Statistical Model Calculations for Neutron Radiative Capture Process

Transcription

1 Statistical Nuclear Physics and its Applications in Astrophysics, Jul. 8-, 2008 Statistical Model Calculations for Neutron Radiative Capture Process T. Kawano T-6 Nuclear Physics Los Alamos National Laboratory

2 Introduction Applications of Statistical Model The Hauser-Feshbach statistical theory has been widely used for nuclear reaction data applications, nuclear reaction rates for astrophysics nuclear data libraries for energy applications however, questions still persist regarding: applicability in the domain of off-stability global inputs level densities, transmission coefficients, γ strength function nuclear deformation effect etc Possible other areas where the HF can be adopted β-delayed or EC neutron emission sequential decay of fission fragments Monte Carlo approach to the HF

3 Hauser-Feshbach on Deformed Nuclei Neutron Reaction on Deformed Nuclei at Low Energies Incorporate Coupled-Channels (CC) method into Hauser-Feshbach formula 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) Nishioka-Weidenmüller-Yoshida, GOE for coupled-channels I will report : Coupled-Channels Hauser-Feshbach (CCHF) method Apply to calculate neutron capture cross sections for actinides (no fissile)

4 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 s-wave (resonance parameters) optical model W s = 2.59 MeV for E n <.3 MeV R = fm (9.6± in Atlas, Mughabghab) S 0 =.3 4 ((.29± 3) 4, ibid) S = ((2.7± 9) 4, ibid) Original SoukhovitskiĩPotential (in the paper) R = 9.57 fm S 0 = S =.80 4 Real

5 U-238 Total Cross Section With Modified Soukhovitskiĩ Potential Total (resonance parameters) I=5keV s-wave p-wave CC calc. 00 I=5keV CC calc. Total Cross Section [b] 0 Total Cross Section [b] Neutron Energy [ev] 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.

6 Detailed Balance for Compound Reaction Neutron Emission Probabilities Transmission CN Usual HF calculation The target is assumed to be a ground state, but the neutron energy is shifted CN Detailed balance The target is at the excited state, coupled to 0+ and 4+ states The same transmission coefficients are used for both entrance and exit channels. Sometimes exit channels are corrected by a factor of + σd /σ R. Solve the Coupled-Channels equations for the excited state, which couples to the negative energy states (super-inelastic).

7 Transmission for Inverse Reactions Transmission Coefficients for the Inverse Channels Transmission Coefficients Ground State st Excited 2nd Excited Neutron Energy [MeV] s-wave Transmission Coefficients Ground State st Excited 2nd Excited 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.

8 Direct Cross Sections Coupled-Channels Calculations for the Excited States Shape Elastic Scattering [mb] GS st 2nd Direct Inelastic Scattering [mb] GS -> st GS -> 2nd st -> GS 2nd -> GS Neutron Energy [MeV] Elastic Scattering Preliminary Neutron Energy [MeV] Inelastic Scattering

9 Capture Cross Section Calculations Model Parameters Optical Potential Modified Soukhovitskiĩ CC calculations are made for the G.S. band 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 00 0 σ 2 = 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]

10 U-238 Results Comparison with ENDF/B-VII (a part of standards evaluation) Capture Cross Section [mb] 0 Capture Cross Section [mb] 0 CC ENDF/B-VII Energy [MeV] CC ENDF/B-VII Energy [MeV] Γ γ =23.36 mev Γ γ =7.83 mev

11 Partial Wave Contribution Decomposition into Each Partial-Wave Contribution 00 Capture Cross Section [mb] 0 CC s-wave p-wave d-wave ENDF/B-VII Energy [MeV]

12 Comparisons with DANCE Data Capture Cross Section [b] 0 Weston (98) Kobayashi (2002) DANCE Radiative Width 40.7 mev Radiative Width 3.2 mev Capture Cross Section [b] 0 DANCE Gayther (977) Vanpraet (986) ENDF/B-VII (640 group) Radiative Width 45 mev 0 00 Neutron Incident Energy [kev] 0 00 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.

13 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 ENDF Decay Lib 0 kev Br-87 Delayed Neutron Spectrum [/MeV] 0.0 ENDF Decay Lib 0 kev Br-88 Q-beta Neutron Separation Energy Delayed Gamma Delayed Neutron Spectrum [/MeV] Delayed Neutron Energy [MeV] ENDF Decay Lib 0 kev Br-89 Delayed Neutron Spectrum [/MeV] Delayed Neutron Energy [MeV] ENDF Decay Lib 0 kev Br-90 Nuclear Structure Model Nuclear Reaction Model Delayed Neutron Energy [MeV] 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.

14 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 Si Excitation Energy [MeV] Sr Excitation Energy [MeV] n 2n 3n 4n 5n 6n n 2n 3n 4n 5n 6n 7n 8n

15 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 Co(p,2γ) 60 Ni, Ep =.9 MeV Voinov (2008) CGM GL CGM LE 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 transition). Data taken from the proc. of Yosemite conference, 2008

16 Concluding Remarks Compound Nuclear Reaction Modeling: Recent Development Improved Hauser-Feshbach model for capture reaction, including nuclear deformation effect, which supports experimental data at LANSCE with DANCE. Detailed balance for inverse channels. Width fluctuation correction still approximation, KKM or NWY needed Extension of the Hauser-Feshbach Model Applications Delayed neutron emission Prompt fission neutron spectra Monte Carlo approach : correlations of particle emissions This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396.

17 U-238 Capture Cross in ENDF/B-VII IAEA Standards Evaluation Though the 238 U capture cross was evaluated at the IAEA standards projects, the results were not included in the standards data library. We re-fitted the HF model calculation to the standards results for ENDF/B-VII. Capture Cross Section [b] Standard ENDF/B-VII JENDL-3.3 Capture Cross Section [b] Standard ENDF/B-VII JENDL Energy [MeV] Energy [MeV]