Chapter 7. Fusion

When two light nuclear particles combine or “fuse” together, energy is released because the product nuclei have less mass than the original particles. Such fusion reactions can be caused by bombarding targets with charged particles, by use of an accelerator, or by raising the temperature of a gas to a high enough level for nuclear reactions to take place. In this chapter we will describe the interactions in the microscopic sense and discuss the phenomena that affect our ability to achieve a practical large-scale source of energy from fusion. Thanks are due to Dr. John G. Gilligan for his comments.

7.1. Fusion Reactions

The possibility of release of large amounts of nuclear energy can be seen by comparing the masses of nuclei of low atomic number. Suppose that one could combine two hydrogen nuclei and two neutrons to form the helium nucleus. In the reactionthe mass-energy difference (by use of atom masses) iswhich corresponds to 28.3 MeV energy. A comparable amount of energy would be obtained by combining four hydrogen nuclei to form helium plus two positrons

This reaction in effect takes place in the sun and in other stars through the so-called carbon cycle, a complicated chain of events involving hydrogen and isotopes of the elements carbon, oxygen, and nitrogen. The cycle is extremely slow, however, and is not suitable for terrestrial application.

In the “hydrogen bomb,” on the other hand, the high temperatures created by a fission reaction cause the fusion reaction to proceed in a rapid and uncontrolled manner. Between these extremes is the possibility of achieving a controlled fusion reaction that uses inexpensive and abundant fuels. As yet, a practical fusion device has not been developed, and considerable research and development will be required to reach that goal. Let us now examine the nuclear reactions that might be used. There seems to be no mechanism by which four separate nuclei can be made to fuse directly, and thus combinations of two particles must be sought.

The most promising reactions make use of the isotope deuterium, , abbreviated D. It is present in hydrogen as in water with abundance only 0.015% (i.e., there is one atom of for every 6700 atoms of ), but because our planet has enormous amounts of water, the fuel available is almost inexhaustible. Four reactions are important:

The fusion of two deuterons—deuterium nuclei—in what is designated the D–D reaction results in two processes of nearly equal likelihood. The other reactions yield more energy but involve the artificial isotopes tritium, , abbreviated T, with the ion called the triton, and the rare isotope , helium-3. We note that the products of the first and second equations appear as reactants in the third and fourth equations. This suggests that a composite process might be feasible. Suppose that each of the reactions could be made to proceed at the same rate, along with twice the reaction of neutron capture in hydrogen

Adding twice this equation to the preceding four, we find that the net effect is to convert deuterium into helium according to

The energy yield per atomic mass unit of deuterium fuel would thus be approximately 6 MeV, which is much more favorable that the yield per atomic mass unit of U-235 burned, which is only 190/235 = 0.81 MeV.

Computer Exercise 7.A permits the exploration of possible nuclear reactions for fusion.

7.2. Electrostatic and Nuclear Forces

The reactions previously described do not take place merely by mixing the ingredients, because of the very strong force of electrostatic repulsion between the charged nuclei. Only by giving one or both of the particles a high speed can they be brought close enough to each other for the strong nuclear force to dominate the electrical force. This behavior is in sharp contrast to the ease with which neutrons interact with nuclei.

There are two consequences of the fact that the coulomb force between two charges of atomic numbers Z¹ and Z² varies with separation R according to Z¹Z²/R². First, we see that fusion is unlikely in elements other than those low in the periodic table. Second, the force and corresponding potential energy of repulsion is very large at the 10⁻¹⁵ m range of nuclear forces, and thus the chance of reaction is negligible unless particle energies are of the order of keV. Figure 7.1 shows the cross section for the D–D reaction. The strong dependence on energy is noted, with σ^DD rising by a factor of 1000 in the range 10 to 75 keV.

Figure 7.1. Cross section for D–D reaction.

Energies in the kilo-electron-volt and million-electron-volt range can be achieved by a variety of charged particle accelerators. Bombardment of a solid or gaseous deuterium target by high-speed deuterons gives fusion reactions, but most of the particle energy goes into electrostatic interactions that merely heat up the bulk of the target. For a practical system, the recoverable fusion energy must significantly exceed the energy required to operate the accelerator. Special equipment and processes are required to achieve that objective.

7.3. Thermonuclear Reactions in a Plasma

A medium in which high particle energies are obtained is the plasma. It consists of a highly ionized gas as in an electrical discharge created by the acceleration of electrons. Equal numbers of electrons and positively charged ions are present, making the medium electrically neutral. The plasma is often called “the fourth state of matter.” Through the injection of enough energy into the plasma, its temperature can be increased, and particles such as deuterons reach the speed for fusion to be favorable. The term thermonuclear is applied to reactions induced by high thermal energy, and the particles obey a speed distribution similar to that of a gas, as discussed in Chapter 2.

The temperatures to which the plasma must be raised are extremely high, as we can see by expressing an average particle energy in terms of temperature, by use of the kinetic relation

For example, even if is as low as 10 keV, the temperature is

Such a temperature greatly exceeds the temperature of the surface of the sun and is far beyond any temperature at which ordinary materials melt and vaporize. The plasma must be created and heated to the necessary temperature under some constraint provided by a physical force. In stars, gravity provides that force, but that is not sufficient on earth. Compression by reaction to ablation is designated as inertial confinement; restraint by electric and magnetic fields is called magnetic confinement. These methods will be discussed in Chapter 14. Such forces on the plasma are required to assure that thermal energy is not prematurely lost. Moreover, the plasma must remain intact long enough for many nuclear reactions to occur, which is difficult because of inherent instabilities of such highly charged media. Recalling from Section 2.2 the relationship pV = nkT, we note that even though the temperature T is very high, the particle density n/V is low, allowing the pressure p to be manageable.

The achievement of a practical energy source is further limited by the phenomenon of radiation losses. In Chapter 5 we discussed the bremsstrahlung radiation produced when electrons experience acceleration. Conditions are ideal for the generation of such electromagnetic radiation because the high-speed electrons in the plasma at elevated temperature experience continuous accelerations and decelerations as they interact with other charges. The radiation can readily escape from the region, because the number of target particles is very small. In typical plasma, the number density of electrons and deuterons is 10¹⁵, which corresponds to a rarefied gas. The amount of radiation production (and loss) increases with temperature at a slower rate than does the energy released by fusion, as shown in Figure 7.2. At what is called the ignition temperature, the lines cross. Only for temperatures greater than that value, 400,000,000 K in the case of the D–D reaction, will there be a net energy yield, assuming that the radiation is lost. In a later chapter we will describe some of the devices that have been used to explore the possibility of achieving a fusion reactor.

Figure 7.2. Fusion and radiation energies.

7.4. Summary

Nuclear energy is released when nuclei of two light elements combine. The most favorable fusion reactions involve deuterium, which is a natural component of water and thus is a very abundant fuel. The reaction takes place only when the nuclei have a high enough speed to overcome the electrostatic repulsion of their charges. In a highly ionized electrical medium, the plasma, at temperatures of the order of 400,000,000 K, the fusion energy can exceed the energy loss because of radiation.

7.5. Exercises

1. Calculate the energy release in amu and MeV from the combination of four protons to form a helium nucleus and two positrons (each of mass 0.000549 amu).
2. Verify the energy yield for the reaction , noting atomic masses (in order) 2.014102, 3.016029, 4.002603, and 1.007825.
3. To obtain 3000 MW of power from a fusion reactor, in which the effective reaction is , how many grams per day of deuterium would be needed? If all of the deuterium could be extracted from water, how many kilograms of water would have to be processed per day?
4. The reaction rate relation nυNσ can be used to estimate the power density of a fusion plasma. (a) Find the speed υ^D of 100 keV deuterons. (b) Assuming that deuterons serve as both target and projectile, such that the effective υ is υ^D /2, find what particle number density would be needed to achieve a power density of 1 kW/cm³.
5. Estimate the temperature of the electrical discharge in a 120-volt fluorescent light bulb.
6. Calculate the potential energy in eV of a deuteron in the presence of another when their centers are separated by three nuclear radii (Note: E^p = kQ¹Q²/R where k = 9 × 10⁹, Q's are in coulombs, and R is in meters).

Computer Exercises

1. Program REACT1 displays the atomic masses for a number of light nuclides that are candidates as fusion projectiles and targets. Run the program and use Print Screen to obtain a paper copy of the table.
2. The reaction energy Q is the difference between masses of products and reactants. Program REACT2 calculates Q for an input of nuclei that might be involved and obtains the approximate distribution of energy between the product nuclei. (a) Test the program by use of the classic D-T reaction, with A1 = 1, Z1 = 1; A2 = 2, Z2 = 1; A3 = 4, Z3 = 2; A4 = 1, Z4 = 0. (b) Try the program with a few other reactions.
3. Program REACT3 surveys the array of light nuclei for potential fusion reactions. Run the program to find reactions with highest reaction energy, those that are neutron-free, and those that would require the lowest temperature on the basis of the product of Z1 and Z2.

7.6 References

Heppenheimer, 1984 T.A. Heppenheimer, The Man-Made Sun, The Quest for Fusion Power 1984 Little, Brown & Co Boston A narrative account of the fusion program of the United States, including personalities, politics, and progress to the date of publication. Good descriptions of equipment and processes.

Herman, 1990 Robin Herman, Fusion: The Search for Endless Energy 1990 Cambridge University Press New York A well-written and interesting account.

Gross, 1985 Robert A. Gross, Fusion Energy 1985 John Wiley & Sons New York A readable textbook. Main emphasis is on magnetic confinement fusion

Duderstadt and Moses, 1982 James J. Duderstadt, Gregory A. Moses, Inertial Confinement Fusion 1982 John Wiley & Sons New York An excellent complement to the book by Gross.

Perspectives on Plasmas Perspectives on Plasmas

http://www.plasmas.org http://www.plasmas.org

All aspects of plasma science and technology All aspects of plasma science and technology.

Harms et al., 2000 A.A. Harms, K.F. Schoepf, G.H. Miley, D.R. Kingdon, Principles of Fusion Energy 2000 World Scientific Singapore Subtitle: An Introduction to Fusion Energy for Students of Science and Engineering.

Miyamoto, 2005 Kenro Miyamoto, Plasma Physics and Controlled Nuclear Fusion 2005 Springer-Verlag New York

Freidberg, 2007 Jeffrey P. Freidberg, Plasma Physics and Fusion Energy 2007 Cambridge University Press New York

Suzanne, 2006 Pfalzner Suzanne, An Introduction to Inertial Confinement Fusion 2006 CRC Press Boca Raton, FL

Theoretical Principles of Plasma Physics Theoretical Principles of Plasma Physics

http://www.plasmaphysics.org http://www.plasmaphysics.org

Highly technical but comprehensive. By Thomas Smid Highly technical but comprehensive. By Thomas Smid.

European Fusion Development Agreement (EFDA) European Fusion Development Agreement (EFDA)

http://www.jet.efda.org http://www.jet.efda.org

About the Joint European Torus (JET) About the Joint European Torus (JET).

Educational Web Site Fusion Energy Educational Web Site Fusion Energy

http://fusioned.gat.com http://fusioned.gat.com

Information on fusion from General Atomic Information on fusion from General Atomic.

Fusion Power Associates Fusion Power Associates

http://fusionpower.org http://fusionpower.org

Source of information on latest technical and political developments Source of information on latest technical and political developments.

Educational Web Site Fusion Energy Educational Web Site Fusion Energy

http://fusioned.gat.com http://fusioned.gat.com

Information on fusion from General Atomic Information on fusion from General Atomic.

Fusion Energy Educational Web Site Fusion Energy Educational Web Site

http://fusedweb.pppl.gov/cpep/chart.html http://fusedweb.pppl.gov/cpep/chart.html

Select Fusion Basics. From Princeton Plasma Physics Laboratory Select Fusion Basics. From Princeton Plasma Physics Laboratory.

Fusion Power Associates Fusion Power Associates

http://fusionpower.org http://fusionpower.org

A foundation that is a valuable source of information on current fusion research and political status, with links to many other sites. Fusion Program Notes appear frequently as e-mail messages A foundation that is a valuable source of information on current fusion research and political status, with links to many other sites. Fusion Program Notes appear frequently as e-mail messages.