The problem with ion convergence

Discuss how polywell fusion works; share theoretical questions and answers.

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jmc
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Post by jmc »

In my previous post I mentioned convergence was hard not impossible.

Firstly there may be a possiblity that much like a bubble adopts a spherical shape due to the fact that it is the most affective way to resist the external pressure of the water, the wiffleball in the beta=1 condition could achieve a shape that is more far spherical then the external magnetic field in the absence of a plasma. By eye this seems to be the case. Whether this spherical looking blob is spherical enough to achieve fusion is another issue that, I believe, has not yet been meassured precisely.

Ramping up the density reduces the number of passes required for fusion the very presence of convergence would reduce that number still further.

I started a thread here:
http://talk-polywell.org/bb/viewtopic.php?t=467
mentioning the possibility of getting a mono-energetic distribution of ions by injecting neutrals from a convergent beam into the region of zero field at fusion energies and ionizing them at the bottom of the potential step.

If you do want an electric field you could shrink the polywell in size down to a debye length. That would mess up your ratio of cusp to total surface area though. I toyed with the idea of a "Polywell matrix", plugging the cusps of one mini-Polywell into the others. If you create a lattice of truncated cubes the interstices in between are octahedral.

I don't agree that Chacon proved that electric fields could be maintained at distances in excess of the Debye length, infact I read the chacon paper that was referred to, the electron densities were 10^18/m^3, the temperatures were 100kV and the diameter of the device was 1cm. Infact in that paper he stated that a Penning trap was a Debye length Machine.

I know that in a tokamak radial electric fields can be sustained at distances far greater that the Debye length, but this is because the two fluid equations mean that not having them would violate the law of conservation of angular momentum. This is not the case in a Polywell.

I would be interested to read of any experimental cases where centrally convergent electric field have been measured over distances far in excess of the Debye length in any plasma system.

Annealing

If the ions slow down at the same radius at the same time, the density would indeed be huge, if they slow down at the same radius at different times the density will not exceed that in the core (unless they are very cold indeed)

There may be thermal expansion of this edge at high density into the regions of lower density, but if the temperature the edge is 1eV say, this free expansion could be quite slow.

I still say the condition required for convergence are very delicate and probably can't be achieved in some cheap and nasty way.

Art Carlson
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Post by Art Carlson »

rnebel wrote:Go ahead and give it a try analytically, but there are some real pitfalls here. Let me give you one example. Suppose you start with a bunch of mono-energetic ions with no angular momentum. All of those ions will have the same turning point near the edge. At that turning point the ion density will be HUGE! So will the collision rate. Is that physical? Probably not. A little thermalization will spread out the density and dramatically drop the collision rate. How much spread is reasonable? I don’t know. This is the reason Luis Chacon did the full bounce averaged calculations. The scary thing about his results were that little changes (like changing to potential well from a square well to a harmonic oscillator) made big changes in the answers.
I don’t know the answer to either of your questions. ...
Then maybe it is worthwhile for me to make an estimate after all. Just the order of magnitude and the scaling. I'm thinking of mono-energetic ions running up an electric potential beach and being reflected back. For the ions moving in either direction,
n*v = Gamma
(the particle flux) is a constant, as is the total energy
W = m*v^2/2 + e*phi = 0,
where phi is the electric potential, defined relative to the position x = 0 where the ions turn around. In terms of the electric field E, we have
phi = x*E,
so
v^2 = 2*e*phi/m = (2*e*E/m)*x.
Next we need the cross section for Coulomb scattering,
sigma = pi*v^-4*e^4*(4pi*epsilon*m)^-2*ln_Lambda,
where ln_Lambda is the Coulomb logarithm, a weak function of the parameters with a value around 15.
Now the fun starts. The collision frequency is sigma*n*v, so the chance of a collision on one bounce is substantial if the integral of the frequency during the bounce is one or more:
I = int[ sigma*n*v*dt ] > 1
I = int[ pi*v^-4*e^4*(4pi*epsilon*m)^-2*ln_Lambda * Gamma * dx/v ]
I = pi*e^4*(4pi*epsilon*m)^-2*ln_Lambda*Gamma * int[ v^-5 * dx ]
I = pi*e^4*(4pi*epsilon*m)^-2*ln_Lambda*Gamma * int[ ((2*e*E/m)*x)^-5/2 * dx ]
I = pi*e^4*(4pi*epsilon*m)^-2*ln_Lambda*Gamma*(2*e*E/m)^-5/2 * int[ x^-5/2 * dx ]
(Are you starting to see why I really didn't want to do this calculation?)
I = (2/3)pi*e^4*(4pi*epsilon*m)^-2*ln_Lambda*Gamma*(2*e*E/m)^-5/2 * [ x^-3/2 ]
Before we calculate a number, let's examine this expression. First, we see that the limit at infinity vanishes, which we like, because it means that we are really talking about a reflection, that doesn't depend to much on conditions far away from the "mirror". Second, we see that the electric field enters in a rather strong way. This (I presume) corresponds to the result of Chacon that the form of the potential is important. (For the same depth and radius, a parabolic well has a much smaller E than a square well.) Finally, we see that the integral diverges. Bummer, but we expected that for the reasons that Rick alreasy pointed out. I propose to deal with this (roughly!) by introducing a lower cut-off for v corresponding to the temperature at the edge or the velocity spead in the center. Let's call the cut-off velocity sqrt(2kT_edge/m), or, in terms of x, the integral extends to
x = (2eE^/m)^-1*v^2 = (2eE/m)^-1*(kT_edge/m) = kT_edge/(eE).
Then we have
I = (2/3)pi*e^4*(4pi*epsilon*m)^-2*ln_Lambda*Gamma*(2*e*E/m)^-5/2 * [ kT_edge/(eE) ]^-3/2
Ugh. We gotta simplify that. First, I'll drop all the numerical factors, although this will lead to a significant overestimate of I because of the (4pi)^-2 (assuming I haven't messed up my formula here). Next I will use the Deby length for the central plasma,
lambda_D^-4 = n^2*e^4*epsilon^-2*kT^-2
and
n = Gamma/v = Gamma*m^1/2*kT^-1/2
and
E ~ (kT/e)/Delta
to write
I ~ e^4*epsilon^-2*m^-2*Gamma*E^-1(e/m)^-5/2 * (kT_edge/e)^-3/2
I ~ (Delta*lambda_D^-4*n^-1)*(T/T_edge)^3/2
(Honestly, I lost a factor of e^5/2 along the way, but it's too late at night to track it down. The units here work out.)
Consequences? As a first guess, let's take Delta~1e-1m, lambda_D~1e-4m, n~1e21, and T/T_edge~1e-2 (due to the lumpy surface). Then we get I~1e-8. I don't offer any guarantees that this number is right. It is much smaller than even I expected. I - or better yet, one of you - should check the calculation by the cold light of day. If it is correct, then the lumpy surfaces will produce so much transverse motion initially and on every subsequent bounce, that annealing doesn't have a snowball's chance in hell of acting to keep ion convergence.

rnebel
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Post by rnebel »

Jmc:

To my knowledge, the Polywell is the only electrostatic machine that operates in the quasi-neutral limit. It’s possible that Lavrent’ev did some similar work to this, but I don’t know the Soviet literature very well. Consequently, I think that the polywell is the only system that has operated in this regime.

I do have 1-D Particle-in-Cell (PIC) simulations which have global electrostatic fields in the quasi-neutral limit. The same is true of the Vlasov-Poisson equilibria generated by Bussard’s people in the early to mid 90s. All of these cases produce electrostatic fields over distances long compared to the Debye length. The way these are produced is by flooding the system with hot electrons and starving it for ions. This forms systems with potential wells comparable to the electron injection energy, and the electron inertia spreads the potentials over the entire radius.


Art:

My own take on the convergence problem is that it is going to be difficult to get a really reliable calculation. The problem is that the answers will probably be very sensitive to the assumptions. I think that the calculation's primary value is to tell you what the high leverage physics is and how to turn the knobs on the machine. My take is that it will be difficult to draw accurate quantitative conclusions.

Art Carlson
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Post by Art Carlson »

rnebel wrote:My own take on the convergence problem is that it is going to be difficult to get a really reliable calculation. The problem is that the answers will probably be very sensitive to the assumptions. I think that the calculation's primary value is to tell you what the high leverage physics is and how to turn the knobs on the machine. My take is that it will be difficult to draw accurate quantitative conclusions.
I know I committed a multitude of sins in my model, but to come back with "it will be difficult to draw accurate quantitative conclusions" is pretty weak when we're talking about eight orders of magnitude.

MSimon
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Post by MSimon »

Art,

It is my contention that we are dealing with a klystron type machine. i.e. it is the bunching that is the main "cause" of annealing. Of course this is based on hunches and some limited simulations. So there may be nothing to my view.

The difficulty with Polywell is that it is a non-equilibrium machine and as such I don't think that we have enough physical understanding to model it with any general equations.

We are in the position of trying to estimate the operation of an internal combustion engine based on a general understanding of fire. Certainly a general understanding of fire is important. However, the proper shape of the combustion chamber is determined by more than just a general understanding of fire. We can get the fuel/air mixture correct. However, that does not tell us enough about how to properly mix the fuel and air to insure "complete" combustion. That is a function of geometry, flow and timing.

Much of the early work on improving the internal combustion engine was doe by trial and error with theories evolving after the fact. Polywell may be the same. At this point we have no idea what simplifying assumptions are useful and which leave out critical details.
Engineering is the art of making what you want from what you can get at a profit.

Art Carlson
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Post by Art Carlson »

Dear MSimon,
You wrote:Art,

It is my contention that we are dealing with a klystron type machine. i.e. it is the bunching that is the main "cause" of annealing. Of course this is based on hunches and some limited simulations. So there may be nothing to my view.

The difficulty with Polywell is that it is a non-equilibrium machine and as such I don't think that we have enough physical understanding to model it with any general equations.

We are in the position of trying to estimate the operation of an internal combustion engine based on a general understanding of fire. Certainly a general understanding of fire is important. However, the proper shape of the combustion chamber is determined by more than just a general understanding of fire. We can get the fuel/air mixture correct. However, that does not tell us enough about how to properly mix the fuel and air to insure "complete" combustion. That is a function of geometry, flow and timing.

Much of the early work on improving the internal combustion engine was doe by trial and error with theories evolving after the fact. Polywell may be the same. At this point we have no idea what simplifying assumptions are useful and which leave out critical details.
When I came to this forum
I wrote:If I can, I will uncover the soft underbelly of the polywell concept and eviscerate it. Then you will all thank me kindly for stopping your waste of time, and we will all move on.
I found that a number of contributors shared this orientation, for example,
tonybarry wrote:May I say that this is our goal too. If the polywell is to work, then it must withstand all criticism. If it is impractical, then the sooner we find out, the better. If it is the Real Deal, then we all benefit.
I thought you also felt this way since
you yourself wrote:When I read your statement ... I was most pleased. So far I have done as much as I have been able to find holes (and fill them). If you can find some holes we haven't it would be most helpful.
A number of contributors have questioned my arguments and rejected them for various reasons that were well-considered, whether right or wrong. When you respond to my physics arguments with statements like "I don't think that we have enough physical understanding to model it with any general equations" and "At this point we have no idea what simplifying assumptions are useful and which leave out critical details.", I must conclude that you will hold to the polywell no matter what objections are brought. Is that what you call "filling holes"?
It would be a somewhat different story if my models were in conflict to experimental evidence. It would still be useful to try to figure out why they were wrong, but at least we would know they are wrong. But the experimental evidence on the polywell is so thin that, while is might be considered suggestive, it is not in contradiction to anything.

Betruger
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Post by Betruger »

The real conclusion will be in the august trials.. At worst, none of the forum readers have conclusive-enough theoretical answers in the interim, but the empirical answer is soon in coming.

kcdodd
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Post by kcdodd »

I will be honest that I don't understand what your equations are showing. I am guessing you're trying to show that the "lumpy surface" factor is larger then annealing. Assuming it is lumpy, and an ion hits a random spot on the surface on each bounce it will get kicked in a completely random direction. However, every ion that it will encounter at the edge was kicked in a random direction as well. So if all this is doing is increasing the lateral motion of ALL ions then probability will still give in to equal radial motion by ion-ion collisions at the edge just as if the kick was caused by a high energy collision in the core. How does the "annealing effect" change depending on what *caused* the lateral motion to begin with?
Carter

tonybarry
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Post by tonybarry »

Hello Art,
I would ask you to continue your investigations into the theoretical underpinnings of polywell. While I do not have the physics knowledge as yet to comment meaningfully, never-the-less I think that (in the absence of experimental results) the theoretical investigations are eminently worthwhile.

Until Rick Nebel provides us with further experimental results, present theory is pretty much all we have to go on. Polywell has not had the benefit of rigourous scientific investigation as far as I can see. This does not mean it is completely without merit. But it does mean that arguments against it have to be answered.

If a working model of polywell achieves its aim of fusing species confinement, it will be because it obeys the laws of physics. Please continue to investigate. I personally appreciate your efforts.

Regards,
Tony Barry

drmike
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Post by drmike »

I ground thru the same formula and got the same integral, then plugged in numbers and got 6.6e-49 * Gamma / (E * kT_edge^3/2). I'm trying to check the units to make sure this makes sense, but have to pretend to be working for a living :D

(e^2/epsilon)^2 is pretty tiny. But I will double check again in a few hours.

MSimon
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Post by MSimon »

Art,

I still agree with the point that you made and that I also made.

However, I have yet to see anything that convinces me one way or the other. Where I am on the spectrum of interest is: hopeful with doubts. I think you are more on the doubtful with doubts side. Good.

As Rick has pointed out: we do not have enough evidence to make a case one way or the other.
Engineering is the art of making what you want from what you can get at a profit.

TallDave
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Post by TallDave »

A number of contributors have questioned my arguments and rejected them for various reasons that were well-considered, whether right or wrong. When you respond to my physics arguments with statements like "I don't think that we have enough physical understanding to model it with any general equations" and "At this point we have no idea what simplifying assumptions are useful and which leave out critical details.", I must conclude that you will hold to the polywell no matter what objections are brought.
Heh, well, sometimes "I don't know" is the only honest answer. That doesn't mean there aren't objections which could in fact rule Polywell out, even given our paucity of data, for anyone with intellectual honesty. We just haven't seen any yet.

Of course, we do in fact need more data to describe some of these processes with enough accuracy to say "this objection is not significant" versus "this objection rules out net power." Throwing around competing theories of what the interior processes of a Polywell looks like is fun, but mostly speculative at this point (which is one reason I don't try to wrestle with calculations like the above much).

drmike
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Post by drmike »

I did a double check and got the following:

E ~ V/R where R is the radius of the MaGrid and V is the potential on it

x ~ R

Gamma ~ n_0*(2*e*V/m)^1/2

Then I ~ 3*ln_lambda*2^-7*e^2*epsilon^-2*n_0*R*V^-2/pi

You need V ~ 10^5 for p-B, but only 10^4 for D-T. For the same density and radius, that is a factor of 100 times easier. Plugging in numbers I get
I ~ 3.6e-27 *n_0*R for p-B or 3.6e-25*n_0*R for D-T.

Not good.

drmike
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Post by drmike »

There is another way to look at this calculation. It is the number of deflections that occur driving ions off center due to coulomb interactions. In that sense it's really good - on any one pass there are very few deflections and ions should stay radial for a good number of passes.

The cross section for p-B is about .1 barns for V ~ 100 kV. If we take the core of radius r_c, the probability of fusion is ~ i*n_0*sigma*r_c with various geometric factors ignored. i is the flux of ions flowing radially into the center.
For r_c ~ .1 m, n_0 ~ 10^20 m^-3, sigma ~ 10^-29 m^2 we get a fusion
rate of 10^-10 * i fusions/sec.

At 8.7MeV per fusion this gives ~ 10^-22 * i Watts. To generate megawatts takes huge flux, or huge density, or both.

But in terms of ion convergence, I'm thinking there's not much of a problem. It's just not likely to have enough fusions to generate net power.

But I need to think about it a lot more.....

scareduck
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Post by scareduck »

drmike wrote:But in terms of ion convergence, I'm thinking there's not much of a problem. It's just not likely to have enough fusions to generate net power.
A victory, of sorts, for Art?

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