Recirculation revisited

Discuss how polywell fusion works; share theoretical questions and answers.

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Art Carlson
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Postby Art Carlson » Tue Jan 06, 2009 9:52 am

KitemanSA wrote:But I don't yet get why we CARE if the "flux tubes" loop back.
I guess my question is, just what is the magnitude of this problem? Is this an exercise in guilding the lily?

It may be only of historical interest. Bussard used to talk about recirculation as if it was electrons traveling along flux loops. I argued that he was dead wrong, but maybe there are circumstances where he could be partly right. I believe the current thinking (e.g. by Rick Nebel) is that "recirculation" is actually a kind of reflection by electric potentials. If the reflection works, then the loops are just an academic exercise.

I don't believe the reflection will work (though I'm not 100% sure), so the loops might become an issue again at some point.

KitemanSA
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Postby KitemanSA » Tue Jan 06, 2009 1:28 pm

Art Carlson wrote:
KitemanSA wrote:But I don't yet get why we CARE if the "flux tubes" loop back.
I guess my question is, just what is the magnitude of this problem? Is this an exercise in guilding the lily?

It may be only of historical interest. Bussard used to talk about recirculation as if it was electrons traveling along flux loops. I argued that he was dead wrong, but maybe there are circumstances where he could be partly right. I believe the current thinking (e.g. by Rick Nebel) is that "recirculation" is actually a kind of reflection by electric potentials. If the reflection works, then the loops are just an academic exercise.

I don't believe the reflection will work (though I'm not 100% sure), so the loops might become an issue again at some point.

Since Dr. B was well aware that the Polywell was an improvement of the Elmore-Tuck-Watson fusor which "recirculates" electrons ONLY with the E-Field, I suspect he used the term the same way as Dr. N.; though he may have also conjectured that some may move more quickly around a tight loop aided by the "flux tube(?)" But since the "reflection" (e.g. recirculation) works well with the ETW fusor, why wouldn't it work with the Polywell?

edited- dark red added

Mike Holmes
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Postby Mike Holmes » Tue Jan 06, 2009 3:13 pm

KitemanSA wrote:guilding the lily?

Is that a typo? Or a very subtle double-entendre?

:-)

If/when we see the experimental data, any chance that it'll give us enough info to decide if this is, in fact, a problem? Or don't we know enough about the data yet? Probably the latter, eh?

Mike

KitemanSA
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Postby KitemanSA » Tue Jan 06, 2009 3:39 pm

Mike Holmes wrote:
KitemanSA wrote:guilding the lily?
Is that a typo? Or a very subtle double-entendre? :-)

I wish I could take credit for a subtle double-entendre, but I don't have a sufficient degree of subtleationness... subtleosity? :roll:

Munchausen
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Postby Munchausen » Sat Jan 10, 2009 6:11 pm

Isn't the WB-7 mostly about testing the efficiency of the electron capture mechanism? Is it a difficult thing to estimate?

Can't we rather safely assume that the electrons are trapped largely as predicted by now? Simply because this line of research have not been abandoned.

Sooner or later there has to be an article in a physics journal somewhere. If a researcher doesn't publish he doesn't exist.

Roger
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Postby Roger » Sun Jan 11, 2009 5:46 pm

By my take.....WB-7 now has an ion gun and some additional instrumentation, and funding to operate the new and improved WB-7, probably to get some new measurements.

If recirc was an issue, I'm not sure we would have seen this sort of activity. If the neutrons are indeed BB, then we've got one hell of a lot of fusion from a very small device, and I cant imagine recirc being an a real issue at that point.

If copious B/B fusion occurs in WB-7, then its far more likely recirc occurs. No?

I think whatever "nuanced" results Dr. Nebel was thinking about, those issues are being addressed in the current work, as in an ion gun, and more instrumentation. I have no doubt that the current work is trying to make WB-7 a bit better, and better understood.
I like the p-B11 resonance peak at 50 KV acceleration. In2 years we'll know.

Solo
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Postby Solo » Wed Feb 04, 2009 8:42 pm

Art_Carlson wrote: I have explained why I don't expect any recycling, and nobody has presented a counter-model in any helpful degree of detail. If you can do that, great!


Ok, let me see if we are on the same page:

Image

The graph is in radius from the center of the machine. The charged magrid produces a field at it's center, but this is reduced by the space charge of predominantly electrons escaping from the cusp. The upscattered ions escape along a thin line straight thru the center of the cusp where the electrons produce the lowest potential path out of the machine. Any ions going "over the hill" are lost to the outside. But any electrons going out the cusp are decellerated and return through the cusp.

Electron particles losses will consist of 1) upscattering and impacting the wall 2) electrons diffusing across the magnetic field until their orbits impact the magrid. Mechanism 1 should be insignificant in terms of both particle and energy loss. This means that the flow of electrons in and out of the cusp is not a total loss mechanism. MEchanism 2) is the primary particle and energy loss mechanism for electrons, representing a current from the ground (e-gun) to the magrid.


The ions are lost by charge exchange or upscattering and loss through the cusps. Ions are lost completely when they exit the cusp (unlike electrons). The ion temperature will have to be several times lower than the well depth so that they are not lost too rapidly. This means a higher magrid voltage for a given plasma temperature, so each ion lost to the wall or each electron lost to the magrid represents more energy lost.

Ok, so do you have any problems with this?

Art Carlson
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Postby Art Carlson » Thu Feb 05, 2009 4:39 pm

That is the picture that polywell advocates have in mind. Let's take a closer look at it. (I've done this before, but I admit it is easy to lose pieces of the argument.) Let's assume your picture of the plasma potential is correct, and that it applies to a flux tube coming out of a point cusp. We will also use some other conditions, like beta = 1 and flux conservation in the sheath. The question I want to ask, semi-quantitatively, is how high the potential on the magrid would have to be, your red tick mark.

By construction, most of the ions are held back by the potential rise. The electrons, in contrast, will be accelerated into this region. By conservation of electron flux, the increased speed is associated with a decreased density, by a moderate factor. We won't be far off if we assume the electron density near the potential peak is half that in the main plasma.
    n = n_0/2
(We won't be able to keep track of all the factors of 2. That gives us an idea of how far we can trust our quantitative results.) In the standard polywell picture, most of the plasma pressure at the edge is due to electrons, and the average energy of the electrons will be a bit more than the well depth phi. Phi will be chosen to give the ions the energy they need to fuse. If we take D-T fuel, but consider that the energy distribution is (presumably) not Maxwellian, we will need about phi = 30 kV. We can now use the beta = 1 condition to relate the density to the magnetic field.
    B^2/(2mu_0) = n_0*e*phi

So far so good. Now I'd like to know how thick this pencil of plasma is. Far from a cusp, the transition from field-free plasma to plasma-free field will take place over a sheath of some thickness. Bussard talks about a sheath with a thickness equal to the electron gyroradius, rho_e = m_e*v / eB, where the electron velocity v is given by (1/2)*m_e*v^2 = e*phi. The end result is
    rho_e^2 = 2*m_e*e*phi / (e^2*B^2)
It is really just about impossible to imagine a thinner sheath, and I'm afraid I will have to insist on taking this as a lower bound. There is good reason to believe that the sheath will actually be several times thicker than this, but I will give it the benefit of the doubt for now. Now draw a circle around the point cusp as large as you dare, i.e. close to the line cusps or at least halfway between the point cusp and the line cusp. This circle will have a circumference a bit less than 2pi*R. Call it 4*R among friends. Make a stubby cylinder out of this circle by extending it through the sheath. It will have an area of
    A = 4*R*rho_e
Because of pressure balance and beta = 1, the magnetic field will be constant along the surface of the palsma, so the field lines passing through the stubby cylinder will map to a pencil pointing along the point cusp with the same area. The radius of this pencil will be roughly
    s = sqrt(R*rho_e)
Notice that the radius of the point cusp must be significantly greater than rho_e, no matter what Bussard says.

Let's start putting this together. What I want is lambda, the line density of charge in the pencil.
    lambda = n*e*A
    = n*e*(4*R*rho_e)
    = 2*n_0*e*(R/rho_e)*rho_e^2
    = 2*(R/rho_e)*e*( B^2/(2mu_0)/(e*phi) )*( 2*m_e*e*phi / (e^2*B^2) )
    = 2 * (R/rho_e) * m_e / (e*mu_0)
    = 2 * (R/rho_e) * (m_e*c^2/e) * epsilon_0
    = (1 MV) * (R/rho_e) * epsilon_0
The last step follows from the fact that the rest mass of the electron in 500 keV in energy units. (If you get a queasy feeling at this point, seeing 1 MV showing up as a characteristic voltage, I understand completely.)

Why do I want this? Because I want to calculate the potential that the magrid would have to have in order to achieve the assumed potential profile. To get that, I use the fact that a cylinder with line charge density lambda and radius s produces a potential at radius R equal to
    V_magrid = (2*lambda/epsilon_0)*ln(R/s)
    = (1 MeV) * (R/rho_e) * (m_e*c^2/e) *ln(R/s)
    = (1 MeV) * (R/s)^2 * ln(R/s)


That should be enough math. We will not want to choose R/s too small, otherwise our cusps will be bigger than our plasma, our electrons won't be confined by the magnetic field, and probably a slew of other problems. The ln term shouldn't bother us too much, but we expect (R/s)^2 to have to be at least 10, so that our magrid voltage relative to the plasma will have to be on the order of 10 MV. Since phi will probably be around 30 kV for D-T fuel, we see that the red tick mark in the potential diagramm will be way off the map.

The analysis of the line cusps is very similar and yields a very similar result, giving me confidence in the robustness of the result.

That is what I call a serious inconsistency in the usual way of looking at the polywell. Please feel free to try to modify the picture to get it to make some sense.

Solo
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Postby Solo » Fri Feb 06, 2009 6:45 am

That seems like a decent treatment to me, that business with mc^2 is very clever. It's counterintuitive that the space-charge voltage is worse than inversely proportional to rho_e. But I think I see why: the area of the (point) cusp goes with 1/B , but the electron density goes with B^2, assuming beta=1. It makes sense that it is proportional to R, though. So it looks like it comes down to a trade off between plasma beta and overkill magrid voltage, which is an energy loss.

One tweak, Dolan's review suggests that n/n_0=0.1 to 0.01, that'd bring us back down to 1000-100 keV range, which is a bit better. I guess we'd get a better picture if we knew what the distribution actually looked like and didn't have to assume an infinite line charge (that's what your formula is for, right?) Then again, if we knew that, we wouldn't be having this discussion...

I'd begun to worry about this space charge issue when I noticed that the toroidal electrostatically-plugged cusp machines all had electrodes at the cusp throats, biased positively relative to the plasma by tens to hundreds of keV, that limited the plasma width to a few millimeters there. They still weren't getting very deep effective potential wells. And of course having the limiting electrodes means that the electrons were being lost by impacting the electrodes. EDIT: yeah, he baselined a driving voltage about 20x the ion and electron temperature, >= 300keV. 20% for the voltage drop, 30% for ion confinement (effective well depth), and 50% for electron confinement (potential between plasma and ground).

I wonder what flux conservation between the various kinds of cusps would tell us? Probably make the picture even more gloomy. The space charge of a sheet in a line cusp is bigger. EDIT: Dolan reviews one experiment with a "barrel cusp" machine: "The barrier height in the point cusp was practically zero, but most of the ion losses occurred through the ring cusp. Point cusp ion losses were probably retarted by the centrifugal and magnetic field gradient forces. " (Plasma Phys. Control. Fusion 36 (1994) 1539-1593. T.J. Dolan "Magnetic electrostatic plasma confinement") I think someone once explained this by flux mapping instead, though.

So do you think putting limiting electrodes around the cusp would be an improvement worth testing? Also, I'm curious what you think defines the flux surface that is the outside of the sheath. What limits the plasma size, basically? Is it that if the sheath moved out further (in flux surface space) it would open up a channel for ions through the cusp? Then the ion loss would shrink the plasma, pulling the sheath back?

Art Carlson
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Postby Art Carlson » Fri Feb 06, 2009 9:14 am

Solo wrote:So it looks like it comes down to a trade off between plasma beta and overkill magrid voltage, which is an energy loss.
Such a high magrid voltage will almost cetainly lead to severe arcing, as well, but even before we start talking about technological problems, if anybody really wants to go down the street of MV magrids, I believe there are some basic physics show stoppers.

Solo wrote:One tweak, Dolan's review suggests that n/n_0=0.1 to 0.01, that'd bring us back down to 1000-100 keV range, which is a bit better.
He must be talking about something else. If electrons reach the throat of the cusp with velocity v, they would need to speed up to 100*v in order to reduce the density by a factor 100. If their energy at the throat is 30 keV, they would need to fall down a potential hill of 300 MeV. Don't go there.

Solo wrote:So do you think putting limiting electrodes around the cusp would be an improvement worth testing?
Dinking with the geometry might help you push down the R/s factor, but what's killing you is the m_e*c^2.

Solo wrote:Also, I'm curious what you think defines the flux surface that is the outside of the sheath. What limits the plasma size, basically? Is it that if the sheath moved out further (in flux surface space) it would open up a channel for ions through the cusp? Then the ion loss would shrink the plasma, pulling the sheath back?
The sheath results from a balance of perpendicular and parallel processes on various time scales. I think of it this way. The electrons cross the field by a step rho_e faster than you can blink an eye. Left alone, they would spread even farther due to collisions, instabilities, field errors, etc. But before they have time to do that, they flow along the field lines out the cusps. Adding ions makes the whole thing more complicated because you need to find self-consistent perpendicular and parallel electric fields, but the idea is similar.

KitemanSA
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Postby KitemanSA » Fri Feb 06, 2009 11:23 am

Solo wrote:
Art_Carlson wrote: I have explained why I don't expect any recycling, and nobody has presented a counter-model in any helpful degree of detail. If you can do that, great!


Ok, let me see if we are on the same page:



Either your graphic above does NOT comport with what I understand the Polywell to be, or I am not understanding your use of the term "potential". My comments are based on reading it as effectively voltage. If you mean voltage, then the graph is wrong. It should drop effectively to zero just inside the magrid, and then go negative (~-80% of MaGrid voltage) to the virtual cathode, then rise to near zero again at the virtual anode.
If you DON'T mean voltage, please excuse the ramblings of a mere mechanical engineer.

Art Carlson
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Postby Art Carlson » Fri Feb 06, 2009 11:58 am

KitemanSA wrote:Either your graphic above does NOT comport with what I understand the Polywell to be, or I am not understanding your use of the term "potential". My comments are based on reading it as effectively voltage. If you mean voltage, then the graph is wrong. It should drop effectively to zero just inside the magrid, and then go negative (~-80% of MaGrid voltage) to the virtual cathode, then rise to near zero again at the virtual anode.
If you DON'T mean voltage, please excuse the ramblings of a mere mechanical engineer.

Potential = voltage is fine. Remember that the zero point of voltage is arbitrary. I don't understand what you think the voltage as a function of radius should look like. Can you draw a picture? Or at least explain what you mean by "virtual cathode" and "virtual anode"? Specifying the minima and maxima in terms of x volts at radius y should also be unambiguous.

KitemanSA
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Postby KitemanSA » Fri Feb 06, 2009 3:02 pm

Art Carlson wrote:Potential = voltage is fine. Remember that the zero point of voltage is arbitrary. I don't understand what you think the voltage as a function of radius should look like. Can you draw a picture? Or at least explain what you mean by "virtual cathode" and "virtual anode"? Specifying the minima and maxima in terms of x volts at radius y should also be unambiguous.


More to come after work, but basically this is my picture of the situation. If the potential at r=magrid equals a positive potential =M, and tails off to ground at r=infinity, then the "well" is a bowl shaped depression below ground inside the MaGrid, and there is a hill in the middle of the bowl. The bottom of the bowl is the virtual cathode, magnitude about 0.8M, the hill is the virtual anode. I am not quite clear whether the peak of the hill goes above ground, but I think it has to. Even without the ions, the electron's voltage zeros out in the middle, no? It would be analogous to there being no "gravity" at the center of Earth. There is, but it pulls in all directions simultaneously. Anyway, if it does go above ground, then it tapers off to ground at r=0. This all of course is predicated on a reasonably spherical polywell.

TallDave
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Postby TallDave » Fri Feb 06, 2009 3:31 pm

Notice that the radius of the point cusp must be significantly greater than rho_e, no matter what Bussard says.


I'm having a little trouble following here. Could you elaborate on this? At a glance, it seems a smaller point cusp would give a very different result, so maybe this is where the discrepancy arises.

If I understand your calculation, you're trying to determine what charge on the Magrid would be needed to pull the electrons out without similar numbers of ions going with them. But if the plasma is near the quasineutral limit, shouldn't it also be trying to spit out electrons?

So do you think putting limiting electrodes around the cusp would be an improvement worth testing?


I think Bussard said he tried that, but it sucked ions out.

Solo
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Postby Solo » Fri Feb 06, 2009 8:12 pm

I should have gone ahead and quoted him:
Dolan wrote:there will probably be about an order of magnitude density ratio between the anode region and the central plasma
n/n_0 ~ 0.1.
The ratio would probably be lowest for cases with a large volume of field-free plasma, narrow anode gaps and low neutral gas pressure; and it could be near unity for cases with spindle cusp magnetic field, wide anode gaps and high neutral gas pressure. Values inferred from data in a variety of experimental conditions range from 0.01 to 1.


Dolan wrote:Moir et al [76] showed that the electron density in the gaps is lower than the magnetically penetrating density n_p by
a factor n/n_p = exp(y) erfc(y^(-1/2)) where y=phi /T_ion, and erfc is the complementary error function. For values of y from 3-6, n/n_p is 0.29-0.22.
[Where n_p = density of electrons that are able to penetrate into the magnetic cusps by geometry and mirror considerations]


Art_Carlson wrote:Dinking with the geometry might help you push down the R/s factor, but what's killing you is the m_e*c^2.

True that! Well, I guess it means Dr. B was right about one thing, though: electrons are easier to confine than ions!

Art_Carlson wrote:But before they have time to do that, they flow along the field lines out the cusps.

That's where I'm getting stuck. What happens then? If recirculation happens by electrostatic field, those electrons are not lost and are returned to the plasma, and the diffusion continues until ... well, I think the actual WB machines probably limit this because the electrons are lost in the coil corners (the center of the line cusp). Somehow they have to be lost though, either they are lost to the wall as you say (ie, recirculation doesn't happen) or else they'll diffuse till they hit the magrid. Do you agree?


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