Microscopic Theory of Nuclear Fission: A Review

Schunck, N.; Robledo, L. M.

doi:10.1088/0034-4885/79/11/116301

Title: Microscopic Theory of Nuclear Fission: A Review

Abstract

This paper reviews how nuclear fission is described within nuclear density functional theory. A distinction should be made between spontaneous fission, where half-lives are the main observables and quantum tunnelling the essential concept, and induced fission, where the focus is on fragment properties and explicitly time-dependent approaches are often invoked. Overall, the cornerstone of the density functional theory approach to fission is the energy density functional formalism. The basic tenets of this method, including some well-known tools such as the Hartree–Fock–Bogoliubov (HFB) theory, effective two-body nuclear potentials such as the Skyrme and Gogny force, finite-temperature extensions and beyond mean-field corrections, are presented succinctly. The energy density functional approach is often combined with the hypothesis that the time-scale of the large amplitude collective motion driving the system to fission is slow compared to typical time-scales of nucleons inside the nucleus. In practice, this hypothesis of adiabaticity is implemented by introducing (a few) collective variables and mapping out the many-body Schrödinger equation into a collective Schrödinger-like equation for the nuclear wave-packet. The region of the collective space where the system transitions from one nucleus to two (or more) fragments defines what are called the scission configurations. The inertia tensor that enters themore »« less

Authors:

Schunck, N. ^[1]; Robledo, L. M. ^[2]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Nuclear and Chemical Science Division
Autonomous Univ. of Madrid (Spain). Dept. de Fisica Teorica

Publication Date:: 2016-10-11

Research Org.:: Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF); Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

OSTI Identifier:: 1341968

Alternate Identifier(s):: OSTI ID: 1328599

Report Number(s):: LLNL-JRNL-680281
Journal ID: ISSN 0034-4885; TRN: US1701535

Grant/Contract Number:: AC52-07NA27344

Resource Type:: Accepted Manuscript

Journal Name:: Reports on Progress in Physics

Additional Journal Information:: Journal Volume: 79; Journal Issue: 11; Journal ID: ISSN 0034-4885

Publisher:: IOP Publishing

Country of Publication:: United States

Language:: English

Subject:: 73 NUCLEAR PHYSICS AND RADIATION PHYSICS

Citation Formats


                    Schunck, N., and Robledo, L. M. Microscopic Theory of Nuclear Fission: A Review.  United States: N. p., 2016. 
Web.  doi:10.1088/0034-4885/79/11/116301.

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                    Schunck, N., & Robledo, L. M. Microscopic Theory of Nuclear Fission: A Review.  United States.  https://doi.org/10.1088/0034-4885/79/11/116301

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                    Schunck, N., and Robledo, L. M. Tue .  
"Microscopic Theory of Nuclear Fission: A Review".  United States.  https://doi.org/10.1088/0034-4885/79/11/116301.  https://www.osti.gov/servlets/purl/1341968.

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@article{osti_1341968,

  title        = {Microscopic Theory of Nuclear Fission: A Review},

  author       = {Schunck, N. and Robledo, L. M.},

  abstractNote = {This paper reviews how nuclear fission is described within nuclear density functional theory. A distinction should be made between spontaneous fission, where half-lives are the main observables and quantum tunnelling the essential concept, and induced fission, where the focus is on fragment properties and explicitly time-dependent approaches are often invoked. Overall, the cornerstone of the density functional theory approach to fission is the energy density functional formalism. The basic tenets of this method, including some well-known tools such as the Hartree–Fock–Bogoliubov (HFB) theory, effective two-body nuclear potentials such as the Skyrme and Gogny force, finite-temperature extensions and beyond mean-field corrections, are presented succinctly. The energy density functional approach is often combined with the hypothesis that the time-scale of the large amplitude collective motion driving the system to fission is slow compared to typical time-scales of nucleons inside the nucleus. In practice, this hypothesis of adiabaticity is implemented by introducing (a few) collective variables and mapping out the many-body Schrödinger equation into a collective Schrödinger-like equation for the nuclear wave-packet. The region of the collective space where the system transitions from one nucleus to two (or more) fragments defines what are called the scission configurations. The inertia tensor that enters the kinetic energy term of the collective Schrödinger-like equation is one of the most essential ingredients of the theory, since it includes the response of the system to small changes in the collective variables. For this reason, the two main approximations used to compute this inertia tensor, the adiabatic time-dependent HFB and the generator coordinate method, are presented in detail, both in their general formulation and in their most common approximations. The collective inertia tensor enters also the Wentzel–Kramers–Brillouin (WKB) formula used to extract spontaneous fission half-lives from multi-dimensional quantum tunnelling probabilities (For the sake of completeness, other approaches to tunnelling based on functional integrals are also briefly discussed, although there are very few applications.) It is also an important component of some of the time-dependent methods that have been used in fission studies. Concerning the latter, both the semi-classical approaches to time-dependent nuclear dynamics and more microscopic theories involving explicit quantum-many-body methods are presented. One of the hallmarks of the microscopic theory of fission is the tremendous amount of computing needed for practical applications. In particular, the successful implementation of the theories presented in this article 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 devoted to discussing the resolution of the HFB equations, especially in the context of very deformed nuclear shapes. In particular, the numerical precision and iterative methods employed to obtain the HFB solution are documented in detail. Finally, a selection of the most recent and representative results obtained for both spontaneous and induced fission is presented, with the goal of emphasizing the coherence of the microscopic approaches employed. In conclusion, although impressive progress has been achieved over the last two decades to understand fission microscopically, much work remains to be done. Several possible lines of research are outlined in the conclusion.},

  doi          = {10.1088/0034-4885/79/11/116301},

  journal      = {Reports on Progress in Physics},

  number       = 11,

  volume       = 79,

  place        = {United States},

  year         = {2016},

  month        = {10}

}

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Title: Microscopic Theory of Nuclear Fission: A Review

Abstract

Citation Formats

Neck influence on fission paths journal, April 2008

On the Gaussian overlap approximation for the collective excitations of odd nuclei journal, January 2015

Microscopic approach of fission dynamics applied to fragment kinetic energy and mass distributions in U 238 journal, February 2005

Fission barriers and the inclusion of axial asymmetry journal, March 1972

The fission barriers in Actinides and superheavy nuclei in covariant density functional theory journal, May 2010

Numerical search of discontinuities in self-consistent potential energy surfaces journal, October 2012

Generalization of Kramers's formula: Fission over a multidimensional potential barrier journal, November 1983

Solution of the Skyrme—Hartree—Fock equations in the Cartesian deformed harmonic oscillator basis II. The program HFODD journal, May 1997

Colloquium : Beta-delayed fission of atomic nuclei journal, October 2013

Potential energy surfaces of actinide nuclei from a multidimensional constrained covariant density functional theory: Barrier heights and saddle point shapes journal, January 2012

Gaussian matrix elements in a cylindrical harmonic oscillator basis journal, July 2009

Surface symmetry energy of nuclear energy density functionals journal, March 2011

Secondary fission barriers in even-even actinide nuclei journal, March 2012

Fission dynamics within time-dependent Hartree-Fock: Deformation-induced fission journal, November 2015

Basis Spline Collocation Method for Solving the Schrödinger Equation in Axillary Symmetric Systems journal, October 1996

Barrier penetration and spontaneous fission in the time-dependent mean-field approximation journal, November 1980

Some properties of three-dimensional harmonic oscillator wave functions journal, February 1970

Local nuclear energy density functional at next-to-next-to-next-to-leading order journal, October 2008

Structure properties of even–even actinides at normal and super deformed shapes analysed using the Gogny force journal, May 2006

Hartree-Fock-Bogolyubov calculations with the D 1 effective interaction on spherical nuclei journal, April 1980

Hartree-Fock-Bogoliubov theory with spin and number projection before the variation: An application to 20Ne and 22Ne journal, December 1984

Exact Tensor Hypercontraction: A Universal Technique for the Resolution of Matrix Elements of Local Finite-Range N -Body Potentials in Many-Body Quantum Problems journal, September 2013

Fission barriers and probabilities of spontaneous fission for elements with Z≥ 100 journal, December 2015

Lipkin translational-symmetry restoration in the mean-field and energy–density-functional methods journal, September 2009

Brownian Shape Motion on Five-Dimensional Potential-Energy Surfaces:Nuclear Fission-Fragment Mass Distributions journal, March 2011

Density functional theory and Kohn-Sham scheme for self-bound systems journal, November 2009

Towards a better parametrisation of Skyrme-like effective forces: A critical study of the SkM force journal, September 1982

Microscopic calculation of Pu 240 scission with a finite-range effective force journal, November 2009

Tensor and tensor-isospin terms in the effective Gogny interaction journal, November 2012

The generator coordinate method and quantised collective motion in nuclear systems journal, January 1987

Application of the finite element method in self-consistent relativistic mean field calculations journal, December 1996

Multidimensionally constrained relativistic mean-field study of triple-humped barriers in actinides journal, January 2015

Computer program for the time-evolution of a nuclear system in relativistic mean-field theory journal, August 1995

Effective density-dependent pairing forces in the T = 1 and T = 0 channels journal, November 1999

Quantification of Uncertainties in Nuclear Density Functional Theory journal, January 2015

Evaluation of the optimal path in ATDHF theory journal, January 1982

Fission Barriers at Finite Temperature: a Theoretical Description with the Gogny Force journal, April 2009

Properties of highly excited nuclei journal, October 1984

Generalization of internal density-functional theory and Kohn-Sham scheme to multicomponent self-bound systems, and link with traditional density-functional theory journal, November 2011

Thermal shape fluctuation effects in the description of hot nuclei journal, September 2003

Microscopic modeling of mass and charge distributions in the spontaneous fission of Pu 240 journal, January 2016

Large superfluidity enhancement in the penetration of the fission barrier journal, April 1974

Pairing correlations. II. Microscopic analysis of odd-even mass staggering in nuclei journal, December 2001

Misfits in Skyrme–Hartree–Fock journal, March 2010

Interband absorption in quantum wires. II. Nonzero-magnetic-field case journal, January 1992

A new kind of natural radioactivity journal, January 1984

Superdeformed rotational bands in the mercury region. A cranked Skyrme-Hartree-Fock-Bogoliubov study journal, September 1994

Local density approximation for proton-neutron pairing correlations: Formalism journal, January 2004

Nuclear masses and deformations journal, June 1966

Tools for incorporating a D-wave contribution in Skyrme energy density functionals journal, February 2015

An introduction to the adiabatic time-dependent Hartree-Fock method journal, October 1984

Statistical properties and stability of hot nuclei journal, April 1985

Time-dependent mean-field approximation for nuclear dynamical problems journal, April 1980

Use of the discrete variable representation basis in nuclear physics journal, May 2013

Temperature dependence of nuclear surface properties journal, September 1983

Simple separable expansions for calculating matrix elements of two-body local interactions with harmonic oscillator functions journal, January 1975

Solution of the Skyrme HF+BCS equation on a 3D mesh journal, September 2005

Hartree-Fock and Thomas-Fermi self-consistent calculations of the 144Nd nucleus at finite temperature and angular momentum journal, April 1985

Nuclear hyperdeformation and the Jacobi shape transition journal, May 2007

Hartree-Fock Calculations with Skyrme's Interaction. I. Spherical Nuclei journal, March 1972

Quadrupole collective inertia in nuclear fission: Cranking approximation journal, November 2011

Analytical proof of the non-uniqueness of the ATDHF path journal, April 1981

Variational approach to collective motion journal, October 1962

Fission properties for r -process nuclei journal, February 2012

Numerical accuracy of mean-field calculations in coordinate space journal, December 2015

Self-Consistent Study of Fission Barriers of Even-Even Superheavy Nuclei conference, January 2005

Neutron-Capture Nucleosynthesis in the First Stars journal, March 2014

Time-dependent hartree-fock theory with Skyrme's interaction journal, September 1975

A consistent microscopic theory of collective motion in the framework of an ATDHF approach journal, June 1978

The nuclear droplet model for arbitrary shapes journal, May 1974

Triaxial Hartree-Fock-Bogolyubov calculations with D 1 effective interaction journal, May 1983

The double-humped fission barrier journal, October 1980

Fission barriers for even-even superheavy nuclei journal, July 2010

Intrinsic-density functionals journal, January 2007

Axially deformed solution of the Skyrme–Hartree–Fock–Bogolyubov equations using the transformed harmonic oscillator basis. The program HFBTHO (v1.66p) journal, April 2005

Fission of rotating fermium isotopes journal, April 2014

Fission properties of the Barcelona-Catania-Paris-Madrid energy density functional journal, November 2013

Neck influence on fission paths
journal, April 2008

On the Gaussian overlap approximation for the collective excitations of odd nuclei
journal, January 2015

Microscopic approach of fission dynamics applied to fragment kinetic energy and mass distributions in ${}^{238}U$
journal, February 2005

Fission barriers and the inclusion of axial asymmetry
journal, March 1972

The fission barriers in Actinides and superheavy nuclei in covariant density functional theory
journal, May 2010

Numerical search of discontinuities in self-consistent potential energy surfaces
journal, October 2012

Generalization of Kramers's formula: Fission over a multidimensional potential barrier
journal, November 1983

Solution of the Skyrme—Hartree—Fock equations in the Cartesian deformed harmonic oscillator basis II. The program HFODD
journal, May 1997

Colloquium : Beta-delayed fission of atomic nuclei
journal, October 2013

Potential energy surfaces of actinide nuclei from a multidimensional constrained covariant density functional theory: Barrier heights and saddle point shapes
journal, January 2012

Gaussian matrix elements in a cylindrical harmonic oscillator basis
journal, July 2009

Surface symmetry energy of nuclear energy density functionals
journal, March 2011

Secondary fission barriers in even-even actinide nuclei
journal, March 2012

Fission dynamics within time-dependent Hartree-Fock: Deformation-induced fission
journal, November 2015

Basis Spline Collocation Method for Solving the Schrödinger Equation in Axillary Symmetric Systems
journal, October 1996

Barrier penetration and spontaneous fission in the time-dependent mean-field approximation
journal, November 1980

Some properties of three-dimensional harmonic oscillator wave functions
journal, February 1970

Local nuclear energy density functional at next-to-next-to-next-to-leading order
journal, October 2008

Structure properties of even–even actinides at normal and super deformed shapes analysed using the Gogny force
journal, May 2006

Hartree-Fock-Bogolyubov calculations with the $D 1$ effective interaction on spherical nuclei
journal, April 1980

Hartree-Fock-Bogoliubov theory with spin and number projection before the variation: An application to 20Ne and 22Ne
journal, December 1984

Exact Tensor Hypercontraction: A Universal Technique for the Resolution of Matrix Elements of Local Finite-Range $N$ -Body Potentials in Many-Body Quantum Problems
journal, September 2013

Fission barriers and probabilities of spontaneous fission for elements with Z≥ 100
journal, December 2015

Lipkin translational-symmetry restoration in the mean-field and energy–density-functional methods
journal, September 2009

Brownian Shape Motion on Five-Dimensional Potential-Energy Surfaces:Nuclear Fission-Fragment Mass Distributions
journal, March 2011

Density functional theory and Kohn-Sham scheme for self-bound systems
journal, November 2009

Towards a better parametrisation of Skyrme-like effective forces: A critical study of the SkM force
journal, September 1982

Microscopic calculation of ${}^{240}{Pu}$ scission with a finite-range effective force
journal, November 2009

Tensor and tensor-isospin terms in the effective Gogny interaction
journal, November 2012

The generator coordinate method and quantised collective motion in nuclear systems
journal, January 1987

Application of the finite element method in self-consistent relativistic mean field calculations
journal, December 1996

Multidimensionally constrained relativistic mean-field study of triple-humped barriers in actinides
journal, January 2015

Computer program for the time-evolution of a nuclear system in relativistic mean-field theory
journal, August 1995

Effective density-dependent pairing forces in the $T = 1$ and $T = 0$ channels
journal, November 1999

Quantification of Uncertainties in Nuclear Density Functional Theory
journal, January 2015

Evaluation of the optimal path in ATDHF theory
journal, January 1982

Fission Barriers at Finite Temperature: a Theoretical Description with the Gogny Force
journal, April 2009

Properties of highly excited nuclei
journal, October 1984

Generalization of internal density-functional theory and Kohn-Sham scheme to multicomponent self-bound systems, and link with traditional density-functional theory
journal, November 2011

Thermal shape fluctuation effects in the description of hot nuclei
journal, September 2003

Microscopic modeling of mass and charge distributions in the spontaneous fission of ${}^{240}{Pu}$
journal, January 2016

Large superfluidity enhancement in the penetration of the fission barrier
journal, April 1974

Pairing correlations. II. Microscopic analysis of odd-even mass staggering in nuclei
journal, December 2001

Misfits in Skyrme–Hartree–Fock
journal, March 2010

Interband absorption in quantum wires. II. Nonzero-magnetic-field case
journal, January 1992

A new kind of natural radioactivity
journal, January 1984

Superdeformed rotational bands in the mercury region. A cranked Skyrme-Hartree-Fock-Bogoliubov study
journal, September 1994

Local density approximation for proton-neutron pairing correlations: Formalism
journal, January 2004

Nuclear masses and deformations
journal, June 1966

Tools for incorporating a D-wave contribution in Skyrme energy density functionals
journal, February 2015

An introduction to the adiabatic time-dependent Hartree-Fock method
journal, October 1984

Statistical properties and stability of hot nuclei
journal, April 1985

Time-dependent mean-field approximation for nuclear dynamical problems
journal, April 1980

Use of the discrete variable representation basis in nuclear physics
journal, May 2013

Temperature dependence of nuclear surface properties
journal, September 1983

Simple separable expansions for calculating matrix elements of two-body local interactions with harmonic oscillator functions
journal, January 1975

Solution of the Skyrme HF+BCS equation on a 3D mesh
journal, September 2005

Hartree-Fock and Thomas-Fermi self-consistent calculations of the 144Nd nucleus at finite temperature and angular momentum
journal, April 1985

Nuclear hyperdeformation and the Jacobi shape transition
journal, May 2007

Hartree-Fock Calculations with Skyrme's Interaction. I. Spherical Nuclei
journal, March 1972

Quadrupole collective inertia in nuclear fission: Cranking approximation
journal, November 2011

Analytical proof of the non-uniqueness of the ATDHF path
journal, April 1981

Variational approach to collective motion
journal, October 1962

Fission properties for $r$ -process nuclei
journal, February 2012

Numerical accuracy of mean-field calculations in coordinate space
journal, December 2015

Self-Consistent Study of Fission Barriers of Even-Even Superheavy Nuclei
conference, January 2005

Neutron-Capture Nucleosynthesis in the First Stars
journal, March 2014

Time-dependent hartree-fock theory with Skyrme's interaction
journal, September 1975

A consistent microscopic theory of collective motion in the framework of an ATDHF approach
journal, June 1978

The nuclear droplet model for arbitrary shapes
journal, May 1974

Triaxial Hartree-Fock-Bogolyubov calculations with $D 1$ effective interaction
journal, May 1983

The double-humped fission barrier
journal, October 1980

Fission barriers for even-even superheavy nuclei
journal, July 2010

Intrinsic-density functionals
journal, January 2007

Axially deformed solution of the Skyrme–Hartree–Fock–Bogolyubov equations using the transformed harmonic oscillator basis. The program HFBTHO (v1.66p)
journal, April 2005

Fission of rotating fermium isotopes
journal, April 2014

Fission properties of the Barcelona-Catania-Paris-Madrid energy density functional
journal, November 2013

Consequences of the center–of–mass correction in nuclear mean–field models
journal, April 2000

Transformation brackets for harmonic oscillator functions
journal, October 1959

Nuclear Properties for Astrophysical and Radioactive-Ion-Beam Applications
journal, July 1997