My experiences with modeling Electron motion.

[mattman]

Hello,

In the 2011 Khackan paper, they argue that there are 2 types of electrons inside Polywell, those in stable orbits and those in unstable orbits.



My thinking is, if we increase the field strength the amount and regions of unstable electrons will decrease.



I would like to show this by getting MATLAB to model electron motion. To model particle motion I planned to use the Lorentz force and Newtons laws of motion.





It is important to realize the total energy of the electron stays constant. Only the electric field does work on the particle. In practice, I did this by creating an excel file and calculating values in X, Y and Z.





This method failed. The reason it failed is the acceleration the electron experiences changes rapidly with time. You cannot ignore this. The solution is to switch to Runge-Kutta formula. I got this formula from: http://www.mare.ee/indrek/ephi/nystrom.pdf Here it is.



Indrek actually made a mistake on the document he had. See if you can catch it. Does anyone see any other problems with this formula? Anyone have experience working with this numerical recipe? I entered it into excel and MATLAB.

[hanelyp]

A couple observations:

- Looks like electrons with a cyclotron radius small compared to mean free path have a stable path, while those with a large cyclotron radius compared to mean free path have a more chaotic path. This goes with the strong magnetic field/low density exterior, canceled magnetic field/higher density interior.

- A chaotic electron path over the interior isn't necessarily bad. So long as the electron transitions to an orderly path as it passes into the outer region, I don't see a problem.

[D Tibbets]

I don't knowwhat you are trying to imply. The Wiffleball is an increase in the area of "chaotic" electron motion- that is electrons with trajectories dependent on electrostatic interactions (collisions), as opposed to magnetic domain gyro motion. The gyro radius inside the Wiffleball border of a test particle is essentially highly elliptical, such that the MFP is shorter than the elliptical major axis distance (or at least the distance to the other side of the machine/ opposing Wiffleball border.), thus magnetic behavior is trivial on the bahavior of the electrons inside the Wiffleball. Your drawings show the ordered magnetized plasma/electrons being moved towards the center, this is essentially a reversal of the Wiffleball process. The Polywell does not work as a magnatized plasma (at least within the Wiffleball border). I think it is accepted that the Wiffleball inflation is the underlying key to the potential of the Polywell in terms of containment, achievable internal density, and reaction volume. Basically the triple product that all fusion schemes must address. Increasing the B field may modify the electron behavior, but it certainly diminishes the Wiffleball, provided the density and KE of the electrons are not increased to compensate. Essentially you are decreasing Beta which is the ratio of the plasma pressure (Temperature or KE and density dependent) to the B field strength . Certainly increasing the B field strength purportedly increases the fusion rate, but it does this through equivalent increases in KE and density. It allows for this increased density * KE / B field relationship with corresponding improved electron confinement that makes input electron power requirements tolerable. Your drawings illustrate this reversed Wiffleball effect (lowering Beta).

If there is any advantage in keeping the electrons in a magnetic domain, it is completely counter to the back bone of Polywell confinement and density and volume considerations. You could of course pump in enough electrons that the low Beta but very high density core might be maintained by boosting the B field proportionately greater relative to the density and KE product. This might reach useful fusion rates but at higher (probably much higher by a factor of perhaps 100) electron input costs. As Nebel said, the Wiffleball expansion (pushing out of the B field border) is an absolute must for the Polywell to succeed. Otherwise the electron cusp losses are always too great.

The 100 number comes from the EMC2 patent application where the low Beta electron cusp confinement is given as ~ 60 passes, while the Wiffleball confinement boosts this to several thousands of passes. Translated to WB6 conditions that would be ~ 2 micro seconds with low Beta cusp confinement vs ~ 200 or more micro seconds with Wiffleball confinement. That is consistent with the reported ~ 0.25 milliseconds confinement times achieved in WB6 (before recirculation contributions). Deviating from near Beta= one conditions is always bad. This means that the Density, temperature and B field must be balanced and the resulting Wiffleball border will always have the same radius. What changes is the internal density and temperature achievable at the practical minimal electron input costs.

Dealing with electron thermalization and angular momentum issues are perhaps separate issues. Some scheme might help this but not at the cost of deviating from Beta=1 conditions.



Dan Tibbets

[mattman]

Dan,

I agree. This is low beta stuff. Who the hell knows what will happen when we extend to 10^12 electrons. This is also a SINGLE electron.


1. There is no implication that "Stable" or "Unstable" ===> "good" or "Bad"


2. I am not sure yet - how this would be connected to the Whiffle Ball.


Actually the concept of "Stable" and "Unstable" or "adiabatic" or "Non-adiabatic" electrons comes from the behavior of Biconic cusps.



The BEST paper that modeled this was: Norton, Roger Van. "The Motion of a Charged Particle near a Zero Field Point." New York: New York University Institute of Mathematical Sciences, 1961. Print.

They showed that the "Stable" or adiabatic electrons behaved like this:



...and that there were "adiabatic" or "Non-adiabatic" electrons like this...



...You can see that erratic behavior happens because the electron has to deal with non-uniform field lines...

...Dr. Khackan Extended this theory to the Polywell in the 2011 paper...




My thinking is that if we increase the magnetic pressure (AKA as we move to higher Beta Ratios) the regions will change...



...For those that don't know, the Beta ratio is...

[D Tibbets]

OK. I think I understand better what you mean by stable and unstable electrons. I think your considerations are based on magnetic field interactions with a single electron(?). This is a collisionless system. My impression is that the Polywell folloows the typical magnetic mirroring / bouncing considerations within narrow limits. These limits is the ignoring of the electrostatic potential on the magrid along with Gauss Law considerations, and a relatively high collision frequency. Both of these aspects modify the single particle/ B field interaction. You mention changes with going to 10^12 electrons per CC or meter cubed(?). In a full scale Polywell the density may be closer to ~ 10^ 16 charged particles per cc.

Within this "ordered(?)" magnetic domain where particles are spiraling along field lines collisions disrupt the behavior, perhaps most significantly through ExB drift mechanisms. The potential on the magrid also effect electrons once they pass to a greater radius than the mid plane of the Magrid. Presumably this electrostatic effect dominates over any charged particle flow along field lines. Once pass the narrowest portion of the cusp I believe the bounce behavior becomes less , um... frequent. The particle may follow the field line to near or through the next cusp before reversing/ bouncing back through the same cusp. This would be bad as it would turn the electron with some residual KE/ velocity back towards the magrid and allow for reenforcing accelerations through the potential on the Magrid, essentially leading to a run away acceleration which would be bad. I think Bussard implied this when he said that the Walls have to be close enough to the Magrid so that the electrons would ground on it before it reached the apex of the field line outside the Magrid.

Back to the potential on the Magrid. This quickly decelerates the electron past the magrid radius and once stopped, then accelerates it back through the same cusp just as it does for e- gun sourced electrons. There may be some issues with electrons mirroring before the cusp mid line is reached, but as the escaping electron was already on permissible passage field lines the reversing should return it back inside the Magrid. Collision effects will modify this but within limits this electrostatic reversal will (I think) greatly dominate over the magnetic ordered behavior. In WB6 this electrostatic reversal was ~ 90% effective. The remaining 10% of electrons that are not recirculated through this electrostatic means (any contribution from magnetic bouncing?) represent electrons that hit non magnetically shielded surfaces (like nubs), or are collisionally up scattered so that they are slowed but not stopped by the Magrid charge/potential. The above mentioned mechanism then becomes important to remove the up scattered electrons before they can be magnetically bounced or looped to the next cusp and reenter with undesired additional KE.

The magnetic ordered electron behavior is important in many ways, but my perspective is that this is not a dominate feature of the Polywell, except for two points. The B field of course determines the loss area size of the cusp - that area where a significant portion of the electrons striking that region will bounce (as in rebound) through the cusp to the outside. The bouncing in a magnetic sense may or may not allow the electron to traverse the cusp. The hard surface bouncing (ricochet) like on a pool table (or inside a Wiffle Ball toy) is a more realistic and significant description because the gyro orbit on the rapidly B field strength changing Wiffleball border is so elliptical (or parabolic) that the behavior is mostly that of the B field acting as a rebounding surface. At low beta or for those electrons that have penetrated deeper into the B field domain things change. This contribution is perhaps illustrated by the ExB losses expected in a Polywell. It is mentioned that ExB losses are ~ 1% of the cusp losses. This suggests that electrons are lost mostly before they become entrapped in the B field and follow 'ordered' behavior.

This is different for ions where ExB drift losses may actually exceed cusp losses, except the ions are electrostatically confined by the potential well so that they are turned before the B field effects are dominate.

All, well almost all, of the important action occurs at the Wiffleball border and inwards where B field effects are trivial- except of course for turning/ rebounding the high speed electrons at the bottom of their potential well. Given enough confinement time the electrons would accumulate dominantly at the Wiffleball border, drift deeper into the Magrid B field through ExB mechanisms and become ordered. But the electrons on average do not last long enough for this to become dominate. They start their life on high speed nearly radial trajectories. On average they do gain angular momentum till they start accumulating on the edge of the Wiffleball- this is the square potential well mentioned by Bussard and others. But, with the introduction of ions the electrons have SOME restoring radial forces (they are tugged along by the ions) allowing for claimed stable parabolic potential wells. As Bussard liked to say, it is a dynamic situation where magnetic fields, electron potential wells, ion potential wells, electron tugging, ion annealing, up scattered electron and up scattered ion preferential removal mechanisms all play a role.

Dan Tibbets