The problem with ion convergence

[Tom Ligon]

Whatever papers you guys turn up, I'm sure Mark Duncan would be willing to post them on Askmar.

[Art Carlson]

Let me summarize my views on this thread.

1. Art’s objections are centered on the assumption that flux surfaces are equipotential surfaces. That’s where the defocusing aberrations come from. That’s a valid assumption for l.t.e. plasmas like Tokamaks, but not IECs. Electron inertia allows gradients along field lines. It’s not clear that there will be any aberrations.

It is true that I am assuming that the flux surfaces are (more or less) equipotential surfaces, and that electron inertia can modify this. By now you should know that I am a friend of quantitative estimates. Let's try to see how much difference electron inertia can make. Ohm's law is essentially force balance on the electrons. Using the convective derivative for dv/dt rather than collisions to balance the electric force, we have:
m*(v dot nabla)v = e grad phi
This is perhaps more intuitive in the integrated form:
m*v^2/2 = e*(Delta Phi)
To significantly change an electric potential, we need to let a significant fraction of the electrons run downhill and then throw them away. If we let them reflect, then the mean velocity is zero. I don't think you really want to do this, Rick. That would mean that the electrons have no confinement. Even if you substitute "10%" for each time I used the word "significant", you are likely to wind up with a figure for the energy loss which is deleterious. Remember that we have a discrepancy of 8 orders of magnitude on the table. (Which may not be right. I haven't double checked my calculation, and I don't think anybody else has either.) Even a little bit of non-sphericity is likely to mess up convergence in short order. While it may not be "clear" to you that there are any aberrations, everything points in that direction. If I made a mistake, it's up to you to point it out or offer an alternative estimate.

I'm not sure what went wrong, but this argument of mine is flakey somewhere and I hereby withdraw it. But I'll be back.

[jmc]

Let me summarize my views on this thread.

1. Art’s objections are centered on the assumption that flux surfaces are equipotential surfaces. That’s where the defocusing aberrations come from. That’s a valid assumption for l.t.e. plasmas like
Tokamaks, but not IECs. Electron inertia allows gradients along field lines. It’s not clear that there will be any aberrations.

I'm sure at low densities this is the case, I find it hard to believe at, say, 10^20 particles per metre cubed collissions won't create local thermal equilibrium.

Which brings me to the point that I have been asking for quite a while and digging up the literature but have found no answer to.... what mechanism is supposed to correct the electron speeds or remove the electrons after they've collided with each other? How much energy will this constant extraction on maxwellized electrons and replacement of non-maxwellian electrons take.

I'm sure that a Polywell power plant will have shots that will last much much longer than an electron-electron collission time, otherwise the ions wouldn't even get a chance to make one pass through the core.

[Art Carlson]

Then maybe it is worthwhile for me to make an estimate after all. Just the order of magnitude and the scaling. I'm thinking of mono-energetic ions running up an electric potential beach and being reflected back.

...

I ~ (Delta*lambda_D^-4*n^-1)*(T/T_edge)^3/2
(Honestly, I lost a factor of e^5/2 along the way, but it's too late at night to track it down. The units here work out.)
Consequences? As a first guess, let's take Delta~1e-1m, lambda_D~1e-4m, n~1e21, and T/T_edge~1e-2 (due to the lumpy surface). Then we get I~1e-8. I don't offer any guarantees that this number is right. It is much smaller than even I expected. I - or better yet, one of you - should check the calculation by the cold light of day. If it is correct, then the lumpy surfaces will produce so much transverse motion initially and on every subsequent bounce, that annealing doesn't have a snowball's chance in hell of acting to keep ion convergence.

I checked my math and it looks good up to the end. (I even found the missing factor of e^5/2.) I think I made two mistakes in the numerical evaluation, though, the worst being "T/T_edge~1e-2" when I obviously wanted to say "T/T_edge~1e+2". Cranking the numbers through again, I get I ~ 1e-3, so the discrepancy now on the table is a mere three orders of magnitude, rather than 8.
It would probably be a good idea to go back and include the numerical factors, rough as they are, and check that the physical values I used are consistent and reasonable. The numerical factor is
(2pi/3)*ln_Lambda*(4pi)^-2*2^-5/2 ~ 15/(96*pi*sqrt(2)) ~ 0.035.
This reduces the signle-bounce collision integral I by a factor of 30, so we are now at 4 to 5 orders of magnitude discrepancy. That gives me some confidence that using different parameters aren't going shift us into a regime where annealing is possible (according to this model, which is the only one that includes lumpy surfaces). Looking at it the other way around, the edge (ion) temperature would have to be about one part in 10^5 of the central temperature, or something less than 1 eV, in order for annealing to work.

[jmc]

Yes around the 1eV mark is sort of what I estimated aswell. I dont think you can ionize all the ions within 1eV of each other, especially when the potential well goes all the way down within a debye sheath.

I toyed around with the idea of accelerating all the ions to the same energy, outside, neutralising them into a monoenergetic neutral beam and reionizing them at the bottom of the almost square well here.

http://talk-polywell.org/bb/viewtopic.php?t=467

Technically I think its possible to creat a neutral beam where the ion energies are within 1eV of each other. If the equipotentials aren't spherical, however it would destroy it.

The only hope for convergence in my opinion is if the wiffleball just so happens to form a spherical "bubble" of zero field in the centre with the immediate magnetic fieldlines around it tracing out almost spherical surfaces. I don't see why this situation would be impossible, though it may well be costly to achieve.

[drmike]

Why does the potential have to be spherical? All that you need is radial currents - net currents. As long as the net transverse motion cancels and the ion energy is within fusion range when the ions get to the core, you'll have fusion. The main trick then is confinement time.

[Art Carlson]

Why does the potential have to be spherical? All that you need is radial currents - net currents. As long as the net transverse motion cancels and the ion energy is within fusion range when the ions get to the core, you'll have fusion. The main trick then is confinement time.

What do you mean by "net radial current"? You had better have just as many ions moving away from the core as moving toward it, i.e. zero radial current.
An isotropic ion velocity distribution will also get you fusion, but if the radial components of the ion velocities are larger than the tangential components, then you get convergence, that is, a density bump in the center, and an enhancement of the fusion rate.

[Art Carlson]

3. This has all been discussed previously in the literature. As near as I can tell, there is nothing new in this thread. Dolan's original comments appear in the August 1993 edition of Fusion Technology. Bussard and Krall's response is in the March 1994 edition of Fusion Technology. I suggest that people who are concerned about this look at the original references.

I'll visit the library tomorrow. Or are these available on-line?

That's it? Dolan points out a serious problem 15 years ago, and nothing happens in the meantime except a bit of handwaving? Bussard and Krall's response is not convincing and partially wrong. You invoke electron inertia, which I agree could makes things a bit better or a bit worse, but you give zero details, not even a cartoon or an order of magnitude estimate. I'm sorry, if you polywell proponents claim that fine tuning to the tune of one part in ten thousand is possible on both the angle of the equipotential surface and the location of the ionization, then the burden of proof lies on you to show that that is more than wishful thinking.

[MSimon]

Art,

There is wishful thinking going on all over the fusion map.

Some of those wishes are cheaper than others. We are learning stuff.

The Tokamak guys think they have an answer to instabilities. The Polywell folks think they have an answer to the mono-energetics problem.

The thing I like about Polywell is that a serious 5 year program should give us a yes or no. What do the Tokamak folks say: building the next one 3X larger will fix it. Or give us the room to fix it. Or it will indicate that we need to build the next one 3X larger. :-)

[rnebel]

to jmc:

I'm sure at low densities this is the case, I find it hard to believe at, say, 10^20 particles per metre cubed collissions won't create local thermal equilibrium.

Which brings me to the point that I have been asking for quite a while and digging up the literature but have found no answer to.... what mechanism is supposed to correct the electron speeds or remove the electrons after they've collided with each other? How much energy will this constant extraction on maxwellized electrons and replacement of non-maxwellian electrons take.

I'm sure that a Polywell power plant will have shots that will last much much longer than an electron-electron collission time, otherwise the ions wouldn't even get a chance to make one pass through the core.


For present generation machines the electron confinement time is less than the electron collision time so thermalization isn't an issue. For reactors, the electron collision time and the confinement time become comparable. Electron distributions are expected to be isotropic, but not thermal. Thermalization is a global process because electron orbits cover the entire interior. If electrons lose their energy (kinetic + potential), they will accumulate near the coil cases until they leave the system. Since the spherical geometry (coupled with conservation of angular momentum) will itself change the distribution functions, l.t.e. doesn't make a lot of sense.

[rnebel]

Art:

If electrons collide and lose all of their angular every time they cycle and all I am looking for is 10:1 convergence (at most, since we both agree that I don't need any convergence) then it seems to me that I have at most a 10% problem. If you want to claim that the collisionality is less, then you should take that up with Nick Krall (see reference 4 in Bussard's note).

[TallDave]

If electrons lose their energy (kinetic + potential), they will accumulate near the coil cases until they leave the system.

Makes sense, since for them that's the bottom of the well, or as close as they can get. I assume leaving the system means cross-field transport loss.

[drmike]

A dynamic system can have radial currents. Ions in = electrons out and they oscillate around each other. If the system is not LTE, it must be in motion some where. I think maintaining non thermal distributions over long time scales will be a good trick.

One of the problems with assuming some formulas are valid when the system is not really thermal ignores the way the formulas were originally derived. I think knowing the full distribution function over space and velocity is needed to get a good handle on how the system will behave under given conditions. It's just a lot easier to do experiments at this point - computers can't follow that many particles in 6D phase space yet.

In a non thermal system, Debye length doesn't make sense. The derivation is based on an exponential distribution of velocity and space. If that kind of uniformity doesn't exist, you have different length scales as a function of position. If 1D models give hints on what to look for, that's great, but full scale experiments are still required to find out if the hints are meaningful.

Speaking of experiments, I want to build a 3 axis hall probe inside a Langmuir probe. Anybody got some references for people who have done that before?

[MSimon]

Speaking of experiments, I want to build a 3 axis hall probe inside a Langmuir probe. Anybody got some references for people who have done that before?

Funny you should mention that. I was going to do a post here on a review of magnetic instruments just published in EDN.

I'm leaving a link here:

http://www.edn.com/contents/images/6578134.pdf

And will also do a separate post. Let me know if you get any ideas.

[drmike]

That's a nice article, but I think his plot on hall sensors needs updating. The Allegro guys have come out with a new device that boasts 10mV/G, so it can easily see 1 Gauss fields. But I notice the plot is from Honeywell...

Too many toys, too little time!

The Allegro 139x looks like the perfect package and device for what I want. But for now I've got a few 132x's that will fit in a bigger ball of metal.

Thanks, I'll read that in more detail later!