In a 1940 Science survey, Enrico Fermi summarized the first quantitative radiochemical survey of uranium fission yields (work with H. L. Anderson and A. V. Grosse), which covered only the heavy fragment group and summed to about half of the expected 100% per-group yield, and listed what remained to be done: measure the light-group yield distribution, identify the missing heavy-group (especially rare-earth) products, improve measurement accuracy, achieve full yield accounting, and develop a theoretical explanation of the observed yield distribution and of the asymmetry of fission. The four experimental items are complete: fission-product yields for all mass chains, including rare-earth products such as the later-discovered promethium, are now measured and maintained in evaluated nuclear data libraries normalized so that chain yields sum to two fragments per fission. The theoretical item survives as the open core. The asymmetry is qualitatively attributed to quantum shell effects of the nascent fragments (spherical shells near doubly magic tin-132, a nucleus whose proton and neutron numbers are both closed, plus deformed and octupole 'pear-shaped' shells at Z = 52-56), and macroscopic-microscopic and density-functional calculations reproduce the main features of measured mass and charge yield curves. However, a complete quantitative, predictive theory of the fission-fragment yield distribution - reproducing widths, fine structure, odd-even effects and systematics across actinide systems and incident neutron energies from nuclear dynamics - is explicitly described by the field as not yet achieved, with active work and unresolved systematic discrepancies in literature through 2026.

Background

Nuclear fission, the division of a heavy nucleus into two fragments of comparable mass, was discovered in 1938-1939 through the identification of barium among the products of neutron-irradiated uranium (Hahn and Strassmann 1939) and was interpreted as the breakup of a charged liquid drop (Meitner and Frisch 1939). In the compound-nucleus picture introduced by Bohr, the absorbed neutron is first assimilated into the target nucleus, forming a short-lived, excited system that then decays; Bohr and Wheeler's liquid-drop theory (1939) successfully accounted for fission energetics and cross sections, but its surface-energy competition favors nearly equal mass divisions and gave no explanation for the observed asymmetric split.

In slow-neutron fission of uranium the fragments fall into two groups: a light group centered near mass number 95 and a heavy group near 140. (Throughout, A denotes the mass number, the total count of protons and neutrons, and Z the proton number.) Each primary fragment is radioactive and decays by a chain of beta emissions toward stability, so a fragment's identity is a fission-product mass chain. The yield of a chain is the percentage of fissions producing a fragment in that chain; because every fission produces one light and one heavy fragment, the chain yields within each group should sum to 100 percent.

The source of this problem is Fermi's September 1940 Science survey 'Reactions Produced by Neutrons in Heavy Elements', which reported the first quantitative yield survey (with Anderson and Grosse; companion paper: 'Branching Ratios in the Fission of Uranium (235)', Phys. Rev. 59, 52 (1941)). Using irradiations with the Columbia cyclotron, chemical separations and activity counting, the survey assigned heavy-group chain yields ranging from about 0.1 to just over 10 percent. The measured heavy-group sum reached only about half of the expected total, attributed to considerable experimental errors and to incomplete chemical investigation, with further heavy-group products, possibly rare earths (the metallic elements around Z = 57-71), expected to await discovery. The survey's unfinished agenda - light-group yields, missing rare-earth products, accuracy, the 100 percent accounting, and a theoretical explanation of the yield distribution and of the asymmetry of fission - is the subject of this record.

The experimental agenda was completed over the following decades. Promethium (element 61), a rare-earth fission product of the anticipated kind, was chemically identified among uranium fission products in 1947 (Marinsky, Glendenin and Coryell). Modern radiochemical and mass-spectrometric programs measure yields for essentially all chains and neutron fields, and the results are maintained as evaluated data libraries (England and Rider's compilation; ENDF/B and JEFF), which supply recommended yields with uncertainties and enforce the normalization that chain yields sum to two fragments per fission.

The theoretical agenda is formulated today as follows. The yield distribution is a curve of yield versus fragment mass (and charge), double-humped for low-energy fission of the actinides (the radioactive elements from actinium upward, Z = 89-103). The accepted qualitative explanation of the asymmetry invokes shell effects: the extra binding that nuclei acquire near closed shells ('magic' proton or neutron numbers), including deformed shells in the fragments and octupole (pear-shaped) correlations, which stabilize asymmetric scission configurations. Quantitative theories must derive the distribution from nuclear dynamics. The main frameworks are: the macroscopic-microscopic method, which combines a liquid-drop macroscopic energy with microscopic single-particle (shell and pairing) corrections, usually paired with Brownian or Langevin shape dynamics - a stochastic random walk of the nuclear shape across the deformation energy surface until scission, the configuration at which the nucleus splits; and density-functional methods, which map the energy surface with Hartree-Fock-Bogoliubov mean-field calculations (the self-consistent mean-field method of nuclear structure, which includes nucleon pairing) and propagate the dynamics with time-dependent extensions (time-dependent Hartree-Fock, i.e. real-time mean-field propagation; the superfluid local density approximation, its pairing-including analogue; or the generator coordinate method, which builds a collective wave packet from mean-field states). Computed primary (pre-neutron) yields must be corrected through a model of prompt neutron evaporation to compare with the post-neutron yields that experiments report. Semi-empirical models such as the GEF code (named from 'general description of fission observables') instead describe fission observables with adjusted parameters.

Reviews of the modern field (Schunck and Robledo 2016; Schmidt and Jurado 2018; Bender et al. 2020; Stetcu et al. 2021) document the present state: the main features of measured fragment distributions are reproduced by several approaches, but a complete microscopic, predictive theory of fission-fragment yields - one of the field's stated grand challenges - remains under construction.

Problem Statement

The source question, from Fermi's 1940 Science survey of neutron-induced reactions in heavy elements, is a five-part agenda left open by the first quantitative radiochemical survey of uranium fission yields: (i) measure the complete yield distribution of the light fragment group; (ii) identify and measure the missing heavy-group elements, especially in the rare-earth region; (iii) improve the accuracy of the yield measurements; (iv) complete the accounting so that the yields within each fragment group sum to 100 percent; and (v) develop a theoretical explanation for the observed yield distribution and for the asymmetry of fission.

Items (i)-(iv) were achieved by the experimental nuclear-chemistry and nuclear-data programs of the following decades: complete, evaluated fission-product yield distributions exist for both fragment groups, including all rare-earth chains, with uncertainties and enforced mass balance. The problem that remains open is item (v) in its modern quantitative form: construct and validate a theory that explains the observed fission-fragment yield distribution in low-energy neutron-induced fission of the actinides (the radioactive elements from actinium upward) - the double-humped asymmetric mass division with its light group near mass 95 and heavy group near mass 140, the widths of the peaks, the fine structure (structure on scales of one or two mass units superimposed on the broad peaks) and odd-even effects (the systematic enhancement of yields for fragments with even proton or neutron numbers) of the yield curves, and their systematics across fissioning systems and incident neutron energies - and that predicts yields for systems and conditions not used in constructing the theory. The asymmetry must emerge in the theory as a consequence of its mechanism, not enter as an input or as a fit to the measured distributions. The theory need not take a prescribed mathematical form, but it must be quantitative. Its computed yields must reproduce, within the uncertainties quoted for them, the ENDF/B-VII.1 evaluated fission-product yields for the two benchmark systems, thermal-neutron-induced fission of 235U and of 239Pu, and must predict, within the same quoted uncertainties and without having been fitted to them, the corresponding ENDF/B-VII.1 yields of the held-out systems thermal-neutron-induced fission of 233U and fission-spectrum-neutron-induced fission of 238U. The named systems are the concrete demonstration of the theory; the broader systematics across fissioning systems and incident neutron energies are the scope that demonstration represents.

References

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