Harold Grad was a huge math nerd. He was a professor at NYU in the fifties and sixties [1, 2]. He specialized in applying advanced math to plasmas. Plasma is a fluid; which also happens to conduct. The math controlling it is a merger of the fluids equations (navier-stokes) and the electricity equations (Maxwell's equations). The two combine to make a whole new field: Magnetohydrodynamics. Harold probed the math; looking for any kind of plasma structure.
Dr. Grads’ favorite geometry was the cusped system. A cusp is a place where two magnetic fields sharply bend and repel one other . One example is two north poles repelling. Cusps are awesome because the plasma (mostly) leaks through the edges and apexes of cusps . This shrinks the total area over which the plasma can be lost. The plasma is contained in a trap with a few small holes. Cusps have two major advantages. First, their fields are bent inward. This is great. Plasma tends to drift into bigger curved paths - so inward bending fields help to push material into the center . That is helpful. Second, cusps have a null point in the center. The null point is a spot with no magnetic field. It is a place where plasma can collect. After Grad had shown the value of cusp geometry, other people latched onto this idea and a whole family of concepts were proposed .
"Free Boundary" Plasmas
Cusp systems can shrink losses, but Harold Grad found a way to go further. In some cases, the plasma can actually plug up the cusps . Plasma is a moving soup of charged material. Its' motion can make its' own magnetic field. This can clash and reject the outside field. This can lead to a plugged cusp. Physically this is a diamagnetic plasma, rejecting the external field. Inside, material can move about, free of the externally applied fields. This system - theoretically - has a sharp boundary with a sheet of electrons moving on its' surface [13, 12, 15]. This is shown below.
Theoretically, this would be the world’s best plasma trap. Not only is material better contained, but the plasma also loses less energy as light .
The Lost Concept:
Free boundary plasma would be awesome; but despite many attempts, the system was never demonstrated [13-15]. There are two good reasons for this. First, high pressure plasmas are very hard to make. Secondly, the effect is hard to measure. By 1980, it had disappeared from most fusion programs. Most teams moved on from the cusp geometries; looping the field lines to make a tokamak. That was situation - until 2014 – when EMC2 reported making a free boundary plasma. It is still early days - but if the full potential of this discovery is realized – someone may win the Nobel Prize.
Sometimes better science comes from using better tools. The new machine is far better than Bussards’ old WB6. The electromagnets or rings were designed better. The first change is its size; WB6 was much bigger. Its’ rings filled six times more space [6,16]. The new machine is much slimmer, but more powerful. This will lead to higher energy densities. The devices are pictured below [3,4,16, 20].
The new model does not link the rings together. They are mounted externally. This leaves more space for plasma recirculation - something both Rider and Bussard stressed as critical [16-18]. Recirculation, means plasma can move without touching metal. Recirculation is also used in the Lockheed Martin concept . Mounting the rings externally also changes everything about how the rings are powered. The old machine formed its’ electromagnets from one long wire. This long wire, snaked its way through all six magnets and ran to one big set of 240 batteries . This wire was over three thousand feet long, overheated and had three ohm of resistance. The new machine broke the power supply up. Six distinct power supplies were used; nine batteries per ring. This is much safer. The wire also has a lower resistance. A diagram of the power supply for one electromagnet is shown below.
The rings are used to create a magnetic geometry. The geometry is custom designed; specific for this application. A comparison of the energy density made by WB6 and this machine are shown below.
Modeling this emitter was not easy. There are nine physical mechanisms that happen simultaneously. I will spare you the math. If you would like to dig into the numbers - you can see them all in this excel file. All the effects that were modelled are shown below; effects are numbered in the order so you can follow along.
First, the current will not spark across the air gap [22, 23]. Electric arching can be modelled using Paschen law. Paschen’s law tells us quickly that sparking will not happen. That means that all the current must move through the plastic sheet. This sheet has some electrical resistance. We can model the resistance this using the equations for a circular sheet of polypropylene [25, 26]. This number allows us to find the electric field inside this sheet. The field is plotted below. As the current moves through, this sheet will vaporize and ionize. This is because the energy needed to break the chemical bonds and fully ionize the plastic is only hundreds of joules [27-29]. Far more energy passes by. So much energy that the sheet probably ionizes instantly. It turns into a cloud of hydrogen and carbon ions.
As the current leaves the plastic and moves down the cathode, it starts a new series of physical effects. The current in the cathode creates a magnetic field. This field can be modelled by treating the cathode like a big wire. You can use the biot-savart law to model the field in a big wire . The resulting magnetic field is plotted below. The cathode has a huge amount of electrons passing through it – which will heat it up. In fact, the metal likely reaches several hundred degrees kelvin [29, 31]. This causes the tungsten to chip away . Overtime, the tungsten cathode degrades. This process is known as thermionic emission. It means that tungsten ions are also mixed into the plasma .
This model tells us that there are electrons, carbon, hydrogen and tungsten ions in the plasma. We need to know how much energy they have as they leave. The emitter makes both an magnetic and electric field which hit the plasma. With both fields in play, it creates a Lorentz force for all charged particles. This is also known as the J X B force . Anything that is positive is pushed outward. This flings ions towards the rings. Anything that is negative is forced backwards. This pushes electrons back into the emitter. Bear in mind, these emitters are really close to the rings. They sit half a centimeter away. The navy had to do it this way. Unlike Bussard’s 2005 experiment - there is no electric field to steer the plasma. The emitters are close, to catch all the particles before they spread out. Before they are lost. This experiment only used two injectors. They could have had many more. That is an important question for the next test. What happens when we have many plasma injectors? An overview of the equations used to model this system is shown below. You can look at all the numbers in this excel file.
All of these effects combine to make a plasma which is mostly hydrogen and carbon. The remaining 6 percent is trace elements. This includes tungsten, oxygen, nitrogen and the electrons. Below is a summary of the plasma composition. These amounts assume everything was injected instantly.
You can estimate the magnetomotive force between the rings by treating them like two circular magnets facing each other. The equation for this is shown below. Below this are the numbers used for WB6 and WB8. The equations predicts 37.6 and 170 newtons for WB6 and WB8 respectively.
1. Park, Jaeyoung. "Polywell Fusion: Electrostatic Fusion in a Magnetic Cusp." Microsoft Research. Microsoft Inc, 22 Jan. 2015. Web. 20 July 2015.
2. Park, Jaeyoung (12 June 2014). SPECIAL PLASMA SEMINAR: Measurement of Enhanced Cusp Confinement at High Beta (Speech). Plasma Physics Seminar. Department of Physics & Astronomy, University of California, Irvine: Energy Matter Conversion Corp (EMC2) url=http://www.physics.uci.edu/seminar/special-plasma-seminar-measurement-enhanced-cusp-confinement-high-beta
3. "Polywell Fusion – Electric Fusion in a Magnetic Cusp" Jaeyoung Park, Friday, December 5, 2014 - 1:00pm to 2:00pm, Physics and Astronomy Building (PAB) Room 4-330, UCLA
4. Talk at University of Wisconsin Madison, Monday, June 16, 2:30 PM room 106 ERB, Jaeyoung Park
5. University of Maryland, Colloquium & Seminars, "Measurement of Enhanced Confinement at High Pressure Magnetic Cusp System", Jaeyoung Park, September 9th 2014
6. "Polywell Fusion Electrostatic Fusion in a Magnetic Cusp", Jaeyoung Park, http://fire.pppl.gov/FPA14_IECM_EMC2_Park.pdf, Tuesday December 16, 2014 Hyatt Regency Capitol Hill 400 New Jersey Avenue NW, Washington, DC 20001
7. Private conversation, Jaeyoung Park, April 2015
8. Linden, Tomas. "Compact Fusion Reactors." CERN Talks. CERN, 25 Mar. 2015. Web. 20 July 2015
9. J. L. Tuck, A new plasma confinement geometry, Nature (London) 187, 863 (1960).
10. Berkowitz, J., K.o. Friedrichs, H. Goertzel, H. Grad, J. Killeen, and E. Rubin. "Cusped Geometries." Journal of Nuclear Energy (1954) 7.3-4 (1958): 292-93. Web. 16 June 2014.
11. McMillan, Brian. "Lecture 8, Slide 20." PX438 Physics of Fusion Power. The University of Warwick, 13 Feb. 2013. Web. 04 Apr. 2013.
12. Park, Jaeyoung, Nicholas A. Krall, and Paul E. Sieck. "High Energy Electron Confinement in a Magnetic Cusp Configuration." In Submission (2014): 1-12. Http://arxiv.org. Web. 13 June 2014.
13. Grad, Harold. "Plasma Trapping in Cusped Geometries." Physical Review Letters 4.5 (1960): 222-23.
14. Haines, M. g. "Plasma Containment in Cusp-shaped Magnetic Fields." Nuclear Fusion 17.4 (1977): 811-58. Web. 18 June 2014.
15. Dolan, Thomas J. “Review Article: Magnetic Electrostatic Plasma Conﬁnement.” Vol. 1539-1593. N.p.: Plasma Physics and Controlled Fusion, 1994. Print.
16. Bussard, Robert W. "The Advent of Clean Nuclear Fusion: Superperformance Space Power and Propulsion." 57th International Astronautical Congress (2006)
17. Rider, Todd H. "A General Critique of Inertial-electrostatic Confinement Fusion Systems." Physics of Plasmas 6.2 (1995): 1853-872. Print.
18. Rider, Todd H. "Fundamental Limitations on Plasma Fusion Systems Not in Thermodynamic Equilibrium." MIT Thesis 1995.
19. Carr, Matthew, and David Gummersall. "Low Beta Confinement in a Polywell Modeled with Conventional Point Cusp Theories." Physics of Plasmas 18.112501 (2011): n. page. Print
20. Duncan, Mark, and Robert Bussard. Should Google Go Nuclear? (Summary). N.d. MS. Should Google Go Nuclear? Www.askmar.com. Mark Duncan, 24 Dec. 2008. Web. 4 Feb. 2013.
21. Moynihan, Matthew. "Taking A Stab At Simulation." The Polywell Blog. N.p., 6 Feb. 2013. Web. 27 June 2015.
22. Private communication, “What was roughly the pressure inside the machine?” Paul Sieck, July 2, 2015
23. Lieberman, Michael A.; Lichtenberg, Allan J. (2005). Principles of plasma discharges and materials processing (2nd ed.). Hoboken, N.J.: Wiley-Interscience. 546. ISBN 978-0471005773. OCLC 59760348
24. Thoma, C., D. R. Welch, and T. P. Hughes. "Ballistic and Snowplow Regimes in J×B Plasma Acceleration." Physics of Plasmas Phys. Plasmas 16.3 (2009): 032103. Web.
25. "How Do You Model the Resistance across a Symmetric Sheet of Plastic, Stretched (think Cernan Wrap) between a Circular Anode and Cathode?" Electricity. Physics Stack Exchange, 13 July 2015. Web. 20 July 2015
26. "Polypropylene Material Information." Polypropylene. Goodfellow, 2015. Web. 20 July 2015.
27. "Ionization Energies of the Elements." Wikipedia. Wikimedia Foundation, n.d. Web. 20 July 2015.
28. "Bond Energies." Chemwiki. University of California Davis, n.d. Web. 20 July 2015.
29. "Joule Heating." Wikipedia. Wikimedia Foundation, n.d. Web. 20 July 2015.
30. NAVE, R. "Biot-Savart Law." Biot-Savart Law. Georgia State University, n.d. Web. 20 July 2015.
31. "The Physical Properties of Tungsten." Wikipedia. Wikimedia Foundation, Web. 20 July 2015.
32. Harbaugh, W. E. "Tungsten Thorinated-Tungsten and Thoria Emitters." (n.d.): n. pag. Web. 20 July 2015.
33. Gunther, Kim. "Model 106381 Electron Gun for Plasma/Fusion Research." Heat Wave Labs Inc. Heat Wave Labs Inc., n.d. Web. 20 July 2015.
34. "The Trapped Plasma Volume." Interview by Jaeyoung Park. Private Communication 24 Aug. 2015
35. Kitsunezaki, Akio. "Cusp Confinement of High-beta Plasmas Produced by a Laser Pulse from a Freely-falling Deuterium Ice Pellet." Physics of Fluids Phys. Fluids 17.10 (1974): 1895. Web.
36. Mare, Indrek. "Cube Polywell Wiffleball Modeling Using Method of Images." http://www.mare.ee/indrek/ephi/images.pdf Indrek's Homepage. Indrek Mare, 2008. Web. 31 Aug. 2015.
37. "Question." Message to John Santarius. 17 Aug. 2015. E-mail.
38. Carr, Matthew, David Gummersall, Scott Cornish, and Joe Khachan. "Low Beta Confinement in a Polywell Modelled with Conventional Point Cusp Theories." Physics of Plasmas Phys. Plasmas 18.11 (2011): 112501. Web.
39. "An Interview With Thomas Ligon on The Polywell." Interview by Thomas Ligon. YouTube. YouTube, 25 May 2009. Web. 31 Aug. 2010.
40. Engelhardt, W. "Is a Plasma Diamagnetic?" Physics Essays 18.4 (2005): 504-13. Web.
41. McGuire, Thomas. "The Lockheed Martin Compact Fusion Reactor." Thursday Colloquium. Princeton University, Princeton. 6 Aug. 2015. Lecture.
42. Tuszewski, M. "Field Reversed Configurations." Nucl. Fusion Nuclear Fusion 28.11 (1988): 2033-092. Web.
43. Park, J. "High-Energy Electron Confinement in a Magnetic Cusp Configuration." Physical Review X. N.p., 11 June 2015. Web. 06 Nov. 2015.
44. Berkowitz, H. H Grad, H Rubin "Magnetohydrodynamic Stability." Journal of Nuclear Energy P/376 (1954): 177-89. Web. 7 Nov. 2015.
45. Private conversation with Dr. Joel Rogers. 30 Sept. 2014.