The machine does not use a bi-modal velocity distribution. We have looked at two-stream in detail, and it is not an issue for this machine. The most definitive treatise on the ions is : L. Chacon, G. H. Miley, D. C. Barnes, D. A. Knoll, Phys. Plasmas 7, 4547 (2000) which concluded partially relaxed ion distributions work just fine. Furthermore, the Polywell doesn’t even require ion convergence to work (unlike most other electrostatic devices). It helps, but it isn’t a requirement.
I know Dr. Nebel has mentioned this paper elsewhere as being definitive for ion behavior in IEC devices. Early on in that paper, we get the following limiting paragraph:
The present research aims to identify efficient regimes of operation of Penning IEC devices. A bounce-averaged Fokker–Planck BAFP model coded as indicated in Ref. 11 [ a work headed to press at the time of publication] has been employed for this purpose. In BAFP, only a single ion species is considered, collisional ion-electron interactions are neglected although space charge interactions between both species are included via the Poisson equation, and electrons are assumed to form a uniform, spherically symmetric cloud.
What I would like to know is this: how could the authors of this piece justifiably ignore the ion-electron collisions? It seems like this is cheating by assuming away one of the bigger energy loss mechanisms.
I thought I'd read somebody on the site recently saying that these collisions were rare, rare enough to drop out of any calculations (by some orders of magnitude). Or was that in reference to something else? Sorry I can't find the comment...
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.
the soup is complicated, but the zeroth order estimate is that ion motion is mostly radial. If you imagine one ion that has no kinetic energy near the MaGrid, it is being repelled by the MaGrid and attracted to the center. So it "falls" towards the center. By the time it gets there, it has huge kinetic energy, but it is at zero potential. So it flys through the center and out towards the MaGrid on the other side.
Now, imagine spheres of ions all doing the same thing. Near the MaGrid, they are very spread out so their interactions are small. As they fall into the center, they get close together, and somewhere along the line will smash into each other and have lots of erratic coulomb interactions.
Other than setting up useful electric fields, the electrons can be ignored. The electron-ion interactions can be pretty much ignored too, at least for initial estimates.
Finally, turn the ion shells into a continuous fluid. Ions are flowing in and out along a mostly radial path. Interactions between them tend to cancel on average, so the radial motion stays radial. Near the MaGrid, the ions tend to force each other into stationary positions (annealing is what I've seen here) and near the center ions tend to smash into each other.
How can you not have bimodal velocity distributions with two species of ions? Any p-B11 elastic collisions (or, for that matter any species1-species2 collisions) are going to transfer momenta, which in turn is going to lead to a bimodal distribution.
Seems to me the best you can do is ensure that all species have the same momenta (which means they'll have different energies and velocities) to start with. After that, there can be no momentum transfer between collisions.
Even with identical ions, a collision that isn't perfectly head-on (ie: a particle's velocity vector doesn't go through the exact centre of its counterpart) will result in a change in momentum for each one. I'm just hoping that since the mass difference between the protons and borons is much greater than the velocity difference, energy transfer will be small.
93143 wrote:Even with identical ions, a collision that isn't perfectly head-on (ie: a particle's velocity vector doesn't go through the exact centre of its counterpart) will result in a change in momentum for each one. I'm just hoping that since the mass difference between the protons and borons is much greater than the velocity difference, energy transfer will be small.
Right. But the (scalar) magnitude of the momentum vectors will be the same, even with oblique collisions. So, if collisions remain concentrated in the center of the machine, you get nothing but radial vectors that have the same magnitude pre- and post-collision, and you preserve the bimodal velocity distribution.
No. Otherwise a Dirac delta function could never relax to a Maxwellian.
Consider a frictionless elastic hard-sphere collision in which two particles moving at identical speeds at right angles to one another come into contact such that the line between their centres is along the axis of motion of particle 1 and perpendicular to that of particle 2. A T-bone collision, if you will.
The collision in this case results in particle 1 stopping cold and particle 2 flying off at a 45 degree angle with sqrt(2) times the speed (ie: its original vector plus the vector of particle 1). This results in conservation of x-momentum, y-momentum, and kinetic energy.
Note how the scalar magnitude of the momentum is not conserved...
Now, in the Polywell we're dealing with Coulomb collisions and not hard-sphere collisions, but the principle is the same. The only fundamental difference is that for a hard-sphere collision, the effective cross section is independent of relative velocity and identical for all moment orders, which makes it easy to visualize.
There are "elastic" collisions, in which kinetic energy is conserved, and there are "plastic" collisions, in which kinetic energy is converted to heat. Collisions between macroscopic objects generally fall somewhere in between the two extremes of perfectly elastic and perfectly plastic. With Coulomb collisions of nucleons, it's generally safe to assume the collisions are perfectly elastic (unless a nuclear reaction occurs).
Of course, with quantum mechanics and such there comes the additional possibility of energy being lost as a photon - hence bremsstrahlung. I understand bremsstrahlung even less well than I understand Cerenkov radiation, so I won't comment further on it.
I've done some initial calculations on this, I calculated the ratio of energy transfer between two different masses at about (4*(m1/m2)) during a collission at the limit where m1 >> m2, since an electron is 3680 times smaller than a deuteron this means that every electron-ion collission confers 1/920th of the energy of an ion-ion collission.
There may be a competing effect, however, if you open up Chen however you can see that the cross section for coloumb collission which an electron presents to an ion is:
sigma= e^4/(16*pi*epsilon^2*m^2*v^4)
we can see here that if the velocity of an electron is zero, or comparable to the velocity of an ion then its lower mass will ensure its cross-section is larger by a factor of (m2/m1)^2 that means that while the mass difference reduces the energy transferred in each collission, the lower mass of the electron ensures that there will be, many, many, more collissions. Infact, if the electrons travel at the same speed as the ions, then the ion-electron collission frequency (related to the energy transfer time as opposed to the electron ion collission time) should be 4*(mi/me) or 3680 times the ion-ion collission freqency, this mean that if the speed of the electrons and ion were comparable the ions would transfer more energy to the electrons then to the ions.
However, you may also notice a v^4 in the denominator of the cross-section, in thermal equilibrium the electrons are moving much faster than the ions and so their cross-section correspondingly reduces, thus in thermal equilibrium if you go through the maths, you will find the ion-ion collission time is about 20 times smaller than the ion-electron collission time, so such an approximation of neglecting ion-electron collission is more or less acceptable.
My concern is that in a BFR fusing boron, I believe the electrons may be maintained at a considerble lower temperature when compared to the ions, this would greatly amplify the power transferred from ions to electrons.
Not only does my reasoning using approximate equations suggest that the overall transfer of energy from ions to electrons is greater than that of ions to ions at low temperatures, experiment seems to suggest it aswell, as the energy of neutral beam atoms injected into a plasma is increased relative to the plasma temperature, they begin to heat the electrons more than the ions.
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.
If anyone wants a copy, there's 11 left on Amazon. Used, of course.