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Reports on Progress in Physics REVIEW Microscopic theory of nuclear fission: a review To cite this article: N Schunck and L M Robledo 2016 Rep. Prog. Phys. 79 116301 Manuscript version: Accepted Manuscript Accepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an ‘Accepted Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors” This Accepted Manuscript is © © 2016 IOP Publishing Ltd. During the embargo period (the 12 month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after the 12 month embargo period. After the embargo period, everyone is permitted to use copy and redistribute this article for non-commercial purposes only, provided that they adhere to all the terms of the licence https://creativecommons.org/licences/by-nc-nd/3.0 Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. View the article online for updates and enhancements. This content was downloaded from IP address 170.106.202.8 on 01/10/2021 at 21:11 Page 1 of 111 AUTHOR SUBMITTED MANUSCRIPT - ROPP-100585.R2 1 2 3 4 5 6 7 Review Article 8 9 10 Microscopic Theory of Nuclear Fission: A Review 11 12 13 N Schunck1, L M Robledo2 14 1 Nuclear and Chemical Science Division, Lawrence Livermore National Laboratory, 15 Livermore, CA 94551, USA 16 17 E-mail: [email protected] 18 2 Departamento de F´ısicaTe´orica,Universidad Aut´onomade Madrid, E-28049 19 Madrid, Spain 20 21 November 2015 22 23 Abstract. This article reviews how nuclear fission is described within nuclear density 24 25 functional theory. A distinction should be made between spontaneous fission, where 26 half-lives are the main observables and quantum tunnelling the essential concept, 27 and induced fission, where the focus is on fragment properties and explicitly time- 28 dependent approaches are often invoked. Overall, the cornerstone of the density 29 functional theory approach to fission is the energy density functional formalism. The 30 basic tenets of this method, including some well-known tools such as the Hartree-Fock- 31 Bogoliubov (HFB) theory, effective two-body nuclear potentials such as the Skyrme 32 and Gogny force, finite-temperature extensions and beyond mean-field corrections, are 33 presented succinctly. The energy density functional approach is often combined with 34 35 the hypothesis that the time-scale of the large amplitude collective motion driving 36 the system to fission is slow compared to typical time-scales of nucleons inside the 37 nucleus. In practice, this hypothesis of adiabaticity is implemented by introducing 38 (a few) collective variables and mapping out the many-body Schr¨odingerequation 39 into a collective Schr¨odinger-like equation for the nuclear wave-packet. The region 40 of the collective space where the system transitions from one nucleus to two (or 41 more) fragments defines what are called the scission configurations. The inertia tensor 42 that enters the kinetic energy term of the collective Schr¨odinger-like equation is one 43 of the most essential ingredient of the theory, since it includes the response of the 44 45 system to small changes in the collective variables. For this reason, the two main 46 approximations used to compute this inertia tensor, the adiabatic time-dependent 47 Hartree-Fock-Bogoliubov and the generator coordinate method, are presented in detail, 48 both in their general formulation and in their most common approximations. The 49 collective inertia tensor enters also the WKB formula used to extract spontaneous 50 fission half-lives from multi-dimensional quantum tunnelling probabilities (For the sake 51 of completeness, other approaches to tunnelling based on functional integrals are also 52 briefly discussed, although there are very few applications.) It is also an important 53 component of some of the time-dependent methods that have been used in fission 54 55 studies. Concerning the latter, both the semi-classical approaches to time-dependent 56 nuclear dynamics as well as more microscopic theories involving explicit quantum- 57 many-body methods are presented. One of the trademarks of the microscopic theory 58 of fission is the tremendous amount of computing needed for practical applications. 59 In particular, the successful implementation of the theories presented in this article 60 requires a very precise numerical resolution of the HFB equations for large values of the collective variables. This aspect is often overlooked, and several sections are AUTHOR SUBMITTED MANUSCRIPT - ROPP-100585.R2 Page 2 of 111 1 2 3 Microscopic Theory of Nuclear Fission: A Review 2 4 5 devoted to discussing the resolution of the HFB equations, especially in the context 6 of very deformed nuclear shapes. In particular, the numerical precision and iterative 7 methods employed to obtain the HFB solution are documented in detail. Finally, a 8 selection of the most recent and representative results obtained for both spontaneous 9 and induced fission is presented with the goal of emphasizing the coherence of the 10 microscopic approaches employed. Although impressive progress has been achieved 11 over the last two decades to understand fission microscopically, much work remains to 12 13 be done. Several possible lines of research are outlined in the conclusion. 14 15 16 PACS numbers: 25.85.-w,25.85.Ca,25.85.Ec,21.60.Jz,21.60.Ev 17 18 19 20 Keywords: Fission, Density functional theory, Hartree-Fock-Bogoliubov, Generator 21 Coordinate Method, Scission, Adiabaticity, Large amplitude collective motion, Time- 22 dependent density functional theory, Fission product yields, TKE, TXE. 23 24 25 26 Submitted to: Rep. Prog. Phys. 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Page 3 of 111 AUTHOR SUBMITTED MANUSCRIPT - ROPP-100585.R2 1 2 3 Microscopic Theory of Nuclear Fission: A Review 3 4 5 1. Introduction 6 7 Experiments on the bombardment of Uranium atoms (charge number Z = 92) with 8 9 neutrons performed by O. Hahn and F. Strassmann in 1938-1939 and published in [1, 2] 10 showed that lighter elements akin to Barium (Z = 56) were formed in the reaction. 11 In February 1939, this observation was explained qualitatively by L. Meitner and O.R. 12 13 Frisch in [3] as caused by the disintegration of the heavy Uranium element into lighter 14 fragments. This tentative explanation was based on the liquid drop model of the nucleus 15 that had been introduced a few years earlier in [4] by N. Bohr. A few months after 16 17 these results, N. Bohr himself, together with J.A. Wheeler formalized and quantified 18 Meitner's arguments in their seminal paper [5]. They described fission as the process 19 20 during which an atomic nucleus can deform itself up to the splitting point as a result 21 of the competition between the nuclear surface tension that favours compact spherical 22 shapes and the Coulomb repulsion among protons that favour very elongated shapes 23 24 to decrease the repulsion energy. They introduced the concepts of compound nucleus, 25 saddle point (the critical deformation beyond which the nuclear liquid drop is unstable 26 against fission) and fissility (which captures the ability of a given nucleus to undergo 27 28 fission), provided estimates of the energy release during the process, of the dependence 29 of the fission cross-section on the energy of incident particles, etc.. Although tremendous 30 progress has been made since 1939 in our understanding of nuclear fission, many of the 31 32 concepts introduced by Bohr and Wheeler remain very pertinent even today. 33 34 35 1.1. Fission in Science and Applications 36 37 In simple terms, nuclear fission is the process during which an atomic nucleus made of Z 38 protons and N neutrons (A = N + Z) may split into two or more lighter elements. The 39 nuclear “fissility" parameter, given by x ≈ Z2=50:88A(1 − ηI2) with I = (N − Z)=A 40 41 and η = 1:7826, is a convenient quantity to characterize the ability of a nucleus to 42 fission as suggested in [6, 7, 8]. In the liquid drop model, the fissility is related to 43 the ratio between the Coulomb and surface energy of the drop. For values of x > 1, 44 45 the drop is unstable against fission, and nuclear fission can then occur spontaneously. 46 This is the case, for instance, in heavy nuclei with large Z values such as actinides or 47 SF 48 transactinides. The process is characterized by the spontaneous fission half-life τ1=2, 49 which is the time it takes for half the population of a sample to undergo fission.
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