Electron injection as an engineering issue

[happyjack27]

We should consider under what conditions an injected electron makes it into a wiffleball.

Shoot it through the X-Cusp.

yes, but that's easier said than done.

i think the point, which i agree with, is that one really has to do some calculus to really map out the parameter space where it gets into vs where it doesn't. the "basin of attraction" so to speak. seems to me like one should simplify it to just a single particle into static e & m fields, and this would be some sort of functional integration/variational calculus/ time-sliced, like feynmann's path integral. similiarly, it might need differeentation under the integral sign: http://mathworld.wolfram.com/LeibnizIntegralRule.html

[KitemanSA]

We should consider under what conditions an injected electron makes it into a wiffleball.

Shoot it through the X-Cusp.

yes, but that's easier said than done.

I would wager it is practically s easy done as said. The magic of a small NO FIELD funnel into but not out of the MaGrid!

[hanelyp]

The problem of an X-cusp is it lets electrons out as easily as it lets them in.

[KitemanSA]

The problem of an X-cusp is it lets electrons out as easily as it lets them in.

First, not likely, IMHO. Second, if recirc. works, not necessarily a problem.

[D Tibbets]

If there is not a substantial B field protecting the metal surfaces of an X cusp- direct impingement of electrons would be bad. Remember the minimum allowable non shielded surface are can not exceed ~ 1 part in 10,000.

Saying electrons would not exit or hit here and yet electrons could be easily injected makes no sense.

Recirculation of electrons works because of the attractive magrid potential (at radii greater than the midplane of the magrid). The magnetic fields keep the electrons from directly accelerating towards and hitting the metal surface. Both recirculating electrons and injected electrons needs this magnetic shielding/ focusing. Even with magnetic shielding these small cusps have minimum separation and ExB drift losses apply here just as they do anywhere else- as evidenced by the compromise spacing in WB6.

Keep in mind that WB5 demonstrated the futility of trying to electrostatically control the cusp with e-guns/ repellars placed to close to the midplane of the cusp. By a parellel reasoning an injection of electrons electrostatically only might work, but only with corresponding degradation of ion confinement. The injection efficiency must be a compromise utilizing the magrid potential along with the magnetic cusp geometry. As pointed out the e- guns in WB6 (and WB7 and WB8.0?) were crude filaments that emitted electrons at low voltage through thermionic emission, and according to the patent application also enhanced by escaping ion bombardment. They are electrostatically accelerated by the hopefully symmetric geometry of the magrid surrounding the cusp, and focused further by the B fields. Any of these electrons too much off axis or with deviation angles too great will mirror back. Bussard had a good handle on the best placement of the e-guns from a radii beyond the cusp. I suspect though that fine tuning of the standoff might make significant differences, especially if combined with a tighter initial beam. A large electron emitting surface perhaps further out that is electrostatically and magnetically focused so that a tight beam 'originates" at the preferred standoff distance may be the best approach. The key point is that these external fields must be far enough beyond the magrid cusps that they do not create conditions similar to WB5. Placing an e- gun very near the midplane of the magrid might indeed inject electrons efficiently through an even small non magnetically shielded hole. The e-gun could also act as a repellar and thus - plug the hole to electrons escaping. As mentioned though this interferes with ion electrostatic confinement as per WB5. This is why Bussard stressed the open machine architecture as being one of the absolute requirements. Electrons must pass through the cusps/ pass beyond the radii of the magrid midplane at high speed. They can then be slowed and reversed by the magrid for recirculation. Any new electrons must also fit this pattern. They need to reach high speed before they approach the midplane cusp. If you argue that placing high voltage e- guns close to the cusp midplane would do this, then I would agree. But this also slows the escaping electrons so that they stop and recirculate here as well. IE: cold electrons again in the cusp= bad!

ps: My impression of the x-cusp is that it is essentially the same as the nubs that connect the magnets in WB6, etc. The nub is made larger and a hole is made in the center of it. Even if larger and thus more wire windings can be incorporated, I suspect the magnetic shielding gains per unit of surface area in this region, is offset by the increased total surface area exposed to ExB drift. Not only the central hole exposure counts but also the surface area on the outside of the X cup. It is essentially like placing two regular nubs close together, essentially doubling the surface area.

Dan Tibbets

[KitemanSA]

If there is not a substantial B field protecting the metal surfaces of an X cusp- direct impingement of electrons would be bad. Remember the minimum allowable non shielded surface are can not exceed ~ 1 part in 10,000.

Saying electrons would not exit or hit here and yet electrons could be easily injected makes no sense.

If there is a pair of substantial B fields that protect the metal surfaces but cancel out where there is no metal, then your objection is moot.

[KitemanSA]

Recirculation of electrons works because of the attractive magrid potential (at radii greater than the midplane of the magrid). The magnetic fields keep the electrons from directly accelerating towards and hitting the metal surface. Both recirculating electrons and injected electrons needs this magnetic shielding/ focusing. Even with magnetic shielding these small cusps have minimum separation and ExB drift losses apply here just as they do anywhere else- as evidenced by the compromise spacing in WB6.

But Dan, there IS no B there for the ExB drift to be created from. Hmm?

[KitemanSA]

Keep in mind that WB5 demonstrated the futility of trying to electrostatically control the cusp with e-guns/ repellars placed to close to the midplane of the cusp. By a parellel reasoning an injection of electrons electrostatically only might work, but only with corresponding degradation of ion confinement.

What has this got to do with ANYTHING I wrote? Woo-woo?

[D Tibbets]

If there is not a substantial B field protecting the metal surfaces of an X cusp- direct impingement of electrons would be bad. Remember the minimum allowable non shielded surface are can not exceed ~ 1 part in 10,000.

Saying electrons would not exit or hit here and yet electrons could be easily injected makes no sense.

If there is a pair of substantial B fields that protect the metal surfaces but cancel out where there is no metal, then your objection is moot.

You just described a magnetic cusp. As the opposing fiends approach the local field strength does cancel out, so assuming percise geometry, wire conductivity, etc. there will be a core parellel to the cusp axis where the B field is zero or very close to zero. This core corridor though is very this. Only the very (?) rare electron will be traveling outward directly down this corridoer exactly perpendicular to the center of the machine. The vast majority will travel on a tangent that crosses the midline of the cusp, but travels deep into the B field on either side. The average electron thus will experiance ExB drift. One can argue that this drift- gyro radius is quite small and can be ignored for the majority of the electrons. But this is not what happens. Otherwise seperating the magnets several gyro radii is required. Also, it is consistent with Nebel,s comments about the nubs being a major heat source in WB7. I don't know what adjustments were made in WB7.1- whether more wires was added in the nubs, they were moved outward (even so far as being wall standoffs).

The eletron MFP may be longer than the cusp length, and the average electron velocity perpendicular to the central cusp B field may be small, but with billions upon billions of electrons traversing the cusp per second there will be many ExB driving collisions that results in originally well behaved electrons gaining more trasverse vectors and walking deeper into the side walls of the cusp, untill a certain amount hit the magrid surface. It is unavoidable. The question is by how much it can be minimized, not if it can be avoided.

Also, keep in mind that ExB is only one form of B field diffusion/ drift. EyB (I think) drift is a collision driven movement of charged particles along a B field line laterally- not penetrating deeper but moving laterally or transversly. In a point cusp I think this movement is moot, but in a line cusp- even one that is highly modified as in the Polywell allows for electrons to slide sideways along the width of the cusp till they might hit a bridging struture like a nub (or X-cusp). In WB 6 the nubs had only 1 wire instead of 200 wire windings so they were essentially unshielded against not only the electrons directly exiting at this point but also from EyB drift electrons. Admittedly having many windings in the nub or X cuspwould help, but never as well as magically having no nubs or standoffs. Electrostatic shielding might help the standoffs as well, so long as they are well outside of the mid plane cusp structures (M.Simon once mentioned this as a reasoned/ pursued (?) application). There is a point cusp in the center of the x-cusp so that may have very little loss as it has strong B fields and small overal are. But, the problem is the other side of the metal tube that makes the connections. Here you have twice the surface are exposed to EyB drift. I'm not sure what is gained by having the nub split in two. You can only load each arm with only 1/2 the magnetic windings, while adding another point cusp that while small in loss area is still an extra cusp.

The line cusp/ equatorial cusp in the opposed biconic mirror machine is highly modified in the Polywell. Instead of one intolerable loss line cusp, you have eight cusps that act nearly like point cusps , but they still have line cusps intercepting the nubs. The close proximity of the magnets here though make for very thin and thus much improved losses, but not totally absent losses. I know you believe the X-cusp nubs makes all of the cusps true point cusps, but I am not convinced if this is true, and if so, if the penalty of more vulnerable surface area to ExB drift and accompanyingly greater electron gyro radius hurts as much as any improvement in EyB losses.

As for injecting electrons through the central null field (or very small B field areas very near the center of the cusp. Of course this would very ExB drift with excellent electron aiming, perfect radial velocities would not mirror those electrons. The problem is that achiving this condition is essentially impossible for a large number of electrons. The beam has to have some diameter, and mutual repulsion, two stream instability, etc. results in very quick dispersion of the electrons. The magnetic field can resist this/ focus the electron beam, but this is a tradoff. Collisions drives ExB drift and thus losses, and also mirroring to some extent. Add to that the problems of space charge buildup in the cusps if the electrons are not transiting this cusp region quickly, and you have the picture of the competing processes that have to be addressed to achieve the best overall compromise that not only allows for good electron and ion confinement with reasonable electron injection efficiency.

Another knob to consider is to introduce the ions deeper in the electron induced potential well. This will keep most of the ions further away from the midplane cusps and any electron buildup space charge there. This though essentially makes for a smaller machine from the ions perspective, while introducing different electron and power input dynamics. A small adaptation in this direction though may add to the best compromise arrangement.

Consider WB5. Assume the ions were introduced at a radius of 9/10ths of the radius to the magrid. The electron repellars/ e-gums attracted the ions as they approached the top of their central virtual cathode potential well and led to ion escape. If the ions were introduced at 8/10ths of the radius this effect would be lessen, but at the cost of smaller effective virtual cathode accelerating potential. You are wasting electron power. But...

Dan Tibbets

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[D Tibbets]

PS: If you could inject a laser like beam of electrons through a cusp, the problem then becomes - what is to prevent them from exiting through the cusp on the opposite side of the machine. That the beam quickly disperses prevents this and (my understanding) is that this lateral dispersion results in the limit of ~ 80-90% of the potential well depth relative to the injection voltage. If you are to have sufficient electrons to drive useful fusion rates, you must have densities that results in collision rates that contributes to the charged particle behavior. This limits what you can do with maintaining beam like behavior for sufficient times to squirt electrons through a cusp. Efforts to improve this squirting must compete with necessary conditions for the machine to work.

This is why I tried to address the power loss from unavoidable (though perhaps modifiable) electron injection inefficiencies. Initially low energy electrons from the e-guns are accelerated to high energy, and by my reasoning ~ 2% of them enter the machine and contribute to its operation. The remaining are completely lost, They apparently hit/ ground on unshielded or inadequacy shielded conductive metal surfaces of the magrid and connecting nubs. If they were reflected and climbed back up their potential well and grounded on the faraday cage, or other peripheral surfaces, then my reasoning is that they have given up their KE back to the Magrid (a direct conversion scheme) and thus do not contribute to the electron power injection losses much. In WB6 the electron losses thus changes from 45 Amps * 12,000 V, to ( ~ 2 Amps*12,000V) + (43 Amps * 12 V). Where the reflected electrons ground makes a big difference. Keep in mind that the mirrored electrons outside the magrid can loop back and forth very many times and thus have opportunity to undergo considerable ExB drift to reach the Magrid surface. The EMC patent application pointed out the importance of intercepting escaped ions before they can loop around. These ions though are accelerated by the magrid outside the magrid radius, the electrons are decellerated as they leave the magrid radius outside the magrid. Thus the electrons looping around the magrid (or back and forth) may have smaller radii than the ions. The faraday cage that intercepts the escaping ions may not catch the average low energy electrons- they stop before reaching the intercepting radius and fall back towards the magrid, regaining their KE but having small chance of piercing the cusp because they are already on mirroring field lines . Subsequent electron - electron collisions would modify few of these electrons to cusp penetrating paths, so the inevitable conclusion is grounding on poorly shielded magrid surfaces, and ExB drift mediated collisions on even well shielded magrid surfaces. These collisions are at the full accelerated voltage and thus represent full losses.

That the electron losses were ~ the parent magrid voltage * the e-gun output suggests that not only was the injection efficiency poor, but that most of the rejected electrons reached the magrid through ExB drift or hitting poorly shielded Magrid structures- like the nubs. Simply moving the Faraday cage in to a radius only slightly greater than the E-Guns, might improve this picture considerably. Of course that introduces other multiple cans of worms.

You cannot shield the magrid proper any better, but the nubs/ standoffs could be shielded better. If they make up a significant portion of the high energy impact losses mild to perhaps significant gains might be obtainable.

So, not only try to improve injection efficiency, but also modify the final fate of rejected electrons as much as the many compromises permit!

Note that this consideration also applies to the direct conversion of the KE of the alpha particles from P-B11 fusion as well. The electrons have to be removed from the environment before they are accelerated by the ion decelerating grids, otherwise you are not gaining any ground. This is still another can of worms that needs consideration in that machine.

Dan Tibbets

[KitemanSA]

You just described a magnetic cusp.

Typical cusps have very strong fields in them, not zero fields. lines, points, they both have field that get squeezed to very high strength. Funny cusps don't Their fields get canceled, not strengthened.

[D Tibbets]

You just described a magnetic cusp.

Typical cusps have very strong fields in them, not zero fields. lines, points, they both have field that get squeezed to very high strength. Funny cusps don't Their fields get canceled, not strengthened.

A cusp is a cusp. You can change the B field strength at the surface of the can and change the seperation between the can surfaces. The magnetic field strength is not squeezed to greater strength, they weaken as you travel away from the can surface. How fast they weaken is determined by the inverse square law if you are talking about a single magnet. With opposing magnets the fall off from a chosen can surface is more rapid. I have not seen a diffenative formula but this fall off will be between the inverse square law and I think an inverse 4th power, depending on relative seperation where the strength is measured.

In any case the cusp is defined by the zero null field in the center. That is why they are called a hole. Of course the gradient, shape and maximal strength of the b fields making up the surrounds of the cusp also play a role as they contribute to the behavior of movingcharged particles in the region.

A strange cusp as conceived by Bussard was an imaginary construct of his inadequate modeling. As he admitted, the mathematical line (which has zero or infinitely small width) he used to define the magnet cans means the opposing magnet cans can be infinitely close together. Going backwards from the inverse square law this might logically imply that the B fields approach infinity. But this is a fallacy. The maximal field strength can only be that generated at the can surface by the contained wire windings*current inside the can.
This was understood by Bussard and infinite fields were never implied or expected. What this simplification did result in was cans of infinite thinness with corresponding surface areas of infinitely small size. This resulted in calculated ExB losses of zero or at least insignificant consequence in his modeling. As he pointed out in his Google talk this error was obvous but not caught untill the work up towards WB6. The vulnerability to the magnet can surfaces adjacent to the cusps to ExB losses was was significant because the dimensions was not an imaginary line. He calculated the number of collisions and thus the ExB drift random walk significance at expected electron energies and B field strengths in the cusps and realized that a minimal separation, and thus there was some real width to the of the magnet cans was essential. He chose separations that allowed for ~ 3-5 gyroradius distance migration due to ExB drift so that ExB losses did not dominate in the total loss calculation in the cusp. The total losses is the sum of ExB losses and other diffusion losses and the hole size. The hole size in the "funny cusp" area was zero where the magnets touched (zero width) and this allowed for the corner cusps to be looked at as point cusps. The true situation approached this but did not match it. There has to be some separation of the magnets due to recognized ExB issues and this results in a real width for the cusp between the magnets.

The cusp hole size is defined by the B field strength and geometry presented to the majority of the traveling electrons. Also assumptions about the motions of the electrons apply. Purely radial electrons vectors versus purly lateral vectors and anything in between determines the cusp losses. I think this is the reason that some have suggested that spinning the plasma will improve the containment, just like it helps to decrease edge instability losses in a Tokamak. It might help in this isolated consideration but it may overly penalize fusion rates. There would be less or no ion confluence.
The cusp losses assoiated with the cusp magnetic structure is not determined by the zero null field located between the opposing magnets. This is present in any cusp by definition*. What is important is the gradient on either side of this central null where the B field builds up to levels where charged particles of interest are mirrored back or alternatly rebounded off of the B field surface on vectors back towards the inside of the machine. Stronger B fields and closer spacing helps this situation. But the spacing has a limit where cusp transit losses become less important than the ExB losses so long as the magnets are real vs imaginary lines.

In short, a cusp is a hole (or slit) with zero B field strength in the center and a gradient B field strength increase as you approach the B field generating structure . If you artificially have zero separation between the magnets, then there is zero width and there is no complete cusp. In theory this is reasonable. In reality, while you can do this there is still a cusp throat where B field strength drops and thus vulnerability to ExB losses increase. In some ways this is similar to considerations about the can surface conformity to the B field . Square cans have surfaces closer to the primary confining surface, thus more vulnerability to ExB drift. They go together and once Bussard realized the significance of the need for magnet can seperation the other was obvious. That is why WB6 incorporated both changes, not just the seperation of square cans. It also led to the understanding that the bridging nubs needed to be as small in diameter as reasonably possible ("dog houses" are bad), should be well outside of the midplane of the magrid radius,and should have some magnetic shielding. The more the shielding the better. WB 6 was a compromise that addressed these issues, but only within the budgetary and time constraints available at the time. WB 8 has presumably improved on these issues(at least in part due to lessons learned with WB7), though electron gun issues were apparently not aggressively addressed in the first iteration of the machine.

Dan Tibbets

[hanelyp]

In an X-cusp the fields cancel.
In a face center point cusp or a line cusp the fields are aligned the wrong direction to contain a charged particle.
A "funny cusp", between the close approach of 2 coils, is something between a point and line cusp.

[happyjack27]

it is geometrically impossible to construct a current loop of non-zero amperage that encloses a finite area, such that an area enclosed by it has zero magnetic flux everywhere.

[D Tibbets]

In an X-cusp the fields cancel.
In a face center point cusp or a line cusp the fields are aligned the wrong direction to contain a charged particle.
A "funny cusp", between the close approach of 2 coils, is something between a point and line cusp.

Of course a cusp of any type has a region between the magnets where the fields cancel. That is what I have been trying to say all along.That cancellation happens midway between the magnets assuming both are of equal strength. If different, then the cancellation will occur offset towards the weaker magnet. In practice one magnet could be so much stronger that the cancellation might occur within the can of the second magnet if the strength differential is to great. Of course that is a dismal result as the cusp then is running directly into metal and has no recirculation or injection purpose.

What seems to be the confusing point is how wide these weak field strength and corresponding geometry holes are. An x cusp, as represented, have close magnets that are up to 1/2 as strong as the regular magnet arrangements. The relative cusp hole size is the result of these two parameters. I think you imply that there is no magnetic shielding in the x-cusp location clear up to the borders of the magnet can surfaces. If so then it is a harmful structure. Even WB6 had one wire through the nubs.

An X- cusp while perhaps point like in loss area is still a combination of two short line cusps, an "X". A true point cusp, at least as I define it, has only acute angles from the central cusp to the borders of the magnetic field generating coils. In this case there are no line cusp extensions intercepting metal surfaces. With a line cusp, even with very strong fields surrounding it (very narrow width), at the end of the line matal is intersected. These corner cusps, x cusps might have the vast majority of their loss area in the region where the magnets are furthest apart. but the line extensions allow EyB (or is it properly called ExY?) drift to allow for the electrons to migrate toowards the ends of the line structures and hit the bridging metal directly or through concurrent ExB drift to hit the magnet cans in the narrowest separation region. These losses are a consequence of competing B field strength, distance to vunerable surfaces. It will always have greater losses than point cusps at equal magnetic strength considerations. The corner cusps losses in WB6 type machines is less because the average separation of the magnets is less and thus the B field strength fall off is smaller, or rather the gradient that makes up the effective walls of the cusp is steeper. It magnetically confines electrons better. This of course also means it rejects more of the electrons approaching from outside at any gien cone of electron vectors. In WB6 I think the face centered point cusps might have been better locations for the E-Guns because of this.

Note that better magnetic confinement need not be the dominate consideration once recirculation is considered. With point cusps nearly almost all of the electrons that escape will at least have the possibility of of recovery (ExB losses may be less and ExY losses are nonexistent). With line cusps the magnetic confinement lost electrons will more likely hit a surface and be lost. A smaller fraction will be available for recirculation. The best compromise needs to consider this. That this is evident comes from Nebel's comment about nub heating in WB7. I have pointed out that this heating loss at the nubs need not only be from electrons escaping internal confinement, but also failed electrons from the e-guns that never successfully penetrated the cusp to the inside of the machine.

The elimination of nubs by using wall standoffs, while not perfect, decreases the significance of ExY drift. The electron may move from the corner region to the closest approach of the magnet cans but without a bridging structure that traverses the line cusp (at least at radii close to the magrid radius where the recirculation action is occurring) direct impingement on the metal surface will be minimized. The electron would only migrate into the neighboring corner region. There is no significant unshielded or minimally shielded surfaces exposed. ExB losses would still need to be considered (why there needs to be at least several gyroradii separation of the magnet cans,) but the ExY (EyB) contributions would be less painful.

Dan Tibbets