Optical potentials and knockout reactions from Green functions treatment Andrea Idini “Connecting Bound States to the Continuum” FRIB, 11-22 March 2018 Andrea Idini
Green functions many-body method 9 6 " 7 8 3 6 5 79: 6 5 79: 8 4 7 6 " Källén–Lehmann ! 34 # + %& = ( 79: + + " 7 + %& # − + 5 spectral 5 7 8 3 9 6 ) 7;: 6 ) 7;: 8 4 6 " representation 7 6 " + ( 7 + + ) 7;: − %& # − + " ) Unperturbed case 1 ! " # + %& = ( ./01 ± %& + − - ) ) Green function self-consistent methods find spectra of the Hamiltonian operator H ( A ) = T � T c.m. ( A + 1) + V + W 15/06/2018 Andrea Idini
Green functions many-body method Dyson Equation % & + '( = % ) & + '( + % ) & + '( Σ ∗ & + '( %(& + '() ‘‘Dressed’’ (with correlation) H ( A ) = T � T c.m. ( A + 1) + V + W Particle Propagator H Σ ∗ = + ‘‘Bare’’ Propagator H 15/06/2018 Andrea Idini
Green functions many-body method Dyson Equation % & + '( = % ) & + '( + % ) & + '( Σ ∗ & + '( %(& + '() ‘‘Dressed’’ (with correlation) Particle Propagator Interaction between the particle and the system Σ ∗ = + (physical choice) Fragments and changes energy of the ‘‘bare’’ state ‘‘Bare’’ Propagator / / 0 - 0 , Σ ,- & + '( = . & − 2 / + '( / 15/06/2018 Andrea Idini
Green functions many-body method Dyson Equation , : + ;< = , = : + ;< + , = : + ;< Σ ∗ : + ;< ,(: + ;<) Σ ∗ Equation of motion = + % + ℏ ' , -, - / ; %, Γ = 2 - − - / + ∫ 5-′′Σ ∗ -, - // ; %, Γ ,(- // , -; %, Γ) ' 2) ∇ + Corresponding Hamiltonian 9 -, - / = − ℏ ' ' + Σ ∗ -, - / ; %, Γ 2) ∇ + Σ corresponds to the Feshbach’s generalized optical potential Escher & Jennings PRC66 034313 (2002) 15/06/2018 Andrea Idini
Hamiltonian method: self consistent Green functions % & + '( = % ) & + '( + % ) & + '( Σ ∗ & + '( %(& + '() Faddeev RPA Particle hole ADC(3) ‘polarization’ HF propagator ADC(1) (ph-RPA) n p Σ ∗ = + Particle-particle (pp-RPA) two-body correlation ‘ladder’ propagator Courtesy of C. Barbieri 15/06/2018 Andrea Idini
(Non) Hamiltonian method: nuclear field theory ansatz Collective Phonon Independent Particle mean field Random Phase Approximation 15/06/2018 Andrea Idini
Self Consistent Green Function Nuclear Field Theory Σ ∗ = + Σ ∗ = Σ ∗ = coupling from Hamiltonian Coupling of physical quantities matrix elements Exploits different truncations Full single valence space ((), &) self-energy particle Vertices (optical potential) structure (ME) Vertices Dyson Σ ∗ (&) Summation Equation Building Constitues Central (&) Blocks Part
How the imaginary part arises in dissipative systems 12 0 "# $ + &' = $ + &' − Σ "# $ + &' * ($) * ($)+ # + " Σ "# $ + &' = ) $ − / * + &' * Complex roots of the Green function Implemented in NFT 1 st Iteration HF Strength FWHM ∝ ' Energy Energy 15/06/2018 Andrea Idini
How the imaginary part arises in dissipative systems 12 0 "# $ + &' = $ + &' − Σ "# $ + &' * ($) * ($)+ # + " Σ "# $ + &' = ) $ − / * + &' * Complex roots of the Green function Implemented in NFT 1 st Iteration HF Convergence, Strength ' → 0 FWHM ∝ ' FWHM ∝ 4+{Σ} Energy Energy Energy 15/06/2018 Andrea Idini
Nucleon elastic scattering Nu E The irreducible self-energy is a nucleon- nucleus optical potential* , - # ,∗ - " * + + ∗ Σ "# % + '( = Σ "# % − / , ± '( Σ 123 , correlated mean-field resonances beyond mean-field / 5 Correlations Σ * Σ 143 *Mahaux & Sartor, Adv. Nucl. Phys. 20 (1991), Escher & Jennings PRC66:034313 (2002) Courtesy of C. Barbieri 15/06/2018 Andrea Idini
%,&∗ ()) - Solve Dyson equation in HO Space, find Σ "," $ Σ %,&∗ (+, + , , )) - diagonalize in full continuum momentum space + / 21 2 %,& + + ∫ 4+ , + ,/ Σ %,&∗ +, + , , ) 2 %,& +′ = E 2 %,& (+) N max %,&∗ ()) Σ "," $ Σ ∗ = + 15/06/2018 Andrea Idini
Knockout Sp Kn Spect ctrosc scopic Factor ors . - 20 1 2,4 . + ∫ 7. 8 . 8- Σ 2,4∗ ., . 8 , ; 1 2,4 .′ = E 1 2,4 (.) - '() Φ *.,. ' !" = $ Φ & Norm of overlap wavefunctions But also the shape of the overlap wavefunction! Collaboration with C. Bertulani 15/06/2018 Andrea Idini
Con Conclusion ons (1 (1) • The non-local generalized optical potential corresponding to nuclear self energy can be calculated in several, different, ways. • Imaginary part can arise spontaneously in non-hamiltonian cases. • Reaction properties calculated from bound state description might differ from effective pure single-particle description. Convergence, Strength ' → 0 FWHM ∝ "#{Σ} Energy 15/06/2018 Andrea Idini
SRG-N 3 LO, Λ = 2.66 fm *+ , + +. O 0. 1. Navràtil, Roth, Quaglioni, Σ " PRC82, 034609 (2010) 15/06/2018 Andrea Idini
NNLO sat 3 / 2 + 5 / 2 + ! + #$ O &. (. +)*+ 200 1 / 2 + δ (deg) 100 0 − 100 0 . 0 2 . 5 5 . 0 7 . 5 10 . 0 12 . 5 15 . 0 E c.m. (MeV) 3 / 2 − 1 / 2 − n p 5 / 2 − 7 / 2 − 200 δ (deg) 100 0 − 100 0 2 4 6 8 10 12 14 16 E c.m. (MeV) 15/06/2018 Andrea Idini
Using the ab initio optical potential for neutron elastic scattering on Oxygen 10 Lister and Sayres, Phys Rev 143, 745 1 0.1 0.01 0.001 0 20 40 60 80 100 120 140 160 180 15/06/2018 Andrea Idini
! " = 2.76 MeV 10 1 0.1 0.01 0 20 40 60 80 100 120 140 160 180 15/06/2018 Andrea Idini
Overlap function - (# - Ψ " # = %∫ '# ( … '# * Φ *,( (# ( , … , # *,( )Φ * ( , … , # * ) # # " " Proton particle-hole gap EM results from A. Cipollone PRC 92 , 014306 (2015) 15/06/2018 Andrea Idini
Kn Knockout Sp Spect ctrosc scopic Factor ors . - 20 1 2,4 . + ∫ 7. 8 . 8- Σ 2,4∗ ., . 8 , ; 1 2,4 .′ = E 1 2,4 (.) - '() Φ *.,. ' !" = $ Φ & Calculated from overlap wavefunctions 100 Oxygen Chain Spectroscopic Factor (%) 80 60 O14 O16 40 O22 20 O24 0 0 5 10 15 20 25 30 Separation Energy (MeV) open circles neutrons, closed protons 15/06/2018 Andrea Idini
Ov Overlap wa wavefunctions Collaboration with C. Bertulani 15/06/2018 Andrea Idini
r 2 ↵ 1 / 2 r 2 ↵ 1 / 2 σ W S σ GF σ W S σ GF C 2 S GF ⌦ ⌦ Nucleus C W S C GF E B qf qf kn kn W S GF [fm − 1 / 2 ] [fm − 1 / 2 ] [mb] (state) [MeV] [fm] [fm] [mb] [mb] [mb] 14 O ( π 1p 3 / 2 ) 8.877 2.856 2.961 6.785 7.172 27.38 28.60 27.19 27.42 0.548 Deviation of quasifree !, !# 5% <1% cross section calculation for different wavefunctions (% &' − % )* )/% )* Collaboration with C. Bertulani 15/06/2018 Andrea Idini
Con Conclusion ons an and Pe Perspectives • We are developing an interesting tool to study nuclear reactions effectively. We have defined a non-local generalized optical potential corresponding to nuclear self energy. • Spectroscopic Factors from ab-initio overlap wavefunctions differ from effective wood saxon. These do not seem to depend much on proton-neutron asymmetry 10 Lister and Sayres, Phys Rev 143, 745 1 0.1 0.01 0.001 0 20 40 60 80 100 120 140 160 180 15/06/2018 Andrea Idini
Wh Why op optical po potentials tials? - Optical potentials reduce many-body 33 10 40 40 n+ Ca p+ Ca complexity decoupling structure E > 100 lab 26 10 40 < E < 100 lab 20 < E < 40 lab contribution and reactions dynamics. [mb/sr] 10 < E < 20 lab 0 < E < 10 19 10 lab - Often fitted on elastic scattering data Ω /d σ 12 10 d (locally or globally) 5 10 - A microscopic model is difficult but worth it 0 50 100 150 0 50 100 150 [deg] θ cm Dickhoff, Charity, Mahzoon, JPG44, 033001 (2017) Koning, Delaroche, NPA713, 231 (2002) Fig. 2. Comparison of predicted neutron total cross sections and experimental data, for nuclides in the Mg–Ca mass region, for the energy range 10 keV–250 MeV. For 15/06/2018 Andrea Idini
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