electron density and distribution

[Robthebob]

Hey it's me again, here to ask questions no one can answer.

Does anyone know what or where i can get more information about electron density and distribution in polywells?

I'm trying to figure out del(P) and E.

[hanelyp]

Potential well distribution, the result of electron and ion distribution, is one detail we'd very much like more information on.

[D Tibbets]

The potential well shape is a contentious issue. But, A. Carlson and Bussard actually agreed that the initial potential well due to electrons was square (Google talk and Carlson's comments on this forum). The complexity comes into play once ions are introduced.Bussard said that due to momentum differences between ions and electrons, as the ions are accelerated inward, they tend to tug electrons along with them. This is not an absolute pairing (like in a tightly coupled plasma) but it does tend to pull the electrons inward- restore the electron radial momentum to a degree. I guess that without this electrons would quickly assume orbits with a lot of angular momentum- ie they accumulate near the Wiffleball border in nearly circular orbits. This is because the electron mutual repulsion will push them outward to the extent that the magnetic containment will allow. The dynamic interaction with the ions though tends to slow/ retard this process so that a stable elliptical well can be maintained through the inward ion tugging and the near radial vector of new and recirculated electrons.

Then there is the issue of the central relative positive virtual anode due to ion convergence and its limiting characteristics.

There are measurements of potential wells in Fusors and these can be pursued in the literature. The best source is probably from the annual or biannual conferences. The most recent:

http://www.aero.umd.edu/sedwick/index.html

Back tracking through the conferences reveals several presentations about the potential well structure.

Dan tibbets

[happyjack27]

fwiw, my simulations showed the a square well, too. basically the electrons made a hollow charged sphere for themselves. though when looking at the electric potential energy by x by y phase space view, it appeared that the inside was rather "noisy". that is, lot's of EM fluctuations. this could have been an artifact of simulation limits and approximations, but it does make some sense: there's a high concentration of very low mass, relatively free-floating spatial charges in the area (namely, electrons). and their energy functions drop off quite nonlinearly in space. So while in the aggregate they should of course follow guass's law, on short time and space scales the energy and charge distribution is much more discontinuous than, say, a space with no charged particles would be.

not sure if this makes much of a difference, just thought it might be interesting to note. from my sims, the core appears to be electrically noisy.

[mattman]

Rob,


1. Electron AND ion density: one particle in 1E-19 M^3. This is from Bussards IAF Paper. He uses this number in his Beta Ratio Calculation and if you plug in the numbers the math checks.



2. Distribution: ions and electrons have bell curves of energy with an average of 2,500 eV. The range is probably ~10 -12.5 KeV to a low of ~tens of eV. This is from Bussards paper also.







==================
Note: Rider argues that ion-to-electron energy transfer is fast. So fast the clouds have the same distribution. Rider wrote a paper on this: http://pop.aip.org/resource/1/phpaen/v2/i6/p1873_s1. In it, he modified a classical equation from Lyman J Spritzer’s book: "the Physics of Fully Ionized gases". I found this expression from pages 127 - 135. The equation I used is on the bottom of page 135:

Time to Equilibrate = [3 * Mass_electron * Mass_Ion * Boltzmann Constant^(3/2)]/ [8 *(2*PI)^(1/2)*density_Electrons*Charge_Ion^2*Charge_electron^2*Elementary Charge*Columb Logarthim]*[(Temp_electron/Mass_electron+Temp_ion/Mass_ion)^(3/2)]


You need to double check this work. I am not sure I used the correct equation. The solution I got indicated the clouds would NEVER equilibrate in temperature, which makes no sense. Because of this, I opted NOT to reprint this work in the Blog Post until I was sure. Lyman's Book is awesome for simple modeling and questions like this. As I understand it, the equation tells you how long it takes the temperature of the ions and electrons to equilibrate. It assumes that both the ion and electron cloud have bell curves of velocities but, different starting temperatures. This equation assumes a cloud of electrons is being “invaded” by cloud of ions – the basic scenario for Polywell fusion.
BTW: are you looking to hire a Polywell researcher??

[MSimon]

From:viewtopic.php?p=98543#p98543

[TallDave]

I think Chacon's paper is still the gold standard on such distributions generally (not to say others haven't done some great work). Joel Roger's papers (presented at IEC conferences) are probably the best simulations of Polywells specifically (again, lots of good work by others).

As Rick said, Chacon basically disproved Rider, who assumed square wells. Apparently you get very different answers with parabolic.

[D Tibbets]

Rob,
...

2. Distribution: ions and electrons have bell curves of energy with an average of 2,500 eV. The range is probably ~10 -12.5 KeV to a low of ~tens of eV. This is from Bussards paper also.

Note: Rider argues that ion-to-electron energy transfer is fast. So fast the clouds have the same distribution. Rider wrote a paper on this: http://pop.aip.org/resource/1/phpaen/v2/i6/p1873_s1. In it, he modified a classical equation from Lyman J Spritzer’s book: "the Physics of Fully Ionized gases". I found this expression from pages 127 - 135. The equation I used is on the bottom of page 135:

Time to Equilibrate = [3 * Mass_electron * Mass_Ion * Boltzmann Constant^(3/2)]/ [8 *(2*PI)^(1/2)*density_Electrons*Charge_Ion^2*Charge_electron^2*Elementary Charge*Columb Logarthim]*[(Temp_electron/Mass_electron+Temp_ion/Mass_ion)^(3/2)]


You need to double check this work. I am not sure I used the correct equation. The solution I got indicated the clouds would NEVER equilibrate in temperature, which makes no sense. Because of this, I opted NOT to reprint this work in the Blog Post until I was sure. Lyman's Book is awesome for simple modeling and questions like this. As I understand it, the equation tells you how long it takes the temperature of the ions and electrons to equilibrate. It assumes that both the ion and electron cloud have bell curves of velocities but, different starting temperatures. This equation assumes a cloud of electrons is being “invaded” by cloud of ions – the basic scenario for Polywell fusion.
BTW: are you looking to hire a Polywell researcher??

Either you are confused, or I am. You seem to imply that the plasma in a Polywell is thermalized. This has been repeatedly denied by Bussard and Nebel(?). The electron average KE in WB6 was ~ 10 KeV and this allowed for a potential well of this depth. The ions are quickly accelerated to this energy when they start at low energy on the outer edge of the potential well. This is straight forward electrostatic acceleration. The ions quickly accelerate to this energy as they fall down the potential well. If the potential well is square the acceleration is very quick, but not instantaneous. If an parabolic potential well is established the ion acceleration is the same, but takes a longer distance.
I've never seen a detailed description of the energy spread in the electron population at a specified radius from the center. Certainly the electrons slow as they approach the center due to mutual repulsion. Even if you started with electrons in perfectly circular orbits at the Wiffleball border, scattering would knock some towards the center. The percentage may be small but not negligible. If parabolic potential well is stable with the presence of ions, then the electron vectors will be more radial. And, of course initial vectors of injected and recirculated electrons will be ~ radial towards the center. The average lifetime of the electrons before loss or recirculation was ~ 200 micro seconds. I understand that the increasing angular momentum from electron- electron collisions occurs much faster (perhaps 10-20 micro seconds). This would imply the peripheral shell of electrons would quickly form and become dominate, but not absolute. The potential well is mostly square but with rounded shoulders. This would still be generally consistent with what Bussard said in his Google talk. But, remember the effects of the ions on the electrons. It is essentially a restoring force inhibiting the angular momentum increase of the electrons. The electron vectors are not purely radial nor maximized in angular momentum and distance from the center. Instead it is in between, it is just right with parabolic average orbits that are long enough in the major radius that statements of electrons fast on the edge and slow in the center is reasonable. This has implications for Bremsstruhlung and other issues.

When you talk about the average energy of electrons and ions in the machine you have to break it down based on the radius from the center. This is why Bussard used terms of electrodynamic effects as opposed to electrostatic effects to help avoid this trap. The potential well for the electrons and ions are in opposite directions. The ion potential well is highest at the edge and lowest in the center, the electron potential well is maximum in the center (electrons KE is maximal at the edge and minimal at the center). You can argue the shape of the well between these limits but the limits are straight forward.

The ion average energy over the entire machine will be ~ 1/2 the acceleration energy (~10,000 volts in WB6). But this means that the average speed/ energy is ~ 10 KeV near the center, and ~ 0 KeV on the edge (this ignores the effect of the central virtual anode). This says nothing about the Maxwellian thermalization spread in the ion energy at any given radius. The ion population is not fully thermalized in this regard. There is a bell curve but it does not reach the extent defined by Maxwellian statistics. This is repeatedly emphasized in the Polywell literature and has multiple implications. The term 'mono energetic' is used, though there is an energy spread (partial thermalization) just not relaxed to it's statistical maximum. Just what is the expected spread is uncertain and depends on various factors- MFP at various radii from the center, and the restoring annealing force on the edge. Perhaps confusingly, there is one region where ion thermalization very quickly reaches the Maxwellian distribution. This is the edge where ion speeds are very slow/ KE is very low. At the top of the potential well at the edge of the Wiffleball the ions are slowing to zero radial speeds and reversing direction. Use an average speed of the ions of 10 eV in this defined annealing zone. Coulomb collision cross section decreasesas the 10^1.75 power of the temperature / KE. This means that in this annealing region the ions collide ~ 20,000 times more frequently that ions near the center which may have ion energies around 10 KeV. Thus the ions in the annealing zone are fully thermalized very quickly (in one pass through this region). But if you look at the distribution of ion energy that has the Maxwellian/ Boltzman distribution with an average KE of, say, 10 eV may have a distribution between ~ 3 to 30 eV. The important comparison is to plug these numbers into the ion energy imparted by the potential well (10,000 volts). Thus on a single pass through the machine the ions will start from the top of the potential well with an energy of 10 +/- 6 to 30 eV. Once accelerated the ion energy distribution will be 10,000 eV +/- 6 to 30 eV. Compare this to a thermalized core plasma with an average KE of 10 KeV, it may have a energy spread of ~ 3 KeV to 300 KeV. But, because of annealing the spread is that from the annealing region, which is less than 1% of the average energy of the ions in the core.This is a relatively tiny thermal spread. This is only an example , the actual numbers may vary, and some assumptions are required- such as the MFP in the higher energy mantle and core regions being close to or greater than the machine diameter. In other words, if the ions do not fully thermalize in one pass across the machine (exclusive of the annealing region), the thermal spread will be reset to a tiny amount each time the ion passes through the annealing region.

For comparison, in this example, if the MFP is 1 mm in the edge annealing region, it might be as much as ~ 20 meters near the core, based purely on the relative Coulomb crossections between the regions.. This is a simplified view but describes the gross considerations.

Keep in mind that thermalization involves the KE spread of the ions. Angular momentum concerns (degree of ion confluence or central focus) is another discussion.

[EDIT] I think I converted wrong. The Coulomb cross section difference between the annealing edge and hot core is 10^1.75 power versus 10,000 ^1.75 power or ~ 60 vs 60,000,000 or a difference of ~ 1 million in the Culoumb collision crossection. In the example, if the MFP in the edge annealing region was 1 mm, in the core (at the same density)the MFP might be !~1 kilometer. The MFP on the edge may be more like 0.01 mm and this would give a core MFP comparable to ~ 10,000 meters. The increased density in the core due to convergence might decrease the core MFP by an optimistic ~ 100-1000 times. Still the core MFP and by extension the mantle MFP would be near to or greater than the machine radius, which enables the natural edge annealing to greatly inhibit thermalization extent per pass through the machine relative to the core ion KE. As Nebel mentioned, WB6 easily meets this condition. A larger, higher B field machine with densities perhaps a thousand times greater makes this condition harder to meet, but increasing the accelerating voltage to ~ 80,000 to 100,000 volts would help some.

Dan Tibbets

[MSimon]

Dan Tibbets,

Tasty. Well done.

[mattman]

Just saw this... I will read all of it and get back to you.

[mattman]

Ok,

Is this Chacon's paper?
http://pop.aip.org/resource/1/phpaen/v7/i11/p4774_s1


Yes, I am saying the electrons are thermalized, AKA their energy distribution is:





I cannot see how the cloud would not thermalize. The particles interact trillions of times each second.


The electrons were at 2,500 eV. Their charge made a drop of 10,000 volts. This heated the ions up to 10,000 eV.


The key is: are the ions and electrons at different temperatures? Rider argued a theoretical: no way. The ion is ~3,600X more massive than the electron. IDK the answer to this.

[hanelyp]

I cannot see how the cloud would not thermalize.

key points:
- both ions and electrons are cycling between regions of higher and lower average kinetic energy, trading between kinetic and electric potential energy.
- the scattering cross section is higher at lower energies.
So particle energy does "thermalize", but to a spread closer to Maxwellian towards the lower energy region.

[mattman]

We need to measure this.

Any model is worthless in the face of data.

=======
For what it is worth, there is a model for thermalization. But it might be worthless for the Polywell.



As I understand it, a Wiener process happens when at each time step a variable “spreads out more”. At each time step, the range of the variable gets larger – while variable’s distribution remains shaped like a bell curve.

You have to make some assumptions: the typical time for electron bumping, the number of electrons and the typical energy exchanged. Here are some numbers:


1. Each bump transfers 3.2E-21 joules of energy, about the energy the average particle has at room temperature. Each transfer of energy will form a basic energy level for the system.

2. Each electron bump takes 320 attoseconds the same amount of time it takes an electron to jump from atom to atom.

3. Assume 1.5E12 Net Electrons.

Here is the math:



Bussard said it took 20 microsecond to reach Beta = 1. In a "worse case" scenario the electrons bumped into one another 62.5 billion times.


===
This is sort of at the edge of what can be estimated. We really need to measure the electron energy spectrum inside the machine. I wish the Navy would publish...

[KitemanSA]

Mattman,
You have to remember that the electrons are not in a simple box. When the machine is working, they tend to travel inward where they lose kinetic energy (temperature) and thus bunch up, spread wise, and then head out where they get reflected and may spread out a bit. In-bunch, out-spread... So the temperature at the center should be fairly low and the bell curve narrow. The same thing happens with the ions but in the opposite sense. It is known as annealing.

[D Tibbets]

Atto seconds? I think you are confusing separate processes. Orbital jumps, transitions within an atom may be on this time scale, but that has little if anything to do with inter atom collisions. The interataction between the atoms (ions or electrons) may occur very rapidly when they are close together, chemical reactions, etc, but you have to account for the time it takes for the particles to get close together. In plasmas as in gasses in general this is often represented as the Mean Free Path (MFP) or plasma collision frequency. This is dependent on the plasma density and temperature. At ~ 1 millionth of an atmosphere this distance/ time to interaction may be a few centimeters to meters as the density drops and the temperature increases. The distance the particle has to travel before a close encounter is derived from the density as mentioned, and also the Coulomb collision cross section which drops rapidly as the temperature increases. This can make a tremendous difference and has been discussed in threads about ion annealing which is perhaps very important.

In WB6 the ion MFP was reported as longer than the diameter of the machine. More precisely the MFP in most of the volume within the Magrid was longer than the diameter of the machine. If there is a central confluence/ focus the MFP shortens because the effective density increases in the center. Also as the ions reach the top of their potential well in the edge /Wiffleball border region their speed (temperature equivalent) approaches zero. Here the MFP becomes very much shorter because the Coulomb collision cross section becomes extremely large. The Coulomb collision cross section scales as the inverse of the temperature to the 1.75 power. Using some numbers from WB6 the electrons at their highest speed was ~ 10 million M/s. The ions maximum speed was closer to ~ 100 thousand M/s. This means that the electron transits the machine in ~ 0.3 M / 10,000,000 M/s =~ 3*10^-7 seconds. If you accept that the electrons slow as they approach the center the average speed would be ~5,000,000 M/s and the transit time would be ~ 6^10-7 seconds, or ~ 1 micro second rounded up. I'm not sure what the average MFP for the electrons was. It could be calculated, or derived from WB6 reported data. Full thermalization would require perhaps 1-5 collisions (MFPs). Reported electron lifetimes were ~ 10,000 passes or ~ 3,000 meters which was covered in ~ 200 micro seconds. If you assume that the the electrons did not fully thermalize in this time, then the MFP minimum was ~ 600 meters. This seems to large, and leaves questions about whether the electrons in WB6 were thermalized. I don't know as I've never seen a good description. But, there are possible restoring forces (impedes thermalization). Interactions with ions, selective removal of the high end of the thermal tail ( the up scattered electrons travel faster, and as they travel faster the MFP increases and the confinement time (the 10,000 passes before an escape through a cusp) decreases. It is a complex dynamic process.

The thermalization of the ions is comparable but not the same due to differences in inertia and of course the claimed annealing. I have never seen mention of annealing for electrons, though the physics are similar to the ions except that the possible annealing region for the electrons would be in the center while for the ions it is on the edge.

As for thermalization, while the particle energies/ temperatures do spread out, it is not an endless process, the distribution of the temperature will eventually (or quickly, depending on your perspective) assume a fixed distribution characterized by Maxwlll Boltzman statistics. The shape of this skewed Bell curve is not symetrical and changes with average temperature. If you look at the Maxwell distribution at 10,000 eV a lot of temperature range is incorporated. If you look at 1 eV average energy, the distribution is very much tighter. This is the basis for the annealing process of the ions on the edge where they slow and turn around. Here the ions will definitely fully thermalize about a low value (~0 eV plus or minus perhaps 10 eV. These ions are then reaccelerated to the 10,000 eV so their energy distribution will be ~ 10,000 eV plus or minus 10 eV. So long as the MFP of the ions in most of the machine (away from the edge) is greater than the diameter of the machine, the energy will continuously be reset on each pass and no or little thermalization (on the scale of the accelerated ions (5 or 10,000eV)) will occur.

Acceleration in the potential well depends on the shape of the potential well as well as the depth. If the potential well for the ions is almost square, the ions will accelerate from 0 eV energy on the edge to full speed in only a tiny distance. By the time you travel from the edge to perhaps 1% of the distance to the center, the ions will have reached their maximum energy, from that point on they are traveling by inertia alone. If the potential well is linier, the ion will reach 1/2 of it's maximal energy 1/2 of the way to the center. If the potential well is elliptical the distance from the center where the ions approach their maximal energy/ speed varies. In the Polywell I believe this elliptical shape is close to a square well so the ions reach near maximum energy near the edge, perhaps ~ 10% of the distance to the center. Things get more complicated with virtual anode formation, etc.

Concerning Mean Free Path, Wikapedia presents some formula, and for those lazy people like me it gives MFP in various environments, like Low Earth Orbit density conditions. These tables of MFP numbers might be misleading though as the temperature dependent Coulomb collision cross sections are not obvious.

ie: in LEO the MFP may be 0.1 meter if the temperature is assumed to be ~ 1000 degrees/ 0.1 eV. But if the temperature at the same density was 1 eV the MFP would be 10^1.75 times greater or ~ 6 meters. And in this example at 10,000 eV the MFP would be ~ 6,000 KM. Perhaps my above electron MFP estimate is not too unreasonable.

Dan Tibbets