Sonja Orrigo Correlation effects and continuum spectroscopy in light exotic nuclei Unbound Nuclei Workshop, Pisa, Italy, November 3-5, 2008

Sonja Orrigo

Correlation effects and continuum spectroscopy in light exotic nucleiCorrelation effects and continuum spectroscopy in light exotic nuclei

Unbound Nuclei Workshop, Pisa, Italy, November 3-5, 2008

http://upload.wikimedia.org/wikipedia/commons/f/f2/Leaning_Tower_of_Pisa.jpg

ContentsContentsContentsContents

Physical scenario: light exotic nuclei

Correlations effects and continuum spectroscopy

DCP correlations: Fano resonances Experimental results on 11Be and 15C via CEX

reactions

Pairing correlations in the continuum Transfer reactions to unbound states: 9Li(d,p)10Li

Summary and conclusions

Why correlation effects in light exotic nuclei?Why correlation effects in light exotic nuclei?Why correlation effects in light exotic nuclei?Why correlation effects in light exotic nuclei?

light

n-r

ich

nucl

ei

D

isso

luti

on

of

shel

l st

ruct

ure

s

I

nfl

uen

ce o

f co

rrel

atio

n d

ynam

ics

Peculiar conditions:

Large charge asymmetry

Weak binding of valence n

Proximity of s.p. continuum

low density (halos)

isovector interaction

open quantum systems

Continuum spectroscopy of light exotic nucleiContinuum spectroscopy of light exotic nucleiContinuum spectroscopy of light exotic nucleiContinuum spectroscopy of light exotic nuclei

Two topics:

• 15C: weakly-bound Sn = 1218 keV

effects due to the DCP correlation dynamics Fano Resonances

• 10Li: n-unbound by 25 keV

effects due to pairing correlations

continuum spectroscopy by one-neutron transfer s.p. excitations

• Dynamical Core Polarization

(DCP) correlations• Pairing correlations

I

mpo

rtan

t to

inve

stig

ate

thei

r

effe

cts

in t

he lo

w-e

nerg

y co

ntin

uum





reactions



Fano ResonancesFano ResonancesFano ResonancesFano Resonances

Fano ResonancesFano Resonances are investigated as a new a new

continuum excitation modecontinuum excitation mode in exotic nuclei

Atomic physics

• H.Feshbach, Ann. of Phys. 5 p. 357 (1958), Ann. of Phys. 19 p. 287 (1962), Ann. of Phys. 43 p. 410 (1967)

• F.H.Mies, Phys. Rev. 175 p. 164 (1968)

• A.F.Starace, Phys. Rev. B 5 p. 1773 (1972)

• A.K.Bhatia and A.Temkin, Phys. Rev. A 29 p. 1895 (1984)

• J.P.Connerade and A.M.Lane, Rep. on Progr. in Phys. 51 p. 1439 (1988)

Hadron physics

N.E.Ligterink, PiN Newslett. 16 p. 400

nucl-th/0203054 (2002)

Solid-state physics

• S.Glutsch, Phys. Rev. B 66 p. 075310 (2002)

Nuclear physics

G.Baur and H.Lenske, Nucl. Phys. A 282

p. 201 (1977)

General phenomenon observed in many different areas of physics

Fano interferenceFano interferenceFano interferenceFano interference

Originally detected in atomic spectra

1960’s, Fano: first model for atomic states

excited in the inelastic scattering e--atoms

Typical for interacting many-body

systems at all scales !

Fano interferencequantum-mechanical interaction betweendiscrete and continuous configurations

asymmetric line shape

Fano Resonances in nuclear physicsFano Resonances in nuclear physicsFano Resonances in nuclear physicsFano Resonances in nuclear physics

BSEC: narrow resonances in the continuum (Ex > Sn)

DCP model: BSEC as quasi-bound core-excited configurations

Experimental signature of the DCP correlationsG.Baur and H.Lenske, Nucl. Phys. A 282(1977)201; H.Lenske et al., Jour. Progr. Part. Nucl. Phys. 46(2001)187

Predicted theoretically by Mahaux and Weidenmüller (1969)C.Mahaux and H.A.Weidenmüller, Shell Model Approach to Nuclear Reactions, North-Holland, Amsterdam (1969)

1st observed BSEC (1980): 13C (stable), Ex = 7.677 MeV (J = 3/2+)H.Fuchs et al., Nucl. Phys. A 343(1980)133

Bound States Embedded in the Continuum (BSEC)

And in exotic nuclei?

In exotic nucleiIn exotic nucleiIn exotic nucleiIn exotic nuclei

n-dripline nuclei:

easily polarizable core

BSEC at low-energy

C-isotopes: presence of

low-energy 2+ core states

good candidates

Importance of a

systematic study

H. Lenske, from HFB & QRPA calculations

Fano Resonances in exotic nucleiFano Resonances in exotic nucleiFano Resonances in exotic nucleiFano Resonances in exotic nuclei

F.Cappuzzello, S.E.A. Orrigo et al., EuroPhys. Lett. 65 p. 766 (2004)

S.E.A. Orrigo et al., Proceedings Varenna 122 p. 147 (2003)

35

30

25

20

15

10

5

0

8.50*

8.50

7.30

7.30*

6.77] 6.4

g.s.

g.s.*

0.77

lab = 14°, 55 keV/ch

0 2 4 6 8 10 12

15C Excitation energy [MeV]

Counts

DCP regime

Single particle regime

S n

0.77*

Fano interference:BSEC – s.p. continuum

8.50*L=8°109 keV/ch

15C Excitation energy [MeV]

Cou

nts

7.30

8.50

1515N(N(77Li,Li,77Be)Be)1515C @ 55 MeVC @ 55 MeV1515N(N(77Li,Li,77Be)Be)1515C @ 55 MeVC @ 55 MeV

1515N(N(77Li,Li,77Be)Be)1515C @ 55 MeVC @ 55 MeV1515N(N(77Li,Li,77Be)Be)1515C @ 55 MeVC @ 55 MeV

F.Cappuzzello, S.E.A. Orrigo et al., EuroPhys. Lett. 65 p. 766 (2004)S.E.A. Orrigo et al., Proceedings Varenna 122 p. 147 (2003);

a) level density

natural parity transitions

0 2 4 6 8 10 12 1415C Excitation energy [MeV]

10 2

10

1

10 2

10 1

dQ

RP

A(

) [

MeV

1]

b) level densityunnatural parity transitions


dQ

RP

A(

) [

MeV

1]

s. p.

s. p.

Sn

Sn

Results of microscopic

QRPA calculations

Ex [MeV] [keV]

0.00 0.03

0.77 0.03

6.77 0.06 < 160

7.30 0.06 < 70

8.50 0.06 < 140

8.50*

8.50

7.307.30*6.77] 6.4

g.s.

g.s.*

0.77

DCP regime

Single particle regime

S n

0.77*


35

30

25 20

15

10

5

0

Counts

lab = 14°55 keV/ch

Strength well reproduced for single particle transitions

(1/2+ g.s., 5/2+ state at 0.77 MeV)

Observed fragmentation for

Ex > 2 MeV not reproduced

15C: Fano Resonances

• Strong competition of mean-field and correlation dynamics

mean-field approaches are no longer appropriate

• Enhanced correlation effects (Dynamical Core Polarization DCP)

new excitation modes involving core-excited configurations (BSEC)

1111B(B(77Li,Li,77Be)Be)1111Be @ 57 MeVBe @ 57 MeV1111B(B(77Li,Li,77Be)Be)1111Be @ 57 MeVBe @ 57 MeV

a) level density

natural parity transitions

0 2 4 6 8 10 12 1411Be Excitation energy [MeV]

dQ

RP

A(

) [

MeV

1]

b) level densityunnatural parity transitions

0 2 4 6 8 10 12 1411Be Excitation energy [MeV]

dQ

RP

A(

) [

MeV

1]

s. p.

s. p.

QRPA calculations

F.C

appu

zzel

lo,

H.L

ens

ke e

t al

.,

Phy

s. L

ett

B 5

16(2

001)

21

7Be detected with the

IPN-Orsay Split-Pole

magnetic spectrometer

DCP regimeSingleparticle

0 1 2 3 4 5 6 7 811Be Excitation energy [MeV]S n

Counts

Strength well reproduced for single particle transitions

(1/2+ g.s., 1/2- state at 0.32 MeV and 5/2+ state at 1.77 MeV)

Observed fragmentation for

Ex > 2 MeV not reproduced

The QPC modelThe QPC modelThe QPC modelThe QPC model

H.L

ens

ke,

J. P

hys

. G

: N

ucl.

Par

t. P

hys.

24

(199

8) 1

429

H.L

ens

ke,

C.M

.Kei

l, N

.Tso

neva

, P

rog

r. in

Par

t. a

nd N

ucl.

Phy

s. 5

3 (2

004)

153

Q

uas

ipar

ticl

e-co

re c

ou

pli

ng

(Q

PC

) m

od

el

(B

ohr

& M

otte

lson

)

DC

P c

orre

latio

ns d

escr

ibed

by

coup

ling

1QP

to

the

core

-exc

ited

conf

igur

atio

ns

1QP

3Q

P

Cor

e ex

cita

tions

: 2Q

P e

xc.

give

n by

QR

PA

Strength fragmentation not reproduced by QRPA DCP effects

3331

1311

HV

VHH

QP

C H

amilt

onia

n of

the

odd

-mas

s sy

stem

:

Jn11H

JC )J (j'33H

3

QP

sta

tes

1

QP

sta

tes

coupled by V13

To study resonances in the low-energy continuum and their line shapes

Theoretical modelTheoretical modelTheoretical modelTheoretical model

3331

1311

HV

VHH

QP

C e

igen

stat

es:

0φ E)(H J

S.E

.A.O

rrig

o, H

.Len

ske

et a

l., P

hys.

Let

t. B

633

(20

06)

469

CJj'JC πππ ;J 'jJ

C

CJ j'

JCJ j'Jεn

JnJ )J (j')E(zε)E(z dεn)E(z φ

s.p. mixing

1QP

3Q

P

E < 0

bound states

E> 0

continuum

BS

EC

(E

C)

Bou

nd c

ore-

exci

ted

stat

es (

E –

EC <

0)

Theoretical modelTheoretical modelTheoretical modelTheoretical model

Coupling of a single particle elastic channel to closed core-excited channels

By projecting the Schrödinger equation onto the 1-QP and 3-QP components

N coupled equations

1QP Channel 1

3QP Channels i = 2, …,

N

0 J V0 εh 'J j'

C131)1(

j

C

cJjj

0 0VJ εhn

13C'i(i)

J j' C jJj c

Ch. 1 1QP

continuum

Ch. i = 2, …, N 3QP states

0ε , ,h 1)1(

j j

0Eεε , ,h (i)C1i'

(i)J j' C

cJj

S.E

.A.O

rrig

o, H

.Len

ske

et a

l., P

hys.

Let

t. B

633

(20

06)

469

Numerical methodsNumerical methodsNumerical methodsNumerical methods

The coupled channels problem is solved in coordinate space

N coupled equations for the radial wave functions

0 (r)u W(r)u N

2 i i,i 1,

212

112

2

i1K

r

1)(

dr

d

0 (r)u W(r)u Kr

1)(

dr

d1i 1,i i,

2i2

ii2

2

1QP Channel 1 (open)

3QP Channels i = 2, …,

N(closed) 2

ii2i Uε 2mK

2i

2i ε 2mk

r < RA

r >> RA

Pot

entia

ls U

i fro

m H

FB

cal

cula

tions

2(i)Ji C

F 2mW 0(i)JC13

(i)J F βJV0F

CC

Tra

nsiti

on f

orm

fac

tors

fro

m

QR

PA

cal

cula

tions

& d

ata

S.E

.A.O

rrig

o, H

.Len

ske

et a

l., P

hys.

Let

t. B

633

(20

06)

469

Numerical methodsNumerical methodsNumerical methodsNumerical methods

r)(Q (r) b (r) b (r)u N

1 mmm im

N

1 mm im i, ii

a

jχr < RA

i = 1, …, N

1) Internal w.f. by solving the NxN eigenvalue problem

r)(k C r)(k(r)u i)(

i11i1 i, i1i

hj r >> RA

2) Asymptotic w.f.

i = 1, …, N

0

0

'

0

0

C

C

C

b

b

b

1

1

1N

12

11

N

2

1

''

j

j

h

h

χ

χ)(Ru)(Ru M

)2( i,M

)1( i,

Matching 2N equations with complex coefficients

dr

d

dr

d )(Ru)(Ru M)2(

i,M)1(

i, AM RR

bm, C1i

2ii2

iii (k)C

12

1j2

k

π4 )k(σ

s i = 1, …, N

S.E

.A.O

rrig

o, H

.Len

ske

et a

l., P

hys.

Let

t. B

633

(20

06)

469

Results for Results for 1515CCResults for Results for 1515CC

Elastic scattering cross section

2112

111 (k)C

12

1j2

k

π4 )k(σ

s

Analytic calculation for 2 ch.

A single excited state of the

14C core: EC = 8.317 MeV

Ui from HFB

V13 is the only free parameter

0

2

4

6

8

10

12

0 5 10 15 20 25 30

s-wave p-wave

d-wave

15C excitation energy [MeV]

11

[m

b]

V13 = 0

S.E

.A.O

rrig

o,

H.L

ens

ke e

t al

., P

hys.

Le

tt.

B 6

33 (

2006

) 46

9

0

2

4

6

8

10

12

0 5 10 15 20 25 30

s-wave p-wave

d-wave V13 0

11 [

mb]


Fano interference

s-, p-, d-waves

Results for Results for 1515CCResults for Results for 1515CC

15C theo. (11 d-wave)

V13 is the only free parameter15C exp. from (7Li,7Be)

Full 5-channels calculation

4 14C states: EC(J) = 6.094(1–),

6.728(3–), 7.012(2+), 8.317(2+) MeV

Ui from HFB

V13 weighted by (i) of 14C(,’)JC

Q

ual

itat

ive

com

par

iso

n:

Eth

. = 6

.67,

7.3

6, 7

.70,

8.9

2 M

eV

th

. = 6

6, 8

0, 1

41,

85 k

eV

Eex

p. =

(6.

77,

7.30

, 8.

50)

0

.06

MeV

ex

p. ≤

160

, 70

, 14

0 ke

V

V13 affects of the resonances

(here V13 = 5 MeV)

S.E

.A.O

rrig

o, H

.Len

ske

et a

l., P

hys.

Let

t. B

633

(20

06)

469

Results Results for for 1717C and C and 1919CCResults Results for for 1717C and C and 1919CC

Sta

te p

aram

eter

s by

QR

PA

V13 is the only free parameter (here 5 MeV)

18 C

sta

tes:

EC(J

) =

1.6

20(2

+),

2.96

7(4+

), 3

.313

(2+),

5.5

02(1

– ) M

eV

16 C

sta

tes:

EC(J

) =

1.7

66(2

+),

3.98

6(2+

), 4

.142

(4+)

MeV

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12

d-wave

11 [

mb]


1717CCSSnn = 0.73 MeV = 0.73 MeV

0

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10 12

d-wave

11

[m

b]


1919CCSSnn = 0.16 MeV = 0.16 MeV

BS

EC

str

uctu

res

mov

e to

war

ds lo

wer

ene

rgie

s w

ith in

crea

sing

the

neu

tron

exc

ess

Incr

ease

d e

ffec

t o

f th

e co

rrel

atio

ns

Systematic study of the evolution of the phenomenon when going towards more n-rich nuclei

S.E.A. Orrigo, H.Lenske et al., Proceedings INPC07, Tokyo

The (The (77Li,Li,77Be) CEX reactionBe) CEX reactionThe (The (77Li,Li,77Be) CEX reactionBe) CEX reaction

Structural properties:

• Single particle isovector excitations

• BSEC and Fano resonancesin the continuum

Reaction dynamics:

• One-step / two-step contributions

• Spin transfer probabilities

N = 1 7He N = 2 11Be N = 3 15C N = 4 19O N = 5 23Ne N = 6 27Mg …

N + 3 n

MA

GN

EX I

PN

-Ors

ay

References:

S.E.A. Orrigo et al., Core excited Fano-resonances in exotic nuclei, Phis.Lett. B 633(2006)469

F.Cappuzzello, S.E.A. Orrigo et al., Excited states of 15C, EuroPhys.Lett. 65(2004)766

F.Cappuzzello et al., Analysis of the 11B(7Li,7Be)11Be reaction at 57 MeV in a Microscopic Approach, Nucl.Phys. A 739(2004)30

S.E.A. Orrigo et al., Spectroscopy of 15C by (7Li,7Be) Charge Exchange Reaction, Proc. “10th Int. Conf. on Nuclear Reaction Mechanisms” , Varenna, Italy, 122(2003)147

C.Nociforo et al., Investigation of light neutron-rich nuclei via the (7Li,7Be) reaction, Acta Physica Polonica, B 34(2003)2387

F.Cappuzzello et al., Excited states of 11Be, Phys.Lett B 516(2001)21

Maximum magnetic rigidity 1.8 T• m

Solid angle 51 msr

E max /E min 1.7

Total energy resolution(target 1 mm2, 90% of full acceptance)

1000

Mass resolution 250

A.Cunsolo et al., NIMA 481 (2002) 48

A.Cunsolo et al., NIMA 484 (2002) 56

E < 30 AMeV

2 < A < 40

E < 25 AMeV

40 < A < 93

Upper bent limits

1919F(F(77Li,Li,77Be)Be)1919O @ 52.4 MeVO @ 52.4 MeV1919F(F(77Li,Li,77Be)Be)1919O @ 52.4 MeVO @ 52.4 MeV

Xfoc [m]

Counts

= 50 msr

Energy byte = ± 27%

PRELIMINARY

g.s.

96 keV19O

E/E ~ 1000

lab = 7° - 19.5°

19.8 keV/ch

Sn = 3.9 MeV





reactions



Single particle dynamics: the relevant energy scale is Sn

• Stable nuclei: Sn~10MeV

→ static MF in the p-h channel + paring for the p-p correlations

• Weakly-bound n-rich nuclei: Sn~few keV-MeV

→ pairing correlations in the p-h channel are also important

Theory of Pairing in the ContinuumTheory of Pairing in the ContinuumTheory of Pairing in the ContinuumTheory of Pairing in the Continuum Extended MF approach for pairing in weakly-bound or unbound nuclei

• 2x2 coupled channel problem described by the Gorkov equationsH.Lenske, F.Hofmann, C.M.Keil, J. Progr. Part. Nucl. Phys. 46(2001)187

Particle channel (open)

Hole channel (closed)

mjn Ee field Pairingr MFrU

qqqq αα

energy) (QP 02

eeE n)p,(q potential chemical

αα

αq

0rv

ru

erUTrΔ

rΔerUT

α

α

αqq†q

qαqq

Similarity between the Pairing and DCP approaches

Particle-stable system: q<0

• e<0 , hole w.f. v decaying exponentially for r»RA

• In the continuum e+>0, particle w.f. u like a scattering wave: 1

u (r) cos F (r) sin G (r)k r

Theory of Pairing in the ContinuumTheory of Pairing in the ContinuumTheory of Pairing in the ContinuumTheory of Pairing in the Continuum

The same type of effects is produced by any types of correlations (e.g., DCP → Fano)

2j j2

2j 1 4(E) sin (E)

2s 1 k

Partial wave elastic scattering cross section:

Resonances in resonances in ℓj

Corr

Observables involving the states u will show a characteristic energy dependence

(e.g., transfer cross sections through S(E))

CorrMF (c) (c)u (r) cos f (r) sin g (r) αα

†α2

α(c)α ugf

4π

mkδtan

Relationship scattering observables – pairing strength

Continuum level density: spectral distribution of particle strength per energy E

= Density of states2j 1 d

S (E)dE

2 2

mkN(k)

2

S.E.A. Orrigo and H. Lenske, submitted to PLB (2008)

Neutron s.p. spectral functions in Neutron s.p. spectral functions in 99LiLiNeutron s.p. spectral functions in Neutron s.p. spectral functions in 99LiLi

Effects of the dynamical correlations (particle-hole coupling) due to the pairing field

• The widths of the hole distributions (E<0) are due to the bound-continuum coupling

• The deeper-lying s-wave levels are coupled more efficiently to the particle continuum

• The 5/2+ d-wave strength is lowered into the bound state sector (intruder component)

• A small amount of 1/2+ and 3/2+ strengths is above the p½ peak

• Dramatic change in dynamics at the n-dripline: the level ordering is not determined by simple MF

E<0 hole sector

E>0 particle sector


Partial wave cross sections Partial wave cross sections for elastic scattering for elastic scattering 99Li+n Li+n Partial wave cross sections Partial wave cross sections for elastic scattering for elastic scattering 99Li+n Li+n

Comparison: full HFB Gorkov-pairing – bare MF calculations

• Pairing gives an attractive self-energy in the p-wave channels

→ 1/2– and 3/2– resonances at very low energy (E<<3MeV ~ threshold for DCP correlations)

• Slight attraction in the 1/2+ channel and repulsion for the d-waves

2j j2

2j 1 4(E) sin (E)

2s 1 k

The structure results are used as input for transfer reaction calculations


Continuum spectroscopy of Continuum spectroscopy of 1010Li by transfer reactionsLi by transfer reactionsContinuum spectroscopy of Continuum spectroscopy of 1010Li by transfer reactionsLi by transfer reactions

Transfer reactions

well established tool for structural studies of bound and exotic nuclei Weakly-bound final state: prevalence of small momentum components

the cross section maximum is at much lower incident energiesH. Lenske and G. Schrieder, Eur. Phys. J 2 (1998) 41

Single-nucleon transfer reactions

as a tool for continuum spectroscopy in exotic nuclei Study of the low-energy s.p. resonances in unbound systems also

Method based on a DWBA approach. Main innovations:

to treat the case of unbound final states and

to calculate the double differential cross section for one-neutron transfer

The model is applied to the 9Li(d,p)10Li reaction to explore the structure of 10Li

10Li is neutron-unbound by 25 ± 15 keV

Why Why 1010Li?Li?Why Why 1010Li?Li?

Crucial for the comprehension of the structure of 11Li as a three body system

(11Li is a Borromean 2-n-halo nucleus) Information on the n-9Li interaction, important for the theoretical models of 11Li

Interest in the structure of 10Li itself: low-lying states are not yet well known

• Ground state: p-wave or s1/2 virtual state at the n+9Li threshold?

• 4 states are expected at low energy with J = 1−, 2− and 1+, 2+

(neutron in 2s1/2 or 1p1/2 unpaired 1p3/2 proton of the 9Li core)

• Two resonances seen at Ex ~ 250 and 500 keV; several resonances at higher Ex

D.R. Tilley et al., Nucl. Phys. A 745 (2004) 155 and refs. therein

Interest in the 9Li(d,p)10Li reaction in inverse kinematics: experiments at MSU @ 20 AMeV (P. Santi at al., Phys. Rev. C 67 (2003) 024606)

resonance at ~ 350 keV with ~ 300 keV experiments at REX-ISOLDE @ 2.36 AMeV (H.B. Jeppesen et al., Phys. Lett. B 642 (2006) 449)

resonance at ~ 380 keV with ~ 200 keV

Theory for transfer reactionsTheory for transfer reactionsTheory for transfer reactionsTheory for transfer reactions

Transfer reaction A(a,b)B (a = b+x, B = A+x) in a DWBA approach [Satchler]

• Optical model Hamiltonians and DW Schroedinger equations:

H = HA + Ha + K + U + V ( = a+A) ; H = HB + Hb + K + U + V = b+B)

(K + U – E) (±) (r, k) = 0 ( = )

• The optical model wave functions (±) (r, k) describe the elastic scattering determined

by the optical potentials U at the channel energies E= E – eA – ea

Hinterberger Menet

d-potentials p-potentials NPA111(1968) PRC4(1971)

Residual interaction V (post)

chosen according to effective self-energy (full HFB Gorkov-pairing), it reproduces B.E., rM, rC of 9Li -36.14

for a fixed energy

• In the post representation for a stripping reaction in which x is transferred from a to B:

F = JB MB sb mb | V | JA MA sa ma =

=jl (Sjl)½ Rjl(rxA)(l s m – m | j )(sb s mb ma – mb | sa ma)(JA j MA MB – MA | JB MB)D(rxb) Ylm*( )

spectroscopic amplitude radial wave function for the transferred particle x

projectile internal function times x-b interaction potential

• Zero-Range Approximation:

D(rxb) = D0 (rx – rb) T = D0 (S)½ (–) | R(rx)Y*( ) (rx – rb) |

(+)

which contains dynamics and structure information

xr̂

xAr̂

Theory for transfer reactionsTheory for transfer reactionsTheory for transfer reactionsTheory for transfer reactions

• First order DWBA transition amplitude:

T = (–) | F |

(+)

F = bB|V|aA form factor 2

MmMmαβ

jβα

ααα

β22

βαj

βα

ββαα

k,kT 12J12j

1

k

k

2π

μ μ

dΩ

dσ

Double differential cross section for one-nucleon transfer to unbound final states

Momentum distribution

(Dynamics: Fourier transform of the wave function)

Spectral function (Structure: probability per energy for finding the particle in state ℓj at energy E)

Theory for transfer reactions Theory for transfer reactions to unbound statesto unbound statesTheory for transfer reactions Theory for transfer reactions to unbound statesto unbound states

Transfer into the continuum:The B = A+x final states are unbound against the reemission of the nucleon x (ex< 0) the overlap form factor oscillates at large distances the DWBA radial integrals converge very slowly

Vincent and Fortune:powerful method of contour integration in the complex radius plane to overcome the convergence problem C.M. Vincent and H.T. Fortune, PRC2(1970); PRC7(1973); PRC8(1973)

)N(q 2π D β22

0

θdΩ

dσ (E)S )D(q

dE dΩ

σd

j β

jβα

jβββ

βα2

S.E.A. Orrigo and H.Lenske,submitted to PLB (2008)

• p1/2 resonance at ER = 400 keV Mainly a potential resonance

• p3/2 resonance at ER = 850 keV Coupled-channels pairing resonance

New feature essential to describe data

obtained by adding a polarization repulsive surface potential (acting in E~250keV

around ER) to reproduce the full HFB Gorkov-pairing results

• The theoretical results include the experimental energy resolution FWHM~250keV

• Good agreement with data (shape and resonances position)

Spectroscopy of Spectroscopy of 1010Li = Li = 99Li+n at the continuum thresholdLi+n at the continuum thresholdSpectroscopy of Spectroscopy of 1010Li = Li = 99Li+n at the continuum thresholdLi+n at the continuum threshold

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4

E [MeV]

d(9Li,10Li)pcm = [98°,134°]

Totaldata1/2+3/2-1/2-5/2+3/2+d

[

mb

/MeV

]d

EAngle-integrated Angle-integrated

cross sectioncross section d(d(99Li,Li,1010Li)p Li)p

@ 2.36 AMeV@ 2.36 AMeV

S.E.A. Orrigo and H. Lenske,

submitted to PLB (2008)

H.B. Jeppesen et al.,

PLB 642(2006)449

ER ~ 380 keV, ~ 200 keV

Angular distributionsAngular distributionsAngular distributionsAngular distributions

• Good agreement with data (no scaling)

• The p1/2-wave is dominant

• As expected, transfer is favoured at low incident energies:

calculations @ 20 AMeV (MSU exp.) → transfer at ER(p1/2) is lowered by a factor of 26

• The measurement of angular distributions is important to identify the 10Li states (ℓ values)

Angular distributionsAngular distributions d(d(99Li,Li,1010Li)p Li)p

@ 2.36 AMeV@ 2.36 AMeV

S.E.A. Orrigo and H. Lenske,

submitted to PLB (2008)

H.B. Jeppesen et al.,

PLB 642(2006)449

ER ~ 380 keV, ~ 200 keV 0.1

1

10

100

0 20 40 60 80 100 120 140 160 180

Totaldata1/2+3/2-1/2-5/2+3/2+

d(9Li,10Li)p

cm [deg.]

d[mb/sr]d

Transfer Transfer 99Li(d,p)Li(d,p)1010Li @ 2.36 AMeV, before folding, Li @ 2.36 AMeV, before folding, cmcm =[98°,134°] =[98°,134°]

Elastic scattering n+Elastic scattering n+99Li (p-wave)Li (p-wave)


0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

p1/2p3/2d R[mb/

MeV]dE

E [MeV]

0

1000

2000

3000

4000

5000

6000

7000

8000

0 0.2 0.4 0.6 0.8 1

p1/2p3/2

E [MeV]

d E[mb/MeV]dE

• Same structure for elastic and transfer:

a physical resonance appears in both

ℓj(E) [°] → Sℓj (E)

Access to the spectroscopic

information by transfer

• However, in transfer there can be

not physical resonances also, due

only to the reaction dynamics part

E [MeV]

0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8 1

Spectral distribution Spectral distribution ↔↔ properties of n- properties of n-99Li interactionLi interactionSpectral distribution Spectral distribution ↔↔ properties of n- properties of n-99Li interactionLi interaction


• Variations by ±10% of the potential

diffuseness c and radius R

IVol = 486.62 (AMeV)·fm3 = constant

• p1/2-wave: ER, strongly sensitive to c

(asymptotic shape of potential, halo tail)

• Scattering length as = 1.69 fm

(c+10%)=1.68 fm, (c-10%)=1.72 fm

(R+10%)=2.21 fm, (R-10%)=1.06 fm

• s1/2-wave: larger sensitivity to R

(no centrifugal barrier)

Information on the residual n-9Li interaction

c, R

c + 10%

c – 10%

R + 10%

R – 10%

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 0.2 0.4 0.6 0.8 1

E [MeV]

Elastic scattering n+Elastic scattering n+99Li (pLi (p1/21/2-wave)-wave)

d E[mb/MeV]dE

SummarySummarySummarySummary

Fano resonances can be expected to be of particular importance for the continuum dynamics

of exotic nuclei

The coupled channels model extends the QPC into the continuum: the interference between

open 1-QP and closed 3-QP ch. gives sharp and asymmetric resonances (→V13)

The calculations performed for 15C, 17C, 19C show increased effects of correlations

Exp. evidence of DCP correlations in the 15C spectra, qualitatively reproduced by theoretical

calculations, and in the 11Be and 19O spectra

Transfer reactions are a powerful tool to do continuum spectroscopy in exotic nuclei

Innovations of the DWBA approach: to treat unbound final states and to calculate d2/ddE

Calculations performed for the d(9Li,10Li)p reaction at ELi = 2.36 and 20 AMeV

10Li continuum: p1/2-resonance at ~400 keV and p3/2-pairing resonance at ~850 keV

in very good agreement with experimental data

Same behaviour of elastic and transfer: same structure S(E)

Correlation: spectral distributions ↔ n-9Li interaction (sensitivity to the halo tail)

Analogy: Configuration Mixing due to

15C continuum = n+14C, BSEC = n+14C* core polarization (DCP)

10Li continuum = n+9Li unbound, (particle-hole) pairing correlations (MF-level)

☺ Thank you for your attention ☺

Pairing in unbound nuclear states explored in terms of an extended MF approach

Paring effects may introduce pronounced structures and shifts in the low-energy

continuum of all the channels

Configuration mixing acts at the MF level, but mechanisms similar to the mixing

due to dynamical correlations

SummarySummarySummarySummary

Documents

Sonja Orrigo Correlation effects and continuum spectroscopy in light exotic nuclei Unbound Nuclei Workshop, Pisa, Italy, November 3-5, 2008