Bussard's bremsstrahlung calculation

[D Tibbets]

That doesn't make any sense to me. Indeed, I'd assert just the opposite: Inside the containment for the electrons, there should be no forces on the ion at all, while outside the containment, Coulomb's Law applies quite nicely.

I agree with you on what happens to the ion outside. But as the ion enters the electron containment should it not have a force on it equal to the electrons that still exist in between it and the center? In essence, the force is going to depend on the distribution of electron densities to some extent. If 99% of the electrons are on the outer edge, then the ion will stop experiencing any force from them the moment it cross that barrier. If 99% are pressed up tight to the center, they will continue exerting force on the ion all the way up to that tight inner ring. The reality likely falling some place in between. Or am I just extrapolating to far from conventional electrostatics?

Recall that a virtual anode might form if the ions are focused to the extent that they will concentrate towards the center, past the region where the electrons dominate, to where the converging ions become the dominate species. At that point the Coulomb forces would be reversed- the ions would slow as they traveled the last bit of distance to the center untill thier inertia was overcome and they reversed or they traveled through the center and started acellerating away from each other and towards the more electron dominate area. I'm guessing that when the ions become concentrated in the center to such a degree that a significant portion of them reverse is when the virtual anode 'blows out the well'. Since the converging ions would be pulling some of the electrons with them, within limits, this might improve the effective electron concentration in the center and perhaps counteract some of the effect of the ions pulling electrons along as they travel outwards.

Dan Tibbets

[bcglorf]

That doesn't make any sense to me. Indeed, I'd assert just the opposite: Inside the containment for the electrons, there should be no forces on the ion at all, while outside the containment, Coulomb's Law applies quite nicely.

I agree with you on what happens to the ion outside. But as the ion enters the electron containment should it not have a force on it equal to the electrons that still exist in between it and the center? In essence, the force is going to depend on the distribution of electron densities to some extent. If 99% of the electrons are on the outer edge, then the ion will stop experiencing any force from them the moment it cross that barrier. If 99% are pressed up tight to the center, they will continue exerting force on the ion all the way up to that tight inner ring. The reality likely falling some place in between. Or am I just extrapolating to far from conventional electrostatics?

Recall that a virtual anode might form if the ions are focused to the extent that they will concentrate towards the center, past the region where the electrons dominate, to where the converging ions become the dominate species. At that point the Coulomb forces would be reversed- the ions would slow as they traveled the last bit of distance to the center untill thier inertia was overcome and they reversed or they traveled through the center and started acellerating away from each other and towards the more electron dominate area. I'm guessing that when the ions become concentrated in the center to such a degree that a significant portion of them reverse is when the virtual anode 'blows out the well'. Since the converging ions would be pulling some of the electrons with them, within limits, this might improve the effective electron concentration in the center and perhaps counteract some of the effect of the ions pulling electrons along as they travel outwards.

Dan Tibbets

I think I'm getting that far. It's still intuitive from basic electrostatics that the force on an individual ion is going to also depend on ion density between it and the core, with virtual anodes and cathodes being just specific cases. My bigger overall question was if it's even accurate of me to understand the forces on an ion that way.

Am I correct to understand that at any given location, the force on an ion is just the net electrostatic force on it from the ions and electrons that exist between it and the center?(assuming the sphere approximation is acceptable) It following that any ions and electrons further from the core in essence cancel one another out for 0 net force. Or am I missing the importance of other effects?

[TheRadicalModerate]

Am I correct to understand that at any given location, the force on an ion is just the net electrostatic force on it from the ions and electrons that exist between it and the center?(assuming the sphere approximation is acceptable) It following that any ions and electrons further from the core in essence cancel one another out for 0 net force. Or am I missing the importance of other effects?

Sounds right to me. The force will be proportional to r inside the sphere, not 1/(r^2), as it is outside the sphere. I had this wrong in a previous post--the force is only 0 in a hollow sphere.

But, in addition to the electron-to-ion attraction, you will need to add in the repulsive force from the virtual anode, which I can't compute (the ion density is non-uniform with radius). And then, of course, there are magnetic and MHD forces to compute, which I really, really, can't compute.

[bcglorf]

The force will be proportional to r inside the sphere, not 1/(r^2), as it is outside the sphere.

Another dumb question from me but that is only if we can assume electron density is uniform within the sphere, right?

Unless I miss my guess aren't we headed toward declaring ion density a function of electron density within the sphere? And of course, electron density within the sphere will more than likely be a function of ion density within the sphere. At which point I think we're in build a simulation mode, which I like because I'm a programmer by trade. Then I look at the processing cycles needed to model things on even a small scale and realize the cheapest simulation is already being built by Nebel. I hate waiting.

[93143]

The average ion velocity/bulk energy aren't proportional to R to any power, at least not without an offset. Think about it - at the radial point of the turnaround, the average ion velocity is ZERO metres per second, implying a differential residence time (seconds per metre) of...?

Are you trying to assert that the residence time doesn't have a limit at R? I'm not buying that one.

Of course not. But there's a singularity there. Obviously the residence time is infinitesimal at the turnaround point, but it's infinitely closer to being finite than anywhere else, because the ions momentarily stop.

Thermal effects will prevent any singularities in a real system. You don't think the density at r=0 is actually infinite, do you? Well then...

I went back and looked at this again. This is basically just the Kepler problem in one dimension (which is where I got the 1/R). But, being pretty much incompetent analytically, I went off and did a cheesy numerical simulation as well. From that I got pretty much the same answer, except it appears that the time in any given shell of dR thickness is proportional to R, not sqrt(R).
(Here's the spreadsheet with the simulation., as an XLS file. Note that, for clarity, I'm not worrying about constants in this. Also note that I'm assuming that the forces on a particle within the wiffle field are 0.)

Sure looks to me like density is considerably higher in the center than it is on the edge.

You forgot to take into account the entry velocity in each shell when computing the residence time. Your calculation assumes acceleration from zero in each shell - not for the velocity, just for the residence time.

Also, this isn't a Kepler problem. The attractive force comes from distributed space charge, so the ions won't keep seeing the whole charge all the way to the core. I did notice you capped the potential at a certain radius, which was good of you...

Here, have some Matlab modelling.

Source code.

The plots labeled 'K' are your clipped Kepler profile. '2' indicates a parabolic well of the form E_k = 1-r^2, and '8' indicates a high-order, somewhat more Carlson-friendly well with E_k = 1-r^8. No attempt has been made to model the formation of a virtual anode. 1001 concentric spherical cells were used for all simulations.

Please note that the collision rates are strongly dependent on velocity and density, and thus the approximations made in the code for the first and last cells do not result in accurate collision rates. The collision rate per cycle is normalized to the maximum value, which is typically the edge value, sometimes by a large margin; it is possible to get a rough idea of how the well shape affects the ratio between central and edge collisionality this way, but the accuracy is not very good. It should nevertheless be noted that the dependence of collisionality on velocity is stronger than its dependence on density, and thus shallower well profiles, where the velocity near the edge stays lower for longer, will have a higher ratio of edge-to-centre collisionality.

Someone correct me if I've said or done something stupid here. It's been a while since my gas kinetics course and I don't have my textbook on me...

Here are the plots from the parabolic well:

To put it another way, if an ion oscillates through the core in a perfectly sinusoidal pattern, it spends about 9% of its time in the outermost 1% of its trajectory in a radial sense, but less than 4% of its time in the next-outermost 1% (which taken as a spherical shell has 98% of the volume of the outermost layer, meaning the average density in this second layer is less than half as high). I'm not proposing that the ion motion is actually sinusoidal, or that it's as locally monoenergetic as implied by this example (the real system will be smeared out somewhat), but it illustrates the point...

I'm pretty much assuming that it is monoenergetic, at least for purposes of this analysis. All I'm trying to show is that the "crowded edge" assumption should be re-examined. I agree that the velocity profile is periodic, but it definitely isn't sinusoidal.

It doesn't have to be sinusoidal; the principle is the same. None of the results presented above have sinusoidal ion paths. It was just an illustrative example.

The R^-2 density profile only applies near the centre. There's definitely a spike there, subject to thermal smearing of course, but the R^-2 profile assumes that the ions are at full speed, so it's invalid once they get away from the core and start to slow down.

That doesn't make any sense to me. Indeed, I'd assert just the opposite: Inside the containment for the electrons, there should be no forces on the ion at all, while outside the containment, Coulomb's Law applies quite nicely.

I see where the problem is. You're thinking like chrismb; that the ball of electrons is smaller than the ball of ions. It's not. They're the same size.

Besides, I said DENSITY profile, not FIELD profile. Collisionless, spherically converging/diverging ions at a constant radial velocity (near the centre, the ion velocity can be approximated as constant) will have a density profile that follows R^-2. That is, the density at the exact centre is theoretically infinite. Thermal smearing will falsify this, of course, but there will still be some sort of density maximum in the centre unless the whole wiffleball thermalizes completely.

My point is, there's ANOTHER density maximum at the edge, and it's more important in a collisional sense than the one in the centre. Not much fusion, but quite a bit of thermalization, and at a reasonably low temperature. Hence annealing.

I assert nothing with regard to the actual potential profile in a wiffleball. My attempt at simulating the coupled electrostatic/collisional process, using a finite-volume representation of the Boltzmann equation, is still in work and may remain so for some time. It looks almost intractable even with one space dimension and two velocity dimensions, and I'm not semi-retired...

[TheRadicalModerate]

You forgot to take into account the entry velocity in each shell when computing the residence time. Your calculation assumes acceleration from zero in each shell - not for the velocity, just for the residence time.

Yup, you're right. Stupid math error on my part. What a delightfully counter-intuitive density profile!

I see where the problem is. You're thinking like chrismb; that the ball of electrons is smaller than the ball of ions. It's not. They're the same size.

I assume that you're including the electrons that escape containment through the cusps and go whipping around the magrid before winding up back in the center. But I'd think that the density profile for those would be completely constant and relatively high up to the containment boundary, then dropping to almost (but not quite) nothing outside it. Is the density of the escapees really high enough to affect ion dynamics?

[93143]

I see where the problem is. You're thinking like chrismb; that the ball of electrons is smaller than the ball of ions. It's not. They're the same size.

I assume that you're including the electrons that escape containment through the cusps and go whipping around the magrid before winding up back in the center. But I'd think that the density profile for those would be completely constant and relatively high up to the containment boundary, then dropping to almost (but not quite) nothing outside it. Is the density of the escapees really high enough to affect ion dynamics?

No - the way I've always understood it is that the ions are supposed to be formed at the edge of the wiffleball, where the magnetic field suddenly goes to zero. This results in the edge of the wiffleball featuring fast electrons being turned around by the magnetic field and slow ions being turned around by the electric field, both at the same radial point.

Given the high field of a power reactor, the ions can in fact be contained by the wiffleball, so I suppose the idea of dropping the ions in from higher up using an ion gun makes a bit more sense - but since the ions are magnetically confined, they still won't leave the ball. In this case there should be a very thin sheath of ions outside the electron ball simply due to the fact that it takes longer for them to turn around when they hit the field.

This second idea seems to break the understanding of annealing I've developed, so I'm not a big fan of it. I'm not sure why they want ion guns. Maybe it has something to do with multiple well formation, so you get annealing regardless of how hot the ions are in the outer layer... or maybe the reduced edge collisionality doesn't matter so much once thermal and virtual-anode effects have done their work on the central spike... I don't know...

In either case, an ion that makes it outside the magrid will be repelled by it and leave, either being collected by the power converter system or smashing into the wall or an electron gun or something. There can be no annealing there.

Also, please note that my "energy exchange rate" plot is actually a collision rate based on energy-exchange cross section, and as such does not actually represent what the title says it does...

[D Tibbets]

Actually, I tink the electron ball is larger than the ion ball. The electrons are susposed to fly outward due to the excess electron repulsion untill they are turned at the Wiffleball border due to the sudden exposure to a strong magnetic field. The ions are purported to stay lower in the potential well so that they do not see the magnetic field. Except of course for upscattered ions, and I'm not sure how ions introduced from outside the magrid will have a a potential well peak so deep within the machine.

Dan Tibbets

[93143]

The proposed ionization scheme I'm familiar with uses electron cyclotron resonance. This would be impossible to pull off inside the wiffleball because there's no magnetic field. You have to do it at the edge, at the very lowest. If you just let neutral-electron collisions do the job, perhaps ionization could happen inside the wiffleball, but there wouldn't be a sharp boundary to the ion profile - there would be at least some that got ionized all the way up at the edge, and some that got considerably further in; the distribution would be very far from monoenergetic. Ion guns are right out; you can't stick an ion gun down inside the wiffleball and expect to have a working power reactor...

Besides, even if you somehow figured out a way to do it, a good chunk of the potential well would be wasted...

[TheRadicalModerate]

The proposed ionization scheme I'm familiar with uses electron cyclotron resonance. This would be impossible to pull off inside the wiffleball because there's no magnetic field. You have to do it at the edge, at the very lowest. If you just let neutral-electron collisions do the job, perhaps ionization could happen inside the wiffleball, but there wouldn't be a sharp boundary to the ion profile - there would be at least some that got ionized all the way up at the edge, and some that got considerably further in; the distribution would be very far from monoenergetic. Ion guns are right out; you can't stick an ion gun down inside the wiffleball and expect to have a working power reactor...

Besides, even if you somehow figured out a way to do it, a good chunk of the potential well would be wasted...

Why wouldn't you just generate your ions external to the magrid and inject them with just enough energy to get them repelled inward instead of outward? Same thing for the electrons, albeit with a fairly hefty amount of energy?

[93143]

Why wouldn't you just generate your ions external to the magrid and inject them with just enough energy to get them repelled inward instead of outward? Same thing for the electrons, albeit with a fairly hefty amount of energy?

Ions: I explained this above. Technically you could fire the ions in just fast enough to overcome the repulsion from the magrid, but there's a space between the magrid and the wiffleball, and the ions will re-accelerate across it. Once they're trapped in the wiffleball, their edge speed is whatever they had when they entered it. This may or may not be a problem.

Electrons: You don't need to start them with any energy at all; you can just drop them in. The magrid is positive and does all the accelerating necessary. This is actually where the "drive voltage" is applied in the system - between the electron emitters and the magrid. ...or did you understand this already, and your wording was just funny?

[TheRadicalModerate]

No - the way I've always understood it is that the ions are supposed to be formed at the edge of the wiffleball, where the magnetic field suddenly goes to zero. This results in the edge of the wiffleball featuring fast electrons being turned around by the magnetic field and slow ions being turned around by the electric field, both at the same radial point.

Confused again. Sorry, everybody, for hijacking this thread for polywell 101. I thought that one-dimensionally, from the outside in, you'd encounter:

1) The outer lobe of the magnetic field generated by the magrid (with field lines that look like one end of an ellipsoid). You'll see a relatively small number of electrons with very low energy here, getting ready to follow a field line back through the magrid on their way back to the wiffle-field.

2) The magrid itself, charged up to the potential needed to get the ions to the bottom of the well at some specific energy, and containing coils that generate the magnetic field lines.

3) The space between the magrid and the wiffle-field. This space has roughly radial magnetic fields lines, with ions oscillating back and forth between the center and some small distance just inside the magrid, and a fairly small number of electrons whipping out and back in along the field lines.

4) The wiffle-field magnetic sheath, which is the other end of the magnetic field line ellipsoid, except it's been flattened at the end due to being expelled by the plasma in inside the sheath.

5) The spherical area inside the wiffle-field sheath, which consists of a) thermalized (but fairly cold) electrons that are bouncing around inside the core, and b) very hot, hopefully monoenergetic ions that come screaming through the wiffle field (because they're heavy and the field strength isn't that high), through the center, and out the other side (unless they fuse).

6) The center of the machine.

Does this gibe with your understanding?

I've always assumed that the core (the volume inside the sheath) was filled with cold, thermalized electrons, but I think that you may be implying that most of the electrons are actually trapped in the sheath, oscillating relatively slowly between cusps and only occasionally escaping through the cusp to whip through the machine. Is this correct?

Given the high field of a power reactor, the ions can in fact be contained by the wiffleball, so I suppose the idea of dropping the ions in from higher up using an ion gun makes a bit more sense - but since the ions are magnetically confined, they still won't leave the ball. In this case there should be a very thin sheath of ions outside the electron ball simply due to the fact that it takes longer for them to turn around when they hit the field.

I don't see how the ions can be magnetically contained. A magnetically contained ion is a thermalized ion--bad.

But this does raise a question: what is the trajectory of an ion through the magnetic sheath? I'm assuming that it gets transversely deflected by some amount, which in turn implies that their trajectory outside the sheath has to be transversely offset just enough so that it will be radial inside the sheath. Otherwise, you can't focus any narrower than the whole interior of the sheath, which is a) going to produce a lousy cross-section and b) is going to thermalize the ions more than they would be otherwise.

[TheRadicalModerate]

Why wouldn't you just generate your ions external to the magrid and inject them with just enough energy to get them repelled inward instead of outward? Same thing for the electrons, albeit with a fairly hefty amount of energy?

Ions: I explained this above. Technically you could fire the ions in just fast enough to overcome the repulsion from the magrid, but there's a space between the magrid and the wiffleball, and the ions will re-accelerate across it. Once they're trapped in the wiffleball, their edge speed is whatever they had when they entered it. This may or may not be a problem.

Ah, I finally understand where we're disagreeing. You're saying that the ions oscillate inside the magnetic sheath, while I was assuming that they oscillate through the sheath, with max R being almost back to the magrid.

If you're correct, then I'm not seeing how there's enough potential difference from the electrons to form a well to oscillate in in the first place. My model for the electron structure inside the wiffle field was that you had the majority of the electrons caught in the sheath, oscillating slowly back and forth between cusps (with some small number escaping through the cusps and whipping around the B field lines through the magrid coils). Then there are obviously non-trivial minorities of electrons that are wandering free inside the sphere and whipping around the field-lines outside the sheath.

Of those, the only ones generating a potential difference are the electrons that are wandering free in the sphere. The ones trapped in the sheath behave like your ever-popular hollow charged shell, as does any positive charge on the magrid. So how do you have a well at all?

Electrons: You don't need to start them with any energy at all; you can just drop them in. The magrid is positive and does all the accelerating necessary. This is actually where the "drive voltage" is applied in the system - between the electron emitters and the magrid. ...or did you understand this already, and your wording was just funny?

I don't think this is right. The positive magrid is attractive to the electrons. You have to fire the electrons into the center at pretty much an energy that's equal to your well potential, or the electrons will neutralize the magrid. The only thing that keeps them from flying back out is that they get caught in the wiffle field.

As for "drive voltage" (or, more accurately "drive current"), this is merely the amount of current required to replace electron losses from the wiffle field. The electrons constituting this current also have to be fired in with an energy equal to the well potential.

[D Tibbets]

No - the way I've always understood it is that the ions are supposed to be formed at the edge of the wiffleball, where the magnetic field suddenly goes to zero. This results in the edge of the wiffleball featuring fast electrons being turned around by the magnetic field and slow ions being turned around by the electric field, both at the same radial point.

Confused again. Sorry, everybody, for hijacking this thread for polywell 101. I thought that one-dimensionally, from the outside in, you'd encounter:

1) The outer lobe of the magnetic field generated by the magrid (with field lines that look like one end of an ellipsoid). You'll see a relatively small number of electrons with very low energy here, getting ready to follow a field line back through the magrid on their way back to the wiffle-field.

2) The magrid itself, charged up to the potential needed to get the ions to the bottom of the well at some specific energy, and containing coils that generate the magnetic field lines.

3) The space between the magrid and the wiffle-field. This space has roughly radial magnetic fields lines, with ions oscillating back and forth between the center and some small distance just inside the magrid, and a fairly small number of electrons whipping out and back in along the field lines.

4) The wiffle-field magnetic sheath, which is the other end of the magnetic field line ellipsoid, except it's been flattened at the end due to being expelled by the plasma in inside the sheath.

5) The spherical area inside the wiffle-field sheath, which consists of a) thermalized (but fairly cold) electrons that are bouncing around inside the core, and b) very hot, hopefully monoenergetic ions that come screaming through the wiffle field (because they're heavy and the field strength isn't that high), through the center, and out the other side (unless they fuse).

6) The center of the machine.

Does this gibe with your understanding?

I've always assumed that the core (the volume inside the sheath) was filled with cold, thermalized electrons, but I think that you may be implying that most of the electrons are actually trapped in the sheath, oscillating relatively slowly between cusps and only occasionally escaping through the cusp to whip through the machine. Is this correct?

Given the high field of a power reactor, the ions can in fact be contained by the wiffleball, so I suppose the idea of dropping the ions in from higher up using an ion gun makes a bit more sense - but since the ions are magnetically confined, they still won't leave the ball. In this case there should be a very thin sheath of ions outside the electron ball simply due to the fact that it takes longer for them to turn around when they hit the field.

I don't see how the ions can be magnetically contained. A magnetically contained ion is a thermalized ion--bad.

But this does raise a question: what is the trajectory of an ion through the magnetic sheath? I'm assuming that it gets transversely deflected by some amount, which in turn implies that their trajectory outside the sheath has to be transversely offset just enough so that it will be radial inside the sheath. Otherwise, you can't focus any narrower than the whole interior of the sheath, which is a) going to produce a lousy cross-section and b) is going to thermalize the ions more than they would be otherwise.

My take-

#1 sounds OK if you are refering to recirculating electrons.

#2 The magrid positive charge serves to acellerate the electrons through a cusp and to the center of the machine (drive energy. You want to avoid this effect with ion guns because because the ions would never penitrate past the positive magrid (unless you give the ion an acellerating energy equal or greater than the magrid potential-with some possible fudge facters), unless you place the ion gun at or just inside the midline of the magrid cases (charged particals do not see electrical potenials from a surrounding sphere) . The problem with this is that the ion guns would be deeper in the cusps and may harm the critical recirculation efforts. Bussard claimed that neutral gas puffers were inneficient and problamatical for small machines, but in larger machines these problems would decrease, so in a power producing Polywell gass puffing may be a better compromise than ion guns.

#3 No. The magnetic field lines are transverse/ almost perpendicular to the center of the machine, except at the cusps where they curve untill they are radial and hopefully this accounts for a very small percentage of the magnetic border as these areas are the 'holes' in the Wiffleball. Once the ions get into or are created inside the Wiffle ball, they idealy never see any magnetic field. They are contained by the electrostatic field setup by the electrons. Only the electrons reach the Wiffleball border where they suddenly see the concentrated magnetic field and turn around. Because there are more electrons in the Wiffle ball, and the electrons spend most of thier time deep within the well , the ions will be confined into a smaller ball that does not reach the Wiffleball border- ie: the ions are electrostatically contained, not magnetically contained. Certainly any ions that do reach the Wiffleball will be contained fairly well except at the cusps which are bigger for them I think. Presumably having too many ions reach the Wiffleball border is bad for several reasons- see arguments by A. Carlson.

#4 OK, this inside flattened area serves to pinch the cusps almost closed, and perhaps allows the magnetic border to be more nearly spherical (good?) while still having a slight convex shape (essential?).

#5 No. The electrons, if thermalized would be a random cloud evenly distributed throughout the Wiffleball, in fact due to mutual repulsion ( they would eventually accumulate near the Wiffleball border till they managed to leak out the cusps. What creates the electrostatic well is the inertia of the electrons- they are injected at high energy (speed) and slow due to the other converging electrons, then pass through the center or reverse and acellerate till the magnetic border is reached. Then the process repeats. Hopefully the electrons stay nearly radial long enough, thermalization would tend to scatter them in all directions, not just radial. As the electrons inevitably thermalize and some of them cool, the potential well would be lost. But, the continous dynamic injection of new monoenergetic high energy radially directed electrons replaces these thermalized electrons which are apparently preferentially leaking out of the cusps; thereby maintaining the negative potential well, that serves to both contain the ions and acellerate them to the center. What is going on in detail and how the electrons and ions are interacting on the small and large scale, and the time scale is what is challenging to understand.

#6 Your conclusions are confused. I've had difficulty concieving how the charged particles interact with the magnetic field, I was traped into thinking the particles were always traped on a magnetic field line and spiraled around it (gyroradius) indefinatly untill it managed to hit something else. But, I believe this only applies to the cool low energy charged particles with resultant tiny gyroradii ( unless you are talking about realitively huge magnetic fields that have very low gradiants between field lines (like Earth's magnetic field)). But in a magnetic field that goes from zero to a large strength in a very short distance (like the Wiffleball border) the strength that the particle sees is greater on one side of it's half orbit (gyroraduis) than the otherside, so instead settling into a circular orbit, it is parabolic. Add to that the absence of the field inside the Wiffleball border and the particle is now traveling free in almost the opposite direction as the original. Or put simply, the charged particles in the conditions in the Polywell see the magnetic field as a wall that it bounces off of. That some of the terms in electromagnetics, like mirroring and bouncing, are used here interchangably doesn't help the confusion.


Dan Tibbets

[93143]

I don't see how the ions can be magnetically contained. A magnetically contained ion is a thermalized ion--bad.

The magnetic field in a power reactor is on the order of 10 T. At 10 T, the gyroradius of a 50 keV boron-11 ion is about 2 mm, and it only gets smaller as the ions get colder. The ions are confined magnetically whether you like it or not. They're also confined electrostatically, of course, but the magnetic field will prevent them from exiting the wiffleball (except at a cusp) even if they have the energy to do so from an electrostatic standpoint.

As for thermalization, magnetic confinement is no worse than electrostatic. Actually, it could be better, since the high-energy tail doesn't leak away and cause the average energy to drop. So long as the fusion rate and the cusp escape rate combined are faster than the global thermalization rate, you're fine.

If you're correct, then I'm not seeing how there's enough potential difference from the electrons to form a well to oscillate in in the first place... the only ones generating a potential difference are the electrons that are wandering free in the sphere. The ones trapped in the sheath behave like your ever-popular hollow charged shell, as does any positive charge on the magrid. So how do you have a well at all?

A Carlson sheath is of finite thickness and thus will generate a finite well, given a global excess of electrons. Is it enough by itself? I believe it won't be, unless the ions have a zero-speed point above the wiffleball, at about the level of the magrid - this could be achieved by ion guns, for instance. Please note that having a zero-speed point outside the wiffleball does NOT mean the ions reach that point at every pass. Since they are magnetically confined, they can't get out that far once they're in the wiffleball, so they have a significant amount of energy at their turnaround point (which is about the same as that of the electrons; the edge of the wiffleball).

There's some uncertainty, actually, about the internal structure of the wiffleball - namely, the shape of the well; whether there's a sheath or whether the excess of electrons goes deeper. But one thing is certain - your picture is impossible. The kinetic energy of a charged particle depends directly on where in the potential well it is, which means you can't possibly have a potential well in a sphere containing uniformly cold electrons. The excess electrons would all flee to the edge and you're back to a sheath. No, if there's a potential well that extends deep into the wiffleball, the electrons are only cold at the bottom of it.

Electrons: You don't need to start them with any energy at all; you can just drop them in. The magrid is positive and does all the accelerating necessary. This is actually where the "drive voltage" is applied in the system - between the electron emitters and the magrid.

I don't think this is right. The positive magrid is attractive to the electrons. You have to fire the electrons into the center at pretty much an energy that's equal to your well potential, or the electrons will neutralize the magrid. The only thing that keeps them from flying back out is that they get caught in the wiffle field.

You drop a cold (~10 eV) electron from 0 V at an electron gun. It sees the charge on the magrid plus the charge of the wiffleball and all it contains. Net result? Well, the magrid is held at (say) 20 kV, which means it holds whatever charge will result in that potential, automatically compensating for the presence of the wiffleball's charge. The electron sees a net positive charge and heads for it.

The (20 keV) electron passes the magrid. Gauss' Law kicks in and the electron suddenly can't see the magrid any more. It just sees the wiffleball. Now it starts to slow down. Since the wiffleball is net negative, the electrons want out. As you yourself said, "The only thing that keeps them from flying back out is that they get caught in the wiffle field." In fact, some of them hit the cusps and DO fly out, but once they clear the magrid, they see the positive charge and head right back in.

As for "drive voltage" (or, more accurately "drive current"), this is merely the amount of current required to replace electron losses from the wiffle field. The electrons constituting this current also have to be fired in with an energy equal to the well potential.

Uh, voltage and current are two different things...

The "energy equal to the well potential" is picked up by the electrons on their trip from the level of the emitters to the level of the magrid. THAT is where you apply the drive voltage - between the emitters and the magrid. Note that basically all of the high-energy electron loss is to the magrid structure, so the current loop does close.

Multiply the drive voltage by the drive current and you get the drive power, which is what you want to minimize. Hence the worries about losing high-energy electrons. No one cares about losing cold electrons off the fuel ionization process, because you haven't pumped tens of keV into each one of those...

Once the ions get into or are created inside the Wiffle ball, they idealy never see any magnetic field. They are contained by the electrostatic field setup by the electrons. Only the electrons reach the Wiffleball border where they suddenly see the concentrated magnetic field and turn around. Because there are more electrons in the Wiffle ball, and the electrons spend most of thier time deep within the well , the ions will be confined into a smaller ball that does not reach the Wiffleball border- ie: the ions are electrostatically contained, not magnetically contained.

This is the only part of your post I have a problem with. You seem to have forgotten that electric fields are conservative. If the ions are electrostatically confined to never even reach the border of the wiffleball, one of three things is true:

1) The ions were formed at low energy inside the wiffleball, significantly below the magnetic boundary. It's possible to run the machine so that this is true for most (not all) of the ions, but it's probably not desirable from an energy distribution standpoint, for reasons I've already mentioned.

2) The ions have lost a substantial chunk of energy to collisions and cyclotron radiation and heating of cold electrons and such. If this is true, the distribution is probably long since thermalized and we're in big trouble.

3) The ions had negative kinetic energy (ie: imaginary velocity) when they were formed.

Note that all three of these options result in only partial utilization of the potential well, so you have to crank the voltage higher for the same result...