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Spectroscopy at the Particle Threshold

H. Lenske

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Agenda:

• Pairing in the continuum• Nuclear Polarizability and Spectral Functions• Continuum spectroscopy and Fano-Resonances• Summary

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Pairing in the Continuum:Quasiparticle Resonances

Extended HFB Theory as Coupled Channels Problem: The Gorkov-

Equations

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(q) 2q n j

n j

Mean-Field Hamiltonian (q p,n):

H U( )2m2j 1 (r) | v (r) |4

jn

qjn

qjnq

qSEq

rvrujr

V

)()(

412)(

:n)p,(qDensity & Field-Pairing

*)()(

21

jmsrvjmsru qj

qj )(|)(~ ; )(|)(~ )()(

EH

H)(

Spectrum of the Gorkov Equation:

Extended HFB Theory: Pairing Self-Energies

• Energy Shifts and Widths• Spectral Functions for particles and holes

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Pairing in Infinite Nuclear Matter

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3

0

3(1 ( ))(

( ) ( , ) u(q;k ) v(q;k ) ~ N(k ) (k , k ) ( ) I ,2

1 ( ) N(k ) (k , k ) I ,( )

( ) ~ e ( ) eF

F SE F F F F SE F F F F

F F SE F F FF

G kG k

F F F

d qk V k q V k k

G k V kG k

k k

)F

Free Space SE (S=0,T=1) Interaction:(Bonn-B Potential)

Pairing is a LOW DENSITY Phenomenon

Pairing in Infinite Nuclear Matter

Pairing Gap Anomal Density

Pairing Correlations in Nuclear Matter

Pairing Gap and Anomal Density in Symmetric Nuclear Matter

Pairing-Field in a Nucleus

RA RA

Neutron Spectrum :

11Li : Continuum HFB Spectral Functions

Dissolution of Shell Structures!

g.s. Densities

g.s. Densities r2 :

2)( |),(|4

12)( revdejrq

qj

jq

jmsrvnpqfor qj

q )(|)(:, )()(

11Li : Continuum HFB g.s.

Densities

Neutron Spectral Functions in 9Li(3/2-): Continuum Admixtures into

the g.s.

Continuum Admixtures!

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Pairing in the Continuum

S. Orrigo, H.L., PLB 677 (2009)

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Pairing Resonances in Dripline Nuclei 9Li+n 10Li

S. Orrigo, H.L., PLB 677 (2009) & ISOLDE newsletter Spring 2010, p.5

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Continuum Spectroscopy at REX-ISOLDE: 10Li=9Li+n d(9Li,10Li)p@2.36AMeV

Data: H. Jeppesen et al., REX-ISOLDE Collaboration, NPA 738 (2004) 511 & NPA 748 (2005) 374.

S. Orrigo, H.L., PLB 677 (2009) & ISOLDE newsletter Spring 2010, p.5

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New experimental results (Dec. 2013): 10Li continuum spectroscopy at TRIUMF

S. Orrigo, M. Cavallo, F. Capppuzzello et al.

1 3 3 5,2 2 2 2

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Spectral Structures by Dynamical Polarization

Beyond the Mean-Field:Short-range Correlations in Nuclear

Matter

PLB483 (2000) 324 NPA723 (2003) 544 NPA (2005)in print

Momentum Distribution n(p) = N(kF) a(w, p) dw

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Nuclear Dynamics…

cJ

Eth [MeV] Eexp [MeV]

2+ 3.220 3.368

1- 6.423 5.960

0+ 6.513 6.179

2- 6.446 6.263

3- 7.372 7.371

4- (9.270)

1+ 7.122

3+ 7.159

0- 7.374

QRPA Response in 10Be

DCP Neutron Spectral Distributions in 11Be

[0+ × 1/2+]: 0.79[2+ × 5/2+]: 0.18

[0+ × 1/2-]: 0.58[2+ × 3/2-]: 0.28

Spectral Distributions in Carbon Isotopes …normalized to sum rule

E1 Dipole E2 Quadrupole

Polarizability of C-Isotopes:HFB+QRPA results

2

1

0

( ) 0

( ) ( )

nn aS E a T

SPS

Multipole polarizabilties coefficients by sum rules:

Longitudinal Momentum Distributions: 17,19C → 16,18C + n

Carbon Target, Elab 900 AMeV•Binding: Correlation Dynamics•17C(5/2+,g.s.)• Sn(the.)=715keV

• C2S(g.s.) = 0.41G(the.): 132 MeV/cG(exp.): 143 ± 5 MeV/cs(-1n,the.): 124 mbs(-1n,exp.): 129± 22 mb

•Binding: Correlation Dynamics•19C(1/2+,g.s.)• Sn(the.)=263keV

• C2S(g.s.) = 0.40G(the.): 69 MeV/cG(exp.): 68 ± 3 MeV/cs(-1n,the.): 192 mbs(-1n,exp.): 233± 51 mb

17C

19C

Hole Spectru

m

Particle Spectru

m

Dynamical Core Polarization:• HFB g.s.:

„3-body renormalized“ G-Matrix

• ph-Interactions:Fermi Liquid Theory

Fano Resonances

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Interactions of Closed and Open Channels:

Fano Resonances

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The Spectral Situation encountered in Atoms, Molecules, Nuclei, and Hadrons

• A closed channel E* is embedded into a continuum of open channels• E* interacts via V(r) with open channels given by scattering states• E* Interacts via V(r) with closed channels, e.g. of (simple) bound states

Bound State Embedded into the Continuum - BSEC

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Examples:

• Atoms: self-ionizing states of multi-electron configuration

• Nuclei: Multi-particle-hole states above threshold

• Mesons: Confined qq-configurations embedded into the continuum of meson-meson scattering states, e.g. (1232), (770), Y‘‘(3770)…

• Baryons: Confined qqq-configurations embedded into the continuum of meson-nucleon scattering states, e.g. (1232), N*(1440), L(1405)…

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Visualizing Quantum Interference in Microscopic Systems:

Asymmetric Fano-Line Shapes of Resonances ( ) (c)d

E i i k ki k

a d b Y

Historically:The famous Silverman-Lassettre data

He(e,e‘)He*(1P) @ 500eV

Note: q must be negative – q=-1.84

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Fano-Resonances in Nuclei

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11 13

22 23

31 32 33

00

b

s

x

H VH H V

V V H

13 '

23 '

' ' 31 32 '

' 0 s.p. motion w.r.t. the g.s.

' 0

( E ) ' ' ' ' 0C

bn n J j c

c

sj J j c

c

j J j c nn

z n V j c z

z n V j c z

z j c V n z d j c V z

Hamiltonian and Wave function

The coupled equations (core nucleus integrated out):

Multi-channel Fano wave function:

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Extension to Several Open Channels

• n=2 open channels• n=2 energetically degenerate solutions with

outgoing flux

2 21 2 ; E E i

i

V W G G G G

( )arctan( )

( ')( ) ''

EE E G E

P EG E dEE E

G

G

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1 21 1 ' 1 '

1 z( , ') sin ( ') cos'

sin ; ( , ') ; c ( , ')E E

PE E E EE E

a b z E E z E E

G G

G G G

1 22 2 ' 2 '0 ; ( ') ; c ( ')E Ea b E E E E G G

G G

Solution 1: fully mixed

Solution 2: continuum mixed

Resonance superimposed on a smoothly varying background!

„Dark States“

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Multi-channel Coupling

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Resonance Scenarios in Nuclear Physics

The Fano-Wave Function:

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, ,*

,

| |1| | | | sin cot| |E E

E E

TM T T

V T

Y

2 2

2

,

2

,*

| cot |= ~1 cot

| | q=

| |

1 ~ | |

s

E

sE

E

q M

TT

TV

s s

s

Reaction Matrix Elements and Formation Cross Section

The (single channel) Fano-Formula:

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Correlation Dynamics in an Open Quantum System: d-wave Fano-Resonances in 15C

G~60…140keV

Sonja Orrigo, H.L., Phys.Lett. B633 (2006)

Xu Cao, H. L., PRL, submitted

DD-Dynamics at Threshold Channel Coupling and the Line Shape of Y(3770)

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0 0D D threshold 3729.7 MeV

D D threshold 3739.2 MeV

3.65

(2s / 3686)

X(3900) ??

q=-2.1±0.6

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Summary

• Dynamics at the particle threshold• Pairing at the dripline/in the continuum• Nuclear polarizabilities• Fano resonances in atomic nuclei • Tools for continuum spectroscopy• Universality of quantum interference

…with contributions by Sonja Orrigo (Valencia) and Xu Cao

(Giessen/Lanzhou)