Asymmetry dependence of Gogny-based optical potential

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Asymmetry dependence of Gogny-based optical potential G. Blanchon, R. Bernard, M. Dupuis, H. F. Arellano CEA,DAM,DIF F-9297 Arpajon, France March 3-6 27, INT, Seattle, USA / 32

Microscopic ingredients for nuclear reaction codes (TALYS): Nuclear level densities γ-ray strength functions Preequilibrium Density and transition density Optical Potential S. Hilaire S. Péru M. Dupuis S. Péru, N. Pillet G. Blanchon TALYS: A.J. Koning, S. Hilaire, M. Duijvestijn, in Proceeding of the International Conference on Nuclear Data for Science and Technology-ND27 (EDP Sciences, Paris, France, 28), pp. 2 24 2 / 32

Ab-initio from bare NN interaction (Beyond) Mean-Field from effective NN interaction Optical Potential Cross Section Phenomenological potentials from data fitting Nuclear Structure Method for scattering N. Vinh Mau, Theory of nuclear structure (IAEA, Vienna) p. 93 (97) 3 / 32

NSM & EDF extended reach Spherical HF ( nuclei) HF+RPA Potential + Integrodifferential Schrödinger G. Blanchon, M. Dupuis, H. Arellano, N. Vinh Mau, Phys. Rev. C (25) G. Blanchon, M. Dupuis, H. Arellano, Eur. Phys J. A, 5 2 (25) 65 Reaction Observables 4 / 32

NSM & EDF extended reach Spherical HFB ( 3 nuclei) HFB+QRPA Potential + Nonlocal Coupled channels Collaboration with R. Bernard and S. Péru (CEA,DAM,DIF) Reaction Observables 4 / 32

NSM & EDF extended reach Deformed HFB ( 6 nuclei) HFB+QRPA deformed potential + Nonlocal Coupled channels Amine Nasri PhD CEA,DAM,DIF Non-local microscopic potentials for calculation of scattering observables of nucleons on deformed nuclei Reaction Observables Collaboration with H. Arellano 4 / 32

NUCLEAR STRUCTURE METHOD FOR SCATTERING 5 / 32

Nuclear Structure Method V = V HF + V RPA Mean Field Target s excitations Limitation: No two-fold charge exchange (n, p, n) and (p, n, p) in the present version of NSM 6 / 32

From NN interaction to the optical potential Field s operator dynamical equation Green s function dynamical equation link between G n, G n and G n+ Dyson equation and Self-energy Optical Potential Hartree-Fock Approximation Hartree-Fock & Random-Phase Approximation 7 / 32

Nuclear Structure Method Optical potential V = V HF + V PP + V RPA 2V (2) Bare Interaction + pp + ph 2 8 / 32

Nuclear Structure Method Optical potential V = V HF + V PP + V RPA 2V (2) Bare Interaction + pp + ph kopkdpzeokdok }{{} 2 Effective Interaction + Im[ ] + ph 2 8 / 32

Nuclear Structure Method Optical potential V = V HF + V PP + V RPA 2V (2) Bare Interaction + pp + ph kopkdpzeokdok }{{} 2 Gogny Interaction + Im[ ] + ph 2 Gogny DS interaction has been designed anticipating the inclusion of particle-vibration couplings 8 / 32

Self-consistency NN interaction NN interaction ρ V HF (ρ) ρ V HF (ρ) + V (ρ) Schrödinger equation Schrödinger equation HF Coordinate representation Treatment of the continuum (SP resonances) RPA Oscillator basis including 5 major shells Excited states up to J = 8 with both parities The resulting potential is nonlocal, complex and energy dependent 9 / 32

Coupling to a single target excited state p+ 4 Ca scattering Coupling to the first state of 4 Ca with E = 9.7 MeV σ R (mb),7,6,5,4,3,2, j=3/2 l= E λ = 2.5 MeV j=5/2 l=3 E λ = 5.65 MeV j=/2 l= E λ = 3.7 MeV DW Coul j=9/2 l=4 E λ = 9.55 MeV 2 4 E inc 6 (MeV) 8 2 Importance of the intermediate single particle resonances Strong impact on reaction cross section Intermediate HF propagator HF + RPA excitation HF phaseshift δ (rad/π) 5/2 2 3/2 j=/2 l= j=/2 l= 4 j=3/2 l= Ca(p,p) Phaseshift j=3/2 l=2 j=5/2 l=2 j=5/2 l=3 j=7/2 l=3 j=7/2 l=4 j=9/2 l=4 j=9/2 l=5 HF unocc occ /2 E (MeV) / 32

NUCLEON SCATTERING OFF N=Z TARGET NUCLEUS / 32

Elastic scattering n/p + 4 Ca Experimental Cross sections Microscopic potential Phenomenological potential Structure Experimental spectroscopy Effective interaction 2 / 32

Integral cross sections p + 4 Ca σ R (mb) n + 4 Ca 8 6 4 2 Exp. KD V HF + V (a) 5 5 2 25 3 35 4 E (MeV) n/p + 4 Ca Compound elastic from Haüser-Feshbach with Koning-Delaroche potential Use of phenomenological width for the excited states of the target. 4 35 Exp. KD V HF + V σ Tot (mb) 3 25 2 (b) 5 2 3 4 E (MeV) 3 / 32

Integral cross sections p + 4 Ca σ R (mb) 8 6 4 2 Exp. KD V HF + V (a) 5 5 2 25 3 35 4 E (MeV) n/p + 4 Ca Compound elastic from Haüser-Feshbach with Koning-Delaroche potential Use of phenomenological width for the excited states of the target. dσ/dω (mb/sr) 4 3 2 σ Tot (.6 Γ) = 247 mb σ Tot (Γ) = 295 mb σ Tot (.4 Γ) = 223 mb Exp. Γ x.6 Γ Γ x.4 n + 4 Ca 4 35 Exp. KD V HF + V 2 4 6 8 2 4 6 8 θ c.m (deg.) Width sensitivity (n+ 4 Ca @ 25 MeV) σ Tot (mb) 3 25 2 (b) 5 2 3 4 E (MeV) 3 / 32

Integral cross sections p + 4 Ca σ R (mb) 8 6 4 2 Exp. KD V HF + V (a) 5 5 2 25 3 35 4 E (MeV) n/p + 4 Ca Compound elastic from Haüser-Feshbach with Koning-Delaroche potential Use of phenomenological width for the excited states of the target. dσ/dω (mb/sr) 4 3 2 σ Tot (.6 Γ) = 247 mb σ Tot (Γ) = 295 mb σ Tot (.4 Γ) = 223 mb Exp. Γ x.6 Γ Γ x.4 n + 4 Ca σ Tot (mb) 4 35 3 25 Exp. KD V HF + V 2 4 6 8 2 4 6 8 θ c.m (deg.) Width sensitivity (n+ 4 Ca @ 25 MeV) Perspective: microscopic determination of energy widths and shifts: 2p-2h coupling (N. Pillet) 2 (b) 5 2 3 4 E (MeV) 3 / 32

Cross section and Analyzing powers n/p+ 4 Ca dσ/dω (mb/sr) σ(θ)/σ Ruth 4 2 8 6 4 2-2 -4 4 2-2 -4-6 -8-2.6 3.29 5.3 5.88 6.5 7.9 9.9 3.9 6.9 9. 2.7 25.5 3.3 4. 2 4 6 8 2 4 6 8 θ c.m. (deg.) 9.86.37 3.49 4.5 5.97 8.57 9.57 2. 23.5 25. 26.3 27.5 3.3 4. 2 4 6 8 2 4 6 8 θ c.m. (deg.) A y (θ) A y (θ).5 -.5 -.5 -.5 -.5 -.5 -.5 9.9. 3.9 6.9 -.5 (a) - 2 4 6 8 2 4 6 8 θ c.m. (deg).5 -.5 -.5 -.5 -.5 -.5 -.5 4.5 5.97 8.57 4. -.5 (b) - 2 4 6 8 2 4 6 8 θ c.m. (deg) NSM (full line) Koning-Delaroche (dashed line) Good agreement with cross section data below 3 MeV. In terms of energy regime, NSM is complementary to g-matrix approaches. Good agreement with analyzing powers data: correct behaviour of the spin-orbit term of the potential. Effective interaction fitted from structure data + fission barriers 4 / 32

Microscopic and phenomenological potentials Experimental Cross sections Microscopic potential Phenomenological potential Structure Experimental spectroscopy Effective interaction 5 / 32

Potential for n + 4 Ca @ MeV NSM potential Non Local Dispersive (NLD) potential fitted on all the available data for 4 Ca ν lj (r, r ) = dˆrdˆr Y m jl (ˆr)V (r, r )Y m jl (ˆr ) NLD V HF + V r 2 Im[ν(r,r)] (MeV/fm) 6 5 4 3 2 2 Im[r r ν(s)] (MeV/fm) 6 4 2-2 6 4 2-2 6 4 l= j=/2 l= j=/2 l= j=3/2 l=2 j=3/2 l=2 j=5/2 l=3 j=5/2 PW 3 4 5 6 7 8 7 6 5 4 2 3 r (fm) 2-2 l=3 j=7/2-3 -2-2 3 l=4 j=7/2-3 -2-2 3 s (fm) l=4 j=9/2-3 -2-2 3 Figure : s = r r NLD: M.H. Mahzoon et al., Phys. Rev. Lett. 2, 6253 (24) 6 / 32

Phenomenological potential and effective interaction Experimental Cross sections Microscopic potential Phenomenological potential Structure Experimental spectroscopy Effective interaction 7 / 32

Phenomenological potential and Gogny interaction Volume integral: J lj V = 4π A dr r 2 dr r 2 ν lj (r, r ) Volume integral 8 7 6 V HF Hartree PB J V (MeV fm 3 ) 5 4 3.5 MeV 2 5 MeV MeV 7 MeV 4 MeV 5 5 2 25 PW PB: Perey Buck optical potential with gaussian non locality and energy independent. 8 / 32

Phenomenological potential and Gogny interaction Volume integral: Total cross section J lj V = 4π A dr r 2 dr r 2 ν lj (r, r ) Volume integral 4 2 n+ 4 Ca.5 MeV 8 7 6 V HF Hartree PB σ T 8 6 4 J V (MeV fm 3 ) 5 4 3 2 5 5 2 25 PW.5 MeV 2 5 MeV MeV 7 MeV 4 MeV 5 5 2 25 PW PB: Perey Buck optical potential with gaussian non locality and energy independent. 8 / 32

Phenomenological potential and Gogny interaction Volume integral: Total cross section J lj V = 4π A dr r 2 dr r 2 ν lj (r, r ) Volume integral 4 2 n+ 4 Ca 5.3 MeV 8 7 6 V HF Hartree PB σ T 8 6 4 J V (MeV fm 3 ) 5 4 3 2 5 5 2 25 PW.5 MeV 2 5 MeV MeV 7 MeV 4 MeV 5 5 2 25 PW PB: Perey Buck optical potential with gaussian non locality and energy independent. 8 / 32

Phenomenological potential and Gogny interaction Volume integral: Total cross section J lj V = 4π A dr r 2 dr r 2 ν lj (r, r ) Volume integral 4 2 n+ 4 Ca 9.9 MeV 8 7 6 V HF Hartree PB σ T 8 6 4 J V (MeV fm 3 ) 5 4 3 2 5 5 2 25 PW.5 MeV 2 5 MeV MeV 7 MeV 4 MeV 5 5 2 25 PW PB: Perey Buck optical potential with gaussian non locality and energy independent. 8 / 32

Phenomenological potential and Gogny interaction Volume integral: Total cross section J lj V = 4π A dr r 2 dr r 2 ν lj (r, r ) Volume integral 4 2 n+ 4 Ca 6.9 MeV 8 7 6 V HF Hartree PB σ T 8 6 4 J V (MeV fm 3 ) 5 4 3 2 5 5 2 25 PW.5 MeV 2 5 MeV MeV 7 MeV 4 MeV 5 5 2 25 PW PB: Perey Buck optical potential with gaussian non locality and energy independent. 8 / 32

Phenomenological potential and Gogny interaction Volume integral: Total cross section J lj V = 4π A dr r 2 dr r 2 ν lj (r, r ) Volume integral 4 2 n+ 4 Ca 3.3 MeV 8 7 6 V HF Hartree PB σ T 8 6 4 J V (MeV fm 3 ) 5 4 3 2 5 5 2 25 PW.5 MeV 2 5 MeV MeV 7 MeV 4 MeV 5 5 2 25 PW PB: Perey Buck optical potential with gaussian non locality and energy independent. 8 / 32

Phenomenological potential and Gogny interaction Volume integral: Total cross section J lj V = 4π A dr r 2 dr r 2 ν lj (r, r ) Volume integral 4 2 n+ 4 Ca 4. MeV 8 7 6 V HF Hartree PB σ T 8 6 4 J V (MeV fm 3 ) 5 4 3 2 5 5 2 25 PW.5 MeV 2 5 MeV MeV 7 MeV 4 MeV 5 5 2 25 PW PB: Perey Buck optical potential with gaussian non locality and energy independent. 8 / 32

Phenomenological potential and Gogny interaction Volume integral: J lj V = 4π A dr r 2 dr r 2 ν lj (r, r ) Volume integral 8 7 6 V HF Hartree PB J V (MeV fm 3 ) 5 4 3.5 MeV 2 5 MeV MeV 7 MeV 4 MeV 5 5 2 25 PW PB: Perey Buck optical potential with gaussian non locality and energy independent. NSM limited to incident energies below about 3 MeV 8 / 32

NUCLEON SCATTERING OFF N>Z TARGET NUCLEUS 9 / 32

σ(θ)/σ Ruth dσ/dω (mb/sr) Cross section and Analyzing powers n/p+ 48 Ca 7 6 5 4 3 2 - -2-3 -4 7 6 5 4 3 2 - -2 (a) n + 48 Ca 2.35 3.5 3.55 4.7 2 4 6 8 2 4 6 8 θ c.m. (deg.) 8.. 2. 4.3 2. 3. p + 48 Ca 6. (b) 7.97.9 6.8-3 2 4 6 8 2 4 6 8 θ c.m. (deg.) A y (θ) A y (θ).5 -.5 -.5 -.5 -.5 -.5 -.5 -.5 -.5 -.5 n + 48 Ca - 2 4 6 8 2 4 6 8 θ c. m. (deg).5 -.5 -.5 -.5 -.5 -.5 - -.5 -.5 8.. 2. 4.3 5.5 p + 48 Ca (d) 2 4 6 8 2 4 6 8 θ c. m. (deg) NSM (full line) Koning-Delaroche (dashed line) Good agreement with cross section data for n + 48 Ca Lack of absorption for p + 48 Ca 2 / 32

TALYS σ Inel (mb) 2 5 5 8 4 5 5 5 n + 4 Ca p + 4 Ca n + 48 Ca p + 48 Ca 2 3 Tot (n,n ) (n,p) (n,np) (n,α) Tot (p,p ) (p,2p) (p,3p) Tot (n,n ) (n,2n) (n,3n) Tot (p,n) (p,2n) (p,3n) (n,p) (n,n ) (n,α) (p,p ) (p,2p) (n,n ) (n,2n) 2 3 2 3 2 3 4 E (MeV) Direct + Pre-equilibrium (p,3p) (n,3n) (p,n) (p,2n) (p,3n) Compound 3 2 3 2 3 2 3 2 NSM potential describes direct components Need for two-fold charge exchange (p, n, p) 2 / 32

3 25 2 5 5 3 25 2 5 5 2 4 6 8-5 -.5 - -.5.5.5 3 25 2 5 5 3 25 2 5 5 2 4 6 8-5 -.5 - -.5.5.5 3 25 2 5 5 3 25 2 5 5 2 4 6 8-5 -.5 - -.5.5.5 3 25 2 5 5 3 25 2 5 5 2 4 6 8-5 -.5 - -.5.5.5 PB-like equivalent of the imaginary NSM potential p + 48 Ca : (, /2) (, /2) (, 3/2) (2, 3/2) Im[V RPA /2] www(x) Im[V RPA /2 ] wwwun(x) Im[V RPA /2 2] wwwdeux(x) Im[V RPA /2 3] wwwtrois(x) Im[V RPA /2 nl.5 ] wwwnl(x) Im[V RPA /2 nl.5 ] wwwnlun(x) Im[V RPA /2 nl.5 ] 2 wwwnldeux(x) Im[V RPA /2 nl.5 ] 3 wwwnltrois(x) Im[V RPA /2 nl 4.5] wwwnl2(x) Im[V RPA /2 nl 4.5] wwwnl2un(x) Im[V RPA /2 nl 4.5] 2 wwwnl2deux(x) Im[V RPA /2 nl 4.5] 3 wwwnl2trois(x) Perey-Buck ansatz : W n/p (r, r ; E) = H(s, β v )Wv n/p (E)f (R, r v, a v ) + 4H(s, β s)ws n/p (E)a sf (R, r s, a s) [ ( )] r r A /3 f (r, r, a) = + exp WS form factor a ( ) H(s, β) = π 3/2 β 3 exp s β Gaussian nonlocality with R = (r + r )/2 and s = r r 22 / 32

PB-like equivalent of the imaginary NSM potential W n/p (r, r ; E) = H(s, β v )Wv n/p (E)f (R, r v, a v ) + 4H(s, β s)ws n/p (E)a sf (R, r s, a s) Fit parameters : r v r s a v a s β v β s.78.254.49 ( 4 Ca).78 ( 48 Ca).44.35. dσ/dω (mb/sr) Check of reliability of the fit : PB-like pot NSM p + 48 Ca @ 3 MeV. 2 4 6 8 2 4 6 8 θ (deg.) 23 / 32

PB-like equivalent of the imaginary NSM potential Magnitudes : 35 35 3 n W v n W s 4 Ca 3 n W v n W s 48 Ca 25 p W v p W s 25 p W v p W s W (MeV) 2 5 W (MeV) 2 5 5 5 5 2 25 3 35 4 45 5 55 6 E-E F (MeV) 5 2 25 3 35 4 45 5 55 6 E-E F (MeV) Lane consistency (already discussed by Osterfeld) Proton/neutron asymmetry of the target yields a bigger proton surface potential than the neutron one Volume imaginary potential decreases with asymmetry for protons and neutrons 24 / 32

NSM/Lane potential Approximate method to recover proton absorption Lane model assumes isospin symmetry in nuclei : V (n/p) = V ± (N Z) 2A V with V n/p (E E n/p F ) in reality, one gets V cc : isospin non-conserving Coulomb corrections in the second order term V cc in the case of 4 Ca V n V p = N Z A V V cc 6 5 data NSM KD NSM/Lane p + 48 Ca @ 2 MeV V n ( 4 Ca) V p ( 4 Ca) V FIT ( 4 Ca) and assuming NSM provides nice results for n+ 48 Ca scattering σ(θ)/σ Ruth 4 3 2 V p NSM/Lane (48 Ca) = 2V FIT ( 48 Ca) VNSM n (48 Ca) 2 4 6 8 2 4 6 8 θ c.m. (deg.) 25 / 32

NUCLEON SCATTERING OFF TARGET WITH PAIRING 26 / 32

NSM for spherical targets with pairing V = V HFB + V QRPA Mean Field Target s excitations 27 / 32

NSM & EDF extended reach Spherical HF ( nuclei) HF+RPA Potential + Integrodifferential Schrödinger G. Blanchon, M. Dupuis, H. Arellano, N. Vinh Mau, Phys. Rev. C (25) G. Blanchon, M. Dupuis, H. Arellano, Eur. Phys J. A, 5 2 (25) 65 Reaction Observables 28 / 32

NSM & EDF extended reach Spherical HFB ( 3 nuclei) HFB+QRPA Potential + Nonlocal Coupled channels Collaboration with R. Bernard and S. Péru (CEA,DAM,DIF) Reaction Observables 28 / 32

HFB equations in coordinate representation d 3 r ( h(rσ, r σ ) (rσ, r σ ) ( ) u(e, r σ ) ( ) ( ) ) E + λ u(e, rσ) σ (rσ, r σ ) h(rσ, r σ ) v(e, r σ = ) E λ v(e, rσ) HFB with DS Gogny interaction in coordinate representation in a box with treatment of the continuum.2.5 "fort.6789" u :2 "fort.679" u :2 "fort.6789" u :3 "fort.679" u :3..5 -.5 -. -.5 NEXT STEP: Dressed mean field & pairing field using QRPA Application to scattering... -.2 5 5 2 25 3 Effect of bound states on the cross section 29 / 32

CONCLUSION 3 / 32

Conclusions Gogny interaction is connected to reaction observables Need for double-charge-exchange component in NSM Results on asymmetry (submitted to EPJA) Study of all doubly-closed-shell target nuclei (in progress) Account of pairing (in progress) 3 / 32

Potential based on effective interaction Nuclear Structure Method N. Vinh Mau, Theory of nuclear structure (IAEA, Vienna 97) p. 93. N. Vinh Mau, A. Bouyssy, NPA 257(2), 89 (976). V. Bernard, N. Van Giai, NPA 327(2), 397 (979). F. Osterfeld, J. Wambach, V.A. Madsen, PRC 23(), 79 (98). K. Mizuyama et al., PRC 86, 463 (22). Y. Xu et al., JPG 4, 5 (24). G. Blanchon, M. Dupuis, H. Arellano and N. Vinh Mau, PRC 9, 466 (25). G. Blanchon, M. Dupuis, H. Arellano, Eur. Phys J. A, 5 2 (25) 65 T. V. Nhan Hao et al., PRC 92, 465 (25). 32 / 32

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