Outline:• towards effective superfluid local density approximation (SLDA) - general remarks• pairing: volume-, mixed- or surface-type - selectivity/resolution of nuclear data• odd-even staggering (OES) in high-spin isomers – new opportunities to study: - blocking of pair correlations - single-particle, time-odd, and residual p-n effects• termination in N~Z and N=Z nuclei in A~50 mass-region - fine tuning of particle-hole field - fine tuning of shell-model interaction• 73Kr – dynamical manifestation of T=0 pn-pairing at high spins?

W. Satuła IFT Univ. of Warsaw

Probing effective NN interaction at band termination

in collaboration withR.A. Wyss, H. Zduńczuk, M. Zalewski, M. Kosmulski, G. Stoicheva, D. Dean, W. Nazarewicz,

H. Sagawa (+exp), A. Bhagwat, J.Meng ...

M. Zalewski/saturdayH. Zduńczuk/poster

Skyrme-force as a particular realization of effective ph interaction

Long-range part of NN interaction

(must be treated exactly!!!)

Fou

rier

local

corre

ctin

gp

ote

ntia

l

infi

nite

nu

mb

er

of e

qu

ivale

nt

eff

ectiv

e th

eorie

s

10(11)parameters spin-orbit

density-dependent

Skyrme interaction: lim aa 0

LEDF: | H |

Slater determinat(number conserving)

Gogny:

20 parameters

- isovector effective mass (GDR sum-rule enhancement)

Fitting the Skyrme force parameters or the nuclear LEDF

Saturation point of symmetric infinite nuclear matter:

+

- saturation density ( ~0.16fm-3)- energy per nucleon (-16 0.2MeV)- incompresibility modulus (210 20MeV)+

+

- isoscalar effective mass (???)

+

Asymmetric infinite nuclear matter and the isovector properties:

- symmetry energy ( 30 2MeV)

- neutron-matter EOS (Wiringa, Friedmann-Pandharipande)

Finite nuclei [masses,radii,sp levels]:- surface properties (semi-infinite nuclear matter)- realistic mean level-density (masses)

entire ZOO of parameterizations !!!

v(k,k’) = g + g2(k-k’)2 + g4(k-k’)4 +.....(I) Low-momentum transfer expansion in particle-particle channel:

TOWARDS EFFECTIVE (local) PAIRING THEORY

...in r-space:co

ntact term

Gaussian Dirac-delta modelleads to Gogny pairing

thre

e-p

oin

t filte

r0

0.5

1.0

1.5

2.0

0 20 40 60 80 100 120 140

exp.th.

neutron number

(

MeV

)

D1S Gogny

• offers excellent agreement with data

S. Hilaire et al. PLB531 (2002) 61

• no cut-off is needed

x (c)2kF

(mc2)>>

1

kF

1000MeV 1MeV

(200MeVfm)21.4fm-1

interparticle distanceHowever, the so called coherence length i.e. spatial extension of the nucleonic Cooper pair :

In this context the use of finite range pairing force can be viewed as rather unnecessary complication.

TOWARDS EFFECTIVE (local) PAIRING THEORY (II) resolution Gogny versus local (DDDI) pp interaction:

E. Garrido et al. PRC60, 064312 (1999)PRC63, 037304 (2001)

DDDI:

Cut-off!!!(otherwise divergent!)

dots represent the Gogny gap

TOWARDS EFFECTIVE (local) PAIRING THEORY (IIIa) in-medium effects towards the SLDA approach:

[Superfluid Local Density Approximation]

anomalous density(pairing tensor)

ultraviolet divergence

Major obstacle in constructing SLDA is:

we use cut-off!!!(usual, but not

at all satisfactorysolution)

In particular, in infinite homogenous system (example):

regular

„regularize” means in practice simply„remove divergent part”

(relate to scattering amplitude; use dimensional regularization; introduce counter-terms [regulators] with explicit cut-off)

;

isolate and regularizedivergent term

LOCAL EFFECTIVE PAIRING THEORY (IIIb) Bulgac-Yu SLDA approach:

A.Bulgac, Y.Yu, PRL88, 042504 (2002)A.Bulgac, PRC65, 051305(R) (2002)

Formally, gap depends onboth the effective (running)coupling constant and on

QP cut-off energy

In fact, for sufficientlylarge Ec gap is cutoff

independent local HFB

Ec~pc2/2mr ropc~& Ec~2/mro

2 ~ 40MeV

interaction distance

110Sn

Ec=20MeV

30MeV

35MeV

40MeV

45MeV50MeV

Eex

p-E

th [

MeV

]

(

MeV

)

2

50CrSLy4

0

0.05

0.10

0.15

0.20

0.25

surfacemixedvolume

1.0 1.5 2.0n [MeV]

0.5

1.01.5

2.0

2.5

3.0

3.5

deformed

spherical

20 40 60 80 100 120

0

0.2

0.4

0.6

20 40 60 80

1428

4050

82 126

(

MeV

)

N,Z

N,Z

nn

1.5

2.0

0.5

1.0

Pairing/resolution (1)

No

sele

ctiv

ity!!!

J.D

ob

acze

wski &

W.N

aza

rew

icz

Pro

g.

of

Th

eor.

Ph

ys.

Su

pp

l. N

o.

14

6 (

200

2)

spherical HFB

2

0.05

0.10

0.15

0.20

EXP

46TiSLy4

2.5

3.0

3.5

4.0

4.5

5.0

1.0 1.5 2.0n [MeV]

Eex

p-E

th [

MeV

]

surfacemixedvolume

deformed

spherical

Hilaire et al. PLB531 (2002) 61

M. Kosmulski – licentiate thesis

Vo(1-(r)/i)

Pairing/resolution (2)

Beaut

iful e

xam

ple

of se

lect

ivity

!!!

Odd-Even Binding Energy Effect in the High-Spin Isomers:Are Pairing Correlations Reduced in Excited States?

A. Odahara, Y. Gono, T. Fukuchi, Y. Wakabayashi,H. Sagawa, WS, W. Nazarewicz, PRC72, 061303(R), (2005)

Stretched configurations:

N=83 E 8.5MeV const.~~ ~~

Hence, the OES:

is similar in GS and HSI!!!!

no blocking???

We expect:

Dracoulis et al. PLB419, 7 (1998)

High-spin isomers - new opportunities to study pairing correlations

Are Pairing Correlations Reduced in Excited States? (II)(

Z)

[MeV

]

0

0.5

1.0

1.5

2.0

2.5

3.0

61 62 63 64 65 66Z

Z

0

0.4

0.8

61 63

(Z

) [M

eV] HF-SLy4

TEfull

GS

HF SkO/HSIHF SLy4/HSI

GSHSIEXP{

{data

h11/2

-2.0-1.5-1.0-0.50.00.5 s1/2

d3/2

d5/2

6464

SkO

SLy4

WS

DIP

M

esp

hole in[402]5/2

-Fermi energy

fixedoccup.

• sp contribution to OES• time-odd terms (within self-consistent models)

Isomerism of the same type!

Oblate shapes at HSI (-0.2)

Nearly spherical GS

Sphericalsp spectrum

EH

SI [

MeV

]

+10%

Strutinsky calculations with pairing:

8

9

10

144 146 148 150143 145 147 149Nd Pm Sm Eu Gd Tb Dy Ho

0.15

0.20

0.25

61 63 65 67Z

p

n| [

MeV

]

DPES+pn

EXPDIPM

DPES

Excitation energy

+15%

Enhanced pairing is needed (see also Xu et al. PRC60,051301 (1999))

Blocking is too strong!!!

pn

(Z

) [M

eV]

1.0

1.5

2.0

1.0

1.5

2.0

61 62 63 64 65 66Z

proj

exp(Z) HSIGS

DIPMDPES

OES/DPES

GAP/OES

Time-oddfields

pairing

time-odd effects (nuclear magnetism)

single-particle effects

residual pn interaction (odd-odd)

This study reveals that many effects can contribute to OES in particular:

Rutz et al. PLB468 (1999) 1

RMF

Is blocking under control?

...and the question is...

E = f7/2

nImax

E( )

E( ) -

d3/2 f7/2

n+1Imax

-1

20d3/2

f7/2

p-h

energy scale(bulk properties)

spin-orbitdominates!!! ~ 0 light

~ ½ heavy nuclei

• the best examples of almost unperturbed sp motion• uniquely defined (in N=Z) • config. mixing beyond mean-field is expected to be mariginal (in particular all pairs are broken)• shape-polarization effects included already at the level of the SHF• time-odd mean-fields (badly known) can be tested

these are ideal for fine tune particle-hole interaction!!!!

Consider the energy difference between stretched (terminating) configurations in A~50 mass region

SM

SkO-0.5

0

0.5

1.0

1.5

Eth

-E

exp [

MeV

]

40Ca 44Ti46V

42Ca44Ca

43Sc44Sc

45Sc45Ti 47V

N=Z

46Ti42Sc

40 42 44 A

1.0

1.4

1.8

ET [

MeV

]

SkO

DZ

Can be evaluated frommirror-symmetric nuclei e.g.

ph

ph

T=0

T=1

centroid

Isospin symmtery „restoration” in

N=Z nuclei:

40Ca from 40K and 42Sc etc.

original HFresult for

ph excitation

Mean-field versus Shell-Model„isospin symmetry restoration” in N=Z nuclei

Shifted by 480keVreduced s-o T=0 pn pairing???

G.Stoitcheva, WS, W.Nazarewicz, D.J.Dean,M.Zalewski, H.Zduńczuk, PRC73, 061304(R) (2006)

J. Terasaki, R. Wyss, and P.H. Heenen PLB437, 1 (1998)

HFB calculations including T=0 and T=1 pairing

48Cr

f7/2 f7/2]4 4

16+

Collective (prolate)rotation

data

Non-collective (oblate) rotation

isoscalarpairing

d3/2 g9/2-1

T=1 collapses

• Skyrme interaction in p-h• DDDI in p-p channel• fully self-consistent theory• no spherical symmetry

no T=0 atlow spins

(termination)

-0.4

-0.2

0

0.2

0.4

44Ti46V

42Ca44Ca

43Sc44Sc

45Sc45Ti 47V

N=Z

46Ti42Sc40Ca

ES

M-

EE

XP [

MeV

]

SM modified SM

Vphph(JT=0)+3dVphph(JT=1)-dd=175keV

Isovector dependence of shell-modelmatrix elements

BFZ mechanism:

N = Z (Tp=1/2):

N = Z (Tp=T-1/2):

E=-3b/4

E=b(T-1/2) /2

The SM overestimates b by ~700keV!!!

particlecontributionTp=T 1/2

E=1/2b[T(T+1)- -Tp(Tp+1)-Th(Th+1)]

single-holecontribution: Th=1/2

+

Bansal & French, Phys.Lett. 11, 145 (1964)Zamick, Phys. Lett. 19, 580 (1965)

„... It is a grave error to assume that the p-h intraction is independent ofisotopic spin...”

... isotopic dependence of p-h interaction can be approximated by a monopole poten-tial vT~bt1

.t2

VT=1-d

Single j-shell J-T SM phenomenology:

bsym = bsym- 4dVT=0+3d

2(2j+1)(2j+3)VT=1,=2-(2j+1)VT=0,=2-2VT=1,J=0

N. Zeldes, Handbook of Nuclear Properties, Clarendon Press, Oxford, 1996, p.13

Low-spin particle-hole intruder statesSM versus modified-SM

3- 3- 3- 3- 3- 3/2+3-3/2+

3/2+

0

0.2

0.4

0.6

0.8

ES

M-E

EX

P [

MeV

]

3-

3-

3/2+

3-

3/2+

3-

3-

3-

3/2+

2-3/2+

42Ca44Ca40Ca

42Sc43Sc 45Sc

44Sc 44Ti46V45Ti

46Ti 47V

SM modified SM

Concluding remarks

Consistent superfluid local density approximation is just behind our doors!

OES in high-spin isomeric states:-new opportunities to study blocking, TO terms, residual-pn, and mean-field (sp-splitting) effects

Termination in N~Z, A~50 nuclei:

Volume-, mixed- or surface-like local pairing- selective nuclear data exist but must be systematically identified (and understood) thrughout the nuclear chart

- excellent laboratory for fine-tuning of ph MF interaction and SM interaction

73Kr – possible fingerprint of enhanced T=0 pairing

Conventional TRS calculations involving only T=1 pairing:

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

E [

MeV

]

(+,+) (,)

0.5 1.0 1.5 [MeV]

0.5 1.0 1.5

5

10

15

20

25

30 (,)73KrIx

[MeV]0.5 1.0 1.5

1qp

5qp

73Kr

positive parity negative parity negative parity

3qp

3qp

1qp

|1qp> = a+(fp)|0>

|3qp> = a+g a+

g a+(fp)|0>

<1qp|E2|3qp> ~ 0(one-body operator)

g

40

fp

73K

r: K

els

all

et

al., Phys.

Rev. C

65

04

43

31

(2

00

5)

(1) 73Kr - a fingerprint of T=0 pairing?R.Wyss, P.J. Davis, WS, R. Wadsworth

Scattering of a T=0 np pair

(fp)g9/2

(fp)g9/2

(fp)g9/2

(fp)g9/2

(fp)g9/2

(fp)g9/2

(fp)g9/2

(fp)g9/2

1qp configuration(fp)( vacuum

g9/2(+) g9/2 (fp)()

3qp configuration

in 73Kr

What makes the 1qp and 3qp configurations alike?

0

5

10

15

20

25

30

0.4 0.8 1.2 1.60.2 0.6 1.0 1.4 [MeV]

I x

73Kr

theory

exp

T=0

00.51.0

[M

eV]

TRS involving T=0 and T=1pairing

(2) 73Kr a fingerprint of T=0 pairing?

-0.5

0.0

0.5

1.0

1.5

2.0

0.5 1.0 1.5 [MeV]

E [

MeV

]

75Rb 3qp

1qp

5

10

15

20

25

30(+,+)

(,+)75RbIx

0.5 1.0 1.5 [MeV]

0.5 1.0 1.5

1qp

3qp

positive parity negative parityall bands

(3) 73Kr a fingerprint of T=0 pairing?

Excellent agreement was obtained in: Tz=1 : 74Kr,76Rb, D. Rudolph et al. Phys. Rev. C56, 98 (1997) Tz=1/2: 75Rb, C. Gross et al. Phys. Rev. C56, R591 (1997) Tz=1/2: 79Y, S.D. Paul et al. Phys. Rev. C58, R3037 (1998)

Conventional TRS calculations involving only T=1 pairingin neighbouring nuclei: