New FAQ - What are Cusps and what kind does a Polywell Have?

[KitemanSA]

Greetings,
The appended is my first cut at a FAQ regarding Cusps.
Please review and comment.
Thanks


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In essence, a cusp is a hole in a magnetic containment field.

What "contains" the charged particle is the portion of the field that is transverse to the particle's direction. In a Polywell system, the desire is to have as spherical a containment field as possible. This is intended to keep the contained electrons moving in as radial a direction as possible. Thus, in Polywell systems, the containment is provided by the part of the field that is tangential to the potential well. Where that tangential portion of the field diminishes to effectively zero, cusps occur.

There are two basic types that are common amoung all magnetic confinement systems and two others that are specific to Polywell systems. They are

Point
Line
Funny
X, or holey X
Point cusps occur at the center of magnetic coils.

Line cusps occur when conductors with currents in opposite directions are placed beside one-another.

Funny Cusps occur when multiple pairs of fields with opposing directions meet at a common vertex. This field configuration is a patented feature of Polywell systems. A funny cusp is generated when two or more conducters with current in opposite directions meet at the vertex and the receed. The current flow must be "in out in out, etc. around the pattern of conductors. The problem here is that there is metal conductor at the vertex where the field is null, so the metal is in the excape path of the electrons. This forms a significant loss path.

X Cusps are the term introduced by a frequent poster to the Talk-Polywell.org forum. They are merely funny cusps where the theoretical meeting point (the vertex) has been replaced by hole left when the conductors are curved away from the vertex before actually meeting; and more specifically, when the conductors form a complete containment of the cusp.

[Art Carlson]

In a Polywell system, the desire is to have as spherical a containment field as possible. This is intended to keep the contained electrons moving in as radial a direction as possible.

This is one of the many contradictions in the descriptions of the polywell. MHD stability - a big selling point - is provided by magnetic field lines bulging into the plasma. Not at all spherical. But ion convergence requires a high degree of sphericity. (The electron motion is claimed by Bussard - pretty clearly correct in this case - to be isotropic, definitely not radial.)

[TallDave]

My understanding was the electron pushback made the group of fields more spherical at beta=1.

I've always pictured it like a bunch of balloons that enclose a space in their middle, which is then pumped full of air. When pressure in the interios space is roughly the same as in the balloons (beta=1), the area inside should be roughly spherical.

[TallDave]

IIRC Bussard says "funny cusps" were what an early reviewer of the design called his "zero field over zero radius" cusps, back in 1987. They are only attainable with magnets of zero radius. The actual machines have semi-line cusps at those points.

It also seems pretty clear from the closed-box machines that electrons are leaving through the line cusps as well. It seems they put the interconnects at the corners because that gave them the smallest intercept area (as oppposed to putting two interconnects around, rather than through, the semi-line-cusp area).

That reminds me, there was talk of supports extending from the walls, but I think those have the same problem: they intersect the field lines (on the outside) so they're going to suck up electrons.

[Art Carlson]

My understanding was the electron pushback made the group of fields more spherical at beta=1.

I've always pictured it like a bunch of balloons that enclose a space in their middle, which is then pumped full of air. When pressure in the interios space is roughly the same as in the balloons (beta=1), the area inside should be roughly spherical.

Maybe you can design your device to do that, but if you can, the plasma will be unstable to MHD modes.

[TallDave]

My understanding was the electron pushback made the group of fields more spherical at beta=1.

I've always pictured it like a bunch of balloons that enclose a space in their middle, which is then pumped full of air. When pressure in the interios space is roughly the same as in the balloons (beta=1), the area inside should be roughly spherical.

Maybe you can design your device to do that, but if you can, the plasma will be unstable to MHD modes.

Why is this so?

My understanding is that the fields in a Polywell or mirror are always convex, in the sense that field strength always increases moving away from the plasma (for plasma-facing fields) in every direction, and should therefore avoid those pesky MHD problems.

[KitemanSA]

IIRC Bussard says "funny cusps" were what an early reviewer of the design called his "zero field over zero radius" cusps, back in 1987. They are only attainable with magnets of zero radius.

If you do the math, the field on either side of the "zero radius magnet" funny cusp goes assymptotically toward infinity then instantaneously drops to zero. When the radius of the magnet is greater than zero, then field still goes assymptotically toward infinity but stops growing when it reaches the radius of the conductor. At that point it drops toward zero at the center. Technically, still a "zero field over zero radius", but the slope to that zero is smaller. I don't see how this changes the statement.

The actual machines have semi-line cusps at those points.

If the WB6 magnets had be sharp cornered square plan-form and had met at the vertex, there would have been a schmeary (aka low slope) funny cusp there. As it was, WB6 had toroidal magnets so at the place the vertex was supposed to be there were two conductors with currents in opposite directions. The conductors happened to bend slowly away from each other.

Line cusps occur when conductors with currents in opposite directions are placed beside one-another

As the magnets move further apart, the line cusps sort of peter out to a ripple in the wiffleball.

[TallDave]

The original patent concept, which provides the basis for the
physics of this type of machine, presumed coil conductors of
zero cross-sectional radius, placed exactly along vertex
edges, with sharp corners where coils came together. This
led to an odd point/radial-line at such corners which had zero
field over zero radius. This was called a “funny cusp“ by the
very first reviewers of the concept (1987). It is, of course,
not attainable with any realistic coil conductors of finite size,
and (as discussed further below) this engineering fact has
profound and dominating consequences for the design of any
machine hoped to be useful and practical for net power
production.

I thought you might want to distinguish between "funny cusps" as an idealized concept versus the actual semi-line cusps that result from real-world limitations. But Bussard tends to use the term sort of interchangeably, as a reference to the same points in an actual machine, so maybe it isn't that important.

[TallDave]

Line cusps occur when conductors with currents in opposite directions are placed beside one-another.

This confuses me too. My understanding of a line cusp is that it's just where two opposing magnetic fields meet.

[Art Carlson]

My understanding was the electron pushback made the group of fields more spherical at beta=1.

I've always pictured it like a bunch of balloons that enclose a space in their middle, which is then pumped full of air. When pressure in the interios space is roughly the same as in the balloons (beta=1), the area inside should be roughly spherical.

Maybe you can design your device to do that, but if you can, the plasma will be unstable to MHD modes.

Why is this so?

My understanding is that the fields in a Polywell or mirror are always convex, in the sense that field strength always increases moving away from the plasma (for plasma-facing fields) in every direction, and should therefore avoid those pesky MHD problems.

Ampere's law in a vacuum requires that the curl of B vanish. Let z be the direction of the field on the surface of the plasma, and y the normal to the surface pointing away from the plasma. The x component of the curl is d(B_y)/dz-d(B_z)/dy, so d(B_y)/dz=d(B_z)/dy. For MHD stability, the field strength must increase moving away from the plasma, i.e. d(B_z)/dy>0. This implies d(B_y)/dz>0 as well. This part is a little tricky - a sketch might help - but this last condition implies that the center of curvature is on the vacuum side.

Using the same logic from the other end, you can also show that a spherical plasma will be MHD unstable.

[D Tibbets]

Question on the concept of the Wiffle Ball. From the above comments can I asume that at least one way to look at the Wiffle Ball is as an arrangement that aproaches but does not reach planer (point between where convex becomes concave) megnetic fields? The closer you can approach this condition while still having slighty convex fields will pinch the cusps to the smallest size possible while still maintaining MHD stability.
Or in the ballon analogy, the pressure in the center is just below the pressure in the ballons, or to make it more complex the pressures match, but the surface tension in the ballons maintain a slight convex shape at Beta=1 (pressure inside= pressure outside). Is there some comparable force in a magnetic field that is radiating out from a curved magnet.

Or, to extend the confusion further- a magnetic field from a plane meets a opposing planar force- the resultant interface would presumably be planer. But, if the opposing force is planer and the magnet is curved would the the interface then be curved? Or, so long as the radius of curvature of the magnetic field is less than the radius of curvature of opposing field then the magnetic field will remain convex at Beta=1. The multiple magnets around the perifery presumably allows these more curved fields per magnet because they are competing with each other. Would this be related to the argument that increased magnetic faces would increase efficiency, ie increased magnets would allow closer approach to the 'planer' or rather spherical shape with consequent smaller cusp openings, while still maintaining MHD stability?

Dan Tibbets

[Art Carlson]

Don't make it so complicated. Just give up ion convergence, which even Rick Nebel says would be nice but is not essential. Then you can let the plasma take whatever exact shape it wants, as long as it is convex, which shouldn't be hard to achieve with polywell coils. [Edit: Dummkopf! That should be conCAVE, i.e. with chunks scooped out.]

[KitemanSA]

Line cusps occur when conductors with currents in opposite directions are placed beside one-another.

This confuses me too. My understanding of a line cusp is that it's just where two opposing magnetic fields meet.

The problem with using "fields" as your term of reference is that "opposition" is position sensitive. If I have two toroidal magnets in the same plane pointing the same direction and bring them close together, I get a line cusp between them. If I have two toroidal magnets positioned along the same axis of rotation, then the fields need to be of opposite sense to get a line cusp in the gap between their coils. In one case "same sense", in the other "opposite sense". In BOTH cases, the current in the parallel conductors are running in opposite directions. It is the single descriptor that I can find that works al the time. Do you have a better one?

[KitemanSA]

Don't make it so complicated. Just give up ion convergence, which even Rick Nebel says would be nice but is not essential. Then you can let the plasma take whatever exact shape it wants, as long as it is convex, which shouldn't be hard to achieve with polywell coils.

I don't understand what your issue with the FAQ is or more specifically how to reword it to make it better. It currently says

In a Polywell system, the desire is to have as spherical a containment field as possible.

which seems to be the part you object to. But as you stated, DrN said it (sphericity) would be nice to have and DrB said he wanted to try two more small scale units to check it out. So based on those, how should I reword that quote? Should it read "practical" vs. "possible"?

Help me here folks!

[Art Carlson]

Don't make it so complicated. Just give up ion convergence, which even Rick Nebel says would be nice but is not essential. Then you can let the plasma take whatever exact shape it wants, as long as it is convex, which shouldn't be hard to achieve with polywell coils.

I don't understand what your issue with the FAQ is or more specifically how to reword it to make it better. It currently says

In a Polywell system, the desire is to have as spherical a containment field as possible.

which seems to be the part you object to. But as you stated, DrN said it (sphericity) would be nice to have and DrB said he wanted to try two more small scale units to check it out. So based on those, how should I reword that quote? Should it read "practical" vs. "possible"?

Help me here folks!

It's easier to criticize than to make constructive suggestions, so I volunteer for the first job. It is especially difficult to figure out what to say when the people you are quoting contradict themselves. In this case, I think you can just leave out those two sentences. That is, just write

What "contains" the charged particle is the portion of the field that is transverse to the particle's direction. Thus, in Polywell systems, the containment is provided by the part of the field that is tangential to the potential well. Where that tangential portion of the field diminishes to effectively zero, cusps occur.

At a later point in the FAQ you might want to address the geometry issue.