All that can go wrong with recirculation

[Solo]

So I'm getting the impression that the interior density is dependent on the exterior electron density, but since ions should not have long lifetimes in the region external to the magrid, the electrons will be un-neutralized and will cause a space charge, which might be unreasonably large.


Ok, I've got an idea here: I imagine that in WB-7 you do see some space charge outside the magrid, in the diverging magnetic field outside the cusps. This will be caused by electrons 'recirculating.' The electrons roughly follow the magnetic field lines, but they loose velocity due to the potential gradient between the magrid and the wall. Then, assuming they are a few dozen eV short of reaching the wall, they turn around and fall back into the throat.

This space charge causes the electrostatic potential gradient to be even greater in this region, sheilding the volume near the wall from the magrid charge. Also, the steeper gradient near the magrid means that the electrons coming out of the cusp should be decelerated in a shorter distance, which will in turn keep the space charge localized closer to the magrid. (It doesn't make sense to me that the electrons should clump like that, you'd think they'd repell each other, but if I'm going wrong, I don't know where.)

Could this scenario explain the electron density outside the machine necessary to support the interior density Nebel is claiming?

EDIT: have a look at this concept, just to see ions jump through a 'recirculation' hoop.

[Art Carlson]

Let me make another try to get a grip.

We want to try to confine ions electrostatically. We try it with grids but find that the losses due to ions hitting the wires are too big. We try it with a bunch of electrons confined by magnetic fields. If we consider perfectly confined electrons, we find that the ions are also perfectly confined (except a bit of Boltzmann slop over the top). Fine. So let's look a bit deeper. Of course the electrons are not perfectly confined, they leak out along point and/or line cusps at a rate that will result in unacceptable power loss. What can we do about it? Charge up our magnetic coils positively with respect to the walls. Then the (few) electrons that try to leak out (along relatively well defined lines and/or planes) are also electrostatically confined. Is that good so far?

Let's finish the iteration. What effect does the fact that some electrons make an excursion out the cusp and back have on the ions? It's hard to say exactly. What I picture is a sheet of electrons, with a thickness of a few electron gyroradii and a density a few times less than the density in the high beta region. I can't see any reason not to expect ions to also leak out, at least until they neutralize the electron density, that is, also in a sheet with similar thickness and density, but with a velocity slower by the square root of the mass ratio. These ions, of course, are lost once they get through the cusp, and take with them their thermal energy plus the magrid voltage. Is this tolerable? I doubt it, at least not if I am correct that the cusps are essentially lines.

A separate comment back on the electrons. I suspect that electrons lost through a cusp cannot be simply reflected back into that cusp. They will probably make it back through the point of highest field, but I think that trapping and detrapping in a cusp depend on the phase of the gyromotion, so that a large fraction of the electrons that came from the high beta region through the cusp will be trapped on field lines when they come back inside. This would be equivalent to non-collisional cross-field diffusion. I haven't figured out yet how to calculate the loss rate due to this mechanism, but I suspect it will be a problem. Have any simulations been done that are capable of investigating this possibility?

Along the same lines, for the line cusps I expect an electric field in the plane of the electron fan. That should push them aside so that they possibly will miss the sweet spot coming back in. Even if the net push per excursion is less than a gyroradius, this effect may limit the effective recycling factor. Grad-B and curvature drifts could still be an issue, too.

[TallDave]

What I picture is a sheet of electrons, with a thickness of a few electron gyroradii and a density a few times less than the density in the high beta region.

Should be 1,000 - 10,0000 times less dense, overall, outside vs. in.

Along the same lines, for the line cusps I expect an electric field in the plane of the electron fan.

I think this will be too small to pull out any significant number of ions. Even if the field extends out there in a very narrow plane/line, the Magrid tends to cancel it out, from the perspective of ions in the interior, and the well should still point the other way even for an ion sitting right on the plane near the edge.

[Art Carlson]

What I picture is a sheet of electrons, with a thickness of a few electron gyroradii and a density a few times less than the density in the high beta region.

Should be 1,000 - 10,0000 times less dense, overall, outside vs. in.

Why? Maybe if you are talking about the volume-average density, but why should the peak density be much less than the density inside? Think of gas escaping from a tank through a pipe.

Along the same lines, for the line cusps I expect an electric field in the plane of the electron fan.

I think this will be too small to pull out any significant number of ions. Even if the field extends out there in a very narrow plane/line, the Magrid tends to cancel it out, from the perspective of ions in the interior, and the well should still point the other way even for an ion sitting right on the plane near the edge.

My comment was referring to the electric field perpendicular to the magnetic field, and I think you are talking about the parallel component.
What is the basis for your thought that the (parallel) electric field will be "too small to pull out any significant number of ions"? In the polywell (and most magnetic confinement devices), the electron gyroradius is close to the Debye length, lambda_D = sqrt(epsilon*kT/(n*e^2)). The electric field due to a sheet of charge a Debye length thick is on the order of n*e*lambda_D = sqrt(epsilon*n*kT), and the electric field discontinuity across such a sheet is sqrt(n*kT/epsilon) = (kT/e)/lambda_D. In a geometry with a scale length R, the potential of the sheet is on the order of (kT/e)*(R/lambda_D), which is huge compared to the voltages in the system. I conclude that the potential of the fan of escaping electrons is not a minor perturbation, but is an effect of zeroth order. Not "too small" at all.

[TallDave]

Why? Maybe if you are talking about the volume-average density, but why should the peak density be much less than the density inside?

Sure, but I think the overall relative densities are going to be more important to the force on the ions.

My comment was referring to the electric field perpendicular to the magnetic field, and I think you are talking about the parallel component.

Hmm? I was thinking of the electric field's gradient. I doubt those little planes can affect it to the extent it points toward the Magrid anywhere. They seem more like shallow ravines on the edges of a deep valley.

In a geometry with a scale length R, the potential of the sheet is on the order of (kT/e)*(R/lambda_D), which is huge compared to the voltages in the system.

But it's still small compared to the potential of the interior electrons, and the Magrid is out there too.

[Art Carlson]

In a geometry with a scale length R, the potential of the sheet is on the order of (kT/e)*(R/lambda_D), which is huge compared to the voltages in the system.

But it's still small compared to the potential of the interior, and the Magrid is out there too.

??? I am working on the assumption that temperatures (or average energies, of both ions and electrons) on the order of the voltages involved, in the range of 100 keV in a reactor, are unavoidable. Are you assuming temperatures of mere tens of eV, or what?

[TallDave]

Hmmm? I guess I'm not sure why that matters, if there are far more electrons on the inside and the Magrid between the ions and the outside. Are you saying you think they're energetic enough that they can climb all the way out, despite the gradient?

[93143]

I would expect the star-shaped potential well to be quite blunted and nearly spherical by the time you get down to the ion formation altitude. This minimizes ion loss pretty much as well as a spherical well.

It's also possible that you could use direct conversion to recover the energy from leaking ions if it turns out to be a problem. In a multi-level p-11B system, you'd need three collectors for this - one for ions, one for low-band alphas, and one for high-band alphas. You might even try to direct-convert the proton stream from a D-D reactor... Ion recovery might or might not improve the power balance and reactor economics, depending on how a lot of things interact and how big a problem ion loss actually turns out to be.

[Art Carlson]

I would expect the star-shaped potential well to be quite blunted and nearly spherical by the time you get down to the ion formation altitude. This minimizes ion loss pretty much as well as a spherical well.

Why would you expect that? The Debye length is very small, so you have to have ions everywhere you have electrons, and the only thing holding the electrons in place is the magnetic field, and the magnetic field is lumpy. Come to think of it, even if you were willing to throw away 99.5% of your fusion power by going to half the radius (a factor of 8 for the volume, and assuming the magnetic field drops as R^-3 - Your power balance will also deteriorate because you will need to apply a higher voltage to get the same well depth.), the shape of the flux surfaces will be lumpy and spikey all the way to the center.

It's also possible that you could use direct conversion to recover the energy from leaking ions if it turns out to be a problem. In a multi-level p-11B system, you'd need three collectors for this - one for ions, one for low-band alphas, and one for high-band alphas. You might even try to direct-convert the proton stream from a D-D reactor... Ion recovery might or might not improve the power balance and reactor economics, depending on how a lot of things interact and how big a problem ion loss actually turns out to be.

When you get within a factor of 2 of breakeven, we can talk about direct conversion. I think you are orders of magnitude away.

[Art Carlson]

Hmmm? I guess I'm not sure why that matters, if there are far more electrons on the inside and the Magrid between the ions and the outside. Are you saying you think they're energetic enough that they can climb all the way out, despite the gradient?

I don't understand your comment. Not even grammatically.
I am assuming T_ion ~ T_electron ~ |electric potential at the center compared to the magrid radius| ~ |electric potential of the magrid relative to the walls| ~ 100 keV. My estimate refers to what happens around the magrid radius. The factor of (R/lambda_D) trumps everything.

[Art Carlson]

Why? Maybe if you are talking about the volume-average density, but why should the peak density be much less than the density inside?

Sure, but I think the overall relative densities are going to be more important to the force on the ions.

My comment was referring to the electric field perpendicular to the magnetic field, and I think you are talking about the parallel component.

Hmm? I was thinking of the electric field's gradient. I doubt those little planes can affect it to the extent it points toward the Magrid anywhere. They seem more like shallow ravines on the edges of a deep valley.

I presented a quantitative estimate that showed the importance of the electric potential resulting from the sheet of electrons. We are past the point of "think", "doubt", and "seem".

[kcdodd]

Art,

I'm a bit confused on your argument. I am assuming the "sheet" you are describing is the result of electrons in the process of escaping and entering the line cusps of the wiffle-ball. Are you arguing that the thickness of that escape"sheet" is on the same order of magnitude as the layer of electrons which forms the "surface" of the wiffle-ball itself?

[Art Carlson]

Along the same lines, for the line cusps I expect an electric field in the plane of the electron fan. That should push them aside so that they possibly will miss the sweet spot coming back in. Even if the net push per excursion is less than a gyroradius, this effect may limit the effective recycling factor. Grad-B and curvature drifts could still be an issue, too.

It is not too hard to quantitatively estimate the size of these effects. I am considering a plane passing through opposing edges of the cube, and consequently also through the diagonals of opposing faces. This is the plane of the line cusps. By symmetry, for all points in this plane, both the electric and magnetic fields will lie in the plane. The fields will be purely radial for the lines passing through the corners of the cube, the midpoints of the sides, and the centers of the faces. I am considering what happens to electrons making an excursion through a cusp and being reflected back toward the cusp, somewhere between the purely radial field lines.

If the electric and magnetic fields are not perfectly aligned, there will be an EXB drift pushing the electrons out of the plane. This drift will be in the same direction on the way out as on the way back in, so it accumulates. The excursion time is about R/v, and the drift velocity is EXB/B^2, so the displacement is delta = (R*EXB)/(v*B^2). If the displacement is greater than an electron gyroradius, then the electrons will miss the hole when they bounce back and not be recirculated, so we should compare the displacement to rho = mv/eB: delta/rho = (R*EXB)/(v*B^2)/(mv/eB) = (R*e*EXB)/(m*v^2*B). R*e*E is the potential difference, which must be comparable to the electron kinetic energy m*v^2/2, so we find that delta/rho is on the order of the cosine of the angle between E and B. How big will that be? We have the symmetry that forces them both to be in the plane we are considering, and also the symmetry that makes them perfectly aligned in 8 directions within this plane, so the misalignment won't be large, but there is no reason to expect it to vanish. Electric fields and magnetic field ultimately have a completely different topology. My guess is a misalignment in the range of 10^-2. This would lead to a maximum recirculation coefficient of 100, far less than the 1000 or 10,000 hypothesized and required.

The calculation for the curvature drift is similar. The drift velocity is m*v^2/(e*B*R_c). Thus delta/rho, which was the cosine of the angle between E and B above, is here the ratio of the device size R to the radius of curvature of the magnetic field R_c. R_c will certainly be much larger than R, but there is no reason to expect it to be infinite. My guess is something like R_c ~ 100*R, again leading to a maximum recirculation around 100. (Note that these effects are independent of each other, so the larger one wins.)

Afterthought: Since I am insisting that this sheet must be quasineutral or at least have a significant ion density nearby, and the curvature drift is in opposite directions for opposite polarities of charge, one must consider the interaction of the electrons and ions. This will be essentially a charge separation resulting in a (large) electric field perpendicular to the sheet, which will push the plasma perpendicular to B within the sheet. I am not sure what the consequences of this would be. It would possibly negate or at least mitigate the curvature effect, leaving the EXB effect to dominate.

[hanelyp]

The Debye length is very small, so you have to have ions everywhere you have electrons, and the only thing holding the electrons in place is the magnetic field, and the magnetic field is lumpy.

The Debye length has been estimated as very small near the cusps. Is it the same in the interior of the plasma away from the cusps? How does "you have to have ions everywhere you have electrons" account for the excess of electrons and the ion energy being short of well depth?

[Art Carlson]

The Debye length is very small, so you have to have ions everywhere you have electrons, and the only thing holding the electrons in place is the magnetic field, and the magnetic field is lumpy.

The Debye length has been estimated as very small near the cusps. Is it the same in the interior of the plasma away from the cusps? How does "you have to have ions everywhere you have electrons" account for the excess of electrons and the ion energy being short of well depth?

lambda_Debye = sqrt((epsilon_0*kT)/(n*e^2))
I expect the density and temperature near the cusps to be similar to that in the center, at least in "beta=1" mode. If anything, the Debye length will be even smaller in the center because of the higher density.
The excess of electrons required to gets tens of kV is only one part in 10^4 (or something like that). It's called "quasineutrality".
The excess in well depth comes about by the potential drop between the central plasma and the magrid. That is, there is a region of near vacuum that has an electric field in it due to the (tiny) excess of electrons in the central region.