Shell model calculations for exotic nuclei with realistic potentials: reliability and predictiveness Luigi Coraggio Istituto Nazionale di Fisica Nucleare - Sezione di Napoli NUSPIN 2017 June 28th, 2017 - GSI, Darmstadt Luigi Coraggio NUSPIN 2017 Workshop
A. Covello (UNINA and INFN) A. Gargano (INFN) N. Itaco (UNINA2 and INFN) T. T. S. Kuo (SUNY at Stony Brook, USA) L. C. (INFN) Luigi Coraggio NUSPIN 2017 Workshop
Part I The theoretical framework Luigi Coraggio NUSPIN 2017 Workshop
Introductory remark What is a realistic effective shell-model hamiltonian ? Luigi Coraggio NUSPIN 2017 Workshop
An example: 19 F 19 F 9 protons & 10 neutrons interacting d3/2 d3/2 spherically symmetric mean d5/2 d5/2 field (e.g. harmonic oscillator) s1/2 s1/2 1 valence proton & 2 valence model space neutrons interacting in a p1/2 p1/2 p3/2 p3/2 truncated model space 16 O s1/2 s1/2 protons neutrons The degrees of freedom of the core nucleons and the excitations of the valence ones above the model space are not considered explicitly. Luigi Coraggio NUSPIN 2017 Workshop
Effective shell-model hamiltonian The shell-model hamiltonian has to take into account in an effective way all the degrees of freedom not explicitly considered Two alternative approaches phenomenological microscopic V NN (+ V NNN ) ⇒ many-body theory ⇒ H eff Definition The eigenvalues of H eff belong to the set of eigenvalues of the full nuclear hamiltonian Luigi Coraggio NUSPIN 2017 Workshop
Workflow for a realistic shell-model calculation Choose a realistic NN potential ( NNN ) 1 Determine the model space better tailored to study the system 2 under investigation Derive the effective shell-model hamiltonian by way of the 3 many-body theory Calculate the physical observables (energies, e.m. transition 4 probabilities, ...) Luigi Coraggio NUSPIN 2017 Workshop
Realistic nucleon-nucleon potential: V NN Strong short-range repulsion Several realistic potentials χ 2 / datum ≃ 1: CD-Bonn, Argonne V18, Nijmegen, ... How to handle the short-range repulsion ? Brueckner G matrix EFT inspired approaches V low − k SRG chiral potentials Luigi Coraggio NUSPIN 2017 Workshop
Realistic nucleon-nucleon potential: V NN Strong short-range repulsion Several realistic potentials χ 2 / datum ≃ 1: CD-Bonn, Argonne V18, Nijmegen, ... How to handle the short-range repulsion ? Brueckner G matrix EFT inspired approaches V low − k SRG chiral potentials Luigi Coraggio NUSPIN 2017 Workshop
Realistic nucleon-nucleon potential: V NN Strong short-range repulsion Several realistic potentials χ 2 / datum ≃ 1: CD-Bonn, Argonne V18, Nijmegen, ... How to handle the short-range repulsion ? k ’ Brueckner G matrix k � 2 EFT inspired approaches � 1 V low − k SRG � 0 chiral potentials Luigi Coraggio NUSPIN 2017 Workshop
Realistic nucleon-nucleon potential: V NN Strong short-range repulsion Several realistic potentials χ 2 / datum ≃ 1: CD-Bonn, Argonne V18, Nijmegen, ... How to handle the short-range repulsion ? k ’ Brueckner G matrix k EFT inspired approaches V low − k SRG ! 1 ! 2 ! 0 chiral potentials Luigi Coraggio NUSPIN 2017 Workshop
Realistic nucleon-nucleon potential: V NN Strong short-range repulsion Several realistic potentials χ 2 / datum ≃ 1: CD-Bonn, Argonne V18, Nijmegen, ... How to handle the short-range repulsion ? Brueckner G matrix EFT inspired approaches V low − k SRG chiral potentials Luigi Coraggio NUSPIN 2017 Workshop
The shell-model effective hamiltonian A-nucleon system Schr¨ odinger equation H | Ψ ν � = E ν | Ψ ν � with A � � ( V NN H = H 0 + H 1 = ( T i + U i ) + − U i ) ij i = 1 i < j Model space d | Φ i � = [ a † 1 a † 2 ... a † � n ] i | c � ⇒ P = | Φ i �� Φ i | i = 1 Model-space eigenvalue problem H eff P | Ψ α � = E α P | Ψ α � Luigi Coraggio NUSPIN 2017 Workshop
The shell-model effective hamiltonian H = X − 1 HX PHP PHQ P H P P H Q ⇒ QHP QHQ 0 Q H Q Q H P = 0 H eff = P H P � � 0 0 Suzuki & Lee ⇒ X = e ω with ω = Q ω P 0 1 H eff 1 ( ω ) = PH 1 P + PH 1 Q ǫ − QHQ QH 1 P − 1 ǫ − QHQ ω H eff − PH 1 Q 1 ( ω ) Luigi Coraggio NUSPIN 2017 Workshop
The shell-model effective hamiltonian Folded-diagram expansion ˆ Q -box vertex function 1 ˆ Q ( ǫ ) = PH 1 P + PH 1 Q ǫ − QHQ QH 1 P ⇒ Recursive equation for H eff ⇒ iterative techniques (Krenciglowa-Kuo, Lee-Suzuki, ...) ′ � ′ � � ′ � � � H eff = ˆ Q − ˆ Q + ˆ ˆ ˆ Q − ˆ ˆ ˆ ˆ ˆ Q · · · , Q Q Q Q Q Q Luigi Coraggio NUSPIN 2017 Workshop
The perturbative approach to the shell-model H eff 1 ˆ Q ( ǫ ) = PH 1 P + PH 1 Q ǫ − QHQ QH 1 P The ˆ Q -box can be calculated perturbatively ∞ ( QH 1 Q ) n 1 � ǫ − QHQ = ( ǫ − QH 0 Q ) n + 1 n = 0 The diagrammatic expansion of the ˆ Q -box j j j h a b a b a b a a b j j j p h p p p h 2 (a) (b) 1 2 h h 1 c d c d c d c c c 1 2 3 4 5 j j j a b a b a b a b h 2 p p h h p p h 1 2 1 p h h p j j j 1 2 3* j j c d c d c d c d 6 7 8 9 h p j j 4 5 Luigi Coraggio NUSPIN 2017 Workshop
The shell-model effective operators Consistently, any shell-model effective operator may be calculated It has been demonstrated that, for any bare operator Θ , a non-Hermitian effective operator Θ eff can be written in the following form: ( P + ˆ Q 1 + ˆ Q 1 ˆ Q 1 + ˆ Q 2 ˆ Q + ˆ Q ˆ Θ eff = Q 2 + · · · )( χ 0 + + χ 1 + χ 2 + · · · ) , where d m ˆ � Q m = 1 Q ( ǫ ) ˆ � , � m ! d ǫ m � ǫ = ǫ 0 ǫ 0 being the model-space eigenvalue of the unperturbed hamiltonian H 0 K. Suzuki and R. Okamoto, Prog. Theor. Phys. 93 , 905 (1995) Luigi Coraggio NUSPIN 2017 Workshop
The shell-model effective operators The χ n operators are defined as follows: (ˆ χ 0 = Θ 0 + h . c . ) + Θ 00 , (ˆ Θ 1 ˆ Q + h . c . ) + (ˆ Θ 01 ˆ χ 1 = Q + h . c . ) , Θ 1 ˆ Q 1 ˆ Θ 2 ˆ Q ˆ (ˆ Q + h . c . ) + (ˆ χ 2 = Q + h . c . ) + (ˆ Θ 02 ˆ Q ˆ Q + h . c . ) + ˆ Q ˆ Θ 11 ˆ Q , · · · and 1 ˆ Θ( ǫ ) = P Θ P + P Θ Q ǫ − QHQ QH 1 P , 1 ˆ Θ( ǫ 1 ; ǫ 2 ) = P Θ P + PH 1 Q ǫ 1 − QHQ × 1 Q Θ Q ǫ 2 − QHQ QH 1 P , d m ˆ d n d m � � Θ m = 1 Θ( ǫ ) 1 ˆ ˆ ˆ � � , Θ nm = Θ( ǫ 1 ; ǫ 2 ) � � d ǫ n d ǫ m m ! d ǫ m n ! m ! � � ǫ = ǫ 0 1 2 ǫ 1 = ǫ 0 ,ǫ 2 = ǫ 0 Luigi Coraggio NUSPIN 2017 Workshop
The shell-model effective operators We arrest the χ series at χ 0 , and expand it perturbatively: One-body operator a a a a a * * h * h X = * p p b b b b b Two-body operator a b a b a b a b a b a b a b h p h 2 X p p = h 1 2 h 1 p c d c d c d c d c d c d c d Luigi Coraggio NUSPIN 2017 Workshop
Our recipe for realistic shell model Input V NN : V low − k derived from the high-precision NN CD-Bonn potential with a cutoff: Λ = 2 . 6 fm − 1 . 75 1 S 0 3 P 0 10 Phase Shift (deg) Phase Shift (deg) 50 0 25 -10 0 -20 0 100 200 300 0 100 200 300 Lab. Energy (MeV) Lab. Energy (MeV) 1 P 1 3 P 1 0 0 Phase Shift (deg) Phase Shift (deg) -10 -10 -20 -20 -30 -30 0 100 200 300 0 100 200 300 Lab. Energy (MeV) Lab. Energy (MeV) H eff obtained calculating the Q -box up to the 3rd order in perturbation theory. Effective operators are consistently derived by way of the the MBPT Luigi Coraggio NUSPIN 2017 Workshop
Part II Reliability Luigi Coraggio NUSPIN 2017 Workshop
Large-scale realistic shell-model calculations Neutron-rich isotopic chains Approaching neutron drip line: Shell-model study of the onset of collectivity at N = 40 L.C., A. Covello, A. Gargano, and N. Itaco, Phys. Rev. C 89 , 024319 (2014) Proton-rich isotopic chains Approaching proton drip line: Enhanced quadrupole collectivity of neutron-deficient tin isotopes L.C., A. Covello, A. Gargano, N. Itaco, and T. T. S. Kuo, Phys. Rev. C 91 , 041301 (2015) Luigi Coraggio NUSPIN 2017 Workshop
Collectivity at N = 40 3 SM B(E2;2 + -> 0 + ) (e 2 fm 4 ) 500 (a) (b) EXP 2.5 400 E(2 + ) (MeV) 2 N=40 300 1.5 200 1 SM 100 0.5 EXP 0 0 20 22 24 26 28 20 22 24 26 28 Z Z ⇒ shell-model study of neutron-rich isotopic chains outside 48 Ca ⇒ Collective behavior framed within the quasi-SU(3) approximate sym- metry ⇒ Two model spaces with 48 Ca inert core, including or not the neutron 1 d 5 / 2 orbital Luigi Coraggio NUSPIN 2017 Workshop
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