** **

** **

**Macroscopic-microscopic approaches**

** **

** **

** **

**Reports available from OSTI’s Information Bridge**

The main influences on the splitting of atoms were understood in at least a rough way almost as soon as nuclear fission was discovered in 1938. An atom’s nucleus is composed of two kinds of nucleons: protons, which all have equal amounts of positive electric charge and thus repel each other, and neutrons, which have no net electric charge and are thus not affected electrically by the protons or each other. If internal electrostatic forces were the only ones that influenced nucleons’ motions, the protons would push each other apart and the neutrons wouldn’t disturb each other at all, so every nucleus would come apart as soon as it formed. But protons and neutrons are also affected by what’s known simply as the “strong force”. Unlike electricity, which acts between charged objects no matter how close together or far apart they get (though it does become weaker as the objects separate), particles only interact through the strong force once they’ve practically come into contact—separated by about a femtometer (10^{-15} m)^{[}^{Wikipedia}^{]} or less. So while electricity pushes protons apart, the strong force pulls contiguous protons and neutrons together. A multi-nucleon nucleus won’t form if it contains only protons: the protons’ electrical repulsion is too strong for the strong force to hold them together. But if there are enough neutrons among a given number of protons, the strong forces can overcome the protons’ repulsion and the nucleons will cohere.

But some stable nuclei can be made unstable enough to split apart. If an energetic particle, such as another nucleon, collides with a nucleus, and the new nucleon’s energy is distributed by the collision to the other nucleons so that they move more rapidly and get far enough apart, the strong force’s attraction may be lessened enough that the protons’ mutual repulsion will drive the nucleus into two (or sometimes more) smaller pieces. Thanks to their protons’ positive charges, the pieces keep repelling each other after they separate, so they fly apart at high speed.

A few other facts are relevant to what happens next to the repelling pieces. To remain within the nucleus, a nucleon can only move so fast in any given direction without escaping the attraction of the other nucleons. In particular, if the nucleon travels along the nuclear surface, the attraction can keep it attached to the surface unless the nucleon surpasses a maximum speed and escapes. This maximum speed will be larger for larger nuclei with a larger turning radius. But an equally energetic nucleon in a smaller nucleus will have less room to turn and will fly out. This can happen with high-energy neutrons from large nuclei once the nuclei split apart. If the smaller nuclei that result from the split aren’t big enough to contain the fastest neutrons, and the neutrons don’t somehow lose enough energy first, those neutrons will escape from the smaller nuclei. The smaller nuclei may also undergo other reactions after the split to produce energetic neutrons that fly away.

If the split atom is one of many similar atoms in a large piece of material, the escaping neutrons may in turn collide with the nuclei of some of the other atoms, causing them in turn to split into mutually repelling smaller nuclei, which may release other neutrons which cause still other large nuclei to split … thus producing a chain reaction. And this chain reaction will produce a lot of heat, in the form of the kinetic energy of all the split pieces pushing each other apart and banging into other atoms themselves—not hard enough to split the other atoms, but enough to spread their kinetic energy around from atom to atom and warm the material up.

Although this picture points out the main features of nuclear fission, it’s not detailed enough to correctly describe exactly how fission reactions proceed, or under what conditions. For one thing, its precise mathematical expression suggests, in the absence of further details, that the smaller nuclei produced by fission of a larger one are most likely to be of equal size. Fission is actually more likely to produce smaller nuclei of very different sizes, one made of about 140 nucleons (before it releases neutrons) and another made of the remaining nucleons.^{[}^{Reference}^{, pp. 152-155]} If enough details are added to the picture to correct such errors, the picture becomes complex enough that deducing what is most likely to happen during fission requires a great deal of computing.

In some respects, nuclei are significantly more complex than, say, our solar system. The motions of the planets are mainly influenced by the sun’s gravity, and to a much smaller degree by each other’s gravity, so accounting for those two influences will suffice to accurately calculate where each planet will be for many centuries into the future. Other influences exist, such as the gravity of other stars light-years away from the solar system, but these are too small to make much difference in how the planets move except over a very long time span. Similarly, such details about the planets as their size and shape have little effect on how they move or influence each other’s motion.

Not so with an atomic nucleus, especially a large one that can be made to release energy by fission. Such nuclei have dozens of mutually and equally attracting nucleons and repelling protons, not just one huge attracting body like the sun or even a few secondary attractors like planets. And while the electrostatic repulsive force depends on electric charge and distance according to a known law analogous to that of gravity’s dependence on mass and distance, the law of the strong force between nucleons is more complex, depends on more variables (such as the nucleons’ relative orientations), and is not even completely known. In addition, the quantum-physical wavelike properties of nucleons significantly affect their interactions, and accounting for these properties involves more mathematical detail than keeping up with where a planet’s center of mass is at any given time.

Despite the complexity, people are still highly motivated to achieve a precise and accurate understanding of how nuclear fission works because of its potential uses, some of which are listed in a 2010 report from Los Alamos National Laboratory and France’s Atomic Energy and Alternative Energies Commission (CEA), which says:

More recently, a renewed interest in this topic has emerged with the resurgence of advanced nuclear reactors and fuels, which involve fast neutron spectra and innovative fuel compositions. Indeed, the next generation of reactors would have to produce (much) more electricity than conventional reactors from the same amount of uranium ore, reduce significantly the amount of nuclear waste produced during the normal operation of the reactors, while maintaining the highest level of safety and security. Accurate fission cross sections^{([}^{Wikipedia}^{;}^{HyperPhysics}^{]—wnw)} for many nuclei, not all observed experimentally, are needed, and only advances in theory and modeling can tackle this challenge.^{[}^{SciTech Connect}^{]}

Recent reports of research sponsored by the Department of Energy show at least two somewhat different general approaches to reckoning with fission’s inherent complexity. One type of approach, known as the “macroscopic-microscopic” method, is to start with a “macroscopic” picture like the one described above that roughly describes the nucleus as a whole with little reference to the individual nucleons’ properties, and then add some “microscopic” details about the behavior of the nucleons as corrections. (In this context, “macroscopic” and “microscopic” are relative terms: in terms of human vision, nuclei and their nucleons are both quite *sub*microscopic, but a nucleus is “macroscopic” compared to the relatively “microscopic” nucleons that it’s made of.) Successive refinements to this approach over the years, particularly those made feasible by advances in computer power, have allowed more and more features of nuclei to be added in so that the resulting pictures of fission agree in increasing detail with experiments, thus increasing researchers’ confidence that the picture might also be accurate for fission reactions yet to be observed, such as those in new reactor designs. In the other type of approach, the “microscopic” method, the nucleus is described in terms of its component protons and neutrons right from the start. However, this approach also involves taking account of certain nuclear phenomena first and adding in others later, as corrections, as advances in computer capabilities make such accounting more practical. Microscopic approaches have become increasingly accurate too.

**Macroscopic-microscopic approaches**

According to most mathematical models^{[}^{Wikipedia}^{]} of nuclear fission, a splitting nucleus changes its shape before it divides. The forces acting on the different parts of the nucleus during this process depend on the nuclear shape and are affected by it. The report on fission theory from Los Alamos National Laboratory and CEA mentioned above (“Recent Advances in Nuclear Fission Theory: Pre- and Post-Scission Physics”^{[}^{SciTech Connect}^{]}) describes a macroscopic-microscopic approach to calculating these forces. A purely macroscopic calculation suggests that a nucleus, as it proceeds toward division, will resist its own elongation more and more, until it lengthens enough for its protons’ mutual repulsion to overcome its nucleons’ mutual attraction and the nucleus splits. Microscopic corrections to this picture suggest that this resistance to elongation doesn’t increase smoothly up to the point of fission, but actually reverses to promote the nucleus’ elongating further for a while, and then changes back to resisting even more elongation until the nucleus does split. Exactly which shapes of the nucleus are associated with forces that push it toward more or less elongation determine how likely it is that the nucleus will actually split. The report describes what preliminary calculations of this sort imply about fission in plutonium isotopes^{[}^{Wikipedia}^{]} 239, 240, 241, and 242, as well as the number and energies of neutrons emitted by fragments from the fission of uranium-235.

As the authors of this report note, “… modern evaluations of nuclear fission data still rely heavily on experimental data, as fission theory and modeling capabilities still remain at the qualitative level for the most part.” Their report, while itself “not tackling the full complexity of the fission problem” is meant to demonstrate progress toward a comprehensive evaluation tool for fission data that would show how different data are related. A later slide presentation by an overlapping set of authors from Los Alamos, the University of New Mexico, Colorado School of Mines, and Lawrence Livermore National Laboratory, “SPIDER: A Predictive Theory for Fission”^{[}^{SciTech Connect}^{]}, describes a new instrument for measuring the energy and velocity of fission-fragment pairs (SPIDER—SPectrometer for Ion DEtermination in fission Research), as well as the extension of an existing mathematical model of fission that will incorporate the new instrument’s measurement data to approximate how fission proceeds in any given nucleus.

The mathematical model is again of the “macroscopic-microscopic” type. The approach to describing the dynamics of a nucleus undergoing fission is to calculate how the forces affecting the nucleus’ overall or “macroscopic” shape depend mainly on that shape and to a lesser degree on details of “microscopic” individual-nucleon behavior, and then calculate how the nucleus’ shape can change as a result of those forces. The shape is described in terms of between 5 and 8 parameters such as its elongation (more precisely, its quadrupole moment^{[Wikipedia;}^{Wikipedia}^{]}), the mass asymmetry of its large and small portions, the portions’ deformation, and where along its length the nucleus begins to divide. In the end, the mathematical model is to relate, for any nucleus, the energy of a fission-inducing neutron to how often each possible combination of fission-fragment charges, masses, kinetic energies, excitation energies, and numbers of released neutrons will result. This model is also a work in progress, and the presentation describes refinements planned for this fiscal year and the next. The planned refinements include matching the model’s description of nuclear mechanics and thermodynamics to relevant data from experiments, and extending the model to predict fission fragments’ electric charges as well as their masses. The investigators also plan to apply the model to more isotopes than uranium-235, uranium-238, and plutonium-239.

Part of the interest in combined macroscopic-microscopic models of fission is that making predictions from existing *microscopic* models requires tens or hundreds of thousands of times more computer time. Yet existing macroscopic-microscopic models have their limitations, as described in the 2007 Lawrence Livermore report “Microscopic Theory of Fission”.^{[}^{SciTech Connect}^{]} One significant problem is that, when fission is induced by high-energy neutrons, the nucleus’ shape cannot be adequately represented in the classical-physics terms used in macroscopic descriptions. While the shape of a nucleus is well described by a single mathematical function of time—the situation presupposed by classical physical theory—when its fission is induced by absorbing a low-energy neutron, a single function of time does not adequately describe the changing shape when fission is induced by a high-energy neutron. For any one function that does partially describe such a nucleus, additional functions of time also have a significant nonzero probability of describing how the nucleus develops. Accounting for multiple functions that each have significant probabilities of describing the same physical system is characteristic of quantum theory.

When high-energy neutrons induce fission, the microscopic individual-nucleon interactions are not simply minor corrections to macroscopic whole-nucleus effects. So a mathematical model that describes fission directly in terms of individual-nucleon interactions, and accounts for multiple state functions at the outset, is expected to be more accurate than a model which treats the nucleus classically and only considers multiple functions when adding individual-nucleon interactions as microscopic corrections. The authors also point out that what happens in fission has a sensitive dependence on microscopic details: for example, the nuclei of fermium-256 and fermium-258 differ by only two neutrons, or about 0.8% of their mass, yet fermium-256 has a spontaneous-fission half-life over 25,000,000 longer than that of fermium-258.^{[}^{It’s Elemental}^{] }The 2007 conference report describes the prediction of hot and cold fission modes and realistic fission-time calculations as recent successes of the microscopic approach.

Some further micrsocopic analyses by the same authors are presented in a mid-2009 workshop report (“The Microscopic Theory of Fission”^{[}^{SciTech Connect}^{]}), a report from Fall 2012 (“Fragment Yields Calculated in a Time-Dependent Microscopic Theory of Fission”^{[}^{SciTech Connect}^{]}), and (with an additional coauthor) a Fall 2012 conference report (“A microscopic theory of low energy fission: fragment properties”^{[}^{SciTech Connect}^{]}). In the first of these reports, the authors describe how a nuclear-shape parameter and the fragment interaction energy were both examined as possible criteria for defining exactly when a nucleus is splitting, with the interaction energy proving more useful; the mathematical model described therein also approximated how frequently different-sized fission fragments are produced, and how many neutrons each fragment releases. In the second report, the authors show how the nuclear fragments’ relative mass-number difference (the same as the “asymmetry parameter” of the SPIDER collaboration report mentioned above^{[}^{SciTech Connect}^{]}) and the fragments’ separation distance turn out to be more relevant for describing fission than the nuclei’s quadrupole and octupole^{[}^{Wikipedia}^{]} moments do, resulting in more accurate predictions of how many resulting fragments will contain a given number of nucleons. The third report describes predictions of the fragments’ kinetic excitation energies that have “reasonable agreement with experiment” but with remaining discrepancies, especially in the excitation-energy calculations.

Given that we have so much left to learn about exactly how nuclei split, how did our present nuclear technology ever become possible in the first place?

We’re somewhat in the position of our forebears in the 18th century, when the modern steam engine was being developed. At first, people had only a rudimentary understanding of steam pressure and its ability to drive machinery, but that was enough to design practical engines and, with further experience, even invent improvements. Early steam technology, in fact, provided something else that raw nature didn’t besides a new power source: it also exhibited thermal processes that were simpler and easier to figure out than natural ones. So whereas Newton was able to approach (and Einstein to refine) the law of gravity largely from observing fairly simple natural phenomena like falling objects and planetary motion, it was the steam engine’s relatively straightforward conversion of heat into mechanical work, not the more complex natural phenomena of weather or metabolism, that gave clear-enough clues for the laws of thermodynamics to be induced. Once that was achieved, not only was it possible to improve the steam engine further, but a more intelligent development of all technologies that involve the transfer of heat and its interconversion with other forms of energy became feasible as well.

The first nuclear reactors were designed when people knew few details of fission beyond how likely it is that a neutron with energy *E* will lose a given fraction of that energy, be absorbed, or set off fission when it collides with a given nucleus, and with what likelihood *N* neutrons of energies *E _{1}*,

*Wikipedia*

- Femtometre
- Cross section (physics): Nuclear physics
- Mathematical model
- Isotope
- Quadrupole moment; Mathematical definition; Generalization: Higher multipoles [example: octupole moment]
- Potential energy; Nuclear potential energy

*Models of the Atomic Nucleus*by Norman D. Cook, © 2006 Springer-Verlag- Nuclear Cross Section (From
*HyperPhyiscs*) - Isotopes of the Element Fermium (From
*It’s Elemental*at Thomas Jefferson National Accelerator Facility)

- Los Alamos National Laboratory
- CEA (Commissariat à l'énergie atomique et aux énergies alternatives)
- University of New Mexico
- Colorado School of Mines
- Lawrence Livermore National Laboratory

**Reports Available through OSTI’s SciTech Connect**

- “Recent advances in nuclear fission theory: pre- and post-scission physics” [Abstract and full text via OSTI’s SciTech Connect]
- “SPIDER: A Predictive Theory for Fission” [Abstract and full text via OSTI’s SciTech Connect]
- “Microscopic Theory of Fission” [Abstract and full text via OSTI’s SciTech Connect]
- “The Microscopic Theory of Fission” [Abstract and full text via OSTI’s SciTech Connect]
- “Fragment Yields Calculated in a Time-Dependent Microscopic Theory of Fission” [Abstract and full text via OSTI’s SciTech Connect]”
- “A microscopic theory of low energy fission: fragment properties” [Abstract and full text via OSTI’s SciTech Connect]”

Prepared by Dr. William N. Watson, Physicist

DoE Office of Scientific and Technical Information