Modeling Some Real Results
Figure 1A) A picture of WB-3 developed by Dr. Bussards’ team from 1998 to 2001. This device had a radius of 10 cm. 5B) a picture of Dr. Khachans’ device developed at the University of Sydney, Australia, in 2009. This device had a width of 6 cm. 5C) a picture of Mark Suppes device built in Brooklyn New York, in 2011.
These similarities are encouraging. It reminds us that the Polywell is not a breakthrough. It is not a miracle discovery. This is the next step in natural progression of US funded research going back 60 years. People are excited because this technology seemed to come from nowhere. That’s not true. The current research is a logical step in a program which stretches back decades – and it has many more steps to go on its' path to a commercial reactor. It may yet still be a bad idea. What would help it along? If the majority of America accepted that this form of nuclear power was viable. If tomorrow 300 million Americans woke up and believed they could see commercial fusion energy in their lifetimes – that would be huge for everyone working on these projects. Please enjoy this review.
Paper: “The dependence of the virtual cathode in a polywell on the coil current and background gas pressure.”
Published: The Physics of Plasmas Journal - May 2010
Figure 2: This is a picture of the Teflon rings used inside the Khachan group’s work. On the left the device is shown against a coffee mug for scale. On the right the device is shown off axis to show the screws and L joints which hold it together.
The bell jar was glass container a foot and a half in diameter and several feet tall. One reason they could not test a larger machine was, because they did not have a larger vacuum chamber. The vacuum chamber pumped down to a pressure of 0.015 torr; compare that to 4E-7 torr in Dr. Bussards' last device . They could not pump down to a lower pressure, because their cathode would not work at a lower pressure. The rings were wired to a 450 volt 2,500 amp power supply. By comparison, Dr. Bussards’ last device reached a voltage of 1,200 with 2,000 amps . This power source was a bank of capacitors which could dump this electricity on the machine for burst, each hundredths of a second long. The capacitors were in series with a rectifier which ensures that direct current got pumped into the rings . Lastly, there was a cathode tube which supplied a beam of electrons. The paper points out that this cathode is – in some ways - an improvement over Bussards’. In fact, they had published an earlier paper focusing entirely on this cathode device . A diagram and picture of the machine they used is included in figures three and four below. This diagram was adopted from presentations, pictures and two papers. It is by no means complete or to scale.
Materials Used in Setup:
Table 1: This is the electrical conductivity and magnetic permeability of materials used near the rings of the Polywell.
Dr. Khachan needed strong, practical materials, which did as little as possible to disrupt the magnetic fields around the machine. At full power, each of the six rings has a field strength of 0.04 Tesla. The rings are placed six centimeters apart from one another. That means the rings are pushing each other apart with two tenths of a Newton force (see appendix). You need screws and L-brackets to hold this thing together. The device must also be held in line with the cathode and the aluminum bar works for this. The aluminum has a magnetic permeability close to vacuum; so from the magnetic fields point of view, the bar is barely there. Aluminum also conducts electricity – but if the bar was held at a uniform voltage, the electrons should have no reason to leak through the bar. The L-joints and screws have similar properties.
Another change is the material used, to make the rings. WB-6 had a smooth metal shell. The lack of edges kept a charge from building up somewhere, leading to arching. The smooth surface also worked well with the swirling magnetic fields. Electrons could ride the B fields and not hit a metal edge on the rings. It is unclear what material would be best suited for the rings. People have mentioned cooled ceramic superconductors as well as tough tungsten carbide to withstand the neutron blowback. Teflon may have a place in the reactor chamber – given its electromagnetic properties. However, in practice Teflon can build up a charge in the chamber as well as brown when exposed to plasma. It is also contains gas pockets which can be hard to vacuum out . This topic is open to discussion.
Let us look at one test from the point of view of the electron. This is a summary of the electrons worldview- for all the detailed modeling which leads to this section, please see the appendix below. We choose to model one test condition from the paper. This condition was: 625 amps through the rings, 15 mTorr background pressure and a 15 KeV electron beam.
The electrons start filling up the Polywell. Modeling fill up is beyond the scope of this post, but some description about how this mechanism is thought to work is provided. It takes some time for the electrons to fill up the center. The data from the paper indicates that this fill up time - depends on the drive current. The paper theorizes that there a “resonance current” for this device. They call it a threshold value. This means the device catches the most electrons at a specific magnetic field strength. If your starting current is too high, it takes time for the current to decay. As the current decays, the magnetic field strength lowers until the “resonance current” is hit. At that point the number of electrons in the center reaches a maximum. At the peak, there were about 10 billion electrons contained in the center during the 625 amp run. If the average lifetime of an electron is 100 microseconds in this machine, the average electron should make 42,000 trips around the rings. By contrast Dr. Bussard estimated that his electrons recirculated about 150,000 times. There are plenty of reasons to expect that Dr. Khachans’ machine would have a lower recirculation rate. This mechanism is just an idea. This being an early Polywell paper – there is fair reason to think much more will be discovered. In the end, the device may or may not have such a “resonance current”.
The graphs really show this “resonance current” condition. When 115 amps is applied, it creates a magnetic field which is too strong. A little time passes, the current dissipates and it hits the specific magnetic field for this device. At this point, the amount of electrons caught peaks; and the voltage dips sharply. If this mechanism is true, then it makes sense that the 625 amp run would take so much longer. It takes more time for the current to dissipate. It should be noted the time to the 625 amp peak was not given – it was estimated using other data. There is strong indication this value is incorrect, but it is drawn in to show to the reader the “resonance current”. Please see the appendix for details.
This is a more refined picture then the magnetic mirror line included in the post: “The debate over electron behavior”. In that post, the cusps were not included. In the regions described above the electron is moving slow compared to changes in the magnetic field. The electron velocity is low, against changes in the magnetic field. This sets up a special region of the magnetic field. In this region the magnetic moment is constant. The moment is a measure of how the electron reacts to the magnetic field. The equation for the moment is shown below.
One of the most critical parameters when examining a magnetic confinement device is the mirror ratio. The mirror ratio is an important measure of how well a device confines its plasma. Rider used a low mirror ratio in his general critique paper . The paper he referenced was “Particle loss rates from electrostatic wells of arbitrary mirror ratios” published in 1984 by Peter Catto. That is a good paper to read when looking at mirror ratios. What Peter was after was a mathematical way to connect a mirror ratio to the loss rate of electrons. One of the bits of information I was hoping to come from Dr. Khachans work was what mirror ratio he measured. Then we can compare what value Rider used to what value Khachan measured. If the Whiffle ball effect is happening, we would expect the ratio to be very high. Unfortunately, this work does not give a ratio. FF Chen provided one definition of the mirror ratio. However since the loss cone definition is slightly different than the paper, it is unclear which equation is correct. I did not have time to clear this up, if someone could assist here, I would appreciate it.
Appendix I – Modeling the 15 KeV, 625 Amp, 15 mTorr, Test
This proportionality says that to the electron, crossing this distance is like to a person traveling seven times to Pluto. It takes roughly 7.2 nanoseconds to reach the rings (see below).
The paper states the electron injection energy was 15 KeV. This means one electron fell down a 15 kilovolt drop to pick up this energy. This is where things get somewhat murky. Was the field 15 kilovolts per meter? If so, this means the electron saw 4500 volts between the cathode and the cage. Was the field 50 kilovolts per meter? If so, the electron saw 15,000 volts between the cathode and the cage. Additionally, from an earlier paper, Dr. Khachan states that the energies the ions experience in his beams are ~30% that of the applied cathode potential . This turns out to be a very important issue. The different possibilities and what they mean are worked out below. In each case, I modeled electron flight using both the electric and magnetic fields it would experience. I assumed a 1 degree difference between the magnetic field vector and the velocity vector. If someone knows what the actual field applied between the cathode and the cage was, please let me know. I compared the magnetic and electric Lorenz forces at the rings edge - and used this to determine if the electron would be "caught" or not.
The simplest way, I found, to model how the velocities of electrons inside the beam change over time, was modeling it like a Wiener process. 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 . There is a simple equation for this process, as well as the resulting bell curve of electron speeds is given below.
The graph above is slightly off, it shows the bell distribution around one average. In reality the bell curve should be spreading out and moving from left to right. That is because earlier in the beam the electrons have a lower average velocity. This predicts that the slowest electrons are traveling 1.4E7 meters per second and the fastest are traveling 3.18E7 meters per second. I applied both speeds to the Lorenz force equation, to see if the magnetic component was stronger than the electrostatic component (see below). At both the high and low speed, the electron should get caught by the device.
For this equation, I assumed that by the time the electrons reach the edge of the rings they feel all six electromagnets. I also assumed only a one degree difference between the velocity and the B fields. That is a low degree difference, for many electrons this would be higher. Oddly, it is the slower electrons which are not “caught” by the Polywell. This is because of the magnetic force depends on the electrons’ velocity and the machines’ magnetic field strength. Too slow of a velocity, and the magnetic force is too low to overcome the electric field driving the beam.
Given that the above prediction is flawed, our estimate for the time to the 625 Amp peak must be taken with a grain of salt. The paper does not provide this value. It is estimated from figure 5b, which shows the time to the minimum for different starting coil currents. This is shown below.
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