Continuum shell model: the unified approach to nuclear structure and - PowerPoint PPT Presentation

Continuum shell model: the unified approach to nuclear structure and reactions Marek Płoszajczak (GANIL) 1. Nuclear theory: Evolution of paradigms 2. Gamow Shell Model - L 2 basis for s.p. resonances 3. Shell Model Embedded in the Continuum 4. Coupled channel formulation of the Gamow Shell Model 5. Complex-symmetric eigenvalue problem in Continuum Shell Model 6. Configuration mixing in weakly bound/unbound states 7 . Continuum coupling correlation energy - ‘Fortuitous’ near-threshold states - Near-threshold collectivization of electromagnetic transtions 8. Unified desciption of structure and reactions in the Gamow Shell Model - p+ 18 Ne excitation function at different angles - p+ 14 O excitation function and spectroscopy of 15 F - Mirror radiative capture cross sections - Role of the non-resonant reaction channels - 40 Ca(d,p) transfer reaction 9. Outlook

N. Michel GANIL J.B. Faes W. Nazarewicz MSU/FRB East Lansing K. Fossez MSU/FRIB J. Rotureau MSU/FRIB S.M. Wang MSU/FRIB J. Oko ł owicz IFJ PAN Krakow Y. Jaganathen IFJ PAN Krakow A.Mercenne Louisiane State Univ. B. Barrett Univ. of Arizona K. Bennaceur Univ. of Lyon G. Papadimitriou. Livermore G. Dong Univ. of Huzhou F. De Oliveira. GANIL O. Sorlin GANIL R.J. Charity Washington Univ., St. Louis L. Sobotka Washington Univ., St. Louis B. Fornal IFJ PAN Krakow

- Network of many-body states coupled via the continuum - Nuclear structure and reactions merge neutrons How to describe the configuration interaction in open quantum systems?

Evolution of paradigms GSM (~2000) SM (~1949) val val core core GSM+RGM CSM/SMEC (~2012) (~1975/~1998) cont val val core core NCGSM (~2013) decay channels NCSM (~2000) decay channels cont NCSM+RGM/NCSMC ( ~2008/~2014) decay channels

Gamow Shell Model ( ) Im k ∑ ∫ ; u n u ˜ u k u ˜ k dk = 1 u i ˜ u n + j = δ ij n L + L − non-resonant bound states continuum resonances ( ) Re k ~ L + ∑ SD k SD k ≅ 1 SD i = u i 1 ... u i A k N. Michel et al, PRL 89 (2002) 042502 Gamow Shell Model R. Id Betan et al, PRL 89 (2002) 042501 [ ] ij = H [ ] ji H → H N. Michel et al, PRC 70 (2004) 064311 No identification of reaction channels Complex-symmetric eigenvalue GSM in this representation is a problem for hermitian Hamiltonian tool par excellence for nuclear structure studies

- Center of mass treatment: Cluster Orbital Shell Model relative coordinates Y. Suzuki, K. Ikeda, PRC 38 (1998) 410 A v A v " % " % 2 V ij + p i p j p i ∑ ∑ H = 2 µ + U i + $ ' $ ' A c # & # & i = 1 i < j “Recoil” term coming from the expression of H in the COSM coordinates. No spurious states

- Center of mass treatment: Cluster Orbital Shell Model relative coordinates Y. Suzuki, K. Ikeda, PRC 38 (1998) 410 Jacobi vs COSM coordinates S.M. Wang et al, PRC 96, 044307 (2017)

Coupled channel formulation of the Gamow shell model ∞ ( ) dr u c r ( ) Im k r 2 ˆ A CS c ∑ ∫ Ψ = r c 0 GSM channel state L − Channel basis: { } = A T , J T ; a P , ℓ P , J int , J P c { } ( ) J A Re k ˆ ) = ˆ J T ⊗ r , ℓ P , J int , J P A CS c ≡ c , r A Ψ T $ & ( % ' M A L + Y. Jaganathen et al, PRC 88, 044318 (2014) K. Fossez et al., PRC 91, 034609 (2015) Entrance and exit reaction channels defined Ÿ à Unification of nuclear structure and reactions Scattering wave functions Ÿ ( ) ⊗ Φ proj p ( ) Ψ GSM A − p are the many-body states Antisymmetry handled exactly Ÿ Core arbitrary Ÿ

L 2 basis for s.p. resonances Resonance Resonance anamnesis ( 2 ) k → κ = R ( ) k W u k ( ) r ( reg ) , H i κ + ( ) r = R = 0 + r + r ( ) ~ H k ( ) ~ H i κ ( reg ) ( reg ) ~ u k ~ u κ Bound states and resonance anamneses form together a discrete subset of the complete set of basis ! { } u n states in Hilbert space ˆ h → ! ˆ p ˆ ! e n ! ! ∑ u n + ˆ h ˆ h = u n p n e n ! ! ! ∑ ˆ p = 1 − u n u n e n = ( res ) = ! 2 κ 2 / 2 µ e n n ! ! ! ! ∑ u i ! ! ∫ u n u n + u k u k = 1 ; u j = δ ij ( ˆ ) ! e n − ! ! e n : ! Discrete states h u n = 0 R + n p ˆ ! ( ) ! Scattering states { } : ! e − ˆ h ˆ e p u = 0 SD i = ! u i 1 ... ! ∑ u i A ⇒ ; SD k SD k ≅ 1 J.B. Faes, M.P., Nucl. Phys. A 800 (2008) 21 k

Shell Model Embedded in the Continuum (SMEC) SM Q – nucleus, localized states CSM P – environment, scattering states H QP H QQ H QQ H PP H PP ( ) − i eff E ( ) V T E H QQ → H QQ ' ( ) = H QQ ( ) E 2V E closed quantum open quantum ( ) − i ) + u QQ E ( SM ( ) = H QQ 2 w QQ E system system hermitian anti-hermitian C. Mahaux, H.A. Weidenmüller, Shell Model Approach to Nuclear Reactions (1969) H.W.Bartz et al, Nucl. Phys. A275 (1977) 111 R.J. Philpott, Nucl. Phys. A289 (1977) 109 K. Bennaceur et al, Nucl. Phys. A651 (1999) 289 J. Rotureau et al, Nucl. Phys. A767 (2006) 13

H Ψ = E Ψ + = H PQ Φ i H QQ Φ i = E i Φ i ( ) ξ = 0 E − H PP ( ) ω i E − H PP + = G P + H PQ Φ i ω i + E Discrete states : ( ) H PQ Φ j = E ij δ ij + w i ω j δ E i E δ E j E Φ i H QQ + H QP G P + Φ i H QP ω j ∑ i c k , Ψ k = Φ i ( ) = E E k E i 1 + E ( ) Scattering solutions : ( ) H PQ Ψ = ξ + Q + G P H QP ξ eff E E − H QQ ( ) non-resonant part resonant part - Shell model and reaction theory reconciled - Coupling of ‘internal’ (in Q) and ‘external’ (in P) states induces effective A-particle correlations

Complex-symmetric eigenvalue problem in Continuum Shell Model (GSM/SMEC) Coupling to the environment of scattering states and decay channels does not reduce to the adjustment of (hermitian) Hamiltonian and leads to new (collective) phenomena - resonance trapping and super-radiance phenomenon - modification of spectral fluctuations - multichannel coupling effects in reaction cross-sections and shell occupancies - anti-odd-even staggering of separation energies in odd-Z isotopic chains - clustering - exceptional points - violation of orthogonal invariance and channel equivalence - matter (charge) distribution (pairing anti-halo effect) - ….

Configuration mixing in weakly bound/unbound states - Analogy with the Wigner threshold 0 + phenomenon for reaction cross- [ ] 6 He g.s. 5 He g.s. ( ) ( ) ⊗ p 3/ 2 S sections SM bound - The interference phenomenon GSM between resonant states and non-  − 1/ 2 ( ) − S 1 n  + 1/ 2 resonant continuum in the vicinity ( ) − S 1 n of the particle emission threshold -S [ ⁵ He] (MeV) 1n 0 + 6 Li T=1 " 5 He g.s. $ ( ) ( ) ⊗ π p 3/2 # % 0 + 6 Li T=1 " 5 Li g.s. $ ( ) ( ) ⊗ ν p 3/2 # % 6 Li T=1 ( ) (MeV) Near-threshold configuration mixing acts differently at the proton and neutron drip lines N. Michel et al., PRC( R) 75, 031301 (2007)

Continuum coupling correlation energy A H QQ eff E { } A ( ) = Re ( ) − H QQ Ψ i E corr ; i E Ψ i Interaction through the continuum Ÿ leads to the formation of the + + 0 0 3 0 4 collective eigenstate (‘aligned state’) which couples strongly to the decay + E corr (MeV) 0 2 0 + channel and carries many of its [ ] ( ) ⊕ π s 1/2 15 F 1/2 + -1 characteristics 1p threshold Aligned state is a superposition Ÿ of SM eigenstates having the same -2 + 16 Ne quantum numbers E OW 0 1 Point of the strongest collectivity Ÿ -3 (centroid of the ‘opportunity -1 -0.5 0 0.5 1 energy window’) is determined by proton energy (MeV) an interplay between the competing forces of repulsion (Coulomb and Okolowicz et al., Prog. Theor. Phys. Suppl. 196 (2012) 230 centrifugal int.) and attraction Fortschr. Phys. 61 (2013) 66 (continuum coupling) à Emergence of new energy scale related to the external configuration mixing via decay channel(s)

Continuum coupling correlation energy A H QQ eff E { } A ( ) = Re ( ) − H QQ Ψ i E corr ; i E Ψ i + + 0 0 3 0 4 + E corr (MeV) 0 2 0 + [ ] ( ) ⊕ π s 1/2 15 F 1/2 + 0 + -1 " $ ( ) ⊕ ν d 5/2 19 O 5 / 2 + # % 1p threshold -2 + 16 Ne E OW 0 1 -3 -1 -0.5 0 0.5 1 proton energy (MeV) J. Okolowicz et al., Prog. Theor. Phys. Suppl. 196 (2012) 230 Fortschr. Phys. 61 (2013) 66 neutron energy (MeV) J. Okolowicz et al., APP B45, 331 (2014) à In contrast to charged particle case, the strong (multi)neutron correlations exist also in heavy nuclei

This generic phenomenon in open quantum systems explains why so many states, both on and off the nucleosynthesis path, exist ‘fortuitously’ close to open channels �� � �� ����� ����� �� � �� 1/ 2 + 6362 � � ���� ���� 13 C+ α 6359 � �� � � � � ���� 4143 �� �� �� 16 O+n ��� ��� � � ��� � � �� ��� 5 / 2 + �� � �� �� 17 O ½ + resonance lying <3 keV above Γ γ branch of 0 + 2 decay to particle- bound state(s) of 12 C forms a seed 13 C+ α threshold enables slow for the synthesis of heavier elements neutron-capture process