14.14 Fission

In spontaneous fission, a very heavy nucleus falls apart into big fragments. If there are two fragments, it is called binary fission. In some cases, there are three fragments. That is called ternary fission; the third fragment is usually an alpha particle. This section summarizes some of the basic ideas.

14.14.1 Basic concepts

What makes fission energetically possible is that very heavy nuclei have less binding energy per nucleon than those in the nickel/iron range, 14.2. The main culprit is the Coulomb repulsion between the protons. It has a much longer range than the nuclear attractions between nucleons. Therefore, Coulomb repulsion disproportionally increases the energy for heavy nuclei. If a nucleus like uranium-238 divides cleanly into two palladium-119 nuclei, the energy liberated is on the order of 200 MeV. That is obviously a very large amount of energy. Chemical reactions produce maybe a few eV per atom.

The liquid drop model predicts that the nuclear shape will become unstable at $Z^2$$\raisebox{.5pt}{$/$}$$A$ about equal to 48. However, only the weirdest nuclei like $\fourIdx{293}{118}{}{}{\rm {Ei}}$ come close to that value. Below $Z$ $\vphantom0\raisebox{1.5pt}{$=$}$ 100 the nuclei that decay primarily through spontaneous fission are curium-250, with $Z^2$$\raisebox{.5pt}{$/$}$$A$ equal to 37 and a half life of 8,300 years, californium-254, 38 and two months, and fermium-256, 39 and less than 3 hours.

Indeed, while the fission products may have lower energy than the original nucleus, in taking the nucleus apart, the nuclear binding energy must be provided right up front. On the other hand the Coulomb energy gets recovered only after the fragments have been brought far apart. As a result, there is normally a energy barrier that must be crossed for the nucleus to come apart. That means that an “activation energy” must be provided in nuclear reactions, much like in most chemical reactions.

For example, uranium has an activation energy of about 6.5 MeV. By itself, uranium-235 will last a billion years or so. However, it can be made to fission by hitting it with a neutron that has only a thermal amount of energy. (Zero is enough, actually.) When hit, the nucleus will fall apart into a couple of big pieces and immediately release an average of 2.4 “prompt neutrons.” These new neutrons allow the process to repeat for other uranium-235 nuclei, making a “chain reaction” possible.

In addition to prompt neutrons, fusion processes may also emit a small fraction of “delayed neutrons” neutrons somewhat later. Despite their small number, they are critically important for controlling nuclear reactors because of their slower response. If you tune the reactor so that the presence of delayed neutrons is essential to maintain the reaction, you can control it mechanically on their slower time scale.

Returning to spontaneous fission, that is possible without the need for an activation energy through quantum mechanical tunneling. Note that this makes spontaneous fission much like alpha decay. However, as section 14.11.2 showed, there are definite differences. In particular, the basic theory of alpha decay does not explain why the nucleus would want to fall apart into two big pieces, instead of one big piece and a small alpha particle. This can only be understood qualitatively in terms of the liquid drop model: a charged classical liquid drop is most unstable to large-scale deformations, not small scale ones, subsection 14.13.1.

14.14.2 Some basic features

While fission is qualitatively close to alpha decay, its actual mechanics is much more complicated. It is still an area of much research, and beyond the scope of this book. A very readable description is given by [35]. This subsection describes some of the ideas.

From a variety of experimental data and their interpretation, the following qualitative picture emerges. Visualize the nucleus before fission as a classical liquid drop. It may already be deformed, but the deformed shape is classically stable. To fission, the nucleus must deform more, which means it must tunnel through more deformed states. When the nucleus has deformed into a sufficiently elongated shape, it becomes energetically more favorable to reduce the surface area by breaking the connection between the ends of the nucleus. The connection thins and eventually breaks, leaving two separate fragments. During the messy process in which the thin connection breaks an alpha particle could well be ejected. Now typical heavy nuclei contain relatively more neutrons than lighter ones. So when the separated fragments take inventory, they find themselves overly neutron-rich. They may well find it worthwhile to eject one or two right away. This does not change the strong mutual Coulomb repulsion between the fragments, and they are propelled to increasing speed away from each other.

Consider now a very simple model in which a nucleus like fermium-256 falls cleanly apart into two smaller nuclear fragments. As a first approximation, ignore neutron and other energy emission in the process and ignore excitation of the fragments. In that case, the final kinetic energy of the fragments can be computed from the difference between their masses and the mass of the original nucleus.

In the fission process, the fragments supposedly pick up this kinetic energy from the Coulomb repulsion between the separated fragments. If it is assumed that the fragments are spherical throughout this final phase of the fission process, then its properties can be computed. In particular, it can be computed at which separation between the fragments the kinetic energy was zero. That is important because it indicates the end of the tunneling phase. Putting in the numbers, it is seen that the separation between the fragments at the end of tunneling is at least 15% more than that at which they are touching. So the model is at least reasonably self-consistent.

Figure 14.28: Simplified energetics for fission of fermium-256. [pdf]
...0,0)[t]{$\fourIdx{132}{50}{}{}{\rm Sn}$}}

Figure 14.28 shows the energetics of this model. Increasing redness indicates increasing energy release in the fission. Also, the spacing between the squares indicates the spacing between the nuclei at the point where tunneling ends. Note in particular the doubly magic point of 50 protons and 82 neutrons. This point is very neutron rich, just what is needed for fission fragments. And because it is doubly magic, nuclei in this vicinity have unusually high binding energy, as seen from figure 14.7. Indeed, nuclei with 50 protons are seen to have the highest fission energy release in figure 14.28. Also, they have the smallest relative spacing between the nuclei at the end of tunneling, so likely the shortest relative distance that must be tunneled through. The conclusion is clear. The logical thing for fermium-256 to do is to come apart into two almost equal fragments with a magic number of 50 protons and about 78 neutrons, giving the fragments a mass number of 128. Less plausibly, one fragment could have the magic number of 82 neutrons, giving fragment mass numbers of 132 and 124. But the most unstable deformation for the liquid drop model is symmetric. And so is a spheroidal or ellipsoidal model for the deformed nucleus. It all seems to add up very nicely. The fragments must be about the same size, with a mass number of 128.

Except that that is all wrong.

Fermium 258 acts like that, and fermium-257 also mostly, but not fermium 256. It is rare for fermium-256 to come apart into two fragments of about equal size. Instead, the most likely mass number of the large fragment is about 140, with only a small probability of a mass number 132 or lower. A mass number of 140 clearly does not seem to make much sense based on figure 14.28.

The precise solution to this riddle is still a matter of current research, but physicists have identified quantum effects as the primary cause. The potential energy barrier that the fissioning nucleus must pass through is relatively low, on the order of say 5 MeV. That is certainly small enough to be significantly affected by quantum shell effects. Based on that idea, you would expect that mass asymmetry would decrease if the excitation energy of the nucleus is increased, and such an effect is indeed observed. Also, the separation of the fragments occurs at very low energy, and is believed to be slow enough that the fragments can develop some shell structure. Physicists have found that for many fissioning nuclei, quantum shell effects can create a relatively stable intermediate state in the fission process. Such a state produces resonances in response to specific excitation energies of the nucleus. Shell corrections can also lower the energy of asymmetric nuclear fissioning shapes below those of symmetric ones, providing an explanation for the mass asymmetry.

Imagine then a very distorted stage in which a neutron-rich, doubly magic 50/82 core develops along with a smaller nuclear core, the two being connected by a cloud of neutrons and protons. Each could pick up part of the cloud in the final separation process. That picture would explain why the mass number of the large fragment exceeds 132 by a fairly constant amount while the mass number of the smaller segment varies with the initial nuclear mass. Whether or not there is much truth to this picture, at least it is a good mnemonic to remember the fragment masses for the nuclei that fission asymmetrically.