Central electron temperature and p-B11 power balance

[TallDave]

I repeat, "no collisional ion-electron interactions". Chacon can't be more accurate on this point because he explicitly ignores it.

Right, someone has already cited Nebel's response on that. I got my copy of Glasstone and Lovberg (Controlled Thermonuclear Reactions, Robert E. Kriger Publishing Co. 1975.) a few days ago but haven't gotten through chapter 4 yet.

But if the electron density were significantly lower than the ion density, you would have to either make the ion density itself very low, or make the device very small

That one I'm not sure about. I think it came up before and there was some disagreement on that point and whether it applies in a non-LTE system.

Chacon is the wrong person to cite in this thread, anyway. He identifies regimes with Q values of 100 and above *for D-T fusion*.

True, p-B11 is very different, but afaik that's the closest model we've got. I agree in principle though; I've said before it may be excessively optimistic to be thinking too much about p-b11 at this point.

EDIT: OK, having read through some part of the Ch. 4 text dealing with the 1/e folding time (the time required for ions' energy to be reduced by a factor of e) also called r, I'm wondering what the time scales involved are in the envisioned device. Is it possible that in a Polywell the fusion timescale is sufficiently fast that the ion cooling isn't a showstopper? The text suggests this is possible: "r might be increased until it was the same order of magnitude as the reaction lifetime."

EDIT: Come to think of it, this probably doesn't matter unless we're looking at a pulsed device.

[Art Carlson]

I think I understood Bussard's expectations differently. I thought he expected the electrons to repel each other out of the very center of the device, resulting in a decrease in electron density at the very center. Ion convergence would weaken that effect, but is it not still possible for the electron density to in fact be lower approaching the core? Or does that even matter? Calculating Brems is over my head, but I understood it to be decreasing with electron and ion density, so could lowered electron density in the core where ions are converging reduce the Brems enough to affect anything?

In a region of lower density, the bremsstrahlung and collisional losses will be decreased by n^2, but so will the fusion power, so you don't gain or lose anything. Why is this so hard to understand?

[bcglorf]

I think I understood Bussard's expectations differently. I thought he expected the electrons to repel each other out of the very center of the device, resulting in a decrease in electron density at the very center. Ion convergence would weaken that effect, but is it not still possible for the electron density to in fact be lower approaching the core? Or does that even matter? Calculating Brems is over my head, but I understood it to be decreasing with electron and ion density, so could lowered electron density in the core where ions are converging reduce the Brems enough to affect anything?

In a region of lower density, the bremsstrahlung and collisional losses will be decreased by n^2, but so will the fusion power, so you don't gain or lose anything. Why is this so hard to understand?

I understand fusion power depending on ion density, just not on electron density. The ion density dead center should be very high if convergence is good. I understood Bussard's claim though to be that the electrons contained in the magrid region would repel each other from the center enough that electron density dead center would be much reduced, without reducing ion density. Is that just an absurd picture?

[Art Carlson]

I understand fusion power depending on ion density, just not on electron density. The ion density dead center should be very high if convergence is good. I understood Bussard's claim though to be that the electrons contained in the magrid region would repel each other from the center enough that electron density dead center would be much reduced, without reducing ion density. Is that just an absurd picture?

All right, then the problem is quasineutrality. Say you've got a region of a particular size, like 10 cm, and you've got a particular voltage available, like 500 keV. These two numbers, when you plug then in to Gauss's law, determine the net charge density that you can have.
phi = (4 pi epsilon_0)^-1 Q / R
rho = Q / (4/3 pi R^3) = (4 pi epsilon_0) * phi / (4/3 pi R^2)
rho = 3 epsilon_0 phi / R^2
(Zn_i-n_e) = 3 epsilon_0 phi / e R^2
(Zn_i-n_e) = 3*(8.85e−12)*(5e5) / (1.602e-19)(1e-2) = 8.3e14 m^-3
Remember that a tokamak has a density of 1e21 m^-3, and that is criticized (e.g. by Nebel) as being much too low. That is the reason that electron density and ion density are tightly coupled for practically any fusion reactor design. If the electron density is low, then the ion density must be low too.

[93143]

The low electron temperature only applies at the centre of the device. The electrons don't spend much time there (nowhere near the amount of time necessary to get significantly heated by the ions in one pass), and they should at least be faster than the ions everywhere even if their energy is a lot lower.

Is it not plausible that the rapidly-cycling non-LTE nature of the plasma causes the energy transfer to behave differently in global terms? Maybe in some parts of the acceleration region (in between the edge and core) the cross-section distribution is favourable to rear-ending of ions by electrons, thus reversing the energy drain... or something...

I'd try to do the math but I haven't got the time and my gas kinetics notes are at school anyway...

[Jboily]

All right, then the problem is quasineutrality. Say you've got a region of a particular size, like 10 cm, and you've got a particular voltage available, like 500 keV. These two numbers, when you plug then in to Gauss's law, determine the net charge density that you can have.
phi = (4 pi epsilon_0)^-1 Q / R
rho = Q / (4/3 pi R^3) = (4 pi epsilon_0) * phi / (4/3 pi R^2)
rho = 3 epsilon_0 phi / R^2
(Zn_i-n_e) = 3 epsilon_0 phi / e R^2
(Zn_i-n_e) = 3*(8.85e−12)*(5e5) / (1.602e-19)(1e-2) = 8.3e14 m^-3
Remember that a tokamak has a density of 1e21 m^-3, and that is criticized (e.g. by Nebel) as being much too low. That is the reason that electron density and ion density are tightly coupled for practically any fusion reactor design. If the electron density is low, then the ion density must be low too.

Art,
This is good news then. assuming we have ions convergence. If I understand correctly, your calculation assume a constant density over the entire volume.
The core radius is where everything happen when we have Ions convergence. With the radius of the core much smaller, would it be allowing for a larger electron-ions density differential by a factor R^2? Would this mean there would be an ions convergence radius allowing a significant effect on the electron heating.
Note that the electrons do not stay in the machine very long, and only a fraction of the population would spend any time in the dense core.

[Art Carlson]

If I understand correctly, your calculation assume a constant density over the entire volume.

The loss processes we are discussing here - radiation (bremsstrahlung) and collisions - are proportional to the square of the density, just like the fusion power is, so Q = P_fusion/P_loss does not depend on the density, and consequently not on any non-uniformities in the density, either.

In a region of lower density, the bremsstrahlung and collisional losses will be decreased by n^2, but so will the fusion power, so you don't gain or lose anything. Why is this so hard to understand?

With the radius of the core much smaller, would it be allowing for a larger electron-ions density differential by a factor R^2?

I am assuming quasineutrality, i.e. electron densities very nearly equal to ion densities (taking the ion charge Z into account). Changing this assumption would certainly change the calculation. But if the electron density were significantly lower than the ion density, you would have to either make the ion density itself very low, or make the device very small - either of which would lead to ridiculously small fusion power levels - or make the voltage very large, which would make the energy balance very unfavorable. You can't get past square one without quasineutrality.

All right, then the problem is quasineutrality. Say you've got a region of a particular size, like 10 cm, and you've got a particular voltage available, like 500 keV. These two numbers, when you plug then in to Gauss's law, determine the net charge density that you can have.
phi = (4 pi epsilon_0)^-1 Q / R
rho = Q / (4/3 pi R^3) = (4 pi epsilon_0) * phi / (4/3 pi R^2)
rho = 3 epsilon_0 phi / R^2
(Zn_i-n_e) = 3 epsilon_0 phi / e R^2
(Zn_i-n_e) = 3*(8.85e−12)*(5e5) / (1.602e-19)(1e-2) = 8.3e14 m^-3
Remember that a tokamak has a density of 1e21 m^-3, and that is criticized (e.g. by Nebel) as being much too low. That is the reason that electron density and ion density are tightly coupled for practically any fusion reactor design. If the electron density is low, then the ion density must be low too.

[Art Carlson]

The electrons ... should at least be faster than the ions everywhere even if their energy is a lot lower.

Bussard wrote in this report:

If the electron energy is so low that the electron speed is comparable to the ion speed, ....

I know it's crazy, but that's Bussard. It's convenient for the analysis because once the electron temperature is more than a factor of sqrt(m_i/m_e) below the ion temperature, the exact value doesn't matter. If you want a higher electron temperature, well, we've been there already. The way to minimize losses is to let the energy run from the ions to the electrons and from there to bremsstrahlung. That calculation gives you Q = 0.57. My result of Q = 0.025 is more pessimistic because it assumes monoenergetic ions (among other things in order to exploit the resonance in cross section). I advise you to stick to thermal plasmas.

Is it not plausible that the rapidly-cycling non-LTE nature of the plasma causes the energy transfer to behave differently in global terms? Maybe in some parts of the acceleration region (in between the edge and core) the cross-section distribution is favourable to rear-ending of ions by electrons, thus reversing the energy drain... or something...

I'd try to do the math but I haven't got the time and my gas kinetics notes are at school anyway...

I gave you the standard analysis, which is pretty solid. When you have developed an alternate kinetic theory of plasmas, you are welcome to present it for critique.

[icarus]

What exactly is the alternate kinetic theory for plasmas that is needed here?

[rcain]

you know, i would love to see a picture/diagram of this - say a cross sectional slice across the core; assumed particle densities, momentum (including angular), charges and B fields, with pdfs' in regions.

i am particularly interested in the dynamics just just around and inside the core shell. don't we have magnetic mirror effects here also to take into account on collisions?

i do take arts point about overall losses and the Lawson criteria. they seem impossible to ague with; thermalisation and Brem may yet overpower us. are you sure you cant scrape together another factor of two or three there Art?

[TallDave]

What exactly is the alternate kinetic theory for plasmas that is needed here?

One that allows for varying electron densities.

Actually, I think EMC2 already has computer simulations for this. Nebel mentioned them a while back.

It's hard to say what more accurately describes reality. I would say it's more likely EMC2's equations are correct for WB-1-7 sized devices (given that they have had access to the experimental data), but how things scale is debatable.

[Art Carlson]

are you sure you cant scrape together another factor of two or three there Art?

I see this as a simple 9-dimensional optimization problem. Since radiation, collisions, and fusion are all local in space, we just need to find the optimal 3-d velocity distribution functions for all three species (protons, boron ions, and electrons). We may not actually be able to get there in a real machine, but if it doesn't work in this ideal case, it cannot work in reality. Furthermore, all the relevant processes depend on the square of the density, so the Q we calculate will not depend on the absolute value of density or any variations in density over space. (Some peolple seem to be having trouble absorbing this point). Of course, the relative densities of the three species must be optimized, within the constraint of quasineutrality. (Another point that isn't sinking in very fast.)

One may question whether the problem has really been solved in this glorious generality, but, despite all criticism, I think Rider and other contributions in the literature have come remarkably close. As an optimistic approximation, you can assume a special distribution for the ions (e.g. monoenergetic, or even counter-streaming beams) and neglect the tendency for the distribution to thermalize. If you do that, the main degree of freedom left is in the electron distribution. You can reduce both bremsstrahlung and collisional coupling if you deplete the low-energy electrons. This is the only loophole I see to reach Q > 1 with p-B11.

The trouble is that electron-electron collsions will tend to fill up a hole in the distribution. Rider calculated this effect for a rather general class of distribution functions and expressed the result as a recirculating power fraction. This fraction turned out to be large even in the most favorable cases, so that you can forget about Q > 1 if (1) the power is recirculated externally (which seems to be what Bussard and Nebel are thinking of, and I can't think of any other way to do it, either), and (2) Rider's parametrized distributions are close to optimum. I think if Rider made any mistake - which hasn't been shown yet - it was in his parameterization. It would be interesting to search for a more favorable electron velocity distribution function, but it will take real physics. The backs of my envelopes are not big enough.

[jmc]

I'd just like to add that in the Chacon and Miley paper, there was very moderate convergence (no greater than five), the size of the device was explicitly limited to the debye length (densities 10^18, radius 1cm), the species as was mentioned before was D-T and the arrangements with the highest Q value were nearly maxwellian.

[TallDave]

Furthermore, all the relevant processes depend on the square of the density, so the Q we calculate will not depend on the absolute value of density or any variations in density over space. (Some peolple seem to be having trouble absorbing this point).

Yes, but some of them depend on ion density, others on electron density.

Also, how would this be affected by POPS improvements?

(2) Rider's parametrized distributions are close to optimum.

Rick mentioned at one point that square wells (such as Rider used) give very different results (i.e., much worse) than "more realistic" parabolic wells such as in the Chacon paper.

[jmc]

Parabolic wells are only "more realistic" if the scalelength of the plasma is less than the Debye length.