This seems to be a common problem. What is the ideal minimum distance between each of the rings? Bussard offers us some hints as to a solution. Below is a quote from his IAF paper. The gyroradius of the electron is what could be needed. I did that calculation here. I would be interested in trying to “balance” the central plane B field with the main axes field. To get the gyroradius, you start by assuming some distance between the rings. From this, the gyroradius of the electron can be found using the method outlined below. The goal would be to see if the initial assumption matched the calculated result.
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“…The spacing between coils should be such that the central plane B field is approximately the same as that of the B field on main face axes. Typically, this may be at minimum the order of a few (5-10) electron gyro radii at the inter-corner field strength, but not larger than this...”
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Find the electron gyroradius:
Imagine an electron. Now, put a magnetic field on it. The electron will not move in any direction. Now “kick” the electron. The electron experiences a Lorentz force. This causes the electron to move in a circle. The direction (counterclockwise or clockwise) will depend on the direction of the magnetic field. The radius of this motion is the electron gyroradius. You can calculate it if you know the strength of the magnetic field and if you know the velocity of the electron perpendicular to that field.
Find the magnetic field:
I can find the magnetic field strength. I solved it for the November 2005, WB-6 tests. I assumed that the minimum distance between the rings was 4.25”, center to center. This was an educated guess. I am going to apply this method and see what minimum distance is needed. If this guess disagrees with the calculated value, either the guess is wrong or the method is flawed. Bussard estimated that at the corners the field was 70 to 100 Gauss [IAF, Page 6]. I calculated 89. Here is how I found it. During those four tests Bussard stated: there was 4,000 amps of current running through each ring [IAF, Page 12]. The “corner” is a point equidistant and centrally located between the three rings, about 5.2” from each ring. Locally, I treated the rings like they were straight wires. Then I used the Biot-Savart law to find the magnetic field. I accounted for 3 rings. The answer was 89 Gauss. This seemed reasonable and nicely agreed with Bussards' estimate.
Find the electron velocity via Energy:
I can also calculate the velocity of the electron at the corner. It is about 5.6% the speed of light. But I cannot know what percentage of this is parallel or perpendicular to the magnetic field. Hence, I cannot outright find the gyroradius. This velocity can be solved by assuming the total energy of an electron is constant. From Bussards paper, he estimated the average electron had 2,500 eV of energy in the center of the device [IAF, Page 12]. The ions need ~10,000 eV to fuse, so this seems reasonable. Incidentally, Rider would argue here. Rider argues energy transfer happens so fast that everything would have to be either 2,500 eV or 10,000 eV. He argues you cannot have two temperatures. Someone should look into this.
Energy In The Center:
I will assume the dead center of the Polywell has nearly no potential energy. This is for the electrons only. The ions see a completely different energy map. This energy has two parts: contributions from the magnetic and electric fields. To have no electric potential means that the electron is surrounded by uniform electrostatic charge in all directions. Additionally, the key to the Polywell is a no magnetic fields in the center, so there is no magnetic potential, either. Hence, the electron has only kinetic energy. This means the electron is moving about 9.9% the speed of light. As the electron moves from the center to the corner, it goes up in potential energy. This is because it is moving into regions where there are denser magnetic fields and lower electric fields. To move, the electron must transfer its kinetic energy into potential energy. If we assume the total energy is constant, then if we know the potential energy at the corner, we will know the electron's kinetic energy. From this we can find the velocity of the electron at the corner.
Surprisingly, the magnetic potential energy at the corner, is tiny. The reason for this is the magnetic moment of an electron is so small. The magnetic moment I used 9.28E-26 Joules/Tesla. There are lots of magnetic moments in the world. There are several moments for the electrons; the value is different for each place an electron could be found and different ways the electron spins. It is not an easy value to solve/estimate. Nominally, the moment is some multiple of the Bohr magneton (9.87E-24 J/T). The magnetic potential energy is the cross product of the moment and the field. This is why the magnetic moment is important. If the moment is higher the magnetic potential energy is higher. But, even if the moment was 3 times the Bohr value and the moment lined up in the highest energy state – the magnetic potential energy would still be tiny. The the range of possible values runs from E-29 to E-25. This is so, so, much smaller than the electric potential - that I just ignored the magnetic component.
The electric potential is easy to find. Remember this is for the electrons, not the ions. Bussard stated that there was a 10 Kv drop in the middle of the machine. I applied Gausses law and found that there were 5.5E11 electrons in the center. I treat this like a point charge. The corner is 0.46 meters away. So the electric potential energy from a point charge at this distance is 1,691 eV. This number makes sense. The electric potential is like a valley between two mountains. At dead center, your in the valley. Move a small distance away and the electron is repulsed from the other electrons in the cloud. This is the mountain peak. The single charge is sitting next to a big pile of electrons, and is repulsed by the columbic force. It is in a dense electric field at that point. So the electron flies away from the center. when it reaches the corner it has 809 eV in kinetic energy. This means the electron is traveling about 5.6% the speed of light at the corner.
Using Some Assumptions:
The overall velocity can be estimated from the kinetic energy. However, it is unknown how much of that runs perpendicular to the B field. I thought about trying to use the Lorentz force. The force will tell you some of the behaviors of the electron at the corner. The problem is you need to know the velocity before you can apply that equation. Dead end. If you watch simulations by HappyJack and Indrek, you can see that the electron is mostly moving back and forth up the field lines. Hence you would expect the perpendicular velocity to be less than 50%. I picked, 50% 30%, 10% and 2% and here is the minimum distance.
% of velocity perpendicular: Minimum distance:
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50% 0.00136”
30% 0.00082”
10% 0.00027”
2% 0.00005”
This distance varies, but it is well below the 4.25” assumed at the start of this calculation.
Seeking the minimum distance between the rings.
Seeking the minimum distance between the rings.
Last edited by mattman on Mon Jun 25, 2012 6:23 pm, edited 15 times in total.
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happyjack27
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Admittedly, some of my reasoning is shaky, but some of my understanding is that the bottom of the electrons potential well is at the edge, not at the center. They have the most KE here, the center has the least KE/ highest PE. So if the potential well is 10 ,000 volts, the electron has ~ 10KeV at the edge/ Wiffleball border. . The Gyro radius is the product of the electron KE devided by the B field strength in that region. The B field strength is considered at the Wiffleball border or just beyond it . I believe this is the value quoted for the B field strength of the machine(it could also be the strength at the surface of the cans, in which case the strength at the Wiffleball border could be mildly less). So, in WB6 the electron gyro radius is calculated from ~ 10KeV / B field strength . Without looking it up I'm uncertain if a constant or power function is involved.
In WB6 the gyro radius consideration is applied to the closest approaches of the magnets- the nubs, or funny cusp region, not the corners or face centered point cusps. This value is multiplied by 3-5 (10 max) to allow for some deeper penetration into the magnetic field due to collisional processes (ExB drift) before the magnet surface is hit. The value is a compromise between minimizing ExB drift losses and limiting cusp losses which increase as the magnets are moved apart. In WB4 the ExB losses were significantly higher due to the lack of space. The square magnet can shape was also significant. I don't know which hurt more , the shape of the can or the lack of spacing. Both probably contributed significantly.
The nubs themselves also contributed, thus the concern with nub heating in WB7. For this reason I speculate that WB7.1 (with modified nubs or nub elimination) performed better to some unknown amount.
Dan Tibbets
In WB6 the gyro radius consideration is applied to the closest approaches of the magnets- the nubs, or funny cusp region, not the corners or face centered point cusps. This value is multiplied by 3-5 (10 max) to allow for some deeper penetration into the magnetic field due to collisional processes (ExB drift) before the magnet surface is hit. The value is a compromise between minimizing ExB drift losses and limiting cusp losses which increase as the magnets are moved apart. In WB4 the ExB losses were significantly higher due to the lack of space. The square magnet can shape was also significant. I don't know which hurt more , the shape of the can or the lack of spacing. Both probably contributed significantly.
The nubs themselves also contributed, thus the concern with nub heating in WB7. For this reason I speculate that WB7.1 (with modified nubs or nub elimination) performed better to some unknown amount.
Dan Tibbets
To error is human... and I'm very human.
In nuclear energy calculations, energy transfer must happen from interactions.
When you are talking about negative ions at one velocity transferring energy to electrons at another velocity, you have coulomb interactions. Imagine a single H+ ion passing through a cloud of e-; furthermore, assume that the cloud is diffuse enough that no collisions occur.
As the ion moves, all of the electrons are attracted to it, and gain speed. But, if the electron cloud is uniform, the forces on the ion are perfectly balanced. Where, then, does the kinetic energy for the electron acceleration come from? The 1eV positive field of the Ion, of course.
Now, the ion is travelling; this means it will get an effect by magnetic forces. This is going to transfer energy away from the Ion into the cloud of electrons.
So, yes, there will be some acceleration of electrons and slowing of ions.
But, the speeds are on the order of a 10^7 m/s so you're talking 10^-8 sec for each pass, which means that these effects are going to be slow acting on the timescale of lifetime in the well.
When you are talking about negative ions at one velocity transferring energy to electrons at another velocity, you have coulomb interactions. Imagine a single H+ ion passing through a cloud of e-; furthermore, assume that the cloud is diffuse enough that no collisions occur.
As the ion moves, all of the electrons are attracted to it, and gain speed. But, if the electron cloud is uniform, the forces on the ion are perfectly balanced. Where, then, does the kinetic energy for the electron acceleration come from? The 1eV positive field of the Ion, of course.
Now, the ion is travelling; this means it will get an effect by magnetic forces. This is going to transfer energy away from the Ion into the cloud of electrons.
So, yes, there will be some acceleration of electrons and slowing of ions.
But, the speeds are on the order of a 10^7 m/s so you're talking 10^-8 sec for each pass, which means that these effects are going to be slow acting on the timescale of lifetime in the well.
Wandering Kernel of Happiness
WizWorm,
Your talking about energy transfer. From the ions to the electrons.
I think this is still a pretty open question.
However, rider did write a paper on this.
We will need to an arguement that beats this.
Here is the citation:
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Rider, Todd H., and Peter J. Catto. "Modification of Classical Spitzer Ion-electron Energy Transfer Rate for Large Ratios of Ion to Electron Temperatures." Phys. Plasmas, American Institute of Physics 2.6 (1995): 1873-885. Web.
Your talking about energy transfer. From the ions to the electrons.
I think this is still a pretty open question.
However, rider did write a paper on this.
We will need to an arguement that beats this.
Here is the citation:
=============
Rider, Todd H., and Peter J. Catto. "Modification of Classical Spitzer Ion-electron Energy Transfer Rate for Large Ratios of Ion to Electron Temperatures." Phys. Plasmas, American Institute of Physics 2.6 (1995): 1873-885. Web.
Also,
====================
“…The spacing between coils should be such that the central plane B field is approximately the same as that of the B field on main face axes. Typically, this may be at minimum the order of a few (5-10) electron gyro radii at the inter-corner field strength, but not larger than this...”
====================
I take that to mean the gyro radius at the corner. Not the little nub spacers between the rings. I can find it for the nubs.
====================
“…The spacing between coils should be such that the central plane B field is approximately the same as that of the B field on main face axes. Typically, this may be at minimum the order of a few (5-10) electron gyro radii at the inter-corner field strength, but not larger than this...”
====================
I take that to mean the gyro radius at the corner. Not the little nub spacers between the rings. I can find it for the nubs.
And Brem requires electrons that are both energetic and dense. It may be there is an intermediate area where this occurs, but I dont see brem in the well or at the perimeter.D Tibbets wrote:Admittedly, some of my reasoning is shaky, They have the most KE here, the center has the least KE/ highest PE.
Dan Tibbets
Admittedly, some of my reasoning is even more shaky than yours is.
I like the p-B11 resonance peak at 50 KV acceleration. In2 years we'll know.
I was talking about the electrons. The top of the potential well for electrons is at the center. The bottom of the electron potential well is at the edge. The opposite applies to the positive ions. A negative ion would have the same potential well direction as electrons, but negative ions are a rarity.
Wisworms comment about ions tugging on electrons individually applies, and this why Bussard mentioned the change in the shape of the potential well once positive ions are introduced. The well goes from square to elliptical (or parabolic. This is a local collisional Coulomb interaction. There is some coupling between them. This is generally a weak componet relative to the space charge, because the coupling is weak. Also momentum is important. The ions have about 60 times the momentum of the electrons, so they can effect the electrons much more than the electrons can effect the ions. Also, momentum transfer between charged particles is poor if the momentum of the particles is different. An ion will exchange momentum with another ion much more than with an electron. And, the reverse is also true.
Between the ions and electrons the space charge interactions dominate.
This is why ions do not have to follow the electrons out of cusps. The space charge of the still excessive electrons inside dominates the effect on the ions. This can be changed though if there are too great of a concentrations outside of or in the cusps. This local space charge can dominate the ions if the ions get too close. This was mentioned in the WB5 results. You can impead electron escape through a cusp with cusp plugging (collection of cold electrons in or very close to the cusp). But this also becomes a space charge attracting the ions. This is one of the important considerations of recirculation. It allows the hot electrons to escape, but then draws them back. The electrons remain hot in the cusp. Hot means they are traveling fast, so they transit the region rapidly, so the time dependent density remains relatively low, so a local space charge does not build up as in cusp plugging.
As for Bremmstrulung. the critical factor is speed. The ions are traveling ~ 60 times slower than the electrons, so a simplified consideration of the ions contribution to the net sppe can almost be ignored. The speed of the electron is the important consideration. If the electrons are significanly slower near the center, bremsstrulung is less because the ions are expected to have the greatest density in this area. And a Coulomb collisionality scales as the square of the density bremsstrulung will be less in this high ion density region if the electrons are moving slower in this region. There is a lot of interplay with degree of ion confluence, potential well shape, thermalization issues, Z, etc. that determines the final results.
Dan Tibbets
Wisworms comment about ions tugging on electrons individually applies, and this why Bussard mentioned the change in the shape of the potential well once positive ions are introduced. The well goes from square to elliptical (or parabolic. This is a local collisional Coulomb interaction. There is some coupling between them. This is generally a weak componet relative to the space charge, because the coupling is weak. Also momentum is important. The ions have about 60 times the momentum of the electrons, so they can effect the electrons much more than the electrons can effect the ions. Also, momentum transfer between charged particles is poor if the momentum of the particles is different. An ion will exchange momentum with another ion much more than with an electron. And, the reverse is also true.
Between the ions and electrons the space charge interactions dominate.
This is why ions do not have to follow the electrons out of cusps. The space charge of the still excessive electrons inside dominates the effect on the ions. This can be changed though if there are too great of a concentrations outside of or in the cusps. This local space charge can dominate the ions if the ions get too close. This was mentioned in the WB5 results. You can impead electron escape through a cusp with cusp plugging (collection of cold electrons in or very close to the cusp). But this also becomes a space charge attracting the ions. This is one of the important considerations of recirculation. It allows the hot electrons to escape, but then draws them back. The electrons remain hot in the cusp. Hot means they are traveling fast, so they transit the region rapidly, so the time dependent density remains relatively low, so a local space charge does not build up as in cusp plugging.
As for Bremmstrulung. the critical factor is speed. The ions are traveling ~ 60 times slower than the electrons, so a simplified consideration of the ions contribution to the net sppe can almost be ignored. The speed of the electron is the important consideration. If the electrons are significanly slower near the center, bremsstrulung is less because the ions are expected to have the greatest density in this area. And a Coulomb collisionality scales as the square of the density bremsstrulung will be less in this high ion density region if the electrons are moving slower in this region. There is a lot of interplay with degree of ion confluence, potential well shape, thermalization issues, Z, etc. that determines the final results.
Dan Tibbets
To error is human... and I'm very human.
The high energy thermalized tail of the ions in a thermalized plasma may be important for most of the fusion, and a significant portion of Bremmstrulung (the ions and electrons have a similar thermal spread). Annealing certainly may limit the high thermal spread of the ions in the Polywell, but for the sub millisecond confinement times in the Pollwell, the high energy ions may not transfer a lot of heat to the electrons, so it may not be a major concern(?).
This decreased annealing contribution might apply to large machines with higher B fields (densities) such that the ion MFP in the mantle region is shorter than the machine radius and thus possibly precluding or at least diminishing effective annealing. Raising the temperature ( such as to 80KeV as proposed by Bussard as a target for D-D fusion) compensates for some of the shortening in the MFP, but not all of it. But, also the ion dwell time before escape or fusion also needs to be factored in. Because up scattered ions would escape the potential well, they may be lost quickly from the magnetic confinement such that they contribute less to progressive up scattering. They aslo fusefaster with the daughter products at much higher energies with corresponding MFP such that they do not further heat the plasma. It may becomes more of a loss issue than a thermalization issue. There may be significant thermalization, but not full thermaization, especially on the upscattered side. I think Bussard addressed this issue in one of his papers.
The low energy ions , and corresponding electrons in a long duration thermalized plasma may be as important. Depending on density issues, etc.
Bussard stated in one of his papers, that Bremsstrulung limits the minimal temperature that permits Q's greater than one. There is a reason why 5 KeV is considered the minimum target for Tokamak fusion. If the average temperature is 5 KeV, the total plasma will be producing Bremsstrulung, but only the high energy tail will be producing most of the fusion. This means that the Bremsstrulung losses are always greater than the Fusion gains at an average thermalized temperature of 5 KeV or less. Bussard also, reveled that for D- He3 fusion this minimum temperature is about 15 KeV. He didn't give the value for D-D fusion but I suppose the minimum temperature would be ~ 7-10 KeV. This is one reason tokamaks can only burn D-T profitably. The penalty of the thermalized plasma associated with bulk Bremsstrulung production overwhelms the fusion gain until the fusion cross section exceeds ~ 0.2 Barns. A mono energetic ion population participates in the fusion process inclusively, not just a modest portion participating. Apparently this consideration that all of ions are participating in both fusion and Bremsstrulung at a given temperature gives an advantage.
The consideration that there may be ion confluence is an additional factor that leads to greater central density (and thus fusion) without the equivalent increased Bremsstrulung radiation (because the electrons are slower in the center). This may not be necessary for D-D profitable fusion in the Polywell, but I think it may be required for successful P-B11 fusion . And even then providing excess hydrogen ion numbers at a ratio of ~ 5-10 to one may be necessary to control the high Bremsstrulung associated with the high Z (5) boron ions.
Note that for D-He3 fusion the the dilution is in the opposite direction. The He3 ions are in excess so that D-He3 reactions dominates over side D-D reactions, in order to suppress neutron production. The Z of 2 for the helium3's leads to more Bremsstrulung than an even mixture , but the lower target temperature and the Bremsstrulung scaling with the near square of the Z allows this reverse dilution to still be profitable.
Dan Tibbets
This decreased annealing contribution might apply to large machines with higher B fields (densities) such that the ion MFP in the mantle region is shorter than the machine radius and thus possibly precluding or at least diminishing effective annealing. Raising the temperature ( such as to 80KeV as proposed by Bussard as a target for D-D fusion) compensates for some of the shortening in the MFP, but not all of it. But, also the ion dwell time before escape or fusion also needs to be factored in. Because up scattered ions would escape the potential well, they may be lost quickly from the magnetic confinement such that they contribute less to progressive up scattering. They aslo fusefaster with the daughter products at much higher energies with corresponding MFP such that they do not further heat the plasma. It may becomes more of a loss issue than a thermalization issue. There may be significant thermalization, but not full thermaization, especially on the upscattered side. I think Bussard addressed this issue in one of his papers.
The low energy ions , and corresponding electrons in a long duration thermalized plasma may be as important. Depending on density issues, etc.
Bussard stated in one of his papers, that Bremsstrulung limits the minimal temperature that permits Q's greater than one. There is a reason why 5 KeV is considered the minimum target for Tokamak fusion. If the average temperature is 5 KeV, the total plasma will be producing Bremsstrulung, but only the high energy tail will be producing most of the fusion. This means that the Bremsstrulung losses are always greater than the Fusion gains at an average thermalized temperature of 5 KeV or less. Bussard also, reveled that for D- He3 fusion this minimum temperature is about 15 KeV. He didn't give the value for D-D fusion but I suppose the minimum temperature would be ~ 7-10 KeV. This is one reason tokamaks can only burn D-T profitably. The penalty of the thermalized plasma associated with bulk Bremsstrulung production overwhelms the fusion gain until the fusion cross section exceeds ~ 0.2 Barns. A mono energetic ion population participates in the fusion process inclusively, not just a modest portion participating. Apparently this consideration that all of ions are participating in both fusion and Bremsstrulung at a given temperature gives an advantage.
The consideration that there may be ion confluence is an additional factor that leads to greater central density (and thus fusion) without the equivalent increased Bremsstrulung radiation (because the electrons are slower in the center). This may not be necessary for D-D profitable fusion in the Polywell, but I think it may be required for successful P-B11 fusion . And even then providing excess hydrogen ion numbers at a ratio of ~ 5-10 to one may be necessary to control the high Bremsstrulung associated with the high Z (5) boron ions.
Note that for D-He3 fusion the the dilution is in the opposite direction. The He3 ions are in excess so that D-He3 reactions dominates over side D-D reactions, in order to suppress neutron production. The Z of 2 for the helium3's leads to more Bremsstrulung than an even mixture , but the lower target temperature and the Bremsstrulung scaling with the near square of the Z allows this reverse dilution to still be profitable.
Dan Tibbets
To error is human... and I'm very human.
For the WB6 it was up to about 12kV. WB7 was supposed to be.about the same. Not sure for WB8. But a pB&J will need several hundred kV.mattman wrote:Guys,
The rings are inside a metal cage. There is a voltage drop, from the cage to the rings. This is meant to hold in electrons.
Does anyone know the voltage, for this drop?