Chapter 6. Fission

Out of the many nuclear reactions known, that resulting in fission has at present the greatest practical significance. In this chapter we will describe the mechanism of the process, identify the byproducts, introduce the concept of the chain reaction, and look at the energy yield from the consumption of nuclear fuels.

6.1. The Fission Process

The absorption of a neutron by most isotopes involves radiative capture, with the excitation energy appearing as a gamma ray. In certain heavy elements, notably uranium and plutonium, an alternate consequence is observed—the splitting of the nucleus into two massive fragments, a process called fission. Computer Exercise 6.A provides a graphic display of the process. Figure 6.1 shows the sequence of events by use of the reaction with U-235. In Stage A, the neutron approaches the U-235 nucleus. In Stage B, the U-236 nucleus has been formed in an excited state. The excess energy in some cases may be released as a gamma ray, but more frequently, the energy causes distortions of the nucleus into a dumbbell shape, as in Stage C. The parts of the nucleus oscillate in a manner analogous to the motion of a drop of liquid. Because of the dominance of electrostatic repulsion over nuclear attraction, the two parts can separate, as in Stage D. They are then called fission fragments, bearing most of the energy released. They fly apart at high speeds, carrying some 166 MeV of kinetic energy out of the total of approximately 200 MeV released in the whole process. As the fragments separate, they lose atomic electrons, and the resulting high-speed ions lose energy by interaction with the atoms and molecules of the surrounding medium. The resultant thermal energy is recoverable if the fission takes place in a nuclear reactor. Also shown in the diagram are the prompt gamma rays and fast neutrons that are released at the time of splitting.

Figure 6.1. The fission process.

6.2. Energy Considerations

The absorption of a neutron by a nucleus such as U-235 gives rise to extra internal energy of the product, because the sum of masses of the two interacting particles is greater than that of a normal U-236 nucleus. We write the first step in the reaction where the asterisk signifies the excited state. The mass in atomic mass units of (U-236)* is the sum 235.043923 + 1.008665 = 236.052588. However, U-236 in its ground state has a mass of only 236.045562, lower by 0.007026 amu or 6.54 MeV. This amount of excess energy is sufficient to cause fission. Figure 6.2 shows these energy relationships. To achieve accurate results, we use many more significant figures and decimals than for typical scoping calculations.

Figure 6.2. Excitation energy caused by neutron absorption.

The preceding calculation did not include any kinetic energy brought to the reaction by the neutron, on the grounds that fission can be induced by absorption in U-235 of very slow neutrons. Only one natural isotope, , undergoes fission in this way, whereas and are the main artificial isotopes that do so. Most other heavy isotopes require significantly larger excitation energy to bring the compound nucleus to the required energy level for fission to occur, and the extra energy must be provided by the motion of the incoming neutron. For example, neutrons of at least 0.9 MeV are required to cause fission from U-238, and other isotopes require even higher energy. The precise terminology is as follows: fissile materials are those giving rise to fission with slow neutrons; many isotopes are fissionable, if enough energy is supplied. It is advantageous to use fast neutrons—of the order of 1 MeV energy—to cause fission. As will be discussed in Chapter 13, the fast reactor permits the “breeding” of nuclear fuel. In a few elements such as californium, spontaneous fission takes place. The isotope , produced artificially by a sequence of neutron absorption, has a half-life of 2.645 y, decaying by alpha emission (96.9%) and spontaneous fission (3.1%).

A small but important amount of spontaneous fission occurs in plutonium-240 in competition with alpha decay.

It may be surprising that the introduction of only 6.5 MeV of excitation energy can produce a reaction yielding as much as 200 MeV. The explanation is that the excitation triggers the separation of the two fragments and the powerful electrostatic force provides them a large amount of kinetic energy. By conservation of mass-energy, the mass of the nuclear products is smaller than the mass of the compound nucleus from which they emerge.

6.3. Byproducts of Fission

Accompanying the fission process is the release of several neutrons, which are all-important for the practical application to a self-sustaining chain reaction. The numbers that appear ν (nu) range from 1 to 7, with an average in the range 2 to 3, depending on the isotope and the bombarding neutron energy. For example, in U-235 with slow neutrons the average number ν is 2.42. Most of these are released instantly, the so-called prompt neutrons, whereas a small percentage, 0.65% for U-235, appear later as the result of radioactive decay of certain fission fragments. These delayed neutrons provide considerable inherent safety and controllability in the operation of nuclear reactors, as we will see later.

The nuclear reaction equation for fission resulting from neutron absorption in U-235 may be written in general form, letting the chemical symbols for the two fragments be labeled F¹ and F² to indicate many possible ways of splitting. Thus

The appropriate mass numbers and atomic numbers are attached. One example, in which the fission fragments are isotopes of krypton and barium, is

Mass numbers ranging from 75 to 160 are observed, with the most probable at approximately 92 and 144 as sketched in Figure 6.3. The ordinate on this graph is the percentage yield of each mass number (e.g., approximately 6% for mass numbers 90 and 144). If the number of fissions is given, the number of atoms of those types is 0.06 as large. Computer Exercise 6.B describes the program YIELD, which calculates the fission yield for several mass numbers.

Figure 6.3. Yield of fission products according to mass number

(Courtesy of T.R. England of Los Alamos National Laboratory).

As a collection of isotopes, these byproducts are called fission products. The isotopes have an excess of neutrons or a deficiency of protons compared with naturally occurring elements. For example, the main isotope of barium is , and a prominent element of mass 144 is . Thus there are seven extra neutrons or four too few protons in the barium isotope from fission, and it is highly unstable. Radioactive decay, usually involving several emissions of beta particles and delayed gamma rays in a chain of events, brings the particles down to stable forms. An example is

The hazard associated with the radioactive emanations from fission products is evident when we consider the large yields and the short half-lives.

The total energy from fission, after all of the particles from decay have been released, is approximately 200 MeV. This is distributed among the various processes as shown in Table 6.1. The prompt gamma rays are emitted as a part of fission; the rest are fission product decay gammas. Neutrinos accompany the beta particle emission, but because they are such highly penetrating particles, their energy cannot be counted as part of the useful thermal energy yield of the fission process. Thus only approximately 190 MeV of the fission energy is effectively available. However, several MeV of energy from gamma rays released from nuclei that capture neutrons can also be extracted as useful heat.

Table 6.1. Energy from Fission, U-235

	MeV
Fission fragment kinetic energy	166
Neutrons	5
Prompt gamma rays	7
Fission product gamma rays	7
Beta particles	5
Neutrinos	10
Total	200

The average total neutron energy is noted to be 5 MeV. If there are approximately 2.5 neutrons per fission, the average neutron energy is 2 MeV. When one observes many fission events, the neutrons are found to range in energy from nearly 0 to more than 10 MeV, with a most likely value of 0.7 MeV. Computer Exercise 6.C discusses calculation of the fission neutron energy distribution according to a semi-empirical formula. We note that the neutrons produced by fission are fast, whereas the cross section for the fission reaction is high for slow neutrons. This fact serves as the basis for the use of a reactor moderator containing a light element that permits neutrons to slow down, by a succession of collisions, to an energy favorable for fission.

Although fission is the dominant process, a certain fraction of the absorptions of neutrons in uranium merely results in radiative capture, according to

The U-236 is relatively stable, having a half-life of 2.34 × 10⁷ y. Approximately 14% of the absorptions are of this type, with fission occurring in the remaining 86%. This means that η (eta), the number of neutrons produced per absorption in U-235 is lower than ν, the number per fission. Thus by use of ν = 2.42, η is (0.86) (2.42) = 2.07. The effectiveness of any nuclear fuel is sensitively dependent on the value of η. We find that η is larger for fission induced by fast neutrons than that by slow neutrons.

The possibility of a chain reaction was recognized as soon as it was known that neutrons were released in the fission process. If a neutron is absorbed by the nucleus of one atom of uranium and one neutron is produced, the latter can be absorbed in a second uranium atom, and so on. To sustain a chain reaction as in a nuclear reactor or in a nuclear weapon, the value of η must be somewhat greater than 1 because of processes that compete with absorption in uranium, such as capture in other materials and escape from the system. The size of η has two important consequences. First, there is a possibility of a growth of neutron population with time. After all extraneous absorption and losses have been accounted for, if one absorption in uranium ultimately gives rise to, say, 1.1 neutrons; these can be absorbed to give (1.1) (1.1) = 1.21, which produce 1.331, etc. The number available increases rapidly with time. Second, there is a possibility of the use of the extra neutron, over and above the one required to maintain the chain reaction, to produce new fissile materials. “Conversion” involves the production of some new nuclear fuel to replace that used up, whereas “breeding” is achieved if more fuel is produced than is used.

Out of the hundreds of isotopes found in nature, only one is fissile, . Unfortunately, it is the less abundant of the isotopes of uranium, with weight percentage in natural uranium of only 0.711 in comparison with 99.3% of the heavier isotope . The two other most important fissile materials, plutonium-239 and uranium-233, are “artificial” in the sense that they are man-made by use of neutron irradiation of two fertile materials, uranium-238 and thorium-232, respectively. The reactions by which is produced are whereas those yielding are

The half-lives for decay of the intermediate isotopes are short compared with times involved in the production of these fissile materials. For many purposes, these decay steps can be ignored. It is important to note that although uranium-238 is not fissile, it can be put to good use as a fertile material for the production of plutonium-239, as long as enough free neutrons are available.

6.4. Energy from Nuclear Fuels

The practical significance of the fission process is revealed by calculation of the amount of uranium that is consumed to obtain a given amount of energy. Each fission yields 190 MeV of useful energy, which is also (190 MeV) (1.60 × 10⁻¹³J/MeV) = 3.04 × 10⁻¹¹ J. Thus the number of fissions required to obtain 1 W-sec of energy is 1/(3.04 × 10⁻¹¹) = 3.3 × 10¹⁰. The number of U-235 atoms consumed in a thermal reactor is larger by the factor 1/0.86 = 1.16 because of the formation of U-236 in part of the reactions.

In one day's operation of a reactor per megawatt of thermal power, the number of U-235 nuclei burned is

Then because 235 g corresponds to Avogadro's number of atoms 6.02 × 10²³, the U-235 weight consumed at 1 MW power is

In other words, 1.3 g of fuel is used per megawatt-day (MW/d) of useful thermal energy released. In a typical reactor, which produces 3000 MW of thermal power, the U-235 fuel consumption is approximately 4 kg/day. To produce the same energy by the use of fossil fuels such as coal, oil, or gas, millions of times as much weight would be required.

6.5. Summary

Neutron absorption by the nuclei of heavy elements gives rise to fission, in which heavy fragments, fast neutrons, and other radiations are released. Fissile materials are natural U-235 and the man-made isotopes Pu-239 and U-233. Many different radioactive isotopes are released in the fission process, and more neutrons are produced than are used, which makes possible a chain reaction and under certain conditions “conversion” and “breeding” of new fuels. Useful energy amounts to 190 MeV per fission, requiring only 1.3 g of U-235 to be consumed to obtain 1 MW/d of energy.

6.6. Exercises

1. Calculate the mass of the excited nucleus of plutonium-240 as the sum of the neutron mass 1.008665 and the Pu-239 mass 239.052157. How much larger is that sum than the mass of stable Pu-240, 240.053807? What energy in MeV is that?
2. If three neutrons and a xenon-133 atom are produced when a U-235 atom is bombarded by a neutron, what is the second fission product isotope?
3. The total kinetic energy of the fission fragments is 166 MeV. (a) What are the energies of each if the mass ratio is 3/2? (b) What are the two mass numbers if three neutrons were released in fission? (c) What are the velocities of the fragments?
4. Calculate the energy yield from the reaction with atomic masses 139.917277 for cesium and 91.919725 for rubidium.
5. The value of η for U-233 for thermal neutrons is approximately 2.30. By use of the cross sections for capture σ^c = 47 barns and fission σ^f = 530 barns, deduce the value of ν, the number of neutrons per fission.
6. A mass of 8000 kg of slightly enriched uranium (2% U-235, 98% U-238) is exposed for 30 days in a reactor operating at heat power 2000 MW. Neglecting consumption of U-238, what is the final fuel composition?
7. The per capita consumption of electrical energy in the United States is approximately 50 kWh/d. If this were provided by fission with 2/3 of the heat wasted, how much U-235 would each person use per day?
8. Calculate the number of kilograms of coal, oil, and natural gas that must be burned each day to operate a 3000-MW thermal power plant, which consumes 4 kg/d of uranium-235. The heats of combustion of the three fuels (in kJ/g) are, respectively, 32, 44, and 50.

Computer Exercises

1. The fission process can be visualized by the computer program FISSION. It shows a neutron approaching a fissionable nucleus and the fragments emerging. Run the program several times, noting the variety of speeds and directions of the particles.
2. Program YIELD calculates the fission yield for several prominent long-lived radionuclides and their precursors by a summing process. Run the program selecting several mass numbers near the peaks of 92 and 144.
3. Program SPECTRUM gives a simple formula for the way fission neutrons are distributed in energy, shows a graph of the distribution, and calculates properties of the curve. Run the program with the menus.

6.7 References

Basics of Nuclear Physics and Fission Basics of Nuclear Physics and Fission

www.ieer.org/reports/n-basics.html http://www.ieer.org/reports/n-basics.html

Decay, binding energy, fission Decay, binding energy, fission. By Institute for Energy and Environmental Research.

Lewin Sime, January 1998 Ruth Lewin Sime, Lise Meitner and the Discovery of Nuclear Fission Scientific American January 1998 Contributions of the scientist who should have received the Nobel Prize80-

Segrè, July 1998 Emilio Segrè, The Discovery of Nuclear Fission Physics Today July 199838-

Discovery of fission Discovery of fission

http://aip.org/history/mod/fission/fission1/03.html http://aip.org/history/mod/fission/fission1/03.html

Includes quotations from the scientists Includes quotations from the scientists. By American Institute of Physics.

Disintegration of Heavy Nuclei Disintegration of Heavy Nuclei

1939 article in 1939 article in Nature by Niels Bohr

Google: liquid drop model fission Google: liquid drop model fission

National Nuclear Data Center (NNDC) National Nuclear Data Center (NNDC)

http://www.nndc.bnl.gov/masses http://www.nndc.bnl.gov/masses

2003 atomic mass evaluation 2003 atomic mass evaluation. Latest and perhaps last.

The Isotope Project (Lawrence Berkeley National Laboratory) The Isotope Project (Lawrence Berkeley National Laboratory)

http://isotopes.lbl.gov http://isotopes.lbl.gov

Select Fission Home Page for data on fission product yields and spontaneous fission Select Fission Home Page for data on fission product yields and spontaneous fission.

Physical Science Resource Center Physical Science Resource Center

http://www.psrc-online.org http://www.psrc-online.org

Select Browse Resources/Modern Physics Select Browse Resources/Modern Physics. By American Association of Physics Teachers.

Lamarsh and Baratta, 2001 John R. Lamarsh, Anthony J. Baratta, Introduction to Nuclear Engineering 3rd Ed. 2001 Prentice-Hall Upper Saddle River, NJ Update of classic textbook

Bodansky, 2003 David Bodansky, Nuclear Energy: Principles, Practices, and Prospects 2nd Ed. 2003 Springer/AIP Press New York