Annealing: a thought experiment proposal

Discuss how polywell fusion works; share theoretical questions and answers.

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charliem
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Annealing: a thought experiment proposal

Post by charliem »

I've read some posts these last days regarding annealing, what it is and is not, and whether it's a real phenomenon or just a case of wishful thinking. Both positions have their supporters.

A running polywell is a complex system, and so quite difficult to model, and even more to understand intuitively. That's why when trying to get some clue about its functioning I use a much simpler design. I propose it to try determine if "annealing" is at all possible.

The model can be used just as a thought experiment, or can be translated to equations, or to a PIC simulation.

It consist on a rectangular 2D box made of perfectly reflective walls, so that any particle that hits then suffers a perfect elastic collision, and doesn't loose any kinetic energy (and its momentum only varies in direction).

The box is immersed in an external, constant, and uniform electrical field that enters it normal to one of its faces, the anode (I use the left wall) and gets out thought the opposite (the cathode).

To study annealing I'd inject negatively charged particles (all same charge, same mass, same KE) through the anode, their KE enough for them to reach a distance D from it (without getting to the cathode).

Some questions:

1) Will them thermalize throught the whole volume or by regions?

2) How quickly?

3) Will thermalization near a line at distance D from the anode take out most of the lateral momentum of the charges?

4) What about the perpendicular momentum?

5) And what happens if at a certain point we start injecting also positive charges, so that the negatively charged plasma becomes a much more dense quasi-neutral negatively biased one?

D Tibbets
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Post by D Tibbets »

A square or rectangular 2D geometry is different than a spherical geometry. But if you can somehow isolate the the charged particles motion to only two opposing sides it might work. Charged particles moving towards, scattered towards the two open ends may grow to dominate the motions. Why have a electrical moving through the box. Just place a field at the center or in a box some distance inside the first- this would represent either a parabolic potential well or a square potential well which I believe can exist, depending on the stage you are talking about. Keep in mind that the potential well , space charge, biased electrical field is created by the high energy and radial vectors (at least initially) of the electrons. It is the ions that then experience the effects of the potential well- low energy/ speed at the top of their potential well. The top of the potential well for the electrons is near the center.

Creating a biased electrical field by having excess electrons contained inside this box will create the net electrical field. You could substitute an electrical field for the electrons, but both together would only complicate things.

If you have a physics simulator, bouncing balls should (I think) show the effect. Just line up a bunch of balls along a line at the top and let them fall (gravity substitutes for the electrical field). Set collisions to elastic, no dampening allowed . The failure of much of these basic simulations, though is that they may not incorporate the collisionality of the balls vs speed. Also, a linear floor would not represent the spherical geometry, where any deflections at the center (or to a lesser extent the near center) can only be radial- or straight up in the linear model.

In your box model, not only an electrical field is needed, but the relevant calculations of ion conditionality vs energy, and the resolution (calculations per bounce (orbit), must be frequent enough to discriminate behavior over (very?) short distances. Also, I'm uncertain how the density issues in a linear or rectangular volume would change in a spherical model. If you take a 1 cm slice in the linear model, the density would vary only due to the speed of the particles passing through that slice per unit of time. The volume of the slice would not change. In the spherical model the volume of this slice would increase in volume as you moved away from the center. This would effect the volume in the slice and thus the density. As collisionality increases at near the square of the density, this also needs frequent calculation. For this reason alone, I think the linear approach would not work, unless you introduced some correction for the changing volumes. Would a 2 D circular model work? Possible it would work better, but you would still need to consider the 2D - vs - 3D consequences (Volumes per slice = width X height - vs - volume per slice = width X height x depth.

Dan Tibbets
To error is human... and I'm very human.

charliem
Posts: 218
Joined: Wed May 28, 2008 8:55 pm

Post by charliem »

Hi Dan.

Some time since I posted anything (Art Carlson was still around ... ;) ).
D Tibbets wrote:A square or rectangular 2D geometry is different than a spherical geometry. But if you can somehow isolate the the charged particles motion to only two opposing sides it might work. Charged particles moving towards, scattered towards the two open ends may grow to dominate the motions.
The idea that aims me is not to pursue any kind of approximation to a Polywell, just to study one of its quirks, the so called "annealing", in a system as simple as possible. See that I'm not even talking about electrons or ions, nor magnetic field, and this configuration doesn't includes a potential well.

That's why I choose to use 2D, a rectangular closed box instead of a more complex geometry, an externally applied constant, homogeneous (at least before injecting any charges) E-field, and perfectly reflective walls.

As I visualize it the negative charges injected through the anode can only get to a distance D from it before the E-field forces them to reverse their motion. When they reach the anode they bounce and the cycle repeats.

With very few particles (and total charge) the speed distribution during the first cycles should be very simple, a linear v=k.d (d=distance from the anode) for the module, and just two directions, (1,0) or (-1,0).

But of course the situation gets complicated if we wait a little and/or inject more charge. The build up of space charge modify the E-field, and interactions between charges alter their momentum.

What I'd like to know better is the behavior when we inject much more charge and let time go. ¿Will the particles thermalize throughout the whole volume (filling the chamber), like Josheph Chikva suggested in another thread talking about the PW? ¿Or will the thermalization mainly happen in a layer/region where our "bouncing charges" have very low speed, and so the particles that reach the anode will still be essentially monoenergetic?

If I get to understand clearly the behavior in this regimen, then the next step would be to add charges in pairs positive-negative. The positive ones would bounce on the opposite wall (the cathode) and merge with the negatives in the middle of the machine. ¿Will this destroy the annealing (supposed that it existed in the previous configuration)?

As I read the description of annealing in a Polywell, at least when referred to the electrons, this oversimplified system should behave in a very similar way. If so, a good understanding of it can provide some clues about one of the mechanisms of loss in a PW.

Carlos.

D Tibbets
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Post by D Tibbets »

In short, I have concerns about the potential well nature of your setup, and the different geometry effects on the various volume and density considerations.

Ignoring Gauss Law complications , the turn around point of the ions (this is the species where annealing applies, at least as a dominate effect) in this linear or 2D rectangular geometry represents a volume of x * Y * Z, but this volume is equal irregardless of the depth of the slice you considered from anywhere within the system. With a 2D circle, or especially a 3D sphere this volume defined by the radius from the center and the thickness of the slice changes tremendously. This effects the total population of ions within this slice This is not considered in the rectangular geometry. The effects of density due to transit time (speed) per unit of volume is accounted for, but the units of volume relative to other slices is not accounted for.

The annealing process is a localized thermalizing process that competes with the global thermalizing process - actually it would be better stated as a competition of the annealing area- slice against the rest of the volume of the sphere at a lesser radius than the annealing edge slice.
The process in all regions is determined on the time dependent density (how many ions are transiting per unit of time), the volume of the slice, and the temperature dependent Coulomb collisionality of the ions in that slice.

The definition of the slices is arbitrary. For accurate assessment, you may integrate thousands of slices. But basically, three regions can be considered. The core, and the local collisions - both KE scattering and angular momentum scattering properties; the mantle, where the MFP (at least in small machines) is greater than the distance across this region- slice. And the edge where the MFP is extremely shorter, the colisionality is greatly increased, and the resultant thermalization distribution is some defined fraction of the average KE of the ions in the core. Monoenergetic conditions is a relative term, not an absolute. I generally consider 1-10% as my arbitrary and purely self serving definition of the thermalized KE spread in the edge annealing region compared to the core.

To achieve annealing this restoring action must occur faster than the global thermalization process in the rest of the machine. My experience (mostly from watching a simulation) is that it takes ~ 3-5 collisions to approach Maxwell thermalization distributions. This means that only this number of collisions in the core and mantle can occur per pass of the ion. Most of these collisions would occur in the core (due to colisionality and density, despite the smaller volume) This introduces up and down scattering, but angular momentum collisions is less of a concern do to geometry considerations of these collisions occurring near the center of a sphere. The edge annealing must 'reset' this spread to a narrow range on each pass of the vast majority of the ions. This is where the increased density of a narrow edge region multiplied by the disproportionate volume of this possibly thin region is adequate to the task.
Without this spherical geometry volume advantage, the edge annealing process would have a much more difficult time thermalizing the vast majority of the ions to a relatively narrow spread on each and every pass.
I don't know if a non spherical geometry could demonstrate the effect. If the correction is not fully applied on each pass the global thermalization would dominate, and if there was any measurable benefit, it would probably wash out in only a few passes- perhaps in only a few microseconds, instead of multiple milliseconds.
If you can measure a small delay in global thermalization with your setup, you should be able to extrapolate to a spherical model.
But, how do you set a baseline? If your system slightly delays thermalization. What do you compare it to? A box filled with a monoenergetic plasma but no potential well? I'm not sure but I suspect you have to be careful to be consistent . As Bussard said, the dynamics has to be considered, static considerations can be misleading.

In my limited perspective, I have no doubt of the mechanics of local thermalization which seems straight forward. What I'm uncertain of is the effects of high grade confluence- collisionality does scale as the ~ square of density. If this core density is high enough the ions could fully thermailze around the higher average energy. . This may not be all bad, depending on the relative effects on bremsstrulung and other considerations.
Also, as the machine size grows the thermalizing process in the mantle region becomes more pronounced. Full annealing may become impossible, but even partial annealing may still have some benefit. Once the MFP in the mantle became much greater than 3-5 passes, annealing while still occurring to an extent would become mostly meaningless. This condition is easily met in 3- meter machines. Upping the drive energy to increase the fusion rate vs the corresponding lower Coulomb conditionality in the core and mantle mitigates this somewhat, but not completely.

A final consideration is the shape of the potential well. A square potential well may behave differently than an parabolic well in terms of the speed of the ions at given distances within a container. This would effect the temperature defined effect on collisionality and thus the region volume and location choices needed to match certain conditions.

Dan Tibbets
To error is human... and I'm very human.

charliem
Posts: 218
Joined: Wed May 28, 2008 8:55 pm

Post by charliem »

Very interesting post. Thanks Dan.
D Tibbets wrote:Ignoring Gauss Law complications , the turn around point of the ions (this is the species where annealing applies, at least as a dominate effect) in this linear or 2D rectangular geometry represents a volume of x * Y * Z, but this volume is equal irregardless of the depth of the slice you considered from anywhere within the system. With a 2D circle, or especially a 3D sphere this volume defined by the radius from the center and the thickness of the slice changes tremendously. This effects the total population of ions within this slice This is not considered in the rectangular geometry. The effects of density due to transit time (speed) per unit of volume is accounted for, but the units of volume relative to other slices is not accounted for.
Ok, I agree. The problem is that if we consider a model too close to reality it gets easily intractable, for calculus and even for simulation. That's why I started asking a much easier question. ¿Is at all possible to maintain a constant locally [almost] mono-energetic distribution in a much much simpler machine?

Of course that the answer to that question cannot be applied to a PW, but it's a starting point.

There are too many unknowns in a polywell. For instance we are talking about "cores", when we are not even sure that it is possible to create one in this machine at significant densities (aside from just a pure geometric definition).

¿What happens if the inner density gradient is much smaller that our hope? ¿Would that kill annealing? If the answer is no then perhaps the machine could still work, but if it is yes ...

¿And what about focussing? Seems that for annealing to work, smaller, denser cores would be better (minimum angular momentum spread) but the worse for KE dispersion (and, of course, focussing would also affect attainable reaction rates).
D Tibbets wrote:The annealing process is a localized thermalizing process that competes with the global thermalizing process - actually it would be better stated as a competition of the annealing area- slice against the rest of the volume of the sphere at a lesser radius than the annealing edge slice.
Ok again.
D Tibbets wrote:The process in all regions is determined on the time dependent density (how many ions are transiting per unit of time), the volume of the slice, and the temperature dependent Coulomb collisionality of the ions in that slice.

The definition of the slices is arbitrary. For accurate assessment, you may integrate thousands of slices. But basically, three regions can be considered. The core, and the local collisions - both KE scattering and angular momentum scattering properties; the mantle, where the MFP (at least in small machines) is greater than the distance across this region- slice. And the edge where the MFP is extremely shorter, the colisionality is greatly increased, and the resultant thermalization distribution is some defined fraction of the average KE of the ions in the core. Monoenergetic conditions is a relative term, not an absolute. I generally consider 1-10% as my arbitrary and purely self serving definition of the thermalized KE spread in the edge annealing region compared to the core.

To achieve annealing this restoring action must occur faster than the global thermalization process in the rest of the machine.
Very good description.
D Tibbets wrote:If you can measure a small delay in global thermalization with your setup, you should be able to extrapolate to a spherical model.
But, how do you set a baseline? If your system slightly delays thermalization. What do you compare it to?
Good point. I've been asking myself that very same question for a while. I wonder if this effect could stabilize the non thermal distribution through time, and not just give it a little more but a lot more. In the end is just a question of how long compared with the expected mean life of ions inside the machine.
D Tibbets wrote:In my limited perspective, I have no doubt of the mechanics of local thermalization which seems straight forward. What I'm uncertain of is the effects of high grade confluence- collisionality does scale as the ~ square of density. If this core density is high enough the ions could fully thermailze around the higher average energy. This may not be all bad, depending on the relative effects on bremsstrulung and other considerations.
I fail to see how that could not represent a problem. I wonder if there is a way to externally control focussing. If it exists should be possible to search for the sweet point that allows maximum reaction rate with minimum losses and thermalization.

D Tibbets
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Post by D Tibbets »

If you have not already done so several references here discuss thermalization issues

http://www.askmar.com/Fusion.html

http://www.askmar.com/Fusion_files/EMC2 ... etimes.pdf



This paper below by N. Krall works through some details. Unfortunately the link no longer works. I've only found abstracts of this recently.

http://www.askmar.com/Fusion_files/EMC2 ... ration.pdf
COLLISIONAL RELAXATION OF THE
NON-MAXWELLIAN PLASMA DISTRIBUTION
IN A POLYWELL

As far as as core 'partial thermalization' not necessarily being bad. This is some handwaving to emphasize the many compromises involved in the Polywell that involves both ions and electrons.

Dan Tibbets
To error is human... and I'm very human.

charliem
Posts: 218
Joined: Wed May 28, 2008 8:55 pm

Post by charliem »

Thanks for the references.

I read those papers some time ago, and I think that I'm going to give them a good re-read.

Can't say it was an easy task the first time. Hope the second is a bit less difficult ... ;-)

I have a copy of the paper by Krall that you mentioned. If you want it just send me a PM with an email address and I'll send it to you.

D Tibbets
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Joined: Thu Jun 26, 2008 6:52 am

Post by D Tibbets »

charliem wrote:Thanks for the references.

I read those papers some time ago, and I think that I'm going to give them a good re-read.

Can't say it was an easy task the first time. Hope the second is a bit less difficult ... ;-)

I have a copy of the paper by Krall that you mentioned. If you want it just send me a PM with an email address and I'll send it to you.
Thanks, I also have it.
As far as simulation, simple particle in cell(?) codes may be innadicuate. According to Chacon, Fokker- Plank (sp?) approaches is needed for modeling the PolywelL. That may also apply to annealing issues (?).

Dan Tibbets
To error is human... and I'm very human.

TallDave
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Post by TallDave »

FWIW, here's the patent comments on electron and ion annealing.

http://www.freepatentsonline.com/y2008/0187086.html
One might raise questions concerning the ability of the device to maintain its quasi-monoenergetic energy distributions among the ion and electron populations. These are, of course, driven by the dynamic injection of fast electrons, and their subsequent loss to structures. One might be concerned that if electrons live sufficiently long in the machine they could become Maxwellianized (thermalized) and develop high energy loss distributions. However, this has been found not to be the case. The same arguments have been found for the ions, as well. Detailed analyses show that Maxwellianization of the electron population will not occur, during the lifetime of the electrons within the system. This is because the collisionality of the electrons varies so greatly across the system, from edge to center. At the edge the electrons are all at high energy where the Coulomb cross-sections are small, while at the center they are at high cross-section but occupy only a small volume for a short fractional time of their transit life in the system. Analysis shows that this variation is sufficient to prevent energy spreading in the electron population before the electrons are lost by collisions with walls and structures.

Similarly, for ions, the variation of collisionality between ions across the machine, before these make fusion reactions, is so great that the fusion reaction rates dominate the tendency to energy exchange and spreading. Ions spend less than 1/1000 of their lifetime in the dense, high energy but low cross-section core region, and the ratio of Coulomb energy exchange cross-section to fusion cross-section is much less than this, thus thermalization (Maxwellianization) can not occur during a single pass of ions through the core. While some up- and down- scattering does occur in such a single pass, this is so small that edge region collisionality (where the ions are dense and “cold”) anneals this out at each pass through the system, thus avoiding buildup of energy spreading in the ion population. Both populations operate in non-LTE modes throughout their lifetime in the system. This is an inherent feature of these centrally-convergent, ion-focussing, driven, dynamic systems, and one not found (or even possible) in conventional magnetic confinement fusion devices.
Given the claims of "excellent" confinement, development of high loss regimes due to electron thermalization is presumably not a problem in WB-8 (though this might be an issue for larger machines where electron lifetimes are longer). Ion annealing is still something of an open question given that we don't know the WB-8 output power, but then getting ion focus is not as critical.
n*kBolt*Te = B**2/(2*mu0) and B^.25 loss scaling? Or not so much? Hopefully we'll know soon...

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