's output...
Number One Son got better with practice, and other companies dusted off their jigs, began second-sourcing. This was fortunate as, several years later, the Kobe quake took the premises off-line for six months. Old guy and son escaped unhurt, but the workshop and vacuum system took a long, long time to scrub and purge back to 'clean room'...
Um, not in 20A league, but I changed and aligned a lot of spectrophotometer deuterium lamps...
Slightly OT: For a while, the *very best* deuterium lamps were hand-made by an elderly craftsman in Kobe, Japan. Well, IIRC, he had to go into hospital for a prostate op, and his son lacked the knack, could produce quantity *and* quality. Supplies of several essential lamps dropped to a trickle, which is how we learned that our 'independent' suppliers were merely re-badging the Sensei's output...
Number One Son got better with practice, and other companies dusted off their jigs, began second-sourcing. This was fortunate as, several years later, the Kobe quake took the premises off-line for six months. Old guy and son escaped unhurt, but the workshop and vacuum system took a long, long time to scrub and purge back to 'clean room'...
ladajo, to answer your questions (sort of) the spacing in the WB6 was to acommidate the electrons gyroradius around the magnetic field lines, so that they do not hit the magrid casing on thier way by. This is not quite the same as arcing to non magnetically shielded surfaces (like the vacuum vessel wall).
Concerning magnets, the microwave donut magnets used were ceramic and I'm guessing may have a few hundred Gauss strength at most. To go higher I'd probably have to go to rare earth magnets, with strengths up to a few thousand Gauss(?). I've not contimplated that due to the marginal mechanical strength of my bailing wire approach, and subsuquent power suply adventures.
I have tried smaller magnets from RadioShack. These magnets are probably ~ 1/4th to 1/3rd the strength of the microwave magnets. That setup did show the spikey beast, though it was less well defined. Also showed the plasma (glow discharge) wrapping around the magnet along the field lines- hitting the outward facing pole. If these were electromagnets of similar strength I believe this would demonstrate recirculation between seperate cusps, at least in these modest conditions, perhaps 1-2 thousand volts, 5-20 ma current, and ~50 Gauss (?). This may be a trivial effect, and even more trivial in a true Polywell where the wiffleball distorts the magnetic fields(electron currents over a thousand times higher). Also, keep in mind that what I am seeing is the visible glow from recombinations. I'm not sure how that relates to the free electrons and ions distributions.
http://www.fusor.net/board/view.php?bn= ... 1255649204
The glow discharge is in air (not deuterium) with plenty of contaminates present.
Dan Tibbets
Concerning magnets, the microwave donut magnets used were ceramic and I'm guessing may have a few hundred Gauss strength at most. To go higher I'd probably have to go to rare earth magnets, with strengths up to a few thousand Gauss(?). I've not contimplated that due to the marginal mechanical strength of my bailing wire approach, and subsuquent power suply adventures.
I have tried smaller magnets from RadioShack. These magnets are probably ~ 1/4th to 1/3rd the strength of the microwave magnets. That setup did show the spikey beast, though it was less well defined. Also showed the plasma (glow discharge) wrapping around the magnet along the field lines- hitting the outward facing pole. If these were electromagnets of similar strength I believe this would demonstrate recirculation between seperate cusps, at least in these modest conditions, perhaps 1-2 thousand volts, 5-20 ma current, and ~50 Gauss (?). This may be a trivial effect, and even more trivial in a true Polywell where the wiffleball distorts the magnetic fields(electron currents over a thousand times higher). Also, keep in mind that what I am seeing is the visible glow from recombinations. I'm not sure how that relates to the free electrons and ions distributions.
http://www.fusor.net/board/view.php?bn= ... 1255649204
The glow discharge is in air (not deuterium) with plenty of contaminates present.
Dan Tibbets
To error is human... and I'm very human.
To sum up, spheroids have interesting, purely geometrical, focal properties (wavefront/particle-path concentration via internal reflections) and maybe they might have some practical physics value for future variations of the Polywell concept. I'm glad this was all in writing, because I don't know how to pronounce "foci". :) [Old-style smilie from an old-style engineer.]
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alexjrgreen
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