Plasma Focus Fusion

Idel Waisberg
March 21, 2013

Submitted as coursework for PH241, Stanford University, Winter 2013

Fusion Fundamentals

Fig. 1: Plasma Focus Reactor and Professor Nikolay Filippov (Kurchatov Institute of Atomic Energy, Moscow, Russia) (Source: Wikimedia Commons)

Fusing two atomic nuclei together can cost or release energy, depending on the mass of the resulting nucleus. For nuclei more massive than Fe-56, fusion costs energy, and therefore fission is used for mass-energy production. For nuclei ligher than Fe-56, the mass of the fused nucleus is smaller than the sum of the separate nuclei, and the excess energy is released according to Einstein's E = Δmc2. This energy is released as heat (primarily carried by neutrons in ordinary fusion reactors) and can be used to boil water to move turbines and generate electricity.

In order to fuse two nuclei, very high energies are needed because the Coulomb barrier must be overcome. Because this barrier is directly proportional to the atomic number Z of each of the two fusing nuclei, the fuel used in fusion reactors usually consists of a combination of hydrogen isotopes (protium, deuterium and tritium) to be fused into helium isotopes. The canonical fusion reaction in a reactor is the fusion of deuterium and tritium into helium, with the release of one neutron: [1]

2D + 3T → 4He + n + 17.6 MeV

Because of the high temperatures required for fusion, the atoms in the fuel are highly ionized and form a plasma. Plasma physics plays therefore a major role in the design of fusion reactors. In order to operate a fusion reactor continuously, the energy losses must be overcome by keeping the temperature of the fusion products sufficiently high. The Lawson criterion expresses the minimum product neE (called triple product) required to sustain fusion for a given reactor design, where ne is the plasma electron density, T is the plasma temperature and τE is the confinement time (rate of energy loss). [1] Leading fusion reactor designs use magnetic fields (tokamak) or high power laser (inertial confinement) to confine the plasma.

The requirements for plasma confinement and minimum fuel temperature have postponed the commercial use of fusion reactors for power production. In this report, aneutronic fusion is investigated as an alternative reactor design to mainstream hydrogen isotope fusion, with emphasis on the Dense Plasma Focus (DPF) reactor.

What Is Aneutronic Fusion?

A fusion reaction is called aneutronic if less than 1% of the released energy is carried by neutrons (D-T reactions, for example, can release up to 80% of their energy in neutrons). [1] As a consequence, a larger fraction of the released energy is carried by charged particles, which can be used to produce electricity directly (without the need of steam turbines), leading to a drastic reduction in cost. Charged particles in motion can be used to produce electricity directly through inductive or electrostatic techniques. Out of the aneutronic fusion reactions with larger cross section, a particularly interesting one involves the fusion of a proton with boron to produce helium: [1]

p + 11B → 3 4He + 8.7 MeV

A small amount of neutrons are produced by side reactions forming nitrogen and carbon, but neutrons only carry around 0.1% of the net energy in a fusion reactor employing boron fuel. Another advantage of the lower neutron yield is the reduction of problems associated with neutron radiation (damage to the reactor, radioactivity etc). In D-T reactions, the neutron flux is about 100 times higher than that of current fission reactors, posing a problem for reactor material design and radiation safety.

Despite its benefits, aneutronic fusion has a major downside to D-T fusion: the requirements associated with the Lawson criterion are much more difficult to achieve. The main technical challenges to aneutronic fusion are discussed below, with a particular emphasis on the Dense Plasma Focus reactor, after a brief overview of the reactor design.

Dense Plasma Focus Reactor: Principles and Design

Instead of trying to stabilize and confine the plasma as in an ordinary fusion reactor, the Dense Plasma Focus takes advantage of the plasma natural instability to release a large amount of energy in a very short time. Rough values are 100 MW/cm3 of input power and 2 × 109 TW/cm3. [2]

The core of the reactor is composed of a central metal electrode (anode), surrounded by a hollow insulating cylinder and by outer metal rods (cathode). The anode and cathode should be made of a metal transparent to X-rays (see below), e.g. Beryllium. This structure is enclosed in a vacuum chamber, which is filled with fusion fuel p-11B fusion in gas state. A charged bank of electrical capacitors (called Marx generator) is attached to the electrodes. When the switch of the bank is closed, it releases a very strong current, going from the anode to the cathode across the electrical insulator, which ionizes the gas into the plasma state. The plasma current sheath interacts with its own magnetic field, via Lorentz force, and will cause the plasma to collapse inward (according to Lenz's Law) and increase its density. By the time the plasma sheath gets to the end of the anode rod, a plasmoid is formed and the energy is focused in a very small area (a few microns across). When the plasmoid gets dense enough, radiation can escape and the resulting fall-off in the current can create a time-varying magnetic field, and hence an electrostatic potential, which accelerates a beam of electrons and a beam of positive ions at opposing ends. The electrons can collide with the nuclei and provide sufficient energy to provide fusion. The life-time of the plasmoid is very small (tens of nanoseconds), but it is enough to ignite a fusion reaction. [2]

The high-energy, few-nanosecond pulsed ion beam can be fed to a metal coil, inducing currents to be used to charge a capacitor bank with very rapid diamond switches. [2]

Main Technical Challenges and Prospects for Improvement

There are two main technical challenges to aneutronic fusion: minimum temperature for fusion ignition and energy loss due to X-ray emission.

The number of nucleons in the boron nucleus is greater than in the hydrogen isotopes, which results in a higher Coulomb barrier that requires a higher temperature to ignite fusion, as well as more strict requirements for confinement time and power density. In the case of p-11B, the optimum temperature is around 300 keV (more than a billion degrees Celsius, around 6-10 times higher than the required temperature for D-T fusion), the confinement time must be 500 times higher and the power density can be 2500 times lower than D-T in regular conditions. [3,4] Therefore, achieving and sustaining fusion is a major challenge for higher atomic number nuclei fuels.

The main source of energy loss for most aneutronic fusion reactions is X-ray emission from Bremsstrahlung. Because X-ray emission is roughly proportional to Z2, where Z is the atomic number of the ion, aneutronic fusion has much greater X-ray energy loss than D-T fusion. The electrons in a plasma will, in general, be at a higher temperature than the ions. The collision of the electron with the ions generates X-ray radiation (~10-30 keV). This radiation, called bremsstrahlung, is produced by the de-acceleration of charged particles when deflected by another charged particle. The reactor is usually optically thin for X-rays in this energy range, with most of the radiation being absorbed by the shielding (steel) and therefore being lost. The ratio between the net fusion power and the energy lost by X-ray radiation (called power ratio) is an important figure of a fusion reactor. For D-T fusion, the power ratio for ordinary magnetic fields is around 140, whereas for p- 11B fusion it is usually not above 0.57. [4] These ratios are upper limits, since there will be in general other sources of energy loss (impurities in the plasma fuel, fuel products releasing energy directly to electrons and not entirely to fusion ions etc).

A possible solution for the high energy losses by X-ray emission in aneutronic fusion reactors may lie in an intriguing quantum mechanical effect, which has been described in the extremely high magnetic fields of neutron stars. [5] In very high magnetic fields (~ megatesla), this effect may suppress the energy transfer from ions to electrons (which reduces the electron temperature and therefore X-ray emission, which is proportional to Te2) enough to make the power ratio of a fusion reactor greater than 1. The electrons follow a helicoidal orbit around the direction of the magnetic field. As the strength of the magnetic field increases, the electron's gyro-radius decreases, and so does its velocity. Because the minimum gyro-radius is the electron wavelength, there is a minimum velocity that the electron must have in this magnetic field line and, therefore, a minimum momentum that can be imparted to the electron by the ions in the plasma. Therefore, the transfer of energy from ions to electrons is reduced and so is X-ray emission. [4]

Strong magnetic fields enhance cyclotron radiation (radiation emitted by moving charges deflected by magnetic fields), which can be even higher than Bremsstrahlung. However, with a sufficiently dense plasma, the cyclotron frequency can be made smaller than twice the plasma frequency, which traps the cyclotron radiation inside the plasmoid to be reabsorbed. [1]

Furthermore, the X-rays that do get emitted can be reabsorbed by Thompson scattering of electrons. The X- rays can be directed to a collection of thin foils (thin so that the ionized electron is not reabsorbed). A large grid of foils is needed to absorb most of the energy since X-rays are highly penetrating. [2]


Dense Plasma Focus is a promising reactor design for achieving aneutronic fusion. Recent theoretical and experimental advances have risen the question of whether DPF could be the key to practical fusion power production. [2,6] The dense plasma focus can also be used for applications such as X-ray lithography, and in the case of higher neutron yield fuels, as a neutron source for medical applications or nuclear weapons.

© Idel Waisberg. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


[1] A. A. Harms et al., Principles of Fusion Energy: An Introduction to Fusion Energy for Students of Science and Engineering (World Sci., 2000).

[2] E. J. Lerner et al., "Fusion Reactions From > 150 keV Ions in a Dense Plasma Focus Plasmoid," Phys. Plasmas 19, 032704 (2012).

[3] T. H. Rider, "Fundamental Limitations on Plasma Fusion Systems Not in Thermodynamic Equilibrium," Phys. Plasmas 4, 1039 (1997).

[4] S. Son and N.J. Fisch, "Aneutronic Fusion in a Degenerate Plasma," Phys. Lett. A 329, 76 (2004).

[5] G. S. Miller, E. E. Salpeter and I. Wasserman, "The Deceleration of Infalling Plasma in the Atmospheres of Accreting Neutron Stars. I. Isothermal Atmospheres," Astrophys. J. 314, 215 (1987).

[6] A. Schmidt, V. Tang and D. Welch, "Fully Kinetic Simulations of Dense Plasma Focus Z-Pinch Devices," Phys. Rev. Lett. 109, 205003 (2012).