Magnetic Confinement Fusion

Jason Ginsberg
November 21, 2016

Submitted as coursework for PH240, Stanford University, Fall 2016


Fig. 1: Magnetic confinement of ionized plasma - (Created by Jason Ginsberg)

Nuclear fusion is the process by which two nuclei combine to form a heavier nucleus. [1] The fusion product, while heavier than either of the two starting nuclei, is not as heavy as the sum of the original masses of the starting nuclei. As the relation E = mc2 dictates, this mass discrepancy is not lost but rather converted to and released as energy. Fusion can only occur when the particles involved overcome coulomb repulsion to be bound by the strong force. The coulomb repulsion barrier is most easily penetrated at high temperatures, where quantum tunneling effects are significant, and by nuclei possessing less than 28 protons. Hydrogen isotopes (protium, deuterium, and tritium), in particular, due to their low atomic weight most readily fuse into helium isotopes. Since this reaction ignites at temperatures exceeding 40 million degrees, the helium gas completely ionizes and exists as a plasma. In a deuterium- tritium reaction for example, D (2.014102 amu) + T (3.016050 amu) → 2He4 (4.002603 amu) + 0n1 (1.008665 amu) + 17.58 MeV. [2] The energy released per a kilogram of deuterium is calculated as:

E = 17.58 × 106 eV/atom × 6.022 × 1023 atoms/mole × 1.602 × 10-19 joules/eV
2.014 × 10-3 kg/mole
= 8.42 × 1014 joules/kg

Comparatively, 1 kg of coal produces 28.8MJ; 1 kg of crude oil produces 43.2MJ; and 1 m3 of natural gas produces 37MJ. [3]

Controlled Fusion

For the past half-century, experimentalists have tried to control the nuclear fusion process in an effort to create a copious and near-limitless source of clean energy. Deuterium exists abundantly in water and tritium readily produces from lithium. At present, 1012 tons of deuterium can be extracted from the surface water of the earth. And unlike fission reactors, fusion reactors produce little and short-lasting radioactive waste. The requisite environmental conditions of fusion, however, unsurprisingly make taming the process a nearly impossible task. The Lawson Criterion strictly defines such a minimum condition for a nuclear fusion device to work. In order to reach ignition, a fusion device must exceed a determined triple product of plasma electron density, plasma temperature, and the rate of energy loss. To be self-sustaining, this specific threshold is 3 x 1021 m-3 keV s-1. [1] Beyond satisfying condition, the nuclear reactor must also generate energy. The Q-factor, reflecting this merit, is defined as the ratio:

Q = (energy released by fusion)
(kinetic energy of injected atoms)

If a particular nuclear fusion device fails to exceed a Q-factor of 1, then the device possesses no practical value. More importantly, the ratio of electric energy output to input, Q' can be expressed approximately as 1/4(1+Q). This effective break-even ratio fixes the desired Q value to be at least 2. [4] For a deuterium-tritium reaction to release energy at a level of practical use, the gas must be heated to nearly 100 million Kelvin. At such high temperatures, the plasmatic deuterium and tritium particles move erratically, and most materials fail to confine the ionized gas. [5] The primary solution to confine the plasma for fusion has been to produce a magnetic field. (Fig. 1) Even still, the confinement lasts only a few seconds as the instability of the plasma accrues.


The most advanced of magnetic confinement devices today, the Tokamak, utilizes a combination of toroidal and poloidal confinement fields. One of the major limiting factors in building such a device is material selection. At temperatures exceeding 16 times the interior temperature of the sun (generated by a neutral beam process), materials must withstand operation lest they melt or contaminate the plasma. [6] More complicating is the fact that the superconducting magnetics surrounding the Tokamak must be set at temperatures near absolute zero. The solution to managing the extreme temperature differences has been to surround the plasma vessel in a cryostat vacuum, use a divertor to control exhaust waste gas, and cover the interior walls with a neutron shielding blanket. [7] As of late 2016, the highest Q-value achieved by a Tokamak given these conditions stands at 0.65.


The greatest effort towards actualizing practical magnetic confinement fusion today is the International Thermonuclear Experimental Reactor (ITER) based in France. The objective of ITER is to sustain a Q-factor of 10 (500 MW of fusion power from 50 MW of input power) for 400 seconds and a Q-factor greater than 5 in steady state operation. [8] The plasma containing vessel for ITER will be ten times larger than the current largest operating Tokamak. ITER will not, however, generate usable energy at first, but act instead as an experimental test bed for researchers. The project, though promising, has faced myriad scientific, economic, and technological difficulties. The timetable for ITER goes beyond 2025. And the initial estimated cost has been reevaluated from $5.6 billion in 2006 to a current budget of $16.5 billion. [9] Even if successful, the EIA estimates that fusion production at the time of ITER's completion will cost twice as much as natural gas and coal per kWh, not withstanding a carbon tax. [10] Even if magnetic confinement fusion one day provides an abundant, clean, high-yield, and relatively safe energy source, one significant concern will remain unresolved. ITER would produce a large number of fast neutrons, which can be used to breed fissile fuel. Though Li-6 could be used as a blanket to replenish the tritium in the fusion reaction, only one neutron per fusion reaction will be captured by fertile material. More likely, Th-232 or U-238 will surround the toroidal chamber to produce fissile material. If Th-232 is used the U- 233 generated (or if U-238 is used the Pu-239) is easy to extract and use for weapons. [11] If operational, ITER and other magnetic confinement fusion devices would be vulnerable to bad actors. So it is not only practical and physical but also geopolitical constraints, which restrict the successful operation of ITER.

© Jason Ginsberg. 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] G. McCracken and P. Stott, Fusion: The Energy of the Universe(Academic Press, 2005).

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

[3] L. R. Radovic, Energy and Fuels in Society (McGraw-Hill, 1992).

[4] H. A. Bethe, "The Fusion Hybrid," Physics Today 32, No. 5, 44 (1979).

[5] A.A. Haasz and J.W. Davis, "Hydrogen Retention in and Release from Carbon Materials," Nucl. Fusion Res. 78, 225 (2005).

[6] J. Kates-Harbeck,"Magnetic Nuclear Fusion,"Physics 240, Stanford University, Fall 2010.

[7] J.-L. Duchateau, "Superconducting Magnets for Fusion," CLEFS CEA, Commissariat à l'Energie Atomique, No. 56 (Winter 2007-208), p. 12.

[8] "Summary of the ITER Final Design Report," International Atomic Energy Agency, IAEA/ITER EDA/DS/22, July 2001.

[9] L. Hickman, "Fusion Power: Is It Getting Any Closer?" The Guardian, 23 Aug 11.

[10] "Annual Energy Outlook 2013," U.S. Energy Information Administration, DOE/EIA-0383(2013), April 2013.

[11] H. A. Bethe, "The Fusion Hybrid," Physics Today 32, No. 5, 44 (1979).