Anyone have the equation for estimating hole size?

[mattman]

Hello,

Anyone have the equation for estimating hole size in a sheath of plasma inside the polywell?

I have read a couple of papers which estimate it, one example is:

Hole^2 = 2* Sheath Thickness * Radius of Spindle Cusps

[hanelyp]

My understanding was cusp radius was roughly the particle gyroradius in the magnetic field through the cusp.

[D Tibbets]

Title: High Energy Electron Confinement in a Magnetic Cusp Configuration: Progress Toward a Polywell Fusion Reactor
Authors: Jaeyoung Park1, Nicholas Krall2, Paul E. Sieck1, Dustin A. Offermann1, Michael Skillicorn1, Andrew Sanchez1, Kevin Davis1, Eric Alderson1 and Giovanni Lapenta3


Excerpt from Park's paper. I think this may be what you are looking for. My understanding is that Grad proposed that the cusp loss diameter/ radius? at Beta = 1 was the gyro radius for the charged particle. As mentioned this is still not good for ions, but for electrons it is presumably good. Bussards scheme of a non neutral plasma with an electron excess implies that the electrons are magnetically contained with the cusp losses dominating all of the magnetic associated loss pathways (such as cusp loss, ExB losses, and a few others) for electrons. Only ions that are upscattered have an oppertunity to escape the electrostatic potential well and subsequently escape the SECONDARY ion magnetic cusp confinement. A prime example of this is the escape of higher energy fusion produced ions. Because of the small percentage of up scattered ions the ion cusp losses will be slower (not stopped entirely). If annealing occurs this may be impeaded even more. The electron cusp losses are thus the dominate loss mechanism. Presumably, the electron cusp losses can be improved to acceptable levels while still being the dominate loss concern.

High Beta cusp confinement by Dan Tibbets, on Flickr

PS: #$%X@%^%ing photo server. I think I finally got the right link information...

Dan Tibbets

[mattman]

DAN,

I can use this equation. Here is where I am at in the development process.


----
Part 4: Modelling Field Free Plasmas

Introduction:

The paper lists the plasma inside the cusp confinement zone as having a 5000 cm^3 volume [12]. That is pretty big (the rings hold a volume of 9500 cm^3). A sphere of that size nearly fills the entire ring structure. Dr. Park said that the team came to this volume using theory. The team also looked at photos of the plasma to verify it. Photos do not disprove or support that volume [34]. You can see that from looking yourself. The cusp confined plasma is (probably) the glowing gas in these time lapse images [1].



It is important to realize what is happening here. The space inside the rings has been split up. It has become two regions. Inside the cloud is a high pressure plasma - with no fields. This is a field free plasma. It has some shape, volume, density and pressure. There does not seem to be any easy way to estimate or measure these properties. Outside the plasma is a high pressure magnetic field - with no plasma in it. The field should be “bunched up” around this shape. It has a local field density – which would be useful to know – because we could estimate the hole sizes from this [35,14].



Several people have proposed ways to model this system. Indrek Mare used the method of images to find the force fields around plasma [36]. Dr. John Santarous thinks that this problem can be solved with kinetic theory [37]. In his thesis, Matt Carr suggested the plasma be treated as lying inside a flux surface [38]. I decided the fastest way to solve this problem was to guess shapes and find the data that they would produce. By predict the flux loop and x-ray data for different shapes – and compare that to raw data from the paper – we could better understand the real shape of the plasma.

Cusp Shape:

The plasma shape is fitted to the cusp geometry; like a hand to a glove [XXX]. Cusps are places where two magnetic fields butt up against each other. They represent weak points in the magnetic geometry – chinks in the armor - where the plasma can bulge outwards. The polywell has three kinds of cusps, shown below. The line cusp runs along the seam between two electromagnets. There are twelve inside the machine. Funny cusps run along the corners. The point cusp lies in the middle of one electromagnet.



This geometry give us some preconceived ideas about the plasmas’ shape. If the plasma moves just along the axis: the shape becomes a 6 point star. Add in the corner cusps – and the shape becomes a 14 point star. This is the most commonly assumed plasma shape [39]. In his PhD thesis, Matt Carr added in the line cusps to this model. [38, 19]. His solution looked like a 26 point, spikey cube. Carrs’ model shoved all the plasma inside a flux surface. This was a surface of constant flux - a plane which has a uniform magnetic field crossing it. The last shape to model is a sphere. Bussard proposed that (at full pressure) the plasma would press outward into a sphere with holes in it. A sphere with holes which looks like a Whiffle ball. I will model large and small versions of these shape, shown below.

[D Tibbets]

I speculate that the glowing plasma in the Mini B is mostly the cool plasma that resulted from the plasma guns firing on the left and the right. The video (sort of) shows the wave front from each plasma gun meeting just right of center (timing not perfect) then starting to rotate due to tilt instability (?). I don't know how much you can derive from the pictures. It was an extremely messy injection of plasma from vaporizing and heating a thin plastic sheet. I suspect there was much neutral gas, recombinations, etc. going on that makes defining a plasma border iffy. The blast was not intended (my interpretation) to show a visual of plasma behavior. It was a cheap and messy way of injecting much pressure into the magnetically confined space in order to push towards Beta=1. I think the actual best was Beta= ~0.7 according to Dr Parks. It allowed for the clever diagnostics that allowed for showing the improved confinement of hot electrons through the resultant hot electron Bremsstruhlung. Anything beyond that is probably suspicious in this grossly messy machine.

Also, keep in mind that even with a good Wiffleball high Beta border, the electrons and to a lesser extent the ions will have some collisions on the border (or already be upscattered) so that they enter the magnetic dominated domain and undergo further ExB diffusion through the B field. This will create a more diffuse visual glow from mostly recombination / charge exchange reactions in these regions. With this machine especially, there would be a large amount of neutral gas around so there would be plenty of opportunities for visual glow discharge from recombination, etc in the ExB dominated magnetifc field region. I would not be at all surprised that WB 8/ 7/ 6, or Mini B before plasma gun firing images shows borders better, even though they may never have been at as high of a Beta. Now if the X ray detection was in an imaging mode, not just a detection mode, the captured image would be much more interesting. Perhaps arranging a hundred detectors around the machine (with windows) would have been revealing as you would be seeing the distribution of hot electrons, not the cold plasma/ gas mix (as with the visual imaging). This may have taken only a few million dollars and a year or two more to accomplish... Dr Bussard, etel stressed that ExB diffusion was present, but that it accounted for only 1-10% of losses compared to the cusp losses. I've never known if this was before or after electron recirculation considerations.

PS: I don't think there was any significant electron recirculation in Mini B and I wonder about WB 8. The grounded magrid would preclude this unless there was enough room outside the magnets for looping to the next cusp, and also very little structure outside the magrid to intercept electrons. In this case further exterior magnets might alleviate this defeciency in recovery of electrons, or neutral plasma as in the Lockheed design's case.

Dan Tibbets

[D Tibbets]

As for the plasma volume within a Wiffleball border. I recall Dr Parks said ~ 50% of the radius. That is consistant with ~ 5000/ 1000CC once the the additionale volume in the spikes are considered. I don't know if Dr Parks was referring to the Mini B at Beta=~0.7 or if he was referring to a Beta of 1.

Keep in mind that my interpretation, at least, is that the sort of spherical growth consumes the the spikes as it expands, the spikes do not balloon out/ widen(at least not much), nor do they pinch. Basically, the more parellel the B field lines are to the outward plasma pressure the less they are effected as the pressure is due to moving charged particles pushing against the B field as they are turned. The gyroradius is determined only by the vector component that is perpendicular to the B field lines. A particle travelling parallel to the B field lines exerts no pressure. I suppose it ties in with the gyro radius size holes left once Beta = one is achieved.

PS: I was confused (of course now I am enlightened ) about P-B 11 fusion alphas exiting cusps without hitting the magnets. The gyro radius is much too wide to fit through the cusp space between magnets. The enlightenment concerning Dr Nebel's comment about the alphas bouncing around about a thousand times before exiting a cusp , assuming the B field and machine radius was great enough for the perpendicular vector energy of the alphas to not result in the ion reaching the magrid on one gyro radius excursion. Also, assuming most fusions occur in the center the alphas would have mostly radial vectors. This means their gyro radius would be near their total energy in areas away from cusps as they would be hitting the B field almost perpendicularly. There is little Coulomb collisions for the alphas though, so the alphas retain their radial vectors till they hit a cusp. Here the vector is approaching parallel paths relative to the B field lines, so the resulting gryro radius is smaller and presumably small enough to pass through the cusp without hitting the magnet cans , preserving the opportunity for exterior direct conversion of the alphas KE.

Dan Tibbets

[ladajo]

Matt,
You should be looking at the recombination zone outside the tori. This will give you a better estimate for the sheath shape and size.
Take a look again at the photos. Notice top right, you can see a sheath extending from the top front corner in a spike conical shape. More importantly, notice how the lower edge curves in a concave manner down and in, then down and out the center of the left side torus.
You can make a good visual approximation of the curve, and extrapolate for the full structure. Some more math will lead you to a better understanding of the pressures/balances when you consider that Beta is (for argument's sake) "0.7".

[mattman]

Guys,

Let me read all these responses and get back to you. At present - I am not sure how exactly to solve this problem. I have generated STL files with the 3D shapes, and I have some software which can simulate the fields around these shapes - and works with STL files. Indrek's method of images looks to be the most promising.

[mattman]

A couple of points:

1. The Beta is One.

2. 5000 cm^3 is a educated guess. It being a guess, I feel AOK making my own guesses about volume and shape.

3. The chamber effectively holds 8,927 cm^3 (I know I said 10,077 earlier but if you lop off the corners), hence 5000 cm^3 seems really big.

4. I am interested in the field right along the surface of the cusp confined volume.



5. I can treat the plasma like a superconductor undergoing the meissener effect. That is a material EM software is acquainted with.

[D Tibbets]

One point I have not heard discussed is where the Wiffleball (not fully formed at Beta=0.7) border was in relation to the decimating magnetic field measuring coxaxial cables inside two of the magnets. Once the Wiffleball border was/ is pushed out to that radii the cables are an immediate sink for energetic charged particles. I doubt any expansion past that point was possible because the losses here would be very much more than the ~ 1/10,000th of the total surface area that could be allowed to be outside the magnetic shielding (Dr Bussards estimate). The 0.7 Beta may have been limited by this geometry / exposure radius dictated by the coaxial cables. I recall that Dr Parks reported about a 40% reduction of the B field at that radius. How that relates to the Beta of 0.7 is unknown. Presumably the various runs at different B field strengths might shed some light on the issue. I do not recall Dr Parks commenting on this.

In designing the experiment Dr Parks presumably wanted the coaxial cables at a radius that would show the B field being at least partially excluded, yet not at such a small radius that only a small Beta would be possible before losses would become overwhelming due to the exposed cables. As such, I wonder how close to the best compromise he envisioned was reached

Dan Tibbets