Does magnetic geometry determine the scaling of a Polywell fusor? Does magnetic geometry determine the scaling of a Polywell fusor?
Forgive imprecise terminology here - by "magnetic geometry" I mean the configuration of the magnets, the configuration that creates the "Christmas star" geometry of the electron field in the fusor.
On to my question... I've read that the rate of fusion scales at $r^5$, with $r$ being the radius of the reaction chamber. Is this because of the aforementioned magnetic geometry? Or is it because of something more essential, like Bremsstrahlung, or something else?
 A: The scaling you refer to is derived (though not in great detail) in for instance papers [0], [1] and [2] below.
Since the polywell approach relies on a very particular kind geometry for its proposed operation, it could be said that the magnetic geometry is determining the scaling, but that would be circumventing the question. 
Fusion power scales as $P_f \propto B^4R^3$. For the polywell type devices $B\propto R$, and hence $P_f \propto R^7$. One main difference between the polywell approach and regular magnetic confinement devices, is that the polywell mainly aims to confine the electrons magnetically. The electrons then create a confining electrostatic potential to capture the ions. This means that much smaller magnetic fields are needed, and that the main loss of energy is assumed to be through electron losses, which scales as $P_l \propto R^2$, the area of the device. This gives a total fusion gain scaling of $G = P_f/P_l \propto R^5$.
As for the particular polywell design used, the most common so far is the cubic, but some of papers have mentioned octahedral and dodecahedral configurations. It is believed that a more spherical configuration will yield better stability, as well as a larger "active" core region compared to the overall size of the device, but the scaling does not rely on the particular polyhedral geometry employed.
It should be stressed that this scaling relies on a lot of as yet unproven assumptions, based on very little (so far) empirical experience from operating this kind of device.

References:
[0]: Bussard, R. W., "Some Physics Considerations of Magnetic Inertial Electrostatic Confinement," Fusion Technology, vol. 19 (1991)
[1]: Bussard, R. W., "Inherent Characteristics of Fusion Power Systems,"
Fusion Technology, vol. 26 (1994)
[2]: Bussard, R. W., "The Advent of Clean Nuclear Fusion," 57th International Astronautical Congress (2006)
A: I do not trust Dr Bussards scaling.  He did not have enough data to make those scaling claims.  The University of Sydney (Gummersall, 2013) scaled some factors in their simulations:
A. Current in the rings (Amps or AmpTurns)
B. Size of the rings  (Meters)
C. Energy of electrons  (KeV)
But, they were looking at how many electrons were trapped - not fusion rate.  They found some bounds for these machines: 

I have not had time to fully digest this plot.  It compares the current in the rings against the energy of the material.  The higher the current the stronger "the container" trying to hold in plasma.  The higher the energy the more plasma is trying to leave.  These bounds appear to show where containment can happen.  On the face of it, it makes sense.  Too much energy or too weak of a container, and the plasma leaks.
I do not think anyone has scaling understood yet.  If they do, they have not published (aka the Navy).  
A new textbook "Inertial electrostatic confinement (IEC) fusion fundamentals and applications" by Dr. George Miley, offers some basic scaling for fusors (from page 209):
Fusion rate ~ Ions heading towards the inner cage^2
By contrast, polywell scaling is very different.  It is based on the same input, the rate of ions flying inward.  On page 376, Miley looks at fusion scaling in polywells:

Fusion rate rises, as more ions fly into the center --- by a factor of 5?  That is a serious claim.  I am still working through this to try and understand it.  It is based on simulations, so I am very skeptical.  Also, why should it be so different than scaling in fusors?
For source material Miley references two papers, the first which I need to get: Tzonev IV, DeMora JM, Miley GH (1996) "Effect of large ion angular momentum spread and high current on inertial electrostatic structures" and Chacon L, Barnes DC, Knoll DA (1998) "An implicit boundary-averaged Fokker-Plank BAFP code to model spherical interial electrostatic confinement fusion systems."
Cheers,
http://thepolywellblog.blogspot.com/
