## plasma density in magnetic bottle

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

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Abstractness
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### plasma density in magnetic bottle

Simple question: what's the density distribution of a plasma cloud confined in a magnetic bottle?
On the plane perpendicular to the central axis, Is it normally distributed or is it uniformly distributed inside a circle or is it something more complicated?

hanelyp
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Joined: Fri Oct 26, 2007 8:50 pm

### Re: plasma density in magnetic bottle

How strong is the magnetic field?
How hot is the plasma?
How does plasma pressure relate to magnetic pressure? This is at least in part a function of magnetic bottle type.
The daylight is uncomfortably bright for eyes so long in the dark.

kcdodd
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Location: Austin, TX

### Re: plasma density in magnetic bottle

It's typically bell shaped peaked in the middle. H-mode in a tokamak has a more steep profile at the very edge called the pedestal due to a transport barrier, then bell shaped toward the middle. Either way it really has to have only one maximum on the macroscopic average since transport tends to smear everything out, and it has to go towards zero as you go out, which limits the options. However, plasmas will have turbulence at different scales caused by this gradient, so there will be all kinds of smaller scale time-dependent density features.
Carter

Abstractness
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### Re: plasma density in magnetic bottle

D Tibbets
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### Re: plasma density in magnetic bottle

Temperature distribution will eventually match Maxwell- Boltzman distributions, a bell shaped curve that is distorted somewhat. If there are mechanisms that selectively removes some of the ions or electrons based on their energy or temperature, then the final Maxwellian distribution will never be fully reached, though it may come close. It depends on a number of restoring forces, initial conditions and survival time within the plasma. It can become very complex. An example of a restoring force may be an active process fed by energy input, or it may be a passive process that is inherent in the plasma. Polywell Annealing is an example of the latter.

Density distribution is a different matter. It depends on the shape of the plasma, whether there is any energy gradients, etc. An example would be the ion density in a Polywell, based on two features. The presence of a potential well means that the charged particles undergo acceleration forces dependent on their location within the plasma. In the Polywell the near spherical geometry combined with the high speed ions in the core (small radius), and slow speed on the edge (large radius from center) changes the dwell time of the ion within any selected volume. This decreases (if the particles are high energy relative to the average) the time dependent density in that sub volume relative to the overall plasma volume. This also ties in with local pressure when the overall volume is constant. The near spherical geometry of the Polywell combined with an assumed confluence or focus of ion vectors towards the center would tend to increase the ion density in the center. The effects oppose each other. The final results is a compromise. It depends on the shape of the potential well, the persistence of radial vectors over fully random vectors , etc. Also, the ions can be bunched through POPS effects such that the ions arrive in the center in bunches. Sometimes the density may be relatively high and sometimes relatively low. This can have profound effects on fusion rates.

Tokamaks have plasma in a torus shape flying around the circumference of the torus continually. This can lead to centrifugal (or is that centripetal?) concentrating of charged particles towards he outer radius of the torus. Often (if not always) this needs to be controlled with another layer of complexity.

No matter what shape of the bag of plasma, there are many considerations that govern the behavior both globally and locally, thus the great difficulty in making predictions with computer models.

Dan Tibbets
To error is human... and I'm very human.

D Tibbets
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### Re: plasma density in magnetic bottle

Another consideration about plasma density magnetically confined is that it can never be homogenous or equal density through out, at least in a magnatized plasma. The charged particles will rapidly or slowly migrate across the magnetic barrier untill they have almost all exited the confined volume. This ExB diffusion process implies that there will always be a density gradient through the plasma from the inside towards the outside. This process is collision driven ( just like gas diffusion) and is unavoidable. The difference is that large (heavy) charged particles (like ions) do this diffusion through a magnetic bottle faster than small particles (like electrons). This may be the reverse of the picture in typical uncharged gas diffusion through a barrier where the smaller particles diffuse faster. This is because the small particles are moving faster, but with charged particles in a magnetic field the larger particles diffuse faster because they leap a further distance per collision. This is why Bussard quoted that magnetic fields were no 'darn' good at confining ions. It is what helps to drive magnatized plasma machines like a Tokamak to large sizes. The Polywell uses a trick to prevent this from being the limiting factor in plasma confinement. It does this by restraining the ion population away from the magnetic field through injecting a excess of electrons so that the ions are confined primarily by the potential well/ electrostatic forces. The electrons still suffer from ExB diffusion at least on the edge of the plasma, but because of their small mass they diffuse through the magnetic field much slower. According to the patent application this changes the limiting ExB losses from the Polywell plasma relative to neutral magnatized plasmas (like a Tokamak) to such an extent that it only contributes a small percentage of the losses, even with much higher average densities. This allows in part for smaller and more dense plasmas, which has profound effects on machine size and probably cost.

Dan Tibbets
To error is human... and I'm very human.

Abstractness
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### Re: plasma density in magnetic bottle

ok thanks. I looked a bit into magnetohydrodynamics. The thing I didn't find was how to calculate the electric resistivity of a plasma. Do you know how to calculate the electric resistivity of a plasma based on temperature and density ?

happyjack27
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Joined: Wed Jul 14, 2010 5:27 pm

### Re: plasma density in magnetic bottle

for my sims, i just use the vacuum permittivity aka permittivity of free space. I figure in these things it's so close to vacuum, it'll suffice. not to mention i'd have to integrate the permittivity over space....

anycase, i don't have an answer, but just from looking it up on wikipedia, it looks more complicated than at least i presumed; it's a a function of a complex number (like resistance in circuits, the imaginary part being capacitance) http://en.wikipedia.org/wiki/Permittivity

i believe that's what you meant by resistance - permittivity. at the particle level, "resistance" is not an intrinsic property, but a consequence of f=ma, electromagnetic forces, and statistical mechanics, and possibly ionizatoin/deionization. in any case resistance is the rate at which (coherent) kinetic energy of charged particles gets converted into heat (= incoherent kinetic energy of charged particles). and that's entirely different than permittivity, so my answer is mostly useless for that. (although permittivity would play a pivotal role in the conversion of energy.)

mattman
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### Re: plasma density in magnetic bottle

I do not know how to calculate electron cloud resistance. I could not find clear equations, or measured constants.

It seemed odd - think about it. If I had a cloud of electrons and put a magnetic field across it, they should "resist" the field. The same way a block of wood, or stone or plastic should. This would be part of the "plasma pressure". I am sure their are constants for, say, glass - so why not an electron cloud?

Dr. Lyman Spitzers 1960's book, has the best bare-bones analysis and equations for plasma. Very good book. It has allot of basic math to get at questions like this. Look there. Spitzer may have tackled this, by assuming a bell curve of e-speeds, estimating the mini B-field made, approximating the angle with the external field and finally predicting a resistance.

I looked into two physical mechanisms for e-cloud resistance. The first was the magnetic fields made by electrons moving. The second was electrons behaving like mini-magnets due to their quantum spin. As you may guess, the forces made by electron quantum spin were insignificant.

My force calculations are here - http://thepolywellblog.blogspot.com/201 ... ments.html - at the end in appendix A.

I also wrote out the equations for magnetic mirrors, from Dr. Richard Fitzpatrick's notes at the University of Texas. I crudely adopted them to the polywell.

Here - http://thepolywellblog.blogspot.com/201 ... avior.html

But, this work is still pretty crude. You will probably want (and can get) better analysis.

happyjack27
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Joined: Wed Jul 14, 2010 5:27 pm

### Re: plasma density in magnetic bottle

as far as resistance from a pure electron cloud would go, i would think it would be on the order of 1/8000th of a proton cloud, because there's that much less inertia. it's kind of like the difference between throwing a bowling ball through a room full of bowling balls, vs. throwing a bowling ball through a room full of tennis balls. it will go much further in the room full of tennis balls. MUCH further!

however, here we're talking about conductance, so a better analogy might be waves on a water or vibrating strings....

still, i think there's an analagous idea in there somewhere. the electron cloud would have a much faster response speed. sort of like much smaller capacitance value. so that's the imaginary component of resistance (capacitance), at least. not sure how that plays into the real (non-imaginary) part.

but i think if you're just shooting electrons into an electron cloud - it's mostly entropic; what you have is kinda fuzzy-billard-balls. and what you get is a sort of angular diffusion of the momentum.

just thinking out loud here.

prestonbarrows
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### Re: plasma density in magnetic bottle

EDIT: apparently this forum doesn't parse latex, see attached for compiled version

Plasma resistivity is given fundamentally by

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\vec{E} + \vec{V} X \vec{B}= \eta*\vec{j}
where E is the vector electric field, V is the velocity vector, B is the vector electric field, eta is the resistivity, and j is the vector current. Let's consider only unmagnetized plasma (or directly along b-field lines as they would have no affect). Allowing for B-fields requires some fairly hefty MHD theory so i'll skip it here. This simplifies to

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\vec{E}= \eta*\vec{j}
This is directly analogous to Ohm's law in any solid resistor. The resistance is due to Coulomb collisions between electrons and ions.

Now, let's get an expression for the resistivity in terms of basic plasma parameters. Assuming no pressure gradients, the force balance equation is

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m*n_e*d/dt(u_e) = -e*n_e\vec{E} + \vec{R_ei}
where \vec{R_ei} is the momentum gained/lost by electrons through collisions with ions. (The above equation is simply an expression of Newton's F = ma). We can express the momentum term more exactly as

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\vec{R_ei} = -m*n_e \langle \nu_ei \rangle * (u_e-u_i)
where \langle \nu_ei \rangle is the average collisional frequency between electrons and ions and (u_e-u_i) gives their relative velocities. Here we can drop the inertia of the low-mass electrons and express current density as

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\vec{j} = -n_e*e(u_e-u_i)
combining the above gives an expression for the resistivity as

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\eta = \frac{m_e * \langle \nu_ei \rangle}{n_e*e^2}
I'll do some hand-waving here over a page or two of the derivation for the collisional frequency and just give the end result.

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\langle \nu_ei \rangle = \frac{\pi*n_e*Z^2*e^4}{(4*\pi*\epsilon_0)^2*m_e*v_e}
Assuming quasineutrality (n_e = Z*n_i, where Z is the ion charge state) and putting these all together gives the plasma resistivity as

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\eta = \frac{\pi * m_e^{0.5} * Z * e^2 * ln(\Lambda)} {(4*\pi*\epsilon_0)^2 * (k_B*T)^{3/2}}
where \Lambda is the Coulomb logarithm This is also known as the Spitzer resistivity
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Abstractness
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### Re: plasma density in magnetic bottle

Ok thanks for showing me the Spitzer resistivity. It's exactly what I needed. I'm going to try to run a 1-d simulation based on that.

mattman
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Joined: Tue May 27, 2008 11:14 pm

### Re: plasma density in magnetic bottle

FYI: The Coulomb logarithm can be described as...

"Where the Ln() term is the Coulomb logarithm. The Coulomb logarithm is a way to find the mean free path for an ion in a big cloud of ions. Lets say you have a cloud of ions. You know the density, the charge, the temperature of this cloud. You throw in a test ion. The coulomb logarithm is a way to tell how for that test ion could go without smacking into other ions. "

From: http://thepolywellblog.blogspot.com/201 ... ument.html

For Most College students it is merely a number they have to throw into their homework, somewhere between 10 and 20.