Con$nuum shell model for nuclear structure and reac$ons

Transcription

1 Con$nuum shell model for nuclear structure and reac$ons Marek Ploszajczak GANIL, Caen Content: - Introduc$on and recent advances in nuclear theory - Con$nuum Shell Model and Gamow Shell Model - Interplay between Hermi$an and an$- Hermi$an couplings - segrega$on of $me scales - configura$on mixing in weakly bound/unbound states - One- nucleon spectroscopic factors - Gamow Shell Model in Coupled Channel representa$on: - (p,p ), (p/n,g) - Outlook and future challenges CANHP2015

2 Collaborators W. Nazarewicz - NSCL- FRIB MSU, East Lansing and Univ. of Warsaw K. Fossez, Y. Jaganathen, J. Rotureau - NSCL- FRIB MSU, East Lansing J. Okolowicz - Ins$tute of Nuclear Physics, Krakow G. Dong, A. Mercenne, N. Michel - GANIL Y.H. Lam Ins$tute of Modern Physics, Lanzhou G. Papadimitriou - LLNL R.M. Id Betan Physics Ins$tute of Rosario B. Barre[ Univ. of Arizona, Tucson

3 Evolu$on of paradigms - In medium nucleon- nucleon interac$on from basic principles (EFT); 3- body interac$ons - Nuclear shell model for open quantum systems (GSM); structure and reac$ons in the low- energy con$nuum - Ab ini&o many- body theories of structure and reac$ons (GFMC, NCSM/RGM, NCGSM, CC,...) the recent progress in many-body methods surpass the progress in effective interactions

4 Challenges for the theory - How to reconcile shell model with reac$on models? - Weakly bound systems; role of con$nuum How to deal with non- locali$es due to the coupling to the decay channels and an$symmetriza$on - Handling mul$- configura$on effects in reac$on theory; How to reduce many- body to a few- body problem? - Having useful microscopic input: Understanding (op$cal) poten$als from microscopic interac$ons Shell model for open quantum systems Con$nuum Shell Model Hilbert space formulation Gamow Shell Model Rigged-Hilbert space formulation

5 neutrons - Network of many- body systems coupled via the con$nuum: nuclear reac$on and nuclear structure approaches meet together - Emergence of new scale(s) related to the threshold(s) energy

6 Mapping of continuum coupling effects in the SM analysis I. Stefan, et al., PRC 90, (2014) S p = MeV S n =+14.2 MeV υ 16F np υ 16F np υ 16F np υ 16F np [ ] [ ] J π = 0,1 ν0p 1/2 π1s 1/2 J π = 2,3 ν0p 1/2 π 0d 5/2 ( 0 ) = 0.775MeV ( 1 ) = 0.577MeV ( 2 ) = 1.829MeV ( 3 ) = 1.523MeV 1.151MeV = υ np 0.874MeV = υ np 2.011MeV = υ np 1.713MeV = υ np ( ) ( ) ( ) ( ) 16 N 0 16 N 1 16 N 2 16 N 3 S p = MeV S n =+2.49 MeV 40% 10%

7 Breaking of K quantum number in the low-energy continuum of HCN K. Fossez et al., PRC 91, (2015) Bands of isolated resonant states 1.0 EJ [Ry] K=0dominant Kmixed threshold J(J +1)[ hω] Dipolar potential, finite-size effects, repulsive core, polarization effects, quadrupolar interaction described by a pseudo-potential V H tot = p2 e + j2 2m e 2I + V W. R. Garrett, J. Chem. Phys. 77, 3666 (1982)

8 Breaking of K quantum number in the low-energy continuum of HCN K. Fossez et al., PRC 91, (2015) Bands of isolated resonant states 1.0 EJ [Ry] K=0dominant Kmixed threshold J(J +1)[ hω] Dipolar potential, finite-size effects, repulsive core, polarization effects, quadrupolar interaction described by a pseudo-potential V H tot = p2 e + j2 2m e 2I + V W. R. Garrett, J. Chem. Phys. 77, 3666 (1982) Challenge: How SM wave functions (gamma transition probabilities, selection rules, conserved quantum numbers, band structure) are changed by the coupling to the continuum? See the talk by Kevin Fossez

9 Shell model describes atomic nuclei as the closed quantum system i.e. coupling to decay channels are neglected Enrico Fermi Maria Goeppert- Mayer J. Hans D. Jensen

10 Role of boundary condi$ons in universal proper$es of reac$on cross- sec$ons at the threshold E.P. Wigner (1948) To what extent the change in boundary condi$ons at the nuclear surface due to Coulomb wave func$on distor$on in the external region can explain rela$ve displacement of states in mirror nuclei? J.B. Ehrman (1950) The exact coincidence of the energies of different configura$ons makes the ordinary perturba$on theory inadequate, so that special procedures are required U. Fano (1961) Eugene P. Wigner U. Fano

11 Role of boundary condi$ons in universal proper$es of reac$on cross- sec$ons at the threshold E.P. Wigner (1948) To what extent the change in boundary condi$ons at the nuclear surface due to Coulomb wave func$on distor$on in the external region can explain rela$ve displacement of states in mirror nuclei? J.B. Ehrman (1950) The exact coincidence of the energies of different configura$ons makes the ordinary perturba$on theory inadequate, so that special procedures are required U. Fano (1961) Eugene P. Wigner U. Fano Hilbert space includes real- energy bound and sca[ering states Resonances are discarded as unphysical! The resolu$on of this paradox required: - New mathematical concepts: Rigged Hilbert Space ( 1964), - Generalized completeness relation including s.p. bound states, resonances, and scattering states ( 1968) - Gamow Shell Model ( 2002) I.M. Gelfand T. Berggren

12 Two strategies to formulate the continuum shell model SM CSM/SMEC Hilbert space formulation H QQ H H PP GSM GSM-CC rigged Hilbert space formulation H H

13 Continuum shell model: Shell model with the real-energy continuum SM Q nucleus, localized states CSM/SMEC P environment, sca[ering states H QP H QQ H QQ H PP H PP closed quantum system eff H QQ H QQ ' ( E) = H QQ hermi$an ( E) i 2 V ( E )V T ( E) an$- hermi$an ( SM) Open QS solution in Q : Ψ α = b αi Φ i i For bound states: E α ( E) is real and E α ( E) = E For unbound states: physical resonances = poles of S-matrix open quantum system 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 Coupling of internal (in Q) and external (in P) states induces effective A-particle correlations and determines the structure of many-body states

14 Quasi-stationary extension of the Shell Model in the complex k-plane i! t Φ( r,t) = H ˆ Φ( r,t) Φ( r,t) = τ( t)ψ( r) H ˆ Ψ = % e i Γ ( ' & 2 * Ψ τ t ) % % ( ) = exp ' i ' e i Γ & 2 & (( ** )) Ψ(0,k) = 0, Ψ( r!,k) Ol(kr) Ψ( r! r,k) Ol(kr) I l r ( ) + O l ( kr) k n = 2m # e! 2 n i Γ n % $ 2 & Bound states Antibound states Resonances (poles of the S- matrix) k n = iκ n k n = iκ n k n = ±γ n iκ n Only bound states are integrable! * u n = u n u n k n = k 1 ik 2, k 1,k 2 > 0 Euclidean inner product * u n u n = dru n ( r)u n r 0 ( ) $ $ e 2k 2 r r RHS inner product u n u n = dr u * n ( r) u n r 0 ( ) $ $ e 2ir ( k 1 ik 2 ) r

15 Gamow Shell Model: Shell model in the complex k-plane Im( k) n un u n + L + uk u k dk =1; ui u j = δij T. Berggren, Nucl. Phys. A109, 265 (1968) K. Maurin, Generalized Eigenfunction Expansion, Polish Scientific Publishers, Warsaw (1968) L L + Re( k) bound states resonances non-resonant continuum SD i = u i1...u ia SD k SD k 1 k ~ H [ H] ij = [ H] ji Complex-symmetric eigenvalue problem for hermitian Hamiltonian Gamow Shell Model N. Michel et al, PRL 89 (2002) R. Id Betan et al, PRL 89 (2002) N. Michel et al, PRC 70 (2004) Ab initio no-core GSM studies of resonances in A=4,5 systems See the talk of George Papadimitriou

16 Interplay between Hermitian and anti-hermitian couplings is a source of new phenomena - Coalescence of eigenfunctions/eigenvalues (exceptional points); entangled states Example: doublet of 3+ resonances in 8 Be: 19.07(0.27) MeV and 19.23(0.23) MeV - Violation of the orthogonal invariance; significant modification of the Porter-Thomas distribution, Example: Reduced neutron width distribution disagreement with the prediction of the random matrix theory P.E. Koehler et al., PRL 105, (2010) - Segregation of time scales: resonance trapping vs super-radiance phenomenon - Collective phenomena: - Multichannel effects in reaction cross-section and shell occupancies - Instability of SM eigenstates at the channel threshold; near-threshold clustering -

17 Coalescence of eigenfunctions ϕ 2 Bound states (Hermitian problem) ϕ 2 ϕ 1 Level repulsion ϕ 1 ϕ 1 ϕ 2 Level crossing V H Level repulsion Width clustering vs V NH Level clustering Width repulsion Resonances (non-hermitian problem) V H Exceptional point: ϕ 1 = ϕ 2 ϕ 1 = ϕ 1 * Γ ϕ 2 V NH ω = v in + iw ex ϕ 1 E M.R. Zirnbauer et al., Nucl. Phys. A411 (1983) 161 J. Okolowicz et al., Phys. Reports 374 (2003) 271

18 16 Ne CSM/SMEC 1 Experimental signatures CSM/SMEC J π =1 1 2 entangled eigenstates J π =1 1 1 Channels: 15 + [ F( 1/2 1 ) p( j )] J π 15 + [ F( 5/2 1 ) p( j )] J π 15 - [ F( 1/2 1 ) p( j )] J π ZBM + WB interac$on V 0 = MeV fm 3 Δδ l j = 2π J. Okolowicz, et al, PRC 80 (2009)

19 eff H QQ ( + ) ( E) = H QQ + H QP G P Statistical aspects of nuclear coupling to continuum ( E)H PQ H QQ + ( W R +W I ) CSM/SMEC J=0 +, T=0 states in 24 Mg, 1 channel W I s = W X X ij W ij σ X W R W I = i 2 VV T Assuming internal dynamics is governed by the GOE and different channels are equivalent one finds: I P Λ ( W ij ) = W I ij ( Λ 1)/2 K Λ 1 Γ Λ / 2 ( ) ( )/2 W ij I ( ) π 2 Λ 1 ( )/2 The assumption of Gaussian distribution of V i c is justified in a generic case of a single channel. The distribution for both W I and W R is the same.

20 J=0 +, T=0 states in 24 Mg, 10 channels CSM/SMEC I P Λ ( W ij ) = W I ij ( Λ 1)/2 K Λ 1 Γ Λ / 2 ( ) ( )/2 W ij I ( ) π 2 Λ 1 ( )/2 CSM/SMEC (X σ X = (W ) ij ) 2 1/2 ρ = W R I R I ( ij W ij W ij W ij ) σ R σ I - W R and W I obey the same statistical ensembles - M.E. of W R and W I are energy dependent and (often) strongly correlated - Different decay channels are not equivalent - Coupling to continuum breaks the statistical model assumption of orthogonal invariance and channel equivalence for several open channels

21 Segregation of time scales CSM/SMEC J=0 +, T=0 states in 24 Mg, 10 channels S. Drozdz et al, PRC 62, (2000) super- radiant states Γ[ a.u. ] Γ[ a.u. ] E[ a.u. ] intermediate coupling [ ] E a.u. strong coupling trapped states Trapped resonances are narrower than uncoupled resonances closed cavity Quasi-bound states in the continuum open cavity

22 Configuration mixing in weakly bound/unbound states N. Michel et al. PRC 75, (R) (2007) bound- state structure dominates SM ( ) 6 He g.s. bound [ 5 He( g.s. ) p 3 / 2 ] 0+ ( S 1n ) 1/ 2 ( S 1n ) +1/ 2 -S [⁵He] (MeV) 1n Analogy with the Wigner threshold phenomenon for reac$on cross- sec$ons Y(b,a)X :σ ~ k X(a,b)Y :σ ~ k E.P. Wigner, PR 73, 1002 (1948) ( S n ) 1/ 2 for ( S n ) +1/ 2 for S n < 0 S n > 0 Unification of discrete (nuclear structure) and continuum (nuclear reactions) aspects of the nuclear many-body problem

23 Configuration mixing in weakly bound/unbound states N. Michel et al. PRC 75, (R) (2007) bound- state structure dominates SM ( ) 6 He g.s. bound [ 5 He( g.s. ) p 3 / 2 ] 0+ ( S 1n ) 1/ 2 ( S 1n ) +1/ 2 -S [⁵He] (MeV) 1n Analogy with the Wigner threshold phenomenon for reac$on cross- sec$ons Y(b,a)X :σ ~ k X(a,b)Y :σ ~ k E.P. Wigner, PR 73, 1002 (1948) ( S n ) 1/ 2 for ( S n ) +1/ 2 for S n < 0 S n > 0 Unification of discrete (nuclear structure) and continuum (nuclear reactions) aspects of the nuclear many-body problem N. Michel, W. Nazarewicz, M.P., PRC 82, (2010)

24 One-nucleon spectroscopic factors involving weakly- and strongly-bound nucleons A. Gade et al, PRL 93,

25 One-nucleon spectroscopic factors involving weakly- and strongly-bound nucleons Ne CSM/SMEC S n S p J. Okolowicz et al, (2015) A. Gade et al, PRL 93, Less than 15% of the ground state spectroscopic strength shifted to higher excitations - One-nucleon spectroscopic factors are not correlated with the asymmetry of S n and S p separation energies

26 CSM/SMEC p n p n n p - Usually stronger changes of R SF for S n,p close to 0 (threshold effect) - R SF correlates better with S n/p

27 Transfer vs knockout? Gade et al, PRL 93, Jenny Lee et al., PRL 104, (2010) Studies of exo$c nuclei require accurate reac$on models

28 The origin of near-threshold clustering α-cluster states can be found in the proximity of α-particle decay threshold K. Ikeda, N. Takigawa, H. Horiuchi (1968)

29 Why cluster states appear in the proximity of cluster thresholds? Correlations in the nearthreshold states depend on the topology of the nearest branching point Near-threshold clustering/correlations result from the interplay between internal mixing by interactions and external mixing via the decay channel(s) è emergence of new energy scale in the problem! J. Okolowicz, M.P., W. Nazarewicz, Prog. Theor. Phys. Suppl. 196 (2012) 230; Fortschr. Phys. 61 (2013) 66

30 Narrow near-threshold states in the continuum Γ = 36(19)keV Γ = 305keV 15 F Γ = 376keV ( 3500) F. De Grancey, F; de Oliveira, et al. (2015) (1513) Conjecture: The coupling to a nearby decay channel induces correlations in the shell model wave functions which are the imprint of this channel.

31 Instability of SM eigenstates at the channel threshold: Appearance of the aligned state E corr (MeV) Con$nuum coupling correc$on to SM eigenstates E (`) corr;i (E) ' V 2 0 h i h(e) i i Ne [ 15 F( 1/2 + ) πs 1/2 ] 0 + CSM/SMEC 1p threshold proton energy (MeV) - Interaction through the continuum leads to a formation of the collective eigenstate which couples strongly to the decay channel and carries many E TP of its characteristics - The maximum continuum coupling point at E TP is determined by an interplay between the Coulomb/centrifugal interactions, and the continuum coupling E TP - The centroid of the energy window is independent of the continuum coupling strength J. Okolowicz et al., Prog. Theor. Phys. Suppl. 196, 230 (2012) Fortschr. Phys. 61, 66 (2013)

32 12 C α+ 8 Be [ ] 290keV [ ] = 97keV S 3α 12 + C( 0 2 ) S α 8 + Be( 0 1 ) 8 Be α + α Z 1 Z 2 For a given Z 1 Z 2, the centroid of the opportunity window at E TP depends weakly on the charged particle decay channel and parameters of the potential. The continuum-coupling correlation energy and collectivity of the aligned state are reduced with increasing the Coulomb barrier.

33 Conclusion: The mixing of SM eigenstates via the continuum has universal features which explain: (i) the emergence of near-threshold clustering/correlations and optimal energies for its appearance, (ii) a gradual disappearance of charged-particle clustering in heavier nuclei

34 Conclusion: The mixing of SM eigenstates via the continuum has universal features which explain: (i) the emergence of near-threshold clustering/correlations and optimal energies for its appearance, (ii) a gradual disappearance of charged-particle clustering in heavier nuclei What is specific and what is generic in this emerging phenomenon? Specific Energetic order of emission thresholds and absence of stable cluster entirely composed of like nucleons Generic Correlations in the nearthreshold states depend on the nature of the nearby decay threshold Experimental verifica$on of the predicted by this mechanism exo$c clustering(s)/correla$on(s) (e.g. 3 H, 3 He, ) is needed!

35 What about the multineutron clustering? - Correlated four neutrons reported in the disintegration of 14 Be F.M. Marques et al, PRC 65, (2002) - Bound tetraneutron incompatible with our understanding of nuclear forces - Isospin structure of the nuclear force prevents that: S 4n < S 2n /S 1n

36 What about the multineutron clustering? - Correlated four neutrons reported in the disintegration of 14 Be F.M. Marques et al, PRC 65, (2002) - Bound tetraneutron incompatible with our understanding of nuclear forces - Isospin structure of the nuclear force prevents that: S 4n < S 2n /S 1n but The nature of nuclear forces does not preclude the manifestation of tetraneutron correlations in the vicinity of the 4n-emission threshold as a consequence of the collective coupling of many-body resonances via the 4n-emission channel(s).

37 = 2 =1 CSM/SMEC J. Okolowicz, M.P., W. Nazarewicz, APP B45, 331 (2014) Similarity between 2 neutral particle = 0 clustering and charged particle clustering Contrary to the charged-particle correlations, the multi-neutron correlations in channels with l=2,3, can appear both in light and heavy nuclei

38 Unified desciption of structure and reactions in the Gamow Shell Model Direct reactions in Gamow Shell Model Y. Jaganathen, N. Michel, M.P., PRC 89 (2014) Center of mass treatment: Cluster Orbital Shell Model relative coordinates Y. Suzuki, K. Ikeda, PRC 38, 410 (1988) A v 2 A " p H = i $ 2µ + U % v " i' + V ij + p ip j % $ ' # & # & i=1 i< j A c Recoil term coming from the expression of H in the COSM coordinates. No spurious states - Emission channels iden$fied in Ψ GSM (A p) and Ψ GSM ( A) - Sca[ering wave func$ons A p are the many- body states Ψ GSM ( ) Φ proj ( p) - - An$symmetry exactly handled Core arbitrary

39 - Resona$ng Group Method for A- body matrix elements: Ψ GSM;f (A p) Φ proj;f ( p) H Ψ GSM;i ( A p) Φ proj;i ( p) leads to coupled- channel equa$ons with microscopic poten$als - Differen$al elas$c and inelas$c sca[ering cross- sec$ons obtained using effec$ve interac$ons which reproduce spectroscopy/binding of (A) and (A- p) nuclei - Consistency test: E GSM;j E GSM-CC;j channels involving non-resonant scattering states of a target nucleus are essential in weakly-bound targets (S n, S p <3MeV)!

40 p+ 6 H elas;c sca<ering at E CM =21-22 MeV Exp: S.V. Stepantsov et al., PLB 542, 35 (2002) L. Giot et al., PRC 71, (2005) R. Wolski et al., PLB 467, 8 (1999) MSGI interac$on 4 He core, p- valence space 5 He, 6 He, 7 Li energies with respect to 4 He core are well reproduced Channels constructed with discrete states of 6 He: 0 +,2 +

41 p+ 6 H elas;c sca<ering at E CM =21 MeV Exp: S.V. Stepantsov et al., PLB 542, 35 (2002) E GSM;j E GSM-CC;j channels involving non-resonant scattering states of target nucleus are important!

42 p+ 18 Ne excitation function at different angles EXP GSM GSM- CC S p =3.921 MeV S n = MeV S p = MeV S n =20.18 MeV Model space: 0d 5/2, 1s 1/2, 0d 3/2, 1p 3/2, 1p 1/2 Interac$on: FHT finite- range interac$on: V(ij)=V C + V SO + V T + V Coul H. Furutani, H. Horiuchi, R. Tamagaki, PTP 60 (1978) 307; 62 (1979) Complete description of the scattering continuum in 19 Na - Scattering continuum states J=0 +,1 +,2 +, and higher lying (bound) states in 18 Ne are unimportant. GSM and GSM-CC results (almost) identical

43 p+ 18 Ne excitation function at different angles θ CM =105 o GSM-CC θ CM =120.2 o θ CM =135 o θ CM =156.6 o Exp: F. de Oliveira Santos et al., Eur. Phys. J. A24, 237 (2005) B. Skorodumov et al., Phys. Atom. Nucl. 69, 1979 (2006) C. Angulo et al., PRC 67, (2003)

44 6 Li(p,g) radiative capture cross sections 6 Li(p,g) - Model space: 0p 3/2, 0p 1/2, 0s 1/2, 0d 5/2, 0d 3/2 - Interac$on: FHT finite- range interac$on - 1-3% renormaliza$on of channel- channel coupling poten$als to correct the spectra of 6 Li and 7 Be for missing channels involving non- resonant sca[ering states of 6 Li Effec$ve interac$on in p- shell: See the talk of Yannen Jaganathen E1, E2, M1 components included S GSM- CC (0)=88.34 b ev S exp- acc (0)=79+/- 18 b ev N. Michel, G. Dong, et al. (2015)

45 Mirror reaction: 6 Li(n,g) 6 Li(n,g) 1-3% renormaliza$on of channel- channel coupling poten$als E1, E2, M1 components included N. Michel, G. Dong, et al. (2015)

46 Mirror systems/reactions: 7 Be(p,g), 7 Li(n,g) 7 Be(p,g) 7 Li(n,g) S GSM- CC (0)=23.21 b ev S exp- acc (0)=20.9+/- 0.6 b ev E1, E2, M1 components included K. Fossez, N. Michel, et al., PRC 91, (2015)

47 Outlook 1. Comprehensive and validated theory of nuclei on the horizon 2. Shell model treatment of weakly bound/unbound states è unification of nuclear structure and reactions 3. Collectivization of nuclear wave functions can be the result of - internal mixing by interactions: low-energy collective vibrations, rotational states, giant resonances, - external mixing via the decay channel(s): coherent enhancement/suppression of radiation, multi-channel effects in spectroscopic factors and reaction cross-sections, - interplay of internal & external mixing: near-threshold cluster states, halos, modification of spectral fluctuations, breaking of the isospin symmetry, K quantum number non-conservation, Thomas-Ehrmann shift, coalescence of eigenvalues,

48 Future challenges 1. The assessments of the validity of random matrix theory is crucial for the statistical theory of nuclear reactions - Coupling to continuum leads to violation of the orthogonal invariance and deviations from PT distribution - Statistical theory neglects the energy dependence of the Hermitian part of the effective Hamiltonian and the strong correlation between matrix elements of Hermitian and anti-hermitian parts (Ex.: segregation of time scales) 2. The connection of surrogate reactions with wanted reactions (ex. (d,p) with (n,g)) 3. The resolution of transfer vs knockout reaction controversy CANHP2015 Thank You