In another thread there was a debate about the magnitude of the difference in the Coulomb and Fusion crossections and how dominate the Coulomb collisions were. This made it difficult to argue for holding off the thermalization process over the lifetime of the ions before fusion. Generally there was two points argued. The type of scattering collision (angle of deflection) and the rate. With the link to Hyperphysics it was shown that the Coulomb collisionality actually decreases tremendously fast as the temperature increased. The significance of this may not have been obvious.
Thus, I present this graph taken from page 40, of :
The graph illustrates how close the Coulomb crossection comes to the fusion crossection at various temperatures. At ~ 10^1 KeV ( ~ 100 million degrees), the Coulomb collision crossection is indeed many times the fusion crossection. But at ~ 10^2 KeV the Coulomb scattering crossection is only ~ 10 times the fusion crossection for this D-T example. The D-D difference may be up to 1000x, but because the operating temperature may be higher, the actual difference may more closely match that of D-T ratios.
This suggest that at appropriate temperatures, edge annealing, and core convergence needs to fight thermalizing forces (Coulomb collisions) for only a few hundred interactions for each fusion interaction, as opposed to a few million or more scattering interactions that would occur at the expected Tokamak average temperatures of ~ 5 KeV.
The narrow initial energy spread in the Polywell also needs to be taken into account as there are not nearly as many ions in the otherwise useless low energy side of the average temperature that would contribute the vast majority of scattering collisions in a thermalized plasma. It is essentially a win - win situation for the Polywell.
I have wondered why ~ 80 KeV was quoted for an operating D-D Polywell, despite the fusion crossection slope becoming less than ideal. I figured it was a matter of power density, but the dynamic Coulomb collision crossection vs the fusion crossection ratios may be even more important.
Dan, I can see that this graph is plotted for scattering of particles at 10 degrees. What you need to take account of is that, for example, scattering at 1 degree is therefore 100 times that scattering cross section (energy lost is the square of half the sine of the scattering angle, so for small angles energy loss per collision is inversely proportional to the square of the angle) and that small angles of scatter are therefore much more likely.
Also, this is just the nuclear-nuclear elastic collisions, and forgets all the other troubles with the electrons!
So the calculation is a bit more involved than that.
That being said, this is touching on my basic approach; if you could actually keep the deuterons 'hot' after each small angle collision (and let's just assume for a minute that all scatters are 10 degrees) then actually the energy you put in is 1MeV for the ion and 20 lots of 10 degree scatters (which is about 20keV each = 400keV) so supposedly you'll end up with net energy out.
But if it were this easy then just a neutral beam whammying into a hot plasma should make more fusion energy than is the beam energy. Unfortunately, it doesn't because all those millions of tiny angle collisions give little probability of a fusion and plenty of probability for the ion to drop energy down to where fusion is no longer likely.
Keep the beam energy high, and the numbers add up. That's the key to this cookie. Beam energy is nothing without recovering those scattered particles back up to a healthy energy with good fusion cross-section.
chrismb. Small angle deflecting collisions ideed are progressively more commen, but their individual effects are proportionatly smaller, at least when talking about cunulative angular momentum that would decrease confluence. The small angle collisions are small angle collisions because they do not pass as close to each other. I suspect, but cannot defend, that this also translates into less magnitude of uperscattering or downscattering in these small angle collisions. My postulate is that you can choose small or large angle collision interactions, the individual occurrences very, but the cumulative thermalization and and angular momentum changing effects from all of the collisions reach the ~ same net effect, irregardless of the specified sampled collision subset. Ie: you can plug in one value for a selected angle deflection and thus calculate a time dependent scattering magnitude.
As far as up and down scattering. I'm not sure what you mean by energy loss. I suppose all of the collisions are not purely elastic, but for limited times I'm assuming the energy losses are not critical (I am not talking about Bremsstrulung here). Certainly down scattered ions will contribute to increased coulomb collision rates at whatever angle you choose, while upscattered ions will contribute less, and perhaps increase the fusion rate within limits. I think the net effect would be neutral, if the coulomb collision crossection was linier (which it is not).
The downscattered ions would have a magnifying effect. That is why I said (sort of) thermalized plasmas would thermalise newly injected monoenergetic ions faster, possibly much faster. But, starting with a monoenergetic plasma as created dynamically by continous injection of monoenergetic ions will slow this process. Perhaps the advantage involves only a few collisions, but if annealing works, and the mean free path is long enough so that there are only a relatively few large angle collisions or a few more small angle collisions, before the ions reach the annealing zone, then the process can be retarded long enough.
As you pointed out in your reply, the Coulomb collisionality becomes more dominate at lower energies. This plays into the annealing claims, as these slower moving ions will be significantly down scattered ions and/ or the ions reaching the top of the potential well near the Wiffleball border. This greatly accelerates thermalization, but the width of the energy distrbution is small compared to the high energies applied to each ion as it falls down the potential well. Annealing is a thermalizing process that dominates at the lowest energies, the spherical geometry makes this the edge regions. The cooler ions increase the collisionality wherever they are located, but most of these will be in the annealing periphery and this is what actually drives this local, and beneficial thermalization around a low average energy.
An analogy may be two valleys. The first deep valley represents most of the potential well. As the marbles (ions) climb out they slow to some slow average speed +/- some range due to scattering. Most of the marbles reach the crest of the first valley, and fall into a shallow side valley. Here they bounce around, thermalizing with each other. Some may manage to climb out of this shallow valley (or almost level plain, with a up slope at the far end, if you prefer) and escape, but most will fall back into the deep valley after jostling each other for a while. The fast ones cannot get through the crowd, and are slowed. The slow ones are jostled by the faster and are sped up. Because of the crowd (and the relative time they spend close together) the sprinters and crawlers end up as walkers, the speed may vary some, but not much due to the crowding.
Note that the 'crowding' reflects the vastly increased collisionality, not just the particle density.
What about the ions that have been significantly downscattered, are they immune to annealing? Yes, if you consider purely radial plasmas and assume the Annealing zone only exists right at the edge. Actually the annealing zone is continuous near the edge, not just on the edge. The definition would depend on several factors. Basically, where the average ions have slowed enough that the thermalized energy spread around that average is less than the defined/ desired energy spread near the core would be within the annealing zone.
This annealing is claimed to adequately manage upscattering and perhaps to a smaller extent downscattering , but only for the needed lifetimes of the ions before fusion or escape. My impression is that the angular momentum scattering may be less well controlled, thus admitted uncertainty about how well confluence (central focus) will be maintained.
The point of my first post is that these are certainly challenges, but not seemingly impossible challenges that the veiwpoint of starting with a globally thermalized, relatively cool plasma throughout the machine would suggest.
