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

CoH3: 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, En > 1 keV (above resolved resonance region) Provide complete information of nuclear reactions

reaction cross sections σ energy and angular distributions of secondary particles dσ

dE , dσ dΩ

γ-ray production cross sections σ(∗, xγ) 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,nα), . . .

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

• riginal 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

CoH3, 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 CoH3

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 + 58Ni

100 200 300 400 500 600 700 800 900 1000 5 10 15 20

58Ni(n,p) Cross Section [mb]

Neutron Incident Energy [MeV] ENDF/B-VII.1 JENDL-4.0 CoH3 50 100 150 200 5 10 15 20

58Ni(n,α) Cross Section [mb]

Neutron Incident Energy [MeV] ENDF/B-VII.1 JENDL-4.0 CoH3 100 200 300 400 500 600 700 800 900 1000 5 10 15 20

58Ni(n,np+d) Cross Section [mb]

Neutron Incident Energy [MeV] ENDF/B-VII.1 JENDL-4.0 CoH3 20 40 60 80 100 120 10 15 20

58Ni(n,2n) Cross Section [mb]

Neutron Incident Energy [MeV] ENDF/B-VII.1 JENDL-4.0 CoH3

Code Exmaples

Angular Distributions, n + 58Ni at En = 3 MeV

Legendre Coefficients Given by Blatt-Biedenharn Formalism

1 10 100 1000 30 60 90 120 150 180 Differential Cross Section [mb/dΩ] C.M. Angle [deg] Shape Elaastic Elastic (n,n1) (n,n2) (n,n3) 1 2 3 4 5 6 30 60 90 120 150 180 Differential Cross Section [mb/dΩ] C.M. Angle [deg] (n,p0) (n,p1) (n,p2) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 30 60 90 120 150 180 Differential Cross Section [mb/dΩ] C.M. Angle [deg] (n,α0) (n,α1) (n,α2)

BL = π k2 1 2(2I + 1)(2L + 1)

• Z2ℜ (1 − S l1s1J1)(1 − S l2s2J2)∗

Coupled-Channels Optical Model and Hauser-Feshbach Transmission Coefficients

Detailed Balance in Compound Reaction

Compound State Ground State Excited State Absorption Process D e c a y P r

• c

e s s Coupled States

TJ

(0) , TJ (1), TJ (2)

TJ

(0) , TJ (1), TJ (2)

Direct Process Compound State Ground State Excited State Absorption Process D e c a y P r

• c

e s s Coupled States

TJ

(n) = TJ (0)(E - En)

TJ

(0)

Generalized transmission coefficient Eliminate direct reaction flux from absorption T (n)

l j =

• JΠ
• c

gJc       1 −

• c′

|

• S JΠ

cc′

• |2

      

c∈n

Replacement of Tl j In standard Hauser-Feshbach code, Tl j for the excited states are approximated by the ground-state Tl j and shifted T (n)

l j (E) = T (0) l j (E − E(n) x )

Coupled-Channels Optical Model and Hauser-Feshbach Transmission Coefficients

Shifted Single-Channel Tl j and Coupled-Channels Tl j

Tl j for the 1st Excited State of 238U s-wave p-wave

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 1 1.5 2 Transmission Coefficient, L=0 C.M. Energy [MeV] 1st level, J=1/2 shfted GS 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.5 1 1.5 2 Transmission Coefficient, L=1 C.M. Energy [MeV] 1st level, J=3/2 J=1/2 shifted GS, J=3/2 J=1/2

Soukhovitskii et al. (2005) potential

Coupled-Channels Optical Model and Hauser-Feshbach EW Transformation

Moldauer and Engelbrecht-Weidenm¨ uller

Diagonalizing Satchler’s P Matrix Pab = δab −

• c

S ac

• S ∗

bc

• (UPU†)αβ = δαβpα

σfl

ab

=

• αβ

U∗

αaU∗ βb

• UαaUβb + UαaUβb(1 − δαβ)
• ×
• | ˜

S αβ|2 + U∗

αaU∗ βbUαaUβb

˜ S αα ˜ S ∗

ββ

• ˜

S αα ˜ S ∗

ββ

• ≃ ei(φα−φβ)

2 να − 1 1/2 2 νβ − 1 1/2 σαβ φα = tan−1 ˜ S αα

10-6 10-5 10-4 10-3 10-2 10-1 20 40 60 80 100 120 140 GOE |<SaaSbb

*>|

Sum of Transmission Coefficients 0.1 1 10 100 1 2 3 4 5 (a) Ratio of |<SaaSbb

*>|

Sum of Transmission Coefficients Moldauer (1980) LANL (2014)

Coupled-Channels Optical Model and Hauser-Feshbach EW Transformation

238U Inelastic Scattering Cross Section Implementation of Full EW Transformation into CoH3

500 1000 1500 2000 1 2 3 4 (a) 44.9 keV 2+

238U(n,n’) [mb]

Neutron Incident Energy [MeV] JENDL-4 without EWT with EWT 100 200 300 400 500 1 2 3 4 (b) 148.4 keV 4+

238U(n,n’) [mb]

Neutron Incident Energy [MeV] JENDL-4 without EWT with EWT

[PRC 94, 014612 (2016)]

Multi Particle Emission

Multiple Particle Emission

9.0 12.2 8.6 8.6 6.1 9.3

58Ni 59Ni 58Ni 57Ni 58Co 57Co 55Fe 54Fe

Incident Energy Separation Energy

P r e

• E

q u i l i b r i u m n p a

Multi Particle Emission

Multi-Particle Emission and Exclusive Cross Section

CN (z,p) (z,n) (z,2n) (z,np) (z,2np) (z,d) (z,t) (z,nd) (z,nt) Z,A+1 Z-1,A Z,A-1 Z-1,A-1 Z-1,A-2 Z,A

Z-1 p d t n n p d t

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)

0.01 0.1 1 10 100 1000 10000 5 10 15 20 Energy Spectra [mb/MeV] Secondary Neutron Energy [MeV] Total Preequilibrium Ni59 Ni58 Co58 Fe55 0.01 0.1 1 10 100 1000 10000 5 10 15 20 Energy Spectra [mb/MeV] Secondary Neutron Energy [MeV] Total (n,n’) 2x (n,2n) (n,np) (n,nα)

Φ = φPE + φn′x + φnpx + φ2nx + . . . = ψ1n + ψnp + 2ψ2n + . . .

Subsidiary Codes

CoH3 Subsidiary Codes

CoH3 BeoH FRLDM FroH

stable branch develop branch

BeoH Statistical decay of CN β-delayed neutron and γ-ray Fission neutron, γ-ray, and FPY

[S. Okumura et al. JNST 55, 1009 (2018)]

FroH Microscopic level density

[PRC 64, 024603 (2001)]

FRLDM Finite-range liquid drop model Fission fragment yield, fission barrier