New Era of Nuclear Physics in the Cosmos, the r-process Nucleo-Synthesis RIKEN, Japan, Sep. 25,26, 2008 Statistical-Model and Direct-Semidirect-Model Calculations of Neutron Radiative Capture Process T. Kawano Theoretical Division Los Alamos National Laboratory
Nuclear Reactions in the Fast Energy Range The Hauser-Feshbach statistical theory has been widely used for nuclear reaction data applications, nuclear reaction rates for astrophysics (nucleo-synthesis) nuclear data libraries for energy applications (mainly for neutron) however, questions still persist regarding: applicability in the domain of off-stability global inputs level densities, optical potentials, γ-ray strength function nuclear deformation effect width fluctuation correction Γ aγ b Γ = Γ a Γ b Γ inclusive / exclusive observables partial γ-rays, coincidence, etc. Monte Carlo approach to the HF needed interaction on the excited state (isomer)
Nuclear Data for Astrophysical Applications Nuclear Reaction Rate in Astrophysics Nuclear Reaction Chain A-2 Z-2 Neutron Capture Proton Emission α-particle Emission β Decay A- Z- A- Z-2 A Z A Z- A Z-2 A+ Z+ A+ Z A+ Z- A+2 Z+ A+2 Z Nuclear reactions in astrophysical environments s and r-processes targets are often neutron-rich and unstable neutron capture process is mostly important β-delayed neutron emission and fission Model prediction is crucial for reaction cross section Hauser-Feshbach theory nuclear structure, including mass macro/microscopic model, Hartree-Fock Understanding of the neutron capture process is important to study nucleo-synthesis, which includes models for γ-ray strength function, GDR (Giant Dipole Resonance), nuclear level densities, and spectroscopic factors.
Nucleon Radiative Capture CN and DSD Processes Compound Reaction An incident neutron and a target form a compound nucleus, and it decays. Hauser-Feshbach statistical theory, with width fluctuation. Cross section decreases rapidly when neutron inelastic channels open. Direct/Semidirect Capture Direct transition to one of the unoccupied single-particle state. 0.00 Giant Dipole Resonance (GDR) 0.000 U-238 Capture Cross Section [b] 0. 0.0 Statistical Decay Hauser-Feshbach Hauser-Feshbach + DSD Direct/Semidirect Process 0. 0 Neutron Energy [MeV] DSD becomes important when () incident particle energy is high, or (2) compound capture cross section is small (few resonance, neutron-rich, doubly-closed shell nuclei.)
Photo Emission / Absorption Inverse process of γ-ray emission
Photo Reaction Neutron capture calculations for s and p-process Total reaction cross section is determined by the optical model. The decay width the sum of transmission probabilities for all decay processes. The γ-ray radiation width is calculated from the γ-ray absorption probability (the inverse process) and the final state density. Brink s hypothesis is used. Important to understand γ-ray transmission T E (E), and the level density ρ(e), off-stability.
γ-ray Strength Function and Transmission Standard Lorentzian E γ Γ f E (E γ ) = Cσ 0 Γ 0 0 (Eγ 2 E2 0 )2 + Eγ 2Γ2 0 Generalized Lorentzian, finite value at low energies, energy dependent width f E (E γ ) = Cσ 0 Γ 0 E γ Γ(E γ, T ) (Eγ 2 E2 0 )2 + Eγ 2Γ2 (E γ, T ) + 0.7Γ(E γ = 0, T ) E 3 0 0-6 where C = 8.68 0 8 mb MeV 2, or can be obtained by normalization to an experimental value if available. The γ-ray transmission coefficient is given by T E (E γ ) = 2πE 3 γ f E(E γ ) Strength Function, f(e) 0-7 0-8 0-9 0-0 Generalized Lorentzian Standard Lorenzian 0 5 0 5 20 25 30 E [MeV]
Normalization of γ-ray Strength Function An absolute capture cross section is still hard to predict, however, it can be estimated from average resonance properties. Γ γ D 0 Bn 0 T E (E γ )ρ(e x )de x where D0 is the average s-wave resonance spacing near the neutron binding energy: Γγ is the average γ decay width Compound Nucleus D 0 Γ γ kev mev 57 Fe 25.4 920 9 Zr 6.0 30 239 U 0.0208 23.6 or we just re-normalize the calculated capture cross section to experimental data avaiable by adjusting Γ γ /D 0.
Level Density Parameter Systematics Level Density Parameter [MeV - ] 50 40 30 20 0 a(from D 0 data) a(asymptotic) Least-Squares Fit Washing-out of Shell Effects Shell correction (δw ) and pairing energies ( ) taken from KTUY05 mass formula a = a { + δw U ( e γu )} a = 0.4A + 7.65 0 5 A 2 At low excitation energies, the constant temperature model is used with 0 0 50 00 50 200 250 300 Mass Number T = 48.A 0.88 0.δW obtained from discrete level data of more than 000 nuclei. TK, S. Chiba, H. Koura, J. Nucl. Sci. Technol., 43, (2006)
Hauser-Feshbach on Deformed Nuclei Incorporate Coupled-Channels (CC) method into the HF formula Scattering matrix is no longer diagonal. Inverse channel problem What is the appropriate transmission coefficient for the excited states? Replaced by the one for the ground state (historical) Solve the CC equation for the excited state (detailed balance) Width fluctuation correction when off-diagonal elements exist Moldauer Engelbrecht-Weidenmüller transformation Kawai-Kerman-McVoy (TK, L.Bonneau, A.Kerman, Nice conf. 2007) Our preliminary results showed that KKM gives almost identical results as Moldauer. Nishioka-Weidenmüller-Yoshida, GOE for coupled-channels
Coupled-Channels Potential Parameters Modification to the Global CC Potential of Soukhovitskiĩ, et al. E. Sh. Soukhovitskiĩ, et al., J. Phys. G: Nucl. Part. Phys. 30, 905 (2004). Adjust the imaginary potential to match the energy averaged S-matrix elements from resonance parameters (TK, F.H. Fröhner, NSE, 27, 30 (997)). When the S-matrix elements (resonance and optical model) are obtained, total and reaction cross sections are automatically reproduced. Imag 0-0. -0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9 - s-wave (resonance parameters) optical model 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 W s = 2.59 MeV for E n <.3 MeV R = 9.606 fm (9.6± 0. in Atlas, Mughabghab) S 0 =.3 0 4 ((.29± 0.3) 0 4, ibid) S = 2.07 0 4 ((2.7± 0.9) 0 4, ibid) Original SoukhovitskiĩPotential (in the paper) R = 9.57 fm S 0 = 0.95 0 4 S =.80 0 4 Real
U-238 Total Cross Section With Modified Soukhovitskiĩ Potential 0000 000 Total (resonance parameters) I=5keV s-wave p-wave CC calc. 000 I=5keV CC calc. Total Cross Section [b] 00 0 Total Cross Section [b] 00 0 0. 0 00 000 0000 Neutron Energy [ev] 0-6 0-5 0-4 0-3 0-2 0-0 0 0 Neutron Energy [MeV] With the modified Soukhovitskiĩ Potential, the energy averaged total cross section in the resonance range is well-reproduced. Above.3 MeV, the parameters are the same as the original ones.
Transmission for Inverse Reactions Transmission Coefficients for the Inverse Channels 0.8 0.7 0.8 0.7 Transmission Coefficients 0.6 0.5 0.4 0.3 0.2 0. 0 Ground State st Excited 2nd Excited 0 0.5.5 2 Neutron Energy [MeV] s-wave Transmission Coefficients 0.6 0.5 0.4 0.3 0.2 0. 0 Ground State st Excited 2nd Excited 0 0.5.5 2 Neutron Energy [MeV] p-wave In a usual case, the ground-state transmission is used for all the calculation. However, this might underestimate the (n,n ) cross-section to the 2 + and 4 + states, because the true transmission to the 2 + and 4 + are larger.
Capture Cross Section Calculations Model Parameters Optical Potential Modified Soukhovitskiĩ CC calculations are made for the G.S. band 0 + -2 + -4 + -6 + -8 + coupling Spherical potential for the uncoupled states Level density Ignatyuk level density formula a =34.69 MeV for 239 U (reproduce D 0 =20.3 ev) Shell correction and pairing energies from KTUY mass formula Spin-cut off parameter, Cumulative Number of Levels 000 00 0 σ 2 = 3.47 0 3 U/aA 5/3 Average γ-ray width Γγ =23.36 mev from resonance analysis or, to be adjusted if needed 239 U 0 2 Excitation Energy [MeV]
U-238 Results Comparison with ENDF/B-VII (a part of standards evaluation) 000 000 Capture Cross Section [mb] 00 Capture Cross Section [mb] 00 CC ENDF/B-VII 0 0.00 0.0 0. Energy [MeV] CC ENDF/B-VII 0 0.00 0.0 0. Energy [MeV] Γ γ =23.36 mev Γ γ =7.83 mev
Partial Wave Contribution Decomposition into Each Partial-Wave Contribution 000 Capture Cross Section [mb] 00 CC s-wave p-wave d-wave ENDF/B-VII 0 0.00 0.0 0. Energy [MeV]
Comparisons with DANCE Data Capture Cross Section [b] 00 0 Weston (98) Kobayashi (2002) DANCE Radiative Width 40.7 mev Radiative Width 3.2 mev Capture Cross Section [b] 00 0 DANCE Gayther (977) Vanpraet (986) ENDF/B-VII (640 group) Radiative Width 45 mev 0. 0. 0 00 000 Neutron Incident Energy [kev] 0. 0. 0 00 000 Neutron Energy [kev] When the calculated capture cross section for 24 Am is averaged over the Jezebel spectrum, it gives a 5% higher value than that for ENDF/B-VII.
Direct/Semidirect Nucleon Capture Model Direct Capture Electric dipole radiation transition from optical potential to single-particle state Amplitude T d = C d R nlj r R LJ Semidirect (Collective) Capture Excite GDR, and decay into single-particle state Vibration-particle coupling, V h(r) Amplitude T s = C s R nlj V h(r) R LJ M GDR 2 E γ E GDR + iγ GDR /2
DSD with Hartree-Fock BCS Theory Model Improvements L. Bonneau, T.K., T. Watanabe, S. Chiba, Phys. Rev. C 75, 05468 (2007) spectroscopic factor S ljk previous studies experimental data (often not available for astrophysical calculations) DSD/HF-BCS single-particle occupation probabilities no experimental data needed single-particle wave-function, R ljk (r) previous studies spherical Woods-Saxon, Nilsson model, coupled-channels model to bound states DSD/HF-BCS HF-BCS calculation and decomposition into spherical HO basis consistent treatment for all nuclei from spherical to deformed nuclei
Calculated Results DSD Cross Sections for Spherical and Deformed Cases 208 Pb(n,γ) 238 U(n,γ) Cross Section (mb) 3 2.5 2.5 Drake (97) Bergqvist (972) Csikai (967) DSD(SLy4) all, shifted DSD(SIII) all, shifted CN Cross Section (mb) 3 2.5 2.5 Drake (97) McDaniels (98) DSD(SLy4) DSD(SLy4) spherical DSD(SIII) DSD(SIII) spherical 0.5 0.5 0 0 2 4 6 8 0 2 4 6 8 20 Incident Neutron Energy (MeV) Occupation Probabilities for U-238 0 0 2 4 6 8 0 2 4 6 8 20 Incident Neutron Energy (MeV) Occupation Probability 0.8 0.6 0.4 0.2 0-20 -5-0 -5 0 Single Particle Energy [MeV]
Neutron Capture Off-Stability Sn-22,32 Cross Section (mb) 0 3 0 2 0 0 0 22 Sn(n,γ) Tolstikov (968) Pasechnik (958) Timokhov (989) Trofimov (987) Tolstikov (995) CN+DSD(SLy4) DSD(Sly4) Cross Section (mb) 0 0 0 0-0 -2 CN DSD(SLy4) 32 Sn(n,γ) 0-0 -2 0-0 0 0 0 2 Incident Neutron Energy (MeV) 0-3 0-2 0-0 0 0 0 2 Incident Neutron Energy (MeV) Since global Hartree-Fock-BCS calculations for all nuclides are feasible, this technique will be a powerful tool to estimate the neutron capture rates in the r-process. HF calculation problems for the odd nuclei still exist. Proton capture calculations underway.
Future Challenge and Works in Progress at LANL Applications of the Hauser-Feshbach Model to Different Areas Combining with nuclear structure calculations β-decay and electron capture produce highly excited states of nuclei, and the excitation energies can be larger than the neutron separation energy. Nuclear structure models, such as QRPA, predict final states of β-decay and EC. Compound decay from a daughter nucleus at given Ex, J π. Monte Carlo approach to the Hauser-Feshbach HF gives an integrated cross-section for all possible intermediate transitions. However, recent experiments are sometimes conducted by measuring partial γ- rays, coincidence with a particular γ-ray transition, selected events such as a fixed γ-ray multiplicity. Correlation information is needed.
Application of HF to Delayed Neutron Emission QRPA and Hauser-Feshbach model for Beta-Delayed Neutron Fission Product A,Z+ Beta Decay A-,Z+ Delayed Neutron Delayed Neutron Spectrum [/MeV] 0 0. 0.0 ENDF Decay Lib 00 kev Br-87 Delayed Neutron Spectrum [/MeV] 0 0. 0.0 ENDF Decay Lib 00 kev Br-88 Q-beta Neutron Separation Energy Delayed Gamma Delayed Neutron Spectrum [/MeV] 0.00 0 0.5.5 2 2.5 3 3.5 4 0 0. 0.0 Delayed Neutron Energy [MeV] ENDF Decay Lib 00 kev Br-89 Delayed Neutron Spectrum [/MeV] 0.00 0 0.5.5 2 2.5 3 3.5 4 0 0. 0.0 Delayed Neutron Energy [MeV] ENDF Decay Lib 00 kev Br-90 Nuclear Structure Model Nuclear Reaction Model 0.00 0 0.5.5 2 2.5 3 3.5 4 Delayed Neutron Energy [MeV] 0.00 0 0.5.5 2 2.5 3 3.5 4 Delayed Neutron Energy [MeV] We assume that the excited state after β-decay is a compound state, having a fixed J value, I J I +, where I is the spin of precursor.
Sequential Neutron Emissions Electron Capture State A,Z+ total excitation energy A-,Z+ Neutron Separation Energy A-2,Z+ 2 Neutron2 Separation Energy A-3,Z+ Electron capture produces a highly excited state, which subsequently decays by emitting several neutrons (very fast process). 3 Neutrons Separation Energy Neutron Emission Probability Neutron Emission Probability 0.8 0.6 0.4 0.2 Si-43 0 0 0 20 30 0.8 0.6 0.4 0.2 Excitation Energy [MeV] Sr-8 0 0 0 20 30 Excitation Energy [MeV] n 2n 3n 4n 5n 6n n 2n 3n 4n 5n 6n 7n 8n
Monte Carlo Simulation for Particle Emission Application of Monte Carlo to Hauser-Feshbach Model at LANL β-delayed γ prompt fission neutron spectrum n-γ correlation γ-ray competition γ-ray cascading γ-ray multiplicity dist. Intensity % / Capture / MeV 25 20 5 0 5 59 Co(p,2γ) 60 Ni, Ep =.9 MeV Voinov (2008) CGM GL CGM LE 0 0 2 4 6 8 0 Gamma-ray Energy [MeV] A preliminary result of the γ-ray spectra from proton capture for the multiplicity 3 case (2 γ- rays and the 2 + 0 + transition). Data taken from the proc. of Yosemite conference, 2008
Concluding Remarks Nuclear Reaction Modeling: Recent Development for Astrophysics Compound Nuclear Reaction Improved Hauser-Feshbach model for capture reaction, including nuclear deformation effect, which supports experimental data at LANSCE with DANCE. Direct/Semidirect Nucleon Capture The Hartree-Fock BCS theory for the DSD process has a potential to predict neutron capture cross-sections near the neutron drip-line. Extension of the Hauser-Feshbach Model Applications Beta-delayed and EC neutron emission Prompt fission neutron spectra Monte Carlo approach : correlations of particle emissions