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

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Continuum shell model: the unified approach to nuclear structure and reactions Marek Poszajczak (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

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SLIDE 1

Continuum shell model: the unified approach to nuclear structure and reactions

Marek Płoszajczak (GANIL)

1. Nuclear theory: Evolution of paradigms 2. Gamow Shell Model

  • L2 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+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
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SLIDE 2
  • 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
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SLIDE 3

neutrons

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

How to describe the configuration interaction in open quantum systems?

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SLIDE 4

Evolution of paradigms

core val SM (~1949) NCSM (~2000) core cont val GSM (~2000) cont NCGSM (~2013) core val decay channels CSM/SMEC

(~1975/~1998)

decay channels NCSM+RGM/NCSMC (~2008/~2014) core val decay channels GSM+RGM

(~2012)

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SLIDE 5

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 states resonances non-resonant continuum

SDi = ui1 ...uiA SDk

k

∑

SDk ≅1

  • N. Michel et al, PRL 89 (2002) 042502
  • R. Id Betan et al, PRL 89 (2002) 042501
  • N. 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

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SLIDE 6
  • Center of mass treatment: Cluster Orbital Shell Model relative coordinates
  • Y. Suzuki, K. Ikeda, PRC 38 (1998) 410

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

  • coordinates. No spurious states

H = pi

2

2µ + Ui " # $ % & '

i=1 Av

∑

+ Vij + pipj Ac " # $ % & '

i< j Av

∑

slide-7
SLIDE 7
  • 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)

slide-8
SLIDE 8

Coupled channel formulation of the Gamow shell model

Ψ = dr uc r

( )

r

∞

∫

c

∑

r2ˆ A CS c

GSM channel state

Channel basis: c

{ }= AT, JT;aP,ℓP, Jint, JP { }

ˆ A CS c ≡ c,r

( ) = ˆ

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)

Ÿ Entrance and exit reaction channels defined à Unification of nuclear structure and reactions Ÿ Scattering wave functions are the many-body states Ÿ Antisymmetry handled exactly Ÿ Core arbitrary

Im k

( )

Re k

( )

L+ L−

ΨGSM A − p

( ) ⊗ Φproj p ( )

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SLIDE 9

L2 basis for s.p. resonances

~ Hiκ

+ r

( )

~ uκ

reg

( )

Resonance anamnesis

~ Hk

+ r

( )

~ uk

reg

( )

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

{ }

ˆ h → ! ˆ h = ! un

n

∑

! en ! un + ˆ p ˆ hˆ p ˆ p =1− ! un

n

∑

! un ! en : ! en − ! ˆ h

( ) !

un = 0

Discrete states Scattering states

! e

{ }: !

e − ˆ p ˆ hˆ p

( ) !

u = 0

! un ! un + ! uk

R+

∫

! uk =1

n

∑

;

! ui ! uj =δij SDi = ! ui1...! uiA ⇒ SDk

k

∑

SDk ≅1

;

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

! en =

en en

(res) = !2κ 2 / 2µ

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SLIDE 10

HPP HQQ

SM HQQ HPP HQP CSM Q – nucleus, localized states P – environment, scattering states

closed quantum system

  • pen quantum

system

HQQ →HQQ

eff E

( ) = HQQ

'

E

( ) − i

2V E

( )V T E ( )

hermitian anti-hermitian

Shell Model Embedded in the Continuum (SMEC)

= HQQ

SM

( ) +uQQ E

( )− i

2 wQQ E

( )

  • 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
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SLIDE 11

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 δEiEδE jE

ω j

+

Φi HQP

Ψk = ck

i i

∑

Φi

,

Ek E

( ) = E

Ψ = ξ + Q +GP

+ E

( )HPQ

( )

1 E −H QQ

eff 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

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SLIDE 12

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)

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SLIDE 13
  • 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 threshold

phenomenon for reaction cross- sections

  • The interference phenomenon

between resonant states and non- resonant continuum in the vicinity

  • f 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)
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SLIDE 14

Continuum coupling correlation energy

  • 3
  • 2
  • 1
  • 1
  • 0.5

0.5 1

proton energy (MeV) Ecorr (MeV) 03 + 01 + 02 + 04 +

16Ne

1p threshold

15F 1/2+

( ) ⊕ πs1/2

[ ]

0+

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

Ÿ Aligned state is a superposition

  • f SM eigenstates having the same

quantum numbers

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

Ecorr;i E

( ) = Re

Ψi

A HQQ eff E

( )− HQQ Ψi

A

{ }

slide-15
SLIDE 15

Continuum coupling correlation energy

  • 3
  • 2
  • 1
  • 1
  • 0.5

0.5 1

proton energy (MeV) Ecorr (MeV) 03 + 01 + 02 + 04 +

16Ne

1p threshold

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

Ψi

A HQQ eff E

( )− HQQ Ψi

A

{ }

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SLIDE 16

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

17O

5 / 2+ 1/ 2+

13C+α

6359 6362 4143

16O+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 above

13C+α threshold enables slow

neutron-capture process

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SLIDE 17

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

  • 2n
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SLIDE 18

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

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SLIDE 19

14C

Near-threshold collectivization of electromagnetic transtions

Ÿ Strong collectivization of the B(E2) in 14C from the near-threshold resonance 2+

2 to the ground state 0+ 1

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

SMEC

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SLIDE 20

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 less important for the completeness of channel basis

Sp=3.921 MeV Sn=19.237 MeV Sp=-0.32 MeV Sn=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

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SLIDE 21

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)

GS GSM-CC CC

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SLIDE 22

p+14O excitation function and spectroscopy of 15F

Ex Exp GS GSM GS GSM-CC CC

15F

Ψ 0p1/2 1

[ ]s1/2 2 [ ]

2 = 0.97

Ψ 0p1/2 1

[ ]d5/2 2 [ ]

2 = 0.02

Q2p=129 keV

1 2 3 4 5 Ec.m.(MeV) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 dσ/dΩc.m.(barn/sr)

GSM-CC

  • Exp. data

4.5 4.9 0.01 0.1

GS GSM-CC CC

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

Scattering states are important for the completeness of channel basis

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SLIDE 23

6Li(p,γ) 6Li(n,γ)

Mirror radiative capture cross sections

  • G. Dong, et al., J. Phys. G: Nucl. Part Phys. 44. 045201 (2017)
slide-24
SLIDE 24

Role of the non-resonant reaction channels

Jscat

π

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

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SLIDE 25

20 40 60 80 100 120 140 160 180 Θlab(deg) 10−2 10−1 100 dσ/dΩlab(mb/sr)

GSM-CC

  • Exp. 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

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SLIDE 26

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 of

radiation, 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
  • … …

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