Continuum shell model: the unified approach to nuclearstructure and reactions

Marek Płoszajczak (GANIL)

1. Nuclear theory: Evolution of paradigms2. Gamow Shell Model

- L2 basis for s.p. resonances3. Shell Model Embedded in the Continuum4. Coupled channel formulation of the Gamow Shell Model5. Complex-symmetric eigenvalue problem in Continuum Shell Model6. Configuration mixing in weakly bound/unbound states7. 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+18Ne excitation function at different angles - p+14O excitation function and spectroscopy of 15F - Mirror radiative capture cross sections- Role of the non-resonant reaction channels - 40Ca(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. Okołowicz 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

neutrons

- Network of many-body states coupled via the continuum- Nuclear structure and reactions merge

How to describe the configuration interaction in openquantum systems?

Evolution of paradigms

coreval

SM (~1949)

NCSM (~2000)

core

cont

val

GSM (~2000)

cont

NCGSM (~2013)coreval

decay channels

CSM/SMEC(~1975/~1998)

decay channels

NCSM+RGM/NCSMC (~2008/~2014)

coreval

decay channels

GSM+RGM(~2012)

€

Im k( )

€

Re k( )

€

L+€

L−

€

H→ H[ ]ij = H[ ] ji

€

un

n∑ ˜ u n + uk

L +

∫ ˜ u k dk =1

€

ui ˜ u j = δij;

Complex-symmetric eigenvalue problem for hermitian Hamiltonian

Gamow Shell Model

bound statesresonances

non-resonant continuum

€

SDi = ui1 ...uiA

€

SDk

k∑ SDk ≅1

N. Michel et al, PRL 89 (2002) 042502R. Id Betan et al, PRL 89 (2002) 042501N. Michel et al, PRC 70 (2004) 064311

Gamow Shell Model €

~

No identification of reaction channels GSM in this representation is a tool par excellence for nuclear structure studies

- Center of mass treatment: Cluster Orbital Shell Model relative coordinatesY. Suzuki, K. Ikeda, PRC 38 (1998) 410

“Recoil” term coming from theexpression of H in the COSM

coordinates. No spurious states€

H =pi2

2µ+Ui

"

# $

%

& '

i=1

Av

∑ + Vij +pipjAc

"

# $

%

& '

i< j

Av

∑

- Center of mass treatment: Cluster Orbital Shell Model relative coordinatesY. 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

Ψ = druc r( )r0

∞

∫c∑ r2A CS

c

GSM channel state

Channel basis: c = AT , JT ;aP,ℓP, Jint, JP

A CSc≡ c, r( ) = A ΨT

JT ⊗ r,ℓP, Jint, JP$%

&'MA

JA

Y. Jaganathen et al, PRC 88, 044318 (2014) K. Fossez et al., PRC 91, 034609 (2015)

Entrance and exit reaction channels defined à Unification of nuclear structure and reactions

Scattering wave functions are the many-body states

Antisymmetry handled exactly Core arbitrary

€

Im k( )

€

Re k( )

€

L+€

L−

ΨGSM A− p( )⊗Φproj p( )

L2 basis for s.p. resonances

~ Hiκ+ r( )

~ uκreg( )

Resonance anamnesis

~ Hk+ r( )

~ ukreg( )

Resonance k→κ = R k( )2( )W uk

reg( ),Hiκ+( ) r( ) r=R = 0

Bound states and resonance anamneses form together a discrete subset of the complete set of basis states in Hilbert space

!un

h→ !h = !unn∑ !en !un + php

p =1− !unn∑ !un

!en : !en − !h( ) !un = 0Discrete states

Scattering states !e : !e − php( ) !u = 0

!un !un + !ukR+∫ !uk =1

n∑ ; !ui !uj = δij

SDi = !ui1... !uiA ⇒ SDkk∑ SDk ≅1;

J.B. Faes, M.P., Nucl. Phys. A 800 (2008) 21

!en =enen(res) = !2κ 2 / 2µ

€

HPP

€

HQQ

SM

€

HQQ

€

HPP€

HQP

CSMQ – nucleus, localized statesP – environment, scattering states

closed quantumsystem

open quantumsystem

€

HQQ →HQQeff E( ) = HQQ

' E( ) − i2V E( )VT E( )

hermitian anti-hermitian

Shell Model Embedded in the Continuum (SMEC)

= HQQSM( ) +uQQ E( )− i

2wQQ E( )

C. Mahaux, H.A. Weidenmüller, Shell Model Approach to Nuclear Reactions (1969)

H.W.Bartz et al, Nucl. Phys. A275 (1977) 111R.J. Philpott, Nucl. Phys. A289 (1977) 109K. Bennaceur et al, Nucl. Phys. A651 (1999) 289J. Rotureau et al, Nucl. Phys. A767 (2006) 13

HΨ = EΨ

HQQΦi = EiΦi E −HPP( )ωi+ = HPQΦi E −HPP( )ξ = 0

ωi+ =GP

+HPQΦi

Φi HQQ +HQPGP+ E( )HPQ Φ j = Eijδij + wi ω j δEiEδEjE

ω j+ Φi HQP

Ψ k = cki

i∑ Φi , Ek E( ) = E

Ψ = ξ + Q+GP+ E( )HPQ( ) 1

E −H QQeff E( )

HQP ξ

Discrete states :

Scattering solutions :

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

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) - ….

Complex-symmetric eigenvalue problem in Continuum Shell Model (GSM/SMEC)

-S [⁵He] (MeV)1n

€

−S1n( )−1/ 2

€

−S1n( )+1/ 2

S

€

6He g.s.( ) 5He g.s.( )⊗ p3 / 2[ ]0+

bound

Configuration mixing in weakly bound/unbound states

(MeV)

6Li T=1( )

6Li T=1( ) 5Li g.s.( )⊗ν p3/2"# $%0+

6Li T=1( ) 5He g.s.( )⊗ π p3/2"# $%0+

- Analogy with the Wigner thresholdphenomenon for reaction cross-sections

- The interference phenomenon between resonant states and non-resonant continuum in the vicinity of the particle emission threshold

GSM

SM

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

-3

-2

-1

0

-1 -0.5 0 0.5 1proton energy (MeV)

E corr (

MeV

)

03+

01+

02+

04+

16Ne1p

thr

esho

ld

€

15F 1/2+( )⊕πs1/2[ ]0+

Point of the strongest collectivity (centroid of the ‘opportunity energy window’) is determined by an interplay between the competing forces of repulsion (Coulomb andcentrifugal int.) and attraction(continuum coupling)

Interaction through the continuum leads to the formation of the collective eigenstate (‘aligned state’)which couples strongly to the decay channel and carries many of its characteristics

Okolowicz et al., Prog. Theor. Phys. Suppl. 196 (2012) 230 Fortschr. Phys. 61 (2013) 66

EOW

Aligned state is a superposition of SM eigenstates having the same quantum numbers

à Emergence of new energy scale related to the external configuration mixing via decaychannel(s)

Ecorr;i E( ) = Re Ψ iA HQQ

eff E( )−HQQ Ψ iA

Continuum coupling correlation energy

-3

-2

-1

0

-1 -0.5 0 0.5 1proton energy (MeV)

E corr (

MeV

)

03+

01+

02+

04+

16Ne1p

thr

esho

ld

€

15F 1/2+( )⊕πs1/2[ ]0+

J. Okolowicz et al., Prog. Theor. Phys. Suppl. 196 (2012) 230 Fortschr. Phys. 61 (2013) 66

EOW

19O 5 / 2+( )⊕νd5/2"#

$%0+

neutron energy (MeV)à In contrast to charged particle

case, the strong (multi)neutron correlations exist also in heavy nuclei

J. Okolowicz et al., APP B45, 331 (2014)

Ecorr;i E( ) = Re Ψ iA HQQ

eff E( )−HQQ Ψ iA

This generic phenomenon in open quantum systems explains why so manystates, both on and off the nucleosynthesis path, exist ‘fortuitously’ closeto open channels

17O5 / 2+

1/ 2+

13C+α 63596362

414316O+n

Γγ branch of 0+2 decay to particle-bound state(s) of 12C forms a seed for the synthesis of heavier elements

½+ resonance lying <3 keV above13C+α threshold enables slow neutron-capture process

This generic phenomenon in open quantum systems explains why so manystates, both on and off the nucleosynthesis path, exist ‘fortuitously’ closeto open channels

2n

AAS T<

T>

SMEC

Exp: R.J. Charity et al, PRC 78, 054307 (2008)Th: J. Okolowicz et al., PRC 97, 044303 (2018)

2p

F. De Grancey et al, PLB 758, 26 (2016)

2n

4n?

?

Courtesy of O. Sorlin

14C

Near-threshold collectivization of electromagnetic transtions

Strong collectivization of the B(E2) in 14C from the near-threshold resonance 2+2 to the ground state 0+1

Another example: strong collective B(E1) transition between halo state1/2-1 (Sn=181 keV) and the ground state 1/2+1 in 11Be

SMEC

Unified desciption of structure and reactions in the Gamow Shell Model

p+18Ne excitation function

EXP GSM GSM-CC

Interaction: FHT finite-range interaction:

GSM and GSM-CC results (almost) identical à Scattering states J=0+,1+,2+,… and higher lying (bound) states in 18Ne are lessimportant for the completeness of channel basis

Sp=3.921 MeVSn=19.237 MeV

Sp=-0.32 MeVSn=20.18 MeV

H. Furutani, H. Horiuchi, R. Tamagaki, PTP 60 (1978) 307; 62 (1979) 981

V(ij)=VC + VSO + VT + VCoul

Y. Jaganathen, et al., PRC 89 (2014) 034624

18Ne

19Na

p+18Ne excitation function at different angles

θCM=105o θCM=120.2o

θCM=135o θCM=156.6o

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, 014308 (2003)

GSM-CC

p+14O excitation function and spectroscopy of 15F

Exp GSM GSM-CC

15FΨ 0p1/2 1[ ]s1/2 2[ ]

2= 0.97

Ψ 0p1/2 1[ ]d5/2 2[ ]2= 0.02

Q2p=129 keV

0 1 2 3 4 5Ec.m. (MeV)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

dσ/d

Ωc.

m.(

barn/s

r)

GSM-CCExp. data

4.5 4.9

0.01

0.1

GSM-CC

F. De Grancey, A. Mercenne, et al, PLB 758, 26 (2016)

Scattering states are important forthe completeness of channel basis

6Li(p,γ)

6Li(n,γ)

Mirror radiative capture cross sections

G. Dong, et al., J. Phys. G: Nucl. Part Phys. 44. 045201 (2017)

Role of the non-resonant reaction channels Jscatπ A-1( ) ⊗ jπ

'

n( )"#

$%

Jπ A( )

42Sc

Channels: 40Ca+d 41Ca+p 41Sc+n

Non-resonant channels built of continuum states: 1/2+, 3/2+, 5/2+, 7/2+, 9/2+ 1/2-, 3/2-, 5/2-, 7/2-in 41Sc and 41Ca

0 20 40 60 80 100 120 140 160 180Θlab (deg)

10−2

10−1

100

dσ/d

Ωla

b(m

b/sr)

GSM-CCExp. data

Exp: I. Fodor et al., Nucl. Phys.. 73, 155 (1965)

Th: A. Mercenne, et al., (2018), in preparation

40Ca(d,p) transfer reaction

40Ca(d,p) 41Cag.s. Ed=1.9 MeV

Shell model treatment of weakly bound/unbound states à unification of nuclear structure and reactions- Crucial role of non-resonant reaction channels

Collectivization of nuclear wave functions due to:- internal mixing by interactions: rotational and vibrational states- external mixing via the decay(s) channels: coherent enhancement/suppression ofradiation, multi-channel effects in shell occupancies and reaction cross-sections, …

- interplay of internal and external mixing: near-threshold cluster/correlated states, breaking of the mirror/isospin symmetry, near-threshold collectivization of electromagnetic transitions, exceptional points, anti-odd-even effect in binding energies, …

Future challenges:- how effective NN interactions are modified in weakly-bound/unbound states- !-selection rules for in- and out- band transitions in the resonance bands- new kinds of multi-nucleon correlations and clustering in the vicinity of particle emission thresholds

- effects of exceptional points in nuclear spectroscopy and reactions- … …

Outlook