Central electron temperature and p-B11 power balance

[Art Carlson]

Bussard wrote in this report:

If the electron energy is so low that the electron speed is comparable to the ion speed, ... This latter condition can obtain only ... if the electron/ion collision rates are sufficiently small that no significant collisional heating of electrons can take place during the electron lifetime in the machine.

Nebel wrote (not sure where):

The theory says that you can beat Bremstrahlung, but it's a challenge. The key is to keep the Boron concentration low compared the proton concentration so Z isn’t too bad. You pay for it in power density, but there is an optimum which works. You also gain because the electron energies are low in the high density regions.

Is this for real? Does the canonical design for a p-B11 polywell power reactor have a low electron temperature/energy in the fusion region? It should be trivial to show that those electrons will drain the energy from the ions long before they have time to fuse. I'll post the calculation myself, but I wanted to first verify that this is what the experts are talking about, because, frankly, I think it is a waste of time to even consider operating in such conditions. (Please point out my error so that I can abjectly apologize!)

[Jboily]

Art,
I saw a report somewhere
(can't remember from whom it was right now) calculating the energies transfer. As I recall, most of the transfer occurs within the dense region, where the protons heat the boron, which in turn heat the electrons. The electron energy is then removed from the machine, and recycled (recaptured somehow) back to the proton.
This means the Boron temperature is lower then the proton temperature in a continuous operation, reducing the Bremmstrahlung radiation as a result. As I recall, this recirculation power is of the same order of the fusion power. I think it is reasonable enough, as long as you have a good power recirculation efficiency.

[rcain]

Bussard wrote in this report:

If the electron energy is so low that the electron speed is comparable to the ion speed, ... This latter condition can obtain only ... if the electron/ion collision rates are sufficiently small that no significant collisional heating of electrons can take place during the electron lifetime in the machine.

Nebel wrote (not sure where):

The theory says that you can beat Bremstrahlung, but it's a challenge. The key is to keep the Boron concentration low compared the proton concentration so Z isn’t too bad. You pay for it in power density, but there is an optimum which works. You also gain because the electron energies are low in the high density regions.

Is this for real? Does the canonical design for a p-B11 polywell power reactor have a low electron temperature/energy in the fusion region? It should be trivial to show that those electrons will drain the energy from the ions long before they have time to fuse. I'll post the calculation myself, but I wanted to first verify that this is what the experts are talking about, because, frankly, I think it is a waste of time to even consider operating in such conditions. (Please point out my error so that I can abjectly apologize!)

i think i understand what you aer saying, however, i do not think that 'neutralization' (distribution) will be as 'symmetrical' as you suggest.

sure, we have a 'quasi-neutral' core, but i think there will be finer structure. i suggest it is the fine structure that ensures we have a 'marginal' but 'sufficient' ('optimal' ?) profile across the core.

typically, i would suggest that such a fine-structure (actually still defined at a 'macro' scale would be determined by the charge, velocity and momentum vectors associated with each species, within each region/zone.

is there a model for this anywhere?

Brem is the flip-side cost of such asymmetry (a second distribution function).

thus, there is (likely to be) an optimal region within the combined space, where fusion (ignition?) conditions might obtain.

or am i totally misunderstanding?

ps: '...waste of time to even consider operating in such conditions...' - are you referring to the difficulty in preventing overall 'Maxwellization/Gaussing' of the core zone?

[Art Carlson]

I'll try to answer my own question, trying to be as generous as possible to the polywell, within the assumption that the ion speed is much larger than the speed of the electrons.

I published a similar calculation already for Rostocker's CBFR, where I concluded that the ratio P_fusion/P_fric cannot be more than 0.12, even taking credit for goodies like the 580 keV (CM energy) resonance and spin polarization. P_fric was power loss due to direct collisions between the protons and boron ions, but I also said "The power balance would be at least another factor of three less favorable than this estimate because the coupling of the ions through the electrons would be stronger than the direct coupling if Te < E0/15 = 40 keV." This would be the regime we are discussing here. In fact, we have to deduct another factor of 1.6 because we cannot polarize the fuel ions in a polywell. With P_fusion/P_loss = 0.025, assuming the power lost from the ions can be harvested and recycled with the same efficiency eta as the fusion power, breakeven would require
eta = P_loss / (P_fusion+P_loss) = (1+P_fusion/P_loss)^-1 = 97.5%
Significant net power production would require eta to be at least 99%.

I did not publish the details of this calculation (I wonder if I have notes somewhere?), but it shouldn't be hard to reproduce. According to the NRL Plasma Formulary, p. 32, in this limit the slowing down rate is given by
nu_s[s^-1] = 1.7e-4 mu^0.5 * n_e[cm^-3] * Z^2 * lambda_ie * epsilon[eV]^-1.5
nu_s = (1.7e-4 s^-1 cm^3 eV^1.5) mu^0.5 * n_e * Z^2 * lambda_ie * epsilon^-1.5
for p: mu = 1 and Z =1
for B11: mu = 11 and Z = 5
I'll just take lambda_ie = 10. (Anyone who wants to refine the decimal places is welcome to do so.)
The energy loss rate is twice as fast (due to the v^2 dependence).

The most generous assumption I can make is a colliding beam configuration like the CBFR, although the core of the polywell will have an approximately isotropic distribution (even if it is monoenergetic), so not all the collisions can be at the resonance energy. In that case, the protons have to have 11/12 of the CM energy and the boron ions the other 1/12.
epsilon_p = (11/12)*epsilon_CM
epsilon_B = (1/12)*epsilon_CM
The power lost to collisions per unit volume will be
P_loss = n_p*epsilon_p*2*nu_s(p) + n_B*epsilon_B*2*nu_s(B)
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * Z^2 * lambda_ie * [
n_p * mu_p^0.5 * epsilon_p^-0.5 +
n_B * mu_B^0.5 * epsilon_B^-0.5 ]
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * lambda_ie * [
Z_p^2 * n_p * mu_p^0.5 * epsilon_p^-0.5 +
Z_B^2 * n_B * mu_B^0.5 * epsilon_B^-0.5 ]
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * lambda_ie * epsilon_CM^-0.5 * [ 1.04 * n_p + 275 * n_B ]
Obviously, the power loss from the boron ions will dominate for any reasonable fuel mixture. (This suggests running boron lean.) Dropping the n_p term, setting n_e = n_p, lambda = 10, and epsilon_CM = 580 keV, we arrive at
P_loss = (3.4e-4 s^-1 cm^3 eV^1.5) * n_p * n_B * 10 * (5.8e5 eV)^-0.5 * [ 275 ]
P_loss = (1.23e-3 s^-1 cm^3 eV) * n_p * n_B

That looks good. The next step is to compare this to the fusion power. sigma*v at resonance is (8.1e-16 cm^3 s^-1) (no bonus for spin polarization here), and the energy gain per fusion reaction is 8.7 MeV. That gives
P_fusion = n_p * n_B * (8.7e6 eV) * (8.1e-16 cm^3 s^-1)
= n_p * n_B * (8.7e6 eV) * (8.1e-16 cm^3 s^-1)
= (7.0e-9 s^-1 cm^3 eV) * n_p * n_B
and
P_fusion/P_loss = 5.7e-6

(Do I believe this result? It wouldn't surprise me if I lost a factor of lambda or mu here or there, but the drag on cold electrons is really pretty drastic, so, yeah, I think you will be dead before you hit the ground if you try to use cold electrons to make p-B11 fusion.)

[rcain]

I'll try to answer my own question, trying to be as generous as possible to the polywell, within the assumption that the ion speed is much larger than the speed of the electrons.

I published a similar calculation already for Rostocker's CBFR, where I concluded that the ratio P_fusion/P_fric cannot be more than 0.12, even taking credit for goodies like the 580 keV (CM energy) resonance and spin polarization. P_fric was power loss due to direct collisions between the protons and boron ions, but I also said "The power balance would be at least another factor of three less favorable than this estimate because the coupling of the ions through the electrons would be stronger than the direct coupling if Te < E0/15 = 40 keV." This would be the regime we are discussing here. In fact, we have to deduct another factor of 1.6 because we cannot polarize the fuel ions in a polywell. With P_fusion/P_loss = 0.025, assuming the power lost from the ions can be harvested and recycled with the same efficiency eta as the fusion power, breakeven would require
eta = P_loss / (P_fusion+P_loss) = (1+P_fusion/P_loss)^-1 = 97.5%
Significant net power production would require eta to be at least 99%.

I did not publish the details of this calculation (I wonder if I have notes somewhere?), but it shouldn't be hard to reproduce. According to the NRL Plasma Formulary, p. 32, in this limit the slowing down rate is given by
nu_s[s^-1] = 1.7e-4 mu^0.5 * n_e[cm^-3] * Z^2 * lambda_ie * epsilon[eV]^-1.5
nu_s = (1.7e-4 s^-1 cm^3 eV^1.5) mu^0.5 * n_e * Z^2 * lambda_ie * epsilon^-1.5
for p: mu = 1 and Z =1
for B11: mu = 11 and Z = 5
I'll just take lambda_ie = 10. (Anyone who wants to refine the decimal places is welcome to do so.)
The energy loss rate is twice as fast (due to the v^2 dependence).

The most generous assumption I can make is a colliding beam configuration like the CBFR, although the core of the polywell will have an approximately isotropic distribution (even if it is monoenergetic), so not all the collisions can be at the resonance energy. In that case, the protons have to have 11/12 of the CM energy and the boron ions the other 1/12.
epsilon_p = (11/12)*epsilon_CM
epsilon_B = (1/12)*epsilon_CM
The power lost to collisions per unit volume will be
P_loss = n_p*epsilon_p*2*nu_s(p) + n_B*epsilon_B*2*nu_s(B)
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * Z^2 * lambda_ie * [
n_p * mu_p^0.5 * epsilon_p^-0.5 +
n_B * mu_B^0.5 * epsilon_B^-0.5 ]
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * lambda_ie * [
Z_p^2 * n_p * mu_p^0.5 * epsilon_p^-0.5 +
Z_B^2 * n_B * mu_B^0.5 * epsilon_B^-0.5 ]
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * lambda_ie * epsilon_CM^-0.5 * [ 1.04 * n_p + 275 * n_B ]
Obviously, the power loss from the boron ions will dominate for any reasonable fuel mixture. (This suggests running boron lean.) Dropping the n_p term, setting n_e = n_p, lambda = 10, and epsilon_CM = 580 keV, we arrive at
P_loss = (3.4e-4 s^-1 cm^3 eV^1.5) * n_p * n_B * 10 * (5.8e5 eV)^-0.5 * [ 275 ]
P_loss = (1.23e-3 s^-1 cm^3 eV) * n_p * n_B

That looks good. The next step is to compare this to the fusion power. sigma*v at resonance is (8.1e-16 cm^3 s^-1) (no bonus for spin polarization here), and the energy gain per fusion reaction is 8.7 MeV. That gives
P_fusion = n_p * n_B * (8.7e6 eV) * (8.1e-16 cm^3 s^-1)
= n_p * n_B * (8.7e6 eV) * (8.1e-16 cm^3 s^-1)
= (7.0e-9 s^-1 cm^3 eV) * n_p * n_B
and
P_fusion/P_loss = 5.7e-6

(Do I believe this result? It wouldn't surprise me if I lost a factor of lambda or mu here or there, but the drag on cold electrons is really pretty drastic, so, yeah, I think you will be dead before you hit the ground if you try to use cold electrons to make p-B11 fusion.)

ah, now i understand, to paraphase, you posit::

in polywell (wiffle-ball) regime:

P_fusion/P_fric <= 0.12

(
ie:
P_fric/P_fusion >= 1/0.12 )
}

where P_fric is power loss due to direct collisions between the protons and boron ions

and that this is a dominant power-loss factor, taking into account, it is a term further amplified (by a factor of around 3) due to "...coupling of the ions through the electrons would be stronger than the direct coupling if Te < E0/15 = 40 keV."

and that this is term is further amplified by a factor of around 1.6 due to (similar symmetrical/maxwellian) collissions between boron ions.

yielding a net (in)equality:

P_fusion/P_loss == 0.025

(
ie:
P_fusion/P_loss <= 0.025
)

and this inequality bounds the theoretically achievable efficiency of our regime.

yes?

i see you point.

thanks for the fine links btw. i'll see if a cant find some alternative view. i may be some time.

ps. Nevins, commenting on the same proposal says:

Highly nonthermal systems, like the colliding beam reactor proposed by Rostoker et al. would relax to local thermal equilibrium before a significant amount of fusion power could be produced. Alternatively, the nonthermal ion distribution could be maintained by cycling sufficient power through the system.

everyone seems to recognize the 'shell-like' structure of the plasma core at least in FRC.

q: does he/do you also consider any contributions from microwave/acoustic/pops-like phenomena/artifacts?

(i think, partially, that may be your term: 7.0e-9 s^-1 cm^3 eV am i correct?)

[Art Carlson]

Sorry. I think I cheated you out of 2 orders of magnitude.

...
The most generous assumption I can make is a colliding beam configuration like the CBFR, although the core of the polywell will have an approximately isotropic distribution (even if it is monoenergetic), so not all the collisions can be at the resonance energy. In that case, the protons have to have 11/12 of the CM energy and the boron ions the other 1/12.
epsilon_p = (11/12)*epsilon_CM
epsilon_B = (1/12)*epsilon_CM
The power lost to collisions per unit volume will be
P_loss = n_p*epsilon_p*2*nu_s(p) + n_B*epsilon_B*2*nu_s(B)
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * Z^2 * lambda_ie * [
n_p * mu_p^0.5 * epsilon_p^-0.5 +
n_B * mu_B^0.5 * epsilon_B^-0.5 ]
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * lambda_ie * [
Z_p^2 * n_p * mu_p^0.5 * epsilon_p^-0.5 +
Z_B^2 * n_B * mu_B^0.5 * epsilon_B^-0.5 ]
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * lambda_ie * epsilon_CM^-0.5 * [ 1.04 * n_p + 275 * n_B ]

Obviously, the power loss from the boron ions will dominate for any reasonable fuel mixture. (This suggests running boron lean.)

Just because the boron ions run out of energy doesn't mean fusion stops. On the short time scale I calculated, the borons will slow to a stop, so we can consider what happens next. We then have a system with motionless boron ions being bombarded by protons. The protons are also losing their energy, but not as fast as the boron ions did. (The polywell as commonly preached doesn't look like this -- but maybe it should!)

The equations still apply, but now we want to put all the energy in the protons: epsilon_p = 650 keV and epsilon_B = 0. (There appears to be several percent uncertainty in the energy of the resonance, but that shouldn't bother us.) That gives us
P_loss = (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * lambda_ie * [
Z_p^2 * n_p * mu_p^0.5 * epsilon_p^-0.5 +
Z_B^2 * n_B * mu_B^0.5 * epsilon_B^-0.5 ]
= (3.4e-4 s^-1 cm^3 eV^1.5) * n_e * lambda_ie * n_p * (6.5e5 eV)^-0.5 ]
= (4.2e-6 s^-1 cm^3 eV) * n_e * n_p

In this case we would want to operate boron rich, so we can take the limit n_e = 5*n_B, leading to
P_loss = (2.1e-5 s^-1 cm^3 eV) * n_B * n_p

That looks good. The next step is to compare this to the fusion power. sigma*v at resonance is (8.1e-16 cm^3 s^-1) (no bonus for spin polarization here), and the energy gain per fusion reaction is 8.7 MeV. That gives
P_fusion = n_p * n_B * (8.7e6 eV) * (8.1e-16 cm^3 s^-1)
= n_p * n_B * (8.7e6 eV) * (8.1e-16 cm^3 s^-1)
= (7.0e-9 s^-1 cm^3 eV) * n_p * n_B[/b]

That equation is still good, so we now have
P_fusion/P_loss = (7.0e-9 s^-1 cm^3 eV) / (2.1e-5 s^-1 cm^3 eV) = 3.3e-4
So there you have the 2 orders of magnitude I cheated you out of. The result is still devastating.

(Remember, this is the result when you assume cold electrons. Take hot electrons and you move from "devastating" back to merely "requires unrealistically optimistic assumptions". Also, we are talking specifically about p-B11. With a more reactive fuel, the focus moves away from these completely general problems back to the confinement properties of the polywell.)

[bcglorf]

I hate to ask a stupid question, but as a layman with nothing more than an undergrad physics minor I'm trying to get my head around things still. My understanding had been that Bussard's loss calculations for things like Brem in particular differed from those of Nevins and Rider primarily because of how they modeled particle energies and densities in the system. From my very limited understanding, the virtual anodes and cathodes in theory created a circumstance were electron densities were very low in the core.

From this if I may try people's patience with 2 questions. First, is my understanding even approximately correct? Secondly, would that layering of densities make a significant difference to calculations like those you've laid out?

[TallDave]

I assumed Art's equations were for an LTE quasi-neutral distribution, and thus his parenthetical disclaimer:

(The polywell as commonly preached doesn't look like this -- but maybe it should!)

I think the standard counter-argument is that the Fokker-Planck modelling Chacon did is probably more accurate.

Maybe Rick Nebel will stop by and enlighten us, although I've been hoping he's been called away for something more important...

[Art Carlson]

From my very limited understanding, the virtual anodes and cathodes in theory created a circumstance were electron densities were very low in the core.

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.

[Art Carlson]

I assumed Art's equations were for an LTE quasi-neutral distribution, and thus his parenthetical disclaimer:

(The polywell as commonly preached doesn't look like this -- but maybe it should!)

No, I'm trying to give you every break I can. When the electron velocities are small compared to the ion velocities (our starting point, based on the quotes from Bussard and Nebel), the exact distribution of electron velocities doesn't matter. I assumed a mono-energetic proton distribution because that is the best you can do, even though I don't believe you would ever be able to maintain such a distribution.

I think the standard counter-argument is that the Fokker-Planck modelling Chacon did is probably more accurate.

Are you referring to "Energy gain calculations in Penning fusion systems using a bounce-averaged Fokker–Planck model", Phys. Plasmas 7, 4547 (2000), where the abstract reads

Fusion energy gains for these systems have been calculated in optimistic conditions (i.e., spherically uniform electrostatic well, no collisional ion-electron interactions, single ion species) using a bounce-averaged Fokker–Planck (BAFP) model.

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

Maybe Rick Nebel will stop by and enlighten us, although I've been hoping he's been called away for something more important...

I second both points.

[bcglorf]

From my very limited understanding, the virtual anodes and cathodes in theory created a circumstance were electron densities were very low in the core.

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.

Thanks for that. If I may continue asking stuff I'm sure is obvious to others, I'm still missing some of the explanation that I've read from Bussard. Although the region within the magrid is quasineutral, I understood Bussard to believe that the distribution within that region was less uniform. Something to the general effect of virtual cathode and anode formation resulting in non-uniform electron densities, most importantly making their densities very small dead center. Am I understanding Bussard's explanation correctly? If I am, does it make any difference anyways?

[scareduck]

Art -- I had the same misgivings about the Chacon, Miley paper as well; Nebel responded in this thread as follows:

At the risk of putting my foot in my mouth, the usual answer is that ion-electron collisions are much smaller than ion-ion collisions. It's much easier for particles of the same mass to transfer momentum to one another. This is the same effect as shooting pool with a cue ball that weighs the same as the other balls vs. shooting pool with a heavy cue ball. It's really hard to stop that heavy cue ball (it's just the combination of conservation of momentum and energy).

The best discussion I've seen of this is in chapter 4 of Glasstone and Lovberg (Controlled Thermonuclear Reactions, Robert E. Kriger Publishing Co. 1975.) but I suspect that it is out of print. Generally, calculating collisions of ions with electrons is more complicated than electron-electron collision, ion-ion collisions, or electron-ion collisions. The general rule of thumb is that electron distributions and ion distributions will equilibrate at a much faster rate than they will transfer energy to each other.

[Art Carlson]

Art -- I had the same misgivings about the Chacon, Miley paper as well; Nebel responded in this thread as follows:

At the risk of putting my foot in my mouth, the usual answer is that ion-electron collisions are much smaller than ion-ion collisions. It's much easier for particles of the same mass to transfer momentum to one another. This is the same effect as shooting pool with a cue ball that weighs the same as the other balls vs. shooting pool with a heavy cue ball. It's really hard to stop that heavy cue ball (it's just the combination of conservation of momentum and energy).

The best discussion I've seen of this is in chapter 4 of Glasstone and Lovberg (Controlled Thermonuclear Reactions, Robert E. Kriger Publishing Co. 1975.) but I suspect that it is out of print. Generally, calculating collisions of ions with electrons is more complicated than electron-electron collision, ion-ion collisions, or electron-ion collisions. The general rule of thumb is that electron distributions and ion distributions will equilibrate at a much faster rate than they will transfer energy to each other.

Great thread! I must have missed it the first time around. My first response to Nebel is that a calculation always trumps a rule of thumb. It is true with regards to thermalization, that each species relaxes with itself faster than the two temperatures equilibrate. On the other hand, (very) fast ions (e.g. neutral beam injection) will give up their energy preferentially to the electrons. I agree it is a good idea to compare the order of magnitude of various effects before starting a calculation, but there are many collision processes in many limits, so you have to be sure just what you are talking about before making generalizations.

***

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*. Since the Lawson criterion (triple product required) for p-B11 is 500 times higher than for D-T, there is no indication *based on this work* that p-B11 has a prayer. Not only that, in the last paragraph he says

Preliminary studies of the effects of electron particle and radiative losses indicate that large Q-values are still possible if Penning IEC devices operate with small electron to ion density ratios, ...

That is, the only hope he sees once electron collisions are added is to give up quasineutrality, which is only possible for physically small devices (on the order of 1 mm, not 1m).

***

Another thing occurred to me while reading that thread. For a two-component fuel mix (anything other than D-D or He3-He3), ion collisions will result in rapid spread of the ion energy distribution under the conditions envisioned for a polywell. If both protons and boron ions fall in from the edge of the potential well, the boron ions will have Z=5 times as much energy and sqrt(Z*mu)=sqrt(5*11)=7.4 times as much momentum. In a head-on collision between p and B11, the p will bounce back with over 4 times its initial energy, and the B11 will drop to a third of its initial energy. The idea that collisions among ions just change the directions and not the energies is dead wrong.

[Art Carlson]

Thanks for that. If I may continue asking stuff I'm sure is obvious to others, I'm still missing some of the explanation that I've read from Bussard. Although the region within the magrid is quasineutral, I understood Bussard to believe that the distribution within that region was less uniform. Something to the general effect of virtual cathode and anode formation resulting in non-uniform electron densities, most importantly making their densities very small dead center. Am I understanding Bussard's explanation correctly? If I am, does it make any difference anyways?

I don't always understand Bussard either, so you're not alone. 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. I believe, however, that Bussard envisions a virtual anode (i.e. a small excess of ion density over electron density) at the core of the device due to ion convergence. I believe this will be associated with an increase in both densities, but I have never seen a model detailed enough that I could understand it.

[bcglorf]

Thanks for that. If I may continue asking stuff I'm sure is obvious to others, I'm still missing some of the explanation that I've read from Bussard. Although the region within the magrid is quasineutral, I understood Bussard to believe that the distribution within that region was less uniform. Something to the general effect of virtual cathode and anode formation resulting in non-uniform electron densities, most importantly making their densities very small dead center. Am I understanding Bussard's explanation correctly? If I am, does it make any difference anyways?

I don't always understand Bussard either, so you're not alone. 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. I believe, however, that Bussard envisions a virtual anode (i.e. a small excess of ion density over electron density) at the core of the device due to ion convergence. I believe this will be associated with an increase in both densities, but I have never seen a model detailed enough that I could understand it.

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?