Annealing...?

[chrismb]

Dan, I'm shipping this comment from your post over into its own thread. I think it is significant enough to be a point of reference.

* Annealing is a term chosen by Bussard, etel presumably because they could not find a better term. Annealing means to reduce stresses, imbalances in a substance. Not only in an end product, but during the processing of a product- like repeated annealing of iron as it is hammered into a horse shoe or what ever. In this sense 'annealing' is a descriptive term as the ions are relaxed into a thermalized state at the beggining of each elliptical orbit. Remember the beginning of each orbit is at the edge , not at the center (or the surface of the Earth in your analogy). It is the physics that dictates that this starting energy will be a narrow thermalized cluster about a low energy average. This is a simplistic view of a complex dynamic plasma, but it gives the trends.

It is also improbable, or impossible (?) to maintain for very long. If you pulse some monoenergetic ions int a spherical potential well, the annealing will slow the overall thermalization process in the hotter regions of the well, but will lose out in the end (become a small correction to the overall thermalized spread that is probably very small). But if you maintain a flow of new monoenergetic ions at the edge of the potential well and/ or the ion dwell time is short enough, then this annealing deviation from a perfect statistical thermalized spread will become increasingly significant, and even dominating . The only real argument is the time intervals involved in a given system. A true Maxwll - Boltzmann distribution describes a plasma without borders or any other extrenal forces effecting the plasma.

Dan Tibbets

Dan, are these your own words, or an edited accumulation of other people's comments? You know I think annealing is a fantasy - but nonetheless I await final confirmation of whether there is any evidence for or against....

Nonetheless, the text you've written I find to be balanced and informative, in a way that I've not read anyone else describe before. [Maybe I missed someone else say it - apologies if that is so.]

Yes, indeed, the whole question comes down to the 'dwell time' of ions that they can get to thermalise at the temperature levels of the edge. Remember, the edge is the lowest density and the collisional rates there are therefore even lower than the off-centre collision rates. For example, you should expect more up-/down- scattering at r=0.5 than at the edge, r=1. So why would the annealing at r=1 be able to dominate the theramlisation at r=0.5? This is the reason I have always struggled to accept annealing.

But your commentary, that annealing might therefore be expected to have only a finite lifetime of efficacy at the start of ion injection, is a bit of fresh air in recognising limitations. Maybe I missed comments like that in the past, but I think it is a point that can be expanded on in its own thread.

I remain confident that if you sit and crunch real numbers, the thermalisation at high energies is a greater rate than thermalisation at these lower edge energies. I just don't see how it could possibly be any other way. And thermalisation at high energies scatters particles everywhere in velocity space such that annealing is rendered insignificant.

However, again in the way you've expressed it - the question is not so much whether it will or will not dominate, but whether it might be 'enough' recovery to 'tip the balance' in favour of net energy. I'll buy that as a reasonable point of debate. I still don't think it is true but I will accept that as a 'finite' probability of effect that's worth looking for, whereas the idea that annealing can dominate velocity-space diffusion and be a continuous process looks to me like it would approach a non-finite probability!

[One of the best bits of writing you've done recently, Dan. Keep it concise like this and it has a much bigger impact, I think.]

[93143]

Remember, the edge is the lowest density and the collisional rates there are therefore even lower than the off-centre collision rates.

At the edge, the ions are very slow. This results in a very high residence time. With constant stream mass flux at all stages of the orbit, this means that the density is actually very high at the edge. The residence-time effect dominates over the inverse-square effect because the latter is smoothly declining while the former exhibits something resembling a vertical asymptote.

So there should be a density spike at the edge. It probably isn't as big as the spike in the centre; in fact it's probably significantly smaller (it's not trivial to figure out because poor ion convergence reduces both peaks), but...

Consider the fact that this density spike happens where the ion relative velocities are at a minimum. The high-energy convergence of the core is great for fusion, but not so much for Coulomb cross-section. The 'mantle' still has ions moving past each other very quickly, depending on radius. The point where the ions stop and turn around (loosely speaking) should have the highest ion scattering cross-section in the entire reactor.

So we have a combination of high residence time, high density, and high cross-section. All because the ions are slow.

...

If my 'Langmuir onion' idea is somehow correct, the annealing would happen in each layer, not just at the edge...

[chrismb]

That's several hand-wavy comments after each other.

Would you care to add some numbers to those comments? My aspiration for this thread was to drill down into some real numbers and show this one way or the other.

I think 'a density spike at the edge' and 'very high residence time' are all so vague as to be inconsequential without supporting numbers.

You could argue that ions turning around and therefore pass through zero velocity. With some finite fraction of the ion population all undergoing this turning-around transition, then the density there must be infinite! Welcome Zeno!! This is the Achilles and the tortoise paradox. You're doing your computations wrt distance rather than time and getting a potentially perverse answer.

[93143]

*sigh* Just because there aren't numbers doesn't mean it's hand-waving. We've been through this before:

viewtopic.php?p=21760#21760

You stated that the density is lowest at the edge, while also apparently assuming that the edge is where the low-speed regime occurs. Given the latter, the former is almost certainly not correct. Further, you said that the collision rate would therefore be low, which ignores the effect of relative velocity on Coulomb cross-section and is thus also an unwarranted statement. This is all I was trying to establish. I wasn't trying to prove that annealing happens or that it works as advertised.

Incidentally, the graphs in that post were slapped together quickly and are based on an assumed potential well, not one calculated based on particle densities. I believe that discussion was a large part of what made me realize that the picture of a simple double-well (single virtual cathode containing single virtual anode) is internally inconsistent for high densities. The Carlson sheath idea didn't make sense to me, for the same reason that the Polywell plasma breaks the assumptions underlying Debye theory, so I thought it through and came up with the Langmuir onion (tightly-nested multiple wells; a "poly-well", if you will...). I seem to recall a Japanese paper showing a multiple well with at least a couple of concentric peaks; this would just add more due to the higher density... I tried to come up with a simulation to demonstrate the idea, but it turned out I was either too busy or too lazy... more recently, someone posted a link to a POPS paper that talks about plasma frequency resonance in a way that makes me think I might actually have been onto something...

[D Tibbets]

My post that was quoted is my own words, of course the discription is based from my evolving (and hopefully converging, not diverging path to understanding) view is derived from many sources, mostly posts on this forum.

Basically my view is that there are three processes contibuting. The relative densities, the relative speeds and the Coulomb crossection at these relative densities and velocities. All of course in a spherical geometry. Also, the dwell time in each zone is inversely proportional to the Ke ^2 or the velocity.

I know that if the math is not carefully used you could end up with infinities as mentioned. The old arrow and target question. - If you shoot an arrow at a target, even if it doesn't slow down, it will never reach the target. You can always divide the remaining distance by 2 to get the remaining distance. Repeat endlessly. The distance slices will become increasingly short but never reach zero.

Still, I think the onion analogy is useful. Let me try an example. I will use 6 zones- each representing a radius extending outward from the center. 1= inner core, 2= outer core, 3= inner mantle, 4= outer mantle, 6= inner edge, 6= outer edge .

First assume that the energies- velocities are about an average- that eliminates concerns about dealing with zero. The velocity may be slightly outward or inward at the outer edge, but the average will be some (ideally) velocity near absolute zero. A rather generous assumption considering the heroic efforts that have been applied to trying to cool helium to Almost absolute zero. Also, time is important, as mentioned. The Coulomb crossection is dependent on how long the charged particles are close together as they pass each other, so I will use time in the formula. Also, I assume enough convergence that the real conditions approach my idealized numbers.
I assume that density is proportional to 1/r^2 convergence. The dwell time in that zone 1/ velocity or KE/MV

ZONE 1-
KE= 100
Dwell time = 1
density= 100

Zone 2-
KE= 90
Dwell time = 10
density= 90

etc......

Zone 5-
KE10
Dwell time = 90
density= 20

Zone 6
KE= 0 (note this is an average approximate energy. It is actually a mixture of small (or not so small if referring to an upscattered ion) outward and inward velocities. This zone along with the inner core (#1) zone are the only zones that has ions effectively transiting the zone twice per orbit, at least in this simplistic model.
Dwell time = 100
Density= 10

If only looking at the density and the coulomb crossection it would seem that the colisionality would average out evenly throughout the plasma (density and crossection would cancel each other out).

But the dwell time is 10 times longer in zone 6. Here is where I am uncertain. One view might be that the crossection already incorperates the dwell time (say within the debye cube surrounding the particle), but recall that the collisionality can also be expressed as the collision frequency, which has a time component. (Collisions/ second). If the particle resides in the zone ten times as long then the number of collisions = collisions / second * amount of time in the zone.
Another way of looking at it is that the debye cube is temperature/ velocity dependent and essentially increases at slower speeds. Is there a temperature component used in calculating the Debye length?

In my model (ignore my use of 6 zones, but decimal scale for the changes- it is close enough to show the trends, though it would be less confusing if I had defined 10 zones instead of six) everything is a wash- balances out, until the dwell time is taken into account. This will lead to a slight to strong dominating effect so long as the collision MFP is long enough that complete thermalization does not occur within the interior zones on one pass.

My attrempt at developing a formula:

T= d*c* (1/t)

T= total thermalization time limit in a defined zone
d= local density in a defined zone
c= collision frequency in that zone
t= dwell time in that defined zone (r/R).

If T is reached in the outermost zone (note this is a range near the border, not percisely on the border) it will be thermalized about a low average speed (which would be a relatively narrow spread as a percentage of the potential well induced maximum KE/ velocity- thus the somewhat misleading term of monoenergetic.

Provided the radial nature of the plasma is maintained at some point the total thermalization time limit will be longer than the dwell time in the deeper zones. Initially a good system may only have zone 6 fully thermalizing. As conditions relax the zones would essentially merge into each other till from a thermalization standpoint there would be only one zone. How long this takes is dependant on the average thermalization time if you assume a boundless plasma without any external force applied* and the MFP that is much shorter than the plasma dimensions (it is boundless, afterall ) . And dependent on the zonal localized differential thermalization that can occur if the plasma does not start out at an initial average condition , is subject to a differentiating force, and has a boundry condition where the MFP is close to or exceeds the dimensions of the plasma (Umm... at least the deeper layers of the plasma).


* The external force is the potential well that is driven by the continuous injection (or recirculation) of nearly monoenergetic excess electrons.

This discussion only addresses ion energy thermalization. Angular momentum 'thermalization' is, I think, more complex (Shish! ). And electron thermalization is a whole seperate issue.......

Dan Tibbets