CoH 3 : The Coupled-Channels and Hauser-Feshbach Code Toshihiko - PowerPoint PPT Presentation

CoH 3 : The Coupled-Channels and Hauser-Feshbach Code Toshihiko Kawano Los Alamos National Laboratory Theoretical Division 2018 Symposium on Nuclear Data Tokyo Institute of Technology, 11/29,30, 2018 LA-UR 18-29068

Introduction Statistical Model Code for Compound Nuclear Reactions Statistical Hauser-Feshbach Code with Width Fluctuation Correction A main tool for calculating nuclear reactions for A > 20 , E n > 1 keV (above resolved resonance region) Provide complete information of nuclear reactions reaction cross sections σ energy and angular distributions of secondary particles d σ dE , d σ d Ω γ -ray production cross sections σ ( ∗ , x γ ) etc Essential for prediction of experimentally unknown cross sections reactions on unstable targets or isomeric states

Introduction ELIESE-GNASH Experience in 1990s Written in old-fashioned FORTRAN-IV, 66, 77 difficult to modify, upgrade, maintain difficult to implement new ideas in physics Reaction chain input not so automated (n,n’), (n,p), (n, α ), (n,2n), (n,np), (n,n α ), . . . Limitation in reaction modeling no angular distribution no width fluctuation correction, unsuitable for low incident energy calculations direct reaction channel treated in an approximate way FORTRAN compiler for PC too expensive!

Introduction Definition of π According to A. Koninig, GNASH defines π at many places, such as 3.1415 (6 places), 6.283 (3 places), and so on TALYS defines π only one time Easy to modify if you want to change it

Code Source Code CoH Development History 1992 1.x original versison written in C on 16bit MS-DOS 1995 2.0 totally rewritten, ANSI standard C 2003 2.3 capture, fission, CC, etc included 2008 3.0 Callisto extended to full HF code rewritten in C++ 2010 3.1 Ariel branch off CGM 2012 3.2 Umbriel exclusive energy spectrum 2013 3.3 Titania memory management advanced 2015 3.4 Oberon including mean-field theories 3.5 Milanda CC calc. enhanced

Code Source Code CoH 3 , Quick Glance Hauser-Feshbach-Moldauer theory for Compound Nuclear Reaction 45,000 lines C++ code ∼ 140 C++ source files ∼ 60 header files OOP , ∼ 80 classes defined GNU Autotools package all physical/mathematical constants defined only once internal optical model / coupled-channels solver compound nucleus decay by deterministic or Monte Carlo method exclusive reaction cross sections and spectra [JNST 47 , 462 (2010)]

Code Source Code Modules and Models Employed in CoH 3 Optical Model spherical and deformed (rotational or vibrational model) DWBA for direct inelastic scattering Compound Reaction Moldauer’s width fluctuation correction with LANL parameters [NDS 118 , 183 (2014)] Engelbrecht-Weidenm¨ uller transformation with direct channels Gilbert-Cameron level density [JNST 43 , 1 (2006)] Pre-equilibrium Reaction 2-component exciton model (FKK MSD/MSC still external code) Prompt Fission Neutron Spectrum Madland-Nix model including pre-fission neutrons Direct/Semidirect Capture [PRC 75 , 054618 (2007)] Mean-Field Model (FRDM and Hartree-Fock-BCS) [EPJ 146 , 12004 (2017)]

Code Exmaples Default Calculations for n + 58 Ni 1000 200 ENDF/B-VII.1 ENDF/B-VII.1 900 JENDL-4.0 JENDL-4.0 CoH 3 CoH 3 58 Ni(n,p) Cross Section [mb] 58 Ni(n, α ) Cross Section [mb] 800 150 700 600 500 100 400 300 50 200 100 0 0 0 5 10 15 20 0 5 10 15 20 Neutron Incident Energy [MeV] Neutron Incident Energy [MeV] 1000 120 ENDF/B-VII.1 ENDF/B-VII.1 900 JENDL-4.0 JENDL-4.0 58 Ni(n,np+d) Cross Section [mb] CoH 3 58 Ni(n,2n) Cross Section [mb] 100 CoH 3 800 700 80 600 500 60 400 40 300 200 20 100 0 0 0 5 10 15 20 10 15 20 Neutron Incident Energy [MeV] Neutron Incident Energy [MeV]

Code Exmaples Angular Distributions, n + 58 Ni at E n = 3 MeV Legendre Coefficients Given by Blatt-Biedenharn Formalism 6 0.8 Shape Elaastic (n,p 0 ) (n, α 0 ) (n,p 1 ) (n, α 1 ) Elastic 0.7 1000 (n,n 1 ) (n,p 2 ) (n, α 2 ) 5 Differential Cross Section [mb/d Ω ] Differential Cross Section [mb/d Ω ] Differential Cross Section [mb/d Ω ] (n,n 2 ) (n,n 3 ) 0.6 4 0.5 100 3 0.4 0.3 2 10 0.2 1 0.1 1 0 0 0 30 60 90 120 150 180 0 30 60 90 120 150 180 0 30 60 90 120 150 180 C.M. Angle [deg] C.M. Angle [deg] C.M. Angle [deg] B L = π 1 � Z 2 ℜ � (1 − S l 1 s 1 J 1 )(1 − S l 2 s 2 J 2 ) ∗ � k 2 2(2 I + 1)(2 L + 1)

Coupled-Channels Optical Model and Hauser-Feshbach Transmission Coefficients Detailed Balance in Compound Reaction Compound State T J (0) , T J (1) , T J T J (0) , T J (1) , T J (2) (2) Generalized transmission coefficient Absorption Process D e c a Eliminate direct reaction flux from absorption y P r o c e s s   � � � T ( n )  � S J Π � | 2    1 − |  l j = g Jc   Coupled States cc ′ Excited State      c ′ J Π c c ∈ n Direct Process Ground State Replacement of T l j Compound State T J (n) = T J (0) (E - E n ) T J (0) In standard Hauser-Feshbach code, T l j for Absorption Process the excited states are approximated by the D e c a y ground-state T l j and shifted P r o c e s s T ( n ) l j ( E ) = T (0) l j ( E − E ( n ) x ) Coupled States Excited State Ground State

Coupled-Channels Optical Model and Hauser-Feshbach Transmission Coefficients Shifted Single-Channel T l j and Coupled-Channels T l j T l j for the 1st Excited State of 238 U s-wave p-wave 0.7 0.7 0.6 0.6 Transmission Coefficient, L=0 Transmission Coefficient, L=1 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 1st level, J=3/2 0.1 0.1 J=1/2 1st level, J=1/2 shifted GS, J=3/2 shfted GS J=1/2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 C.M. Energy [MeV] C.M. Energy [MeV] Soukhovitskii et al. (2005) potential

Coupled-Channels Optical Model and Hauser-Feshbach EW Transformation Moldauer and Engelbrecht-Weidenm¨ uller Diagonalizing Satchler’s P Matrix 10 -1 10 -2 � � S ∗ � ( UPU † ) αβ = δ αβ p α P ab = δ ab − � S ac � bc * >| 10 -3 GOE |<S aa S bb c 10 -4 � U ∗ α a U ∗ � � 10 -5 � σ fl ab � U α a U β b + U α a U β b (1 − δ αβ ) = β b αβ 10 -6 � ˜ 0 20 40 60 80 100 120 140 � S αβ | 2 � � Sum of Transmission Coefficients | ˜ + U ∗ α a U ∗ S αα ˜ S ∗ × β b U α a U β b ββ 100 * >| 10 Ratio of |<S aa S bb � 2 � 1 / 2 � 2 � 1 / 2 � ˜ S ∗ � ≃ e i ( φ α − φ β ) S αα ˜ − 1 − 1 σ αβ ββ ν α ν β 1 (a) Moldauer (1980) φ α = tan − 1 ˜ LANL (2014) S αα 0.1 0 1 2 3 4 5 Sum of Transmission Coefficients

Coupled-Channels Optical Model and Hauser-Feshbach EW Transformation 238 U Inelastic Scattering Cross Section Implementation of Full EW Transformation into CoH 3 2000 500 JENDL-4 JENDL-4 without EWT without EWT with EWT with EWT 400 1500 238 U(n,n’) [mb] 238 U(n,n’) [mb] 300 1000 200 500 100 (a) 44.9 keV 2+ (b) 148.4 keV 4+ 0 0 0 1 2 3 4 0 1 2 3 4 Neutron Incident Energy [MeV] Neutron Incident Energy [MeV] [PRC 94 , 014612 (2016)]

Multi Particle Emission Multiple Particle Emission n p P r e - E q u i l i b r i u m a Incident Energy 57 Ni 54 Fe 57 Co 12.2 8.6 9.3 58 Ni 58 Co 58 Ni 55 Fe Separation Energy 9.0 8.6 6.1 59 Ni

Multi Particle Emission Multi-Particle Emission and Exclusive Cross Section Z,A+1 CN Z-1 n n p p (z,p) Z,A Z-1,A (z,n) d d (z,d) (z,np) Z,A-1 Z-1,A-1 (z,2n) t t (z,t) (z,nd) (z,2np) Z-1,A-2 (z,nt) GNASH CoH Nucleus objects for (n,d) and (n,np) channels are different A large number of CN object emerge at high energies

Multi Particle Emission Exclusive Particle Emission Spectrum Inclusive φ ( E ) Exclusive ψ ( E ) 10000 10000 Total Total Preequilibrium (n,n’) Ni59 2x (n,2n) 1000 1000 Ni58 (n,np) Co58 (n,n α ) Energy Spectra [mb/MeV] Energy Spectra [mb/MeV] Fe55 100 100 10 10 1 1 0.1 0.1 0.01 0.01 0 5 10 15 20 0 5 10 15 20 Secondary Neutron Energy [MeV] Secondary Neutron Energy [MeV] Φ = φ PE + φ n ′ x + φ npx + φ 2 nx + . . . = ψ 1 n + ψ np + 2 ψ 2 n + . . .

Subsidiary Codes CoH 3 Subsidiary Codes BeoH Statistical decay of CN β -delayed neutron and γ -ray develop branch Fission neutron, γ -ray, and FPY BeoH [S. Okumura et al. JNST 55 , 1009 (2018)] FroH CoH 3 FroH Microscopic level density FRLDM [PRC 64 , 024603 (2001)] stable branch FRLDM Finite-range liquid drop model Fission fragment yield, fission barrier