Definition of Cusp Confinement.

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D Tibbets
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Definition of Cusp Confinement.

Post by D Tibbets »

What is Cusp Confinement?

There is a confusing (to me) use of the term to describe both basic magnetic trapping at low Beta and at high Beta in the Polywell.

My definition, which is derived from that used in the EMC2 patent appication is that cusp confinement only refer to low Beta confinement in a polyhedral device. This opposed to mirror confinement for an opposed biconic mirror machine. The mechanism as gryro radius dominated particle movements along field lines is the same. What is different is that the basic linear equatorial cusp is compressed and deformed (into two spiky narrow derivatives) by the introduction of additional magnets in the equatorial cusp loss dominated region. It is this modification of the linear equatorial cusp that results in the differences that define "Cusp confinement" from mirror confinement. The mechanisms are the same and only the magnet geometry is changed.

High Beta Cusp confinement is synonymous with Wiffle Ball tm confinement, which I often shorten to one word- Wiffleball confinement just to be different and to distinguish it from the toy. Here the confinement is best / dominantly described as the charged particles turning or rebounding off of the B field border. There is no trapping on a field line, typical mirroring descriptions are inappropriate. The pool table analogy is actually more descriptive. This is fundamentally different from mirror confinement and it's derivative cusp confinement or perhaps better described as pohyhedral cusp confinement (as opposed to biconic mirror cusp confinement).

There is an intense gradient in the B field strength at the Wiffleball border- sometimes described as a sharp border. The mechanism of confinement is thus modified, still it is lorentz force turning, but only a partial orbit is completed before the particle with it's extended parabolic orbit leaves the magnetic domain (that produced by the electromagnets). It is comparable to a comet with a highly parabolic orbit . At perihelion it is turned tightly by the gravity of the Sun, but as it travels outward, it covers a huge major axis orbit distance relatively to the minor axis radius. In the Polywell situation, this large major axis orbit is interrupted either by reaching the opposite side of the machine, or by local inter particle collisions and/ or space charge effects. The magnetically governed movement become insignificant compared to the other interactions to the extent that the particle is effectively not in a magnatized plasma (no memory)- except at the subsequent turn points/ Wiffleball border. This sharp border with Wiffleball confinement changes things. But, simplistically (the Wiffleball analogy) is important because of the geometry change of the edge surface. The volume / radius of the confined plasma increase, while the likelihood of rebounding deep into a cusp goes down- the ratio of volume or radius to loss area improves accordingly.

The magnitude of this effect is uncertain based on descriptions but it seems to be ~ 40 X in the mini- B example, and perhaps ~ the same in the WB6 example. In Mini B confinement was improved from ~ 7 to 300 passes, while in WB6 ~ 60 to several thousand passes was claimed. Note that passes is also a moving target. At low Beta the confined plasma radius may be an eg 40% of the magrid radius. At high Beta this plasma plasma border may be pushed out to eg 80%. Not only is the number of transits increased but so is the distance per transit, the confinement time is the product of the two so confinement time is uncertain. Also an increased radius machine would increase the confinement time further.

PS: As addressed in another thread, the cusp confinement of mini B is poor- appx. the same as biconic opposed mirror machine confinement mentioned in the patent application (5-8 passes). My explanation for the relatively poor cusp confinement in mini-B is due to the wide spacing of the magnets (both absolutely compared to WB6 (2.4 vs 1.0 cm) and relative to machine radius (~1/2 that of WB6)). Add to that the intruding B field measuring wire loops and the large surface area plasma gun structure very near the magrid radius had to seriously harm the confinement. But, it got the job done...

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

mattman
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Re: Definition of Cusp Confinement.

Post by mattman »

Dan,

At low beta - "cusp confinement" never worked.

"Cusp Confinement" only truly works at high beta.

I see the "Whiffle ball" and "Cusped confinement" and the "free-boundary" plasma as all the same idea, different names.

D Tibbets
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Re: Definition of Cusp Confinement.

Post by D Tibbets »

I'm not sure what low Beta cusp confinement will not work means, except that it is indeed inadequate for required electron confinement.

For some reason I cannot find Bussard's description, which I believe was in the Patent application. As I recall, he used the term mirror confinement for the biconic opposed magnet configuration, cusp confinement (low Beta) for the polyhedral geometry,and Wiffleball confinement for high Beta polyhedral cusp confinement.
Perhaps more definitive terms would be mirror cusp confinement , polyhedral cusp confinement (or polyhedral mirror cusp confinement) , and Wiffleball confinement (or high Beta polyhedral cusp confinement).


It is important, I think, to distinguish the Wiffleball cusp confinement as fundamentally different in some ways. Mostly because of the sharp border at the Wiffleball edge where electrons turn in a single loop, no spiraling along the field line, with subsequent mirroring/ bouncing behavior (at least for most of the electrons in the system). The end result is still containment, but mechanistically, it is somewhat different.

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

mattman
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Re: Definition of Cusp Confinement.

Post by mattman »

Cusp Confinement is not well understood. There are 200 papers on it. But we cannot find it on Wikipedia. SOMEONE PLEASE PUT IT ON WIKIPEDIA. Here are a couple of ways that I have heard it expressed.

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Here are the questions I have:

1. How do you model a "bunched up" field mathematically?

2. How does this "zone of no magnetic field" start to form? That maybe the "special sauce" for why we see it now and not before. How does it grow, after it forms?

All this leads to a picture of the plasma and the magnetic field changes from Rider's model system, to something with structure.



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From Magnetized Plasma to Cusp Confined Plasma

mattman
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Re: Definition of Cusp Confinement.

Post by mattman »

I see now that #3 and #4 cannot exist simultaneously. The electrons must move in a curved path to generate the magnetic field

D Tibbets
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Re: Definition of Cusp Confinement.

Post by D Tibbets »

I think your drawings illistrate the situation fairly well. i'm not sure which illistrations you are referring to as #3 and #4

If there is at least one charged particle (with some KE) inside a set of convex magnetic fields there is some outward pressure that pushes back the B fields and compresses them. The field lines (or referring to them as the isophopes (sp?) could also be used. The amount is trivial but real. The Beta would be extreamly small and adiabatic effects would be very trivial. The corkscrewing or gyroradius induced motions would dominate for pratically the entire lifetime of the charged particle. As charged particles are added and/ or KE is increased the B field compressing effects and corresponding Beta increase. The straight line motions of the charged particles become more prevelent during the particles lifetimes. The gyro orbits change from almost circular orbits around a B field line to a highly elliptical (parabolic?) orbit around that reference B field line. Also, that refrence B field line is pushed out and this increases the volume where the adiabatic motions dominate. It is not that the particles are moving in an absolutely straight line, it is moving along a very elongated parabolic curve while inside the Wiffleball border. Since this elongated parabolic curve/ major radius path is greater than the distance to the opposite side the path is close enough to linear (ignoring squiggles from small angle Coulomb collisions or major deflections from large angle Coulomb collisions) that any curve can be mostly ignored (there is an exception that has to do with central focus). Without collisions the slight curvature/ relatively long parabolic major radius, and subsequent magnetic turns near the opposite side of the internal volume could be traced back so there could be a claim of some magnetic memmory. But if the Mean Free path (distance to large angle deflecting Coulomb collisions) is shorter than the cummulative effects of the B fields curves (less than one pass to up to several pass length MFPs) the collisions become dominate in descriptions of the particle motions over the vast majority of the particles lifetime. The exceptions are when the particle is turning through it's minor radius portion of its orbit or when the particle enters a cusp. There the magnetic behavior best describes the particle motions. Even that though is not absolute in a collisional plasma because Coulomb collisions still contribute to the particles motion. This can be very significant, though not the dominate interaction. The prime example of this would be ExB diffusion or transport across/ through a magnetic field. The magnetic effects dominate on the very edge of the confined plasma and in the cusps, everywhere else inside the collision effects dominate.

Why is this important? I think for several reasons. First the cusps- exit holes, become much smaller in comparison to the surface area and especially the volume of the contained plasma- the so called Wiffleball effect. This allows for maintenance of greater plasma densities without requiring increased input power. It is a transition from low Beta to high Beta (up to one). Secondly the internal plasma is not magnetic. If it was I think edge instabilities/ turbulence would be unavoidable and only minimized to acceptable levels with great effort. In the Polywell with it's convex fields on the edge of the plasma, even here the edge instabilities are essentially absent because any eddies would energetically favor the movement of particles back into the lower/ absent magnetic domain towards the center of the machine. There would not be buildups to macro instabilities trying to burst through the confining magnetic field. Finally, the Wiffleball effect that allows for greater maintainable densities with reasonable losses, results in exponential increased fusion rates. This leads to a smaller machine which has all sorts of advantages (and some possible engineering disadvantages). The picture changes further for the better when the plasma is non neutral with a small electron excess, mostly due to ExB issues for electrons versus ions, handy acceleration mechanism for ions, handy method of annealing or suppressing ion thermalization processes due to the roughly spherical potential well, and charged particle cusp losses defined by the electron gyroradius instead of the ion gyroradius. This is because the ions ideally never reach the edge where they would enter the maqnetic domain., again because of the centrally directed potential well (from the ions perspective). Even if the ions enter the magnetic dominate domain at the top of their potential well, the ion KE here is so small that the corresponding ion gyroradius is probably smaller than the gyroradius of the high energy edge electrons.

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

mattman
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Re: Definition of Cusp Confinement.

Post by mattman »

This is a little bit more refined.

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D Tibbets
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Re: Definition of Cusp Confinement.

Post by D Tibbets »

Looks like you are developing a broad interpretation that is both usefull and relatively easy to follow.

I do have some comments though, of course 8)

I prefer showing the magnetic fields pushed out further as Beta approaches one. At Beta =1 the fields are pushed out and flattened towards a spherical geometry and reach this appearance everywhere except the pits of the cusps, and at exactly Beta=1 this reaches the midplane radius of the cusps- the radius of the mid plane of the magrid. The interior of the magrid is almost totally B field free. I don't know how close this border approaches the magnet cans in regions where the fields are almost perpendicular to center. Obvously, if the field is pushed out to where the can surfaces are exposed, the containment is shot (or rather within several electron gyroradii of this surface).

Your comment about initiating the high Beta condition suggests to me that you are thinking it is a either one or the other. Actually even at very low Beta there is an adiabatic region in the center that is due to the fall off of the B fields between the opposing magnets. With a low density and low temperature plasma, this B field free (relatively field free) region in the center is tiny, but within this small space the charged particle motions are adiabatic. Because of the small size and the non abrupt fall off in the electromagnet B fields, the the adiabetic region plays a very small role in describing the charged particle motions.
The growth towards high Beta just a progression of this central field free region volume expansion, accompanied by progressive compaction of the confining electromagnet B field lines. Near Beta =1 this compaction results in the B field gradient being so great that the gyro orbit becomes so elliptical/ parabolic that the particle cannot complete a single elongated orbit before collisional effects (almost ) completely dominates the future motions of the particle- at all times except when the particle is completing the relatively very short half orbit as it turns around on this now well defined edge border. This condition can be reached in any confining magnetic field arrangement. You just need to drive more current into the machine than escapes through a combination of cusps (if present), ExB diffusion, edge instabilities (if present) , and several other modalities. Add to that radiation losses from inside the containment volume (I think Bremsstruhlung is the biggest contributor) that cool the plasma and thus reduces the pressure....

In texts the gyro orbit is often described as a circular spiral. This is a simplification though. For a true circle the B field has to be constant, there can be no gradient. This is of course impossible, but if the gradient is shallow enough in relation to the kinetic energy of the charged particle in question, the circle is approximately correct.

The, Polywell, if successful, may operate at near Beta= one, or actually at some intermediate level, perhaps Beta= 0.8?. Beta= one is not magical, it is just the best limit you can achieve. The closer to it you can get, the less effort you have to expend to maintain the pressure against particle losses, at least in a machine where cusp losses are the dominate loss mechanism. Cooling due to Bremsstruhlung is a separate issue. Related to Bremsstruhlung is cyclotron radiation losses ( radiation due to charged particles spiraling around magnetic field lines). In a Polywell this is a minor concern as it is much less than Bremsstruhlung. In a fully magnatized plasma cyclotron radiation may be a bigger concern(?).

I suppose I should point out that in a magnatized plasma such as in a torus tokamak, the confining B fields are pushed out and replaced with a plasma induced B field oriented by the direction of the plasma circulation. As such the plasma remains magnatized on large scales. This is fundamentally different than the spherically convergent Polywell where particle motions are either radial back and forth with some some degree of angular momentum- but this angular momentum is in random directions, so there is no dominate direction of plasma flow, and thus no gross or macroscopic plasma induced magnetic fields. Some have suggested imparting directional spin to the plasma in a Polywell, perhaps to reduce cusp losses, or with the presumption that edge instabilities are a problem (which they are claimed to not be). This fundamentally changes the behavior of the Polywell. The plasma becomes magnetized itself, with all of the complications and paradyne shifts that entails.

The surface current is, I guess, a term describing the charged particle motions along field lines as they spiral between mirroring points at a near constant radius from the center. With low Beta these entrapped particles may represent a large portion of the motions (current) in the system. In the Polywell though, with high Beta and resultant sharp border this back and forth motion along field lines is greatly reduced, and the resultant edge current sheath is thus reduced, perhaps by considerable amounts. The back and forth radial currents are much more dominate. This is a moving target due to various interactions. In the cusps this current sheath is perhaps more significant, but remember that even here the sharp borders results in particles rebounding back into the field free spaces (perhaps after rebounding several times off the "hard" walls of the cusp funnel) so they spend less time at this edge region. I don't know how much you have to modify edge sheath currents as a part of the dynamics inside the machine, but I suspect it is significant.

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

hanelyp
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Re: Definition of Cusp Confinement.

Post by hanelyp »

I don't see energy distribution and cusp confinement really relating. The energy distribution varying from Maxwellian comes more from the electric potential well, though particles being able to cross the diameter of the plasma with little influence from the confining magnetic field is important.
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mattman
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Re: Definition of Cusp Confinement.

Post by mattman »

Dan,

1. Why would the particles move in a parabolic line in the center? They would move straight... unless they interact amongst themselves.


2. I forgot about the null point. So - the field free region would start from here... and push outward?

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3. What is: "cumulative effects of the B fields curves"? You mean field line density?

4. Dan, on the particle motions...

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I don't see energy distribution and cusp confinement really relating. The energy distribution varying from Maxwellian comes more from the electric potential well, though particles being able to cross the diameter of the plasma with little influence from the confining magnetic field is important.
5. Hanelyp: That depends on if your talking about electrons or ions. I am talking electrons.

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D Tibbets
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Re: Definition of Cusp Confinement.

Post by D Tibbets »

Questions addressed to me:
mattman wrote:Dan,

1. Why would the particles move in a parabolic line in the center? They would move straight... unless they interact amongst themselves.


2. I forgot about the null point. So - the field free region would start from here... and push outward?


3. What is: "cumulative effects of the B fields curves"? You mean field line density?

4. Dan, on the particle motions...
1) I'm splitting hairs. It is difficult to claim that there is absolutely zero magnetic field left. Even if it is infinitesimally small. That is why I used the qualifier of parabloic paths instead of straight lines. The important point is that relatively speaking, the remaining electromagnetic field , if any, is essentially trivial to many orders of magnitude. It can be considered as zero, though I was perhaps confusing with my caution against assuming absolute exclusion.

2) Yes, that is my understanding.

3) If I remember my mental processes at the time, I was referring to the cumulative effects of the charged particle (electron) bouncing from one side to the other and back, etc. Each turn introduces a deflection / turn around that is not exactly 180 degrees. There is a 1/2 spiral, such that if the intercepted B field is oriented perpendicular to the machine center, then a purely radial electron is not reversed straight back to the center but at a small offset . Add that the Wiffleball border- interacting B field is curved convexly towards the center and that multiple rebounds may occur before Coulomb collisions completely obliterate the 'memory'(?) of the magnetic border deflections, they may contribute in some small way to electron vectors. Not all of these magnetic mediated turns have to be defocusing the electrons towards the center, but many or perhaps most are. Think of multiple bank shots on a billiard table. One of Bussard's papers mentions this effect in the context of ions that are upscattered and thus reach radii where they are turned by the B field, and that this impedes the maintenance of a tightly focused - confluence of ions towards the center.

4) Here I think your illistration catches the motions well. I am uncertain how much spiraling there is in the cusps though. Here the B field is also compressed, so the electron may depart the B field domain after only one half orbit. It flies across the cusp and hits the opposite magnetic wall, and again bounces off- again much like billiard balls. After several bounces it may achieve a trajectory that carries it back towards the center of the machine. The mirroring is not really occuring, at least till the point very close to the cusp throat at it's narrowest at the midplane radii of the magrid. I don't know if this makes a difference in the math of confinement, but it represents a different dynamic. Electrons that are collisionally knocked deeper into the magnetic field (ExB diffusion) at the cusps or elsewhere will assume fully captured B field spiraling.

Also, I think the graph , by Rider?, must be miss labeled. The potential well will be set at the desired temperature for fusion, not at the high end thermal tail. This may be confused with the approach in thermalized machines like the tokamak. Assuming deep potential wells can be achieved, the potential well will be close to but below the injection energy of the electrons. In WB 6 this was ~ 80%. 12KV injection energy was used to achieve a 10KV potential well. The remaining energy is essentially lost at the begining,

My understanding is that ion - ion collisional scattering and thermalization is straight forward. But ion - electron KE exchange, or electron-electron exchange is more complex. It also is covered by one of Bussard's papers. In any case, the average temperature of the ions and electrons are about the same. The electrons temperature dictates the potential well depth and through space charge effects (the potential well created by these energetic excess electrons) accelerate the ions to ~ the same energy. Down scattering of ions is not a process that I have much of a handle on, but that it is not prominently discussed makes me think it is not much of an issue. Upscattering of ions may be impeded by annealing. Down scattered ions with their resultant increased Coulomb collisionality may be rapidly reheated by collisions with their hotter neigboring ionsin the interior of the plasma ball. I think this is reflected in the graphs I've seen of Maxwell thermalized gasses. The low energy side is steeper than the high energy side where the energy tails off to very high deviations from the average temperature. It is this high energy tail which participates in most of the fusions in a Tokamak, but it is different in the Polywell. The detramental effects due to Bremsstruhlung radiation may actually increase faster than the leveling off Fusion related to the slope of the fusion cross section curve.

If the Polywell can prevent full thermalization , this high energy tail is suppressed. This changes the competing considerations of fusion cross section and Bremsstruhlung cross section curves. You can move the average temperature more towards desired peak fusion cross section curve without as much Bremsstruhlung pain. This is easy to see with d-T as it has a well defined peak if the fusion cross section curve. It is more subtle with D-D, D-He3or P-B11 cross sections . There is not a fusion cross section peak that quickly fallsof in such an obvious manner as for D-T, but whwn the fusion cross section curve slope becomes less than the slope of the Bremmstruhlung cross section curve, you start losing ground, There is a window of desired energies. For D-T it is possibly about 15-30 KeV average for thermalized plasma, and closer to the peak of 50 KeV average for monoenergetic plasma. For D-D I think the best compromise is about 80-100 KeV average energy for monoenergetic plasma, and 400 KeV for P-B11 (though I am intrigued by the P-B11 resonance peak at ~ 120 KeV). These numbers are center of mass collisional energy. If beam- beam the required energy is only ~ 1/2 of this.

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

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