Loss of potential well in corner cusps
Posted: Sat Jan 02, 2016 7:05 pm
Loss of potential well in corner cusps
Dr Parks has reported that having a positive charge on the magrid results in the potential well- negative space charge droping to zero in the corner cusp areas. As such the electrostatic confinement of ions suffer. He uses this to argue that a positively charged magrid is not a viable option.
I have been ruminating on this and have several considerations that may modify this viewpoint. Then again, it may be flawed thinking.
First, this may be a real measured effect, but misinterpreted. I have invoked Gauss Law multiple times in the past, and it is convenient to consider the magrid as a conductive metal sphere with a radius equal to the midplane of the magrid cans. This actually has two modifiers though. First the holes compromise the absolute manifestation of Gauss Law. It may still be dominate, but needs some fudging. Plasma frequency or other oscillations within the plasma may effect the significance of the hole size. But, here I am emphasizing a possible oversight in the interpretation of the observed results. Bussard, etel considered the magrid as a structure made up of lines- no width. This flaw in reasoning was not appreciated until WB6, where the physical dimensions of the magrid was considered. The ring magnets are not mathmatical one dimensional lines, but three dimensional cans with significant width and height along with length. As such the center of the magnet can radius was not representative for some considerations, but the can surfaces were. This resulted in ExB losses - cross field diffusion of charged particles through the magnetic fields, being a major concern, while before they were mostly ignored. It changed the perspective that the magnets needed to be as closely approximated as possible- the "Funny Cusp " analogy. Before, with line representation of the magnet cans in simulations, there was no surface area, so ExB losses could be ignored, though obviously ExB diffusion was a major concern with ion magnetic losses in any magnetic machine. But by decoupling the ions from magnetic confinement, only the electron ExB diffusion was an issue, and the attitude was apparently that these could be ignored as the electron cusp losses were highly dominate. This was a flaw in reasoning admitted by Bussard, and was not appreciated despite repeated reviews untill WB4 and WB5 results were analyzed. With the significant can surface area to intercept electrons as they diffused through the magnetic fields- well before reaching the radius from the center represented by the mathematical line representing the center of the magnets. This more realistic consideration lead to the spacing of the magnets so that the electrons would have to diffuse through several gyroradii distances before loss to the can surface. Bussard estimated this separation needed to be about 3-5 electron gyroradii. In "Mini -B" this separation is even more, perhaps representing improved electron confinement via cusp confinement without bridging nubs, compared to ExB losses in the cusp regions.
This modified the view that "Funny Cusps" with zero cusp losses (because the cusp was squeezed to near infinate thinness with the magnet cans almost touching) was desirable. It also decreases the impression that the corner cusps were effectively point cusp instead of highly modified continuous line cusps. This led to the appreciation of the need for significant magnet separations and need for minimization for any bridging structures across the cusps in the region that was previously viewed as "Funny Cusps".
This issue of perspective (line representation as opposed to 3 dimensional can perspective) may also apply to the corner cusp space charge consideration that is the concern of this thread.
With a line perspective, the radius from the center to the middle of the magnet cans is considered as the radius of the machine for Gauss Law Considerations. At any radius less than this the charge on the magrid is not seen by any charged particle. But with the appreciation that the magnet can has real 3 dimensional measurements, this perspective needs to be changed. In WB6 the can minor radius (thickness) was ~2.5 cm. This implies that for Gauss Law considerations, the location of the hole starts at the magrid radius (as measured to the magrid mid plane radius) minus 2.5 cm. With some gradation, the charged particle starts seeing the charge on the magrid at this smaller radius.
This has several consequences. With a positively charge magrid, the electron starts decelerating as it passes outward 2.5 cm earlier than may have been appreciated. As such it slows and reverses sooner. As the electrons slow, they accumulate locally (traverse the same distance slower). This local collection of cold electrons create their own space charge that competes with the central space charge that forms the potential well that is of interest for confining ions. This results in ions near these cusps being 'drawn out' of the central potential well. This cold electron cusp plugging may have benefits for electron contaminant, but is harmful for electrostatic ion containment and may harm electron injection. It is similar to what was found with WB5 with it's intentional electron cusp plugging. The same thing would occur with the line representation of the magrid, but this would occur 2.5 cm further out. This is especially significant for a small machine , perhaps less so for a large machine depending on several other considerations.
`Why the corner cusps and not the center face point cusps? This is simply due to inverse square law considerations. In WB6 patterned machines. The radius of the magnets is approximately two times that of the seperation in the corner regions. This means that any fixed charge on the magnet cans will have only ~ 1/4th the effect on charged particles in the cusps. Decelleration and thus turn around distance is greater. The cold electron cusp plugging effect is less. In the patent Bussard, etel stressed that the low voltage electron guns (effectively a cold electron collection) needed to be a at a significantly greater radius from the center than the positively charged magrid (perhaps twice as far) to prevent this cold electron cusp plugging effect from being too harmful. With a grounded magrid and high voltage electron guns located even further away, these concerns are much diminished. This may be why WB8 and Mini B depend on a grounded magrid and distant high voltage electron gun/s. The conclusion that positively charged magrid is not viable for the above considerations, but with out the modifier that the cans are of real dimensions.
The real dimensions are important, not only for the imposition of Gauss Law considerations relative to the radius to the midplane of the magrid (need to subtract several cm or even more in larger machines- though the proportions would remain the same). This effect is at a smaller radius from the center than the narrowest part of the cusps (at the magrid midplane radius) The cold electron cusp plugging is closer to the center, perhaps even less that the actual cusp distance (narrowest portion of the cusp at the magrid midplane distance. I don't know how fast the electrons accelerate (decelerate) as they experience any charge on the magrid cans. The inverse square law considerations coupled to the curvature of the cans play a role. As the electrons cool with Bremsstruhlung losses and down scattering, they may accumulate even more in the regions just inside the cusps (corner cusps mostly) compounding the problem.
The measured space charge reported near the corner cusps as falling to zero may be due to the competing space charges created by the hot injected electrons (considered as a central dominate negative space charge (ion attracting) versus a local near cusp electron accumulation that balances out the central charge so that the local measured potential some distance between the influences falls to zero. The actual measured potential versus radius data would be revealing.
Secondly, what are modifications that may minimize this effect.
If the curving metal cans with a positive charge is decelerating the electrons before the center of the cusps are reached, the same would apply to any ions in the region. The would be repelled back towards the center to an equal degree. At first this might seem to balance out the concerns. The decreased central potential well diminution is balanced by ion repulsion from the magrid as the ions enter the inner 1/2 half of the magrid can radii before the center of the cusp. But, this would not be the case as the plasma is not net neutral. The electron excess creates the central potential well and any local near cusp space charge effects would maintain this ratio (I think), So ion electrostatic containment suffers, as may peak acceleration/ confluence of ions towards the center.
The simplest solution is to ground the magrid and accelerate the electrons with distant high voltage electron guns. This may not be the best solution though due to concerns with electron injection efficiencies. It appears that this is a major and perhaps limiting concern with this approach. Mini B injection efficiency may have been only a few percent. Without significant improvement break even may be unobtainable. An electron beam created distant from the target cusp will spread due to mutual repulsion. High voltages may help as the electrons have less time to spread before the electrons reach the cusp, but current will need to be scaled up even more in a break even machine so beam spread and resultant mirror rejection of electrons at the cusp will probably be worse. Some type of intermediate focusing will probably be needed. This complicates the machine and near machine magnetic and electrostatic picture, and of course also applies to any direct conversion possibilities.
The alternate solution is the positively charged magrid. It should help to focus an electron beam to improve injection efficiency without cluttering up the exterior space, presumably simplifying any direct conversion scheme, etc. The problem (by my reasoning) is mostly the positive charge on the inner portion (facing the center and at a smaller radius than the magrid midplane radius), not the positive charge on the outer can surface. The simplest solution is to only charge the outside portion of the magrid cans. A ribbon of conductive metal located outside the midplane radius of the cans, or perhaps several cm beyond the midplane would serve to accelerate new electrons and focus them into the cusps, while moving the recirculating cold electron nucleus further outward by perhaps 2.5 to 5 cm (using WB6 dimensions). A positively charged wire may even be suspended further above the can surface, though trad offs may become increasingly unattractive.
I have mentioned before that a compromise may be the best solution. A distant electron gun at intermediate voltage coupled to an intermediate voltage on the magrid may give the best compromise between electron injection efficiency, eletron recirculation , and ion containment. Without the appreciation of the magrid cans real dimensions, the charge on the magrid is absolute- none/ grounded, or some selected voltage though out the surface of the can. With this appreciation the voltage on the magrid cans are localized to taylor both injection efficiency and ion containment efficiency (avoiding cold electron plugs near the cusp centers) at the same.
Changes in the geometry of the magrids may also effect this picture.
Dan Tibbets
Dr Parks has reported that having a positive charge on the magrid results in the potential well- negative space charge droping to zero in the corner cusp areas. As such the electrostatic confinement of ions suffer. He uses this to argue that a positively charged magrid is not a viable option.
I have been ruminating on this and have several considerations that may modify this viewpoint. Then again, it may be flawed thinking.
First, this may be a real measured effect, but misinterpreted. I have invoked Gauss Law multiple times in the past, and it is convenient to consider the magrid as a conductive metal sphere with a radius equal to the midplane of the magrid cans. This actually has two modifiers though. First the holes compromise the absolute manifestation of Gauss Law. It may still be dominate, but needs some fudging. Plasma frequency or other oscillations within the plasma may effect the significance of the hole size. But, here I am emphasizing a possible oversight in the interpretation of the observed results. Bussard, etel considered the magrid as a structure made up of lines- no width. This flaw in reasoning was not appreciated until WB6, where the physical dimensions of the magrid was considered. The ring magnets are not mathmatical one dimensional lines, but three dimensional cans with significant width and height along with length. As such the center of the magnet can radius was not representative for some considerations, but the can surfaces were. This resulted in ExB losses - cross field diffusion of charged particles through the magnetic fields, being a major concern, while before they were mostly ignored. It changed the perspective that the magnets needed to be as closely approximated as possible- the "Funny Cusp " analogy. Before, with line representation of the magnet cans in simulations, there was no surface area, so ExB losses could be ignored, though obviously ExB diffusion was a major concern with ion magnetic losses in any magnetic machine. But by decoupling the ions from magnetic confinement, only the electron ExB diffusion was an issue, and the attitude was apparently that these could be ignored as the electron cusp losses were highly dominate. This was a flaw in reasoning admitted by Bussard, and was not appreciated despite repeated reviews untill WB4 and WB5 results were analyzed. With the significant can surface area to intercept electrons as they diffused through the magnetic fields- well before reaching the radius from the center represented by the mathematical line representing the center of the magnets. This more realistic consideration lead to the spacing of the magnets so that the electrons would have to diffuse through several gyroradii distances before loss to the can surface. Bussard estimated this separation needed to be about 3-5 electron gyroradii. In "Mini -B" this separation is even more, perhaps representing improved electron confinement via cusp confinement without bridging nubs, compared to ExB losses in the cusp regions.
This modified the view that "Funny Cusps" with zero cusp losses (because the cusp was squeezed to near infinate thinness with the magnet cans almost touching) was desirable. It also decreases the impression that the corner cusps were effectively point cusp instead of highly modified continuous line cusps. This led to the appreciation of the need for significant magnet separations and need for minimization for any bridging structures across the cusps in the region that was previously viewed as "Funny Cusps".
This issue of perspective (line representation as opposed to 3 dimensional can perspective) may also apply to the corner cusp space charge consideration that is the concern of this thread.
With a line perspective, the radius from the center to the middle of the magnet cans is considered as the radius of the machine for Gauss Law Considerations. At any radius less than this the charge on the magrid is not seen by any charged particle. But with the appreciation that the magnet can has real 3 dimensional measurements, this perspective needs to be changed. In WB6 the can minor radius (thickness) was ~2.5 cm. This implies that for Gauss Law considerations, the location of the hole starts at the magrid radius (as measured to the magrid mid plane radius) minus 2.5 cm. With some gradation, the charged particle starts seeing the charge on the magrid at this smaller radius.
This has several consequences. With a positively charge magrid, the electron starts decelerating as it passes outward 2.5 cm earlier than may have been appreciated. As such it slows and reverses sooner. As the electrons slow, they accumulate locally (traverse the same distance slower). This local collection of cold electrons create their own space charge that competes with the central space charge that forms the potential well that is of interest for confining ions. This results in ions near these cusps being 'drawn out' of the central potential well. This cold electron cusp plugging may have benefits for electron contaminant, but is harmful for electrostatic ion containment and may harm electron injection. It is similar to what was found with WB5 with it's intentional electron cusp plugging. The same thing would occur with the line representation of the magrid, but this would occur 2.5 cm further out. This is especially significant for a small machine , perhaps less so for a large machine depending on several other considerations.
`Why the corner cusps and not the center face point cusps? This is simply due to inverse square law considerations. In WB6 patterned machines. The radius of the magnets is approximately two times that of the seperation in the corner regions. This means that any fixed charge on the magnet cans will have only ~ 1/4th the effect on charged particles in the cusps. Decelleration and thus turn around distance is greater. The cold electron cusp plugging effect is less. In the patent Bussard, etel stressed that the low voltage electron guns (effectively a cold electron collection) needed to be a at a significantly greater radius from the center than the positively charged magrid (perhaps twice as far) to prevent this cold electron cusp plugging effect from being too harmful. With a grounded magrid and high voltage electron guns located even further away, these concerns are much diminished. This may be why WB8 and Mini B depend on a grounded magrid and distant high voltage electron gun/s. The conclusion that positively charged magrid is not viable for the above considerations, but with out the modifier that the cans are of real dimensions.
The real dimensions are important, not only for the imposition of Gauss Law considerations relative to the radius to the midplane of the magrid (need to subtract several cm or even more in larger machines- though the proportions would remain the same). This effect is at a smaller radius from the center than the narrowest part of the cusps (at the magrid midplane radius) The cold electron cusp plugging is closer to the center, perhaps even less that the actual cusp distance (narrowest portion of the cusp at the magrid midplane distance. I don't know how fast the electrons accelerate (decelerate) as they experience any charge on the magrid cans. The inverse square law considerations coupled to the curvature of the cans play a role. As the electrons cool with Bremsstruhlung losses and down scattering, they may accumulate even more in the regions just inside the cusps (corner cusps mostly) compounding the problem.
The measured space charge reported near the corner cusps as falling to zero may be due to the competing space charges created by the hot injected electrons (considered as a central dominate negative space charge (ion attracting) versus a local near cusp electron accumulation that balances out the central charge so that the local measured potential some distance between the influences falls to zero. The actual measured potential versus radius data would be revealing.
Secondly, what are modifications that may minimize this effect.
If the curving metal cans with a positive charge is decelerating the electrons before the center of the cusps are reached, the same would apply to any ions in the region. The would be repelled back towards the center to an equal degree. At first this might seem to balance out the concerns. The decreased central potential well diminution is balanced by ion repulsion from the magrid as the ions enter the inner 1/2 half of the magrid can radii before the center of the cusp. But, this would not be the case as the plasma is not net neutral. The electron excess creates the central potential well and any local near cusp space charge effects would maintain this ratio (I think), So ion electrostatic containment suffers, as may peak acceleration/ confluence of ions towards the center.
The simplest solution is to ground the magrid and accelerate the electrons with distant high voltage electron guns. This may not be the best solution though due to concerns with electron injection efficiencies. It appears that this is a major and perhaps limiting concern with this approach. Mini B injection efficiency may have been only a few percent. Without significant improvement break even may be unobtainable. An electron beam created distant from the target cusp will spread due to mutual repulsion. High voltages may help as the electrons have less time to spread before the electrons reach the cusp, but current will need to be scaled up even more in a break even machine so beam spread and resultant mirror rejection of electrons at the cusp will probably be worse. Some type of intermediate focusing will probably be needed. This complicates the machine and near machine magnetic and electrostatic picture, and of course also applies to any direct conversion possibilities.
The alternate solution is the positively charged magrid. It should help to focus an electron beam to improve injection efficiency without cluttering up the exterior space, presumably simplifying any direct conversion scheme, etc. The problem (by my reasoning) is mostly the positive charge on the inner portion (facing the center and at a smaller radius than the magrid midplane radius), not the positive charge on the outer can surface. The simplest solution is to only charge the outside portion of the magrid cans. A ribbon of conductive metal located outside the midplane radius of the cans, or perhaps several cm beyond the midplane would serve to accelerate new electrons and focus them into the cusps, while moving the recirculating cold electron nucleus further outward by perhaps 2.5 to 5 cm (using WB6 dimensions). A positively charged wire may even be suspended further above the can surface, though trad offs may become increasingly unattractive.
I have mentioned before that a compromise may be the best solution. A distant electron gun at intermediate voltage coupled to an intermediate voltage on the magrid may give the best compromise between electron injection efficiency, eletron recirculation , and ion containment. Without the appreciation of the magrid cans real dimensions, the charge on the magrid is absolute- none/ grounded, or some selected voltage though out the surface of the can. With this appreciation the voltage on the magrid cans are localized to taylor both injection efficiency and ion containment efficiency (avoiding cold electron plugs near the cusp centers) at the same.
Changes in the geometry of the magrids may also effect this picture.
Dan Tibbets