(Pre-Wiffle) B-Field From Cubic Magrid

[D Tibbets]

KitemanSA,
Optimizing for leaking cusps might not be an issue requiring geometry efforts if confinement is excellent as reported by EMC2. Just a thought.

True, but Dr. B seemed to think there would be a significant improvement, so who am I to argue? Could "excellent" be bettered?

The B4 scaling is based on the B field strength facing the plasma center. Cusp losses is a somewhat different issue. It is related to the input current that is required to maintain the plasma pressure that may reach the limit determined by the B field strength facing the center. The cusps (central portions) do not change leakage with Wiffleball inflation, except the entrance into the cusp is pushed outward so that the vulnerable cusp surface area decreases while the total surface area increases. A funnel analogy has been described to illustrate this changing entrance geometry . As the Wiffleball inflates, the cone of the funnel is cut off till only the funnel tube itself (near parallel cusp B field lines ) remains. An hour glass may also be usefull. Mark the glass a certain distance from the center. Determine the collection area at that point on either side of the center. This represents the cusp collection area as the Beta= one condition is approached, then passed.

The B dependant density limits is, I think, related to the cross field transport losses as much as the cusp losses, at least under Wiffleball conditions where the cusp losses (with recirculation) may approach the cross field losses of electrons. Actually, in the patent application there is mention that excellent recirculation may allow for cusp losses to reach ~ 10X of the cross field losses. Theoretically, further cusp loss improvements could reach or become less than cross field losses. But,even if this is possible, thermalization issues may worsen, and even alpha particle cusp losses may become problematic (start heating the plasma). There is a sweat spot, or if you prefer, a Goldilocks situation.

Dan Tibbets

[mvanwink5]

KitemanSA,
Optimizing for leaking cusps might not be an issue requiring geometry efforts if confinement is excellent as reported by EMC2. Just a thought.

True, but Dr. B seemed to think there would be a significant improvement, so who am I to argue? Could "excellent" be bettered?

In putting together a theoretical design, simple coils will make computation easier, and thus predictability. Just a practical consideration.
Best regards,

[KitemanSA]

KitemanSA,
Optimizing for leaking cusps might not be an issue requiring geometry efforts if confinement is excellent as reported by EMC2. Just a thought.

True, but Dr. B seemed to think there would be a significant improvement, so who am I to argue? Could "excellent" be bettered?

In putting together a theoretical design, simple coils will make computation easier, and thus predictability. Just a practical consideration.
Best regards,

. Ever so true, and nothing is so simple as straight lines.

[mvanwink5]

The equation of a circle is r=constant. Not so easy for squares, and in 3 dimensions....

[TheRadicalModerate]

I can add that alternating the coil current in adjacent magnets does not give a symmetrical cusp geometry and behaves much different from the Polywell.

Dan, I'm not talking about alternating polarities on the square faces--I agree that that will be wildly asymmetrical. But I think this is where rectifying the cuboctahedron (truncated cube) becomes important. Take a look:



Note that the (now smaller) square faces are still arranged in a cube-like geometry. So if all the squares have north polarity and all the triangles have south polarity, then the field should be symmetric. Each cube pushes against its opposite cube and each triangle pushes against itsl opposite triangle, but triangles "pull" against their 3 square neighbors and squares "pull" against their 4 triangular neighbors. Indeed, it should look like a quadrupole (except for the minor little detail that it's a 14-pole...).

But note that the gap on the WB-6+ experiments for the funny cusps is pretty much the same triangular shape as it is in the cuboctahedron. It's just that it's not a rectified cuboctahedron:



So, with the rectified shape and real opposite-polarity magnets on the rectified triangles, here's what my intuitive prediction is:

1) The funny cusps would be replaced with real point cusps, through the center of the opposite current loops.

2) The line cusps would be replaced with a single point cusp at the point of closest approach between the magnets on the square faces. (You obviously need to leave a gap to prevent the escaping electrons from hitting the coils.)

Note that just because the faces have different numbers of sides doesn't mean that the magnets can't still be circular. However, they're going to be different diameters, because the circle you can inscribe in an equilateral triangle has a diameter that's 70% of the diameter of a circle inscribed in a square with the same length sides. I don't know whether you have to increase the field strength of the triangular faces to compensate or whether it's OK to have the opposite polarity magnets have weaker fields. After all, all you're trying to do is get rid of the funny and line cusps.

Surely somebody has simulated this, or can do so easily? A 3D picture showing the field lines would be handy here...

[KitemanSA]

The equation of a circle is r=constant. Not so easy for squares, and in 3 dimensions....

In 3D, r=const doesn't help much. With Cartesian, it is a general mess, with polar it is just a mess in the other two D.

And really, modern CAD makes them all simple.

Dr. B seemed to think it worth while testing out. I have to agree with him, especially if you bow the sides to make them conformal to a sphere. This SHOULD result in a more spherical well.

[D Tibbets]

I Am not sure what you (TheRadicalModerate) are pursuing. First, just to confuse things, the funny cusp is not the corner cusp/ It was a name Bussard used to describe the cusps between the magnets, especially where the magnets touched. In his mathmatical modeling he used formulas that treated the magnets as lines. The magnets could almost touch, with an infinitely thin separation and the charged particles passing through these narrow gaps would be very few because the gap was so narrow, and because the mathematicallly representation of lines representing the magnets are not subject to charged particles hitting them due to gryro motion because the lines are infinitely thin and present no crossection for the charged particles to hit. As he mentioned in his Google he did not appreciate this difference between theoretical physics and reality untill he was working on WB5. He thin applied this real limitation in WB6 by spacing the magnet cans several electron gyroradii apart. There is no real funny cusp, though the term seems to have been retained to discribe the cusp near the closest approaches of the real magnets- leading to losses to the nubs in WB6 and 7. The rest of the line cusps broaden and merge to form the roughly triangular cusp that is more like a point cusp rather than long line cusps. This cusp region is what is sometimes portrayed as a virtual magnet with the corresponding point cusp. While admittedly not entirely accurate, I assume it is used because is simplifies the math. That is my understanding.

Again, using the mathematical line model the cusp strength / width on the outside of the magnet rings is greatest if the magnets are as close together as possible, Making the circular rings smaller in their major radii helps strengthen the real face centered point cusps, but the corner (or virtual corner point cusps) are weaker because there is more separation of the magnets. In reality the magnets consist of multiple windings of wires and several additional layers, so the magnet cans have significant thickness. If the B field strength is determined at the can surface , then the line/ funny/ virtual corner cusps derive from this geometry. The difference is that the real face centered point cusps from this can surface also (the inner radii). This requires two seperate values to be plugged into the seperation length when calculating the field dropoff. is is why I have wondered if playing with the cross sectional geometry of the magnet cans and windings may be usefull. Oval to even extended elliptical shaps to the magnet cross section.. I think a straight cylinder with rounded ends may exceed the limit for the field lines to always be convex towards the center.

Along these lines of comparing the consequences of magnet can thickness, by rectifying the truncated cube more than is minimally nessisary has two effects. It helps the central face centered true point cusps, but it also harms the B field strength of the corner cusps. Thereis a compromize needed, As mentioned above, making the cans thicker and/ or more oval may beneficially effect this compromise. WB 4 had 25% minor radii compared to the major radius of the magnetic cans. In WB 6 this was ~ 17%. Not only is there more room for copper windings, etc. , the shape of the can may help so long as the critical convexity can be maintains.

PS: This is one reason why I question the square maggrids. With the corners meeting to form the 'funny cusps'. the corner cusps are larger. With the square magnets lined up side by side, the 'funny cusps' are much longer while the corner cusps are smaller. A hexagon shape may be close to the ideal compromise, and this is consistent with the prediction that higher order polyhedra may be beneficial. There are so many ways the design can be modified in this seemingly simple design, and all of this is before you start talking about designs for charged particle injection and the dynamic plasma behavior within the Wiffleball.

Dan Tibbets

[rjaypeters]

Spherized [edit - larger picture]:

[KitemanSA]

R Jay,
Thanks for the graphics!
Just for clarification, the funny cusps on these are smaller than the line like cusps that now exist with WB-6/8. Thus, the losses may be lower. In addition, the well should be more spherical, potentially increasing the fous of the ions, increasing the density and improving the Q.

But that is what research is all about, no?

[krenshala]

A reply in conjunction with what TheRadicalModerate posted, you can view the dodec, trucated dodec, and rectified dodec images here: http://en.wikipedia.org/wiki/Icosidodecahedron

[edit] couldn't get the [url] tag to work