I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA

I. Bentley and S. FrauendorfDepartment of Physics

University of Notre Dame, USA

Calculation of the Wigner Term in the Binding Energies by Diagonalization of

the Isovector Pairing Hamiltonian

Eground bvolume A bsurface A2 / 3

E shell (N,Z) bCoulomb

Z 2

A1/ 3 cCoulombZZ

A

1/ 3

bsymmetry

| Tz | (| Tz |X)

A

Tz N Z

2

bsymmetry 92MeV , A 100

linear term : ~ 1MeV| Tz |

Important for p/rp process near the N=Z line

Quantify X

Subtract the Coulomb energy

ECoulomb bCoulomb

Z 2

A1/ 3 cCoulombZ

Z

A

1/ 3

ECoulomb

Z A

2bCoulomb

Z

A1/ 3 cCoulomb

1

2

1/ 3

E

Z mirror nuclei

Estrong Eground ECoulomb

A=44

A=56

A=68

2

)1()1(

||2/

||

|)(|

)|(|||2

:chain isobarican Along

zstrongzstrong

z

shellz

z

strong

zshell

zzsymmetrystrong

TETE

T

EXTc

T

E

constTEA

XTTbE

“Experimental” Wigner X

Substantial scatter caused by shell effectsMean value ~1 for A<70Mean value ~4 for 80<A<90

Contains shell effects! Separation is problematic.

Eground bvolumeA bsurfaceA2 / 3

E shell (N,Z) bCoulomb

Z 2

A1/ 3 cCoulombZZ

A

1/ 3

bsymmetry(A)| Tz | (| Tz |X(A))

A

Tz N Z

2P. Moeller et al. Atomic Data and

Nuclear Data Tables 59, 185 (1995):

Phenomenological expression for symmertry energy,

which corresponds to X 1

Phenomenological treatment: Micro-Macro

Micro-Macro with Nilsson potential

Density functionals

Skyrme–Hartree–Fock–Bogoliubov mass formula by N. Chamel, S. Goriely, J.M. Pearson, Nuclear Physics A 812 (2008) 72–98:

Skyrme HFB give parameter dependent values of X, substantially smaller than 1, sensitive to effective mass (Satula, Wyss, Rep. Prog. Phys. 68, 131 (05)

Unsatisfactory!

Relativistic Mean Field gives X approximately 1(Ban et al., Phys. Lett. B 633, 231 (06)

What is the origin of X?

Isovector Proton-Neutron Pairing.

Strength is fixed by isospin invariance of stronginteraction. It gives X approximately 1 by symmetry.(Frauendorf, Sheikh, Nucl. Phys. A 645, 509, (99)

1) Fixing the isovector pairing strength to the standard value for pp, nn pairing, obtained from even-odd mass differences, we quantitatively reproduce the experimental X.

2) Possibilities for implementation into density functional approaches (ongoing)

There is a well founded mechanism, which has to be there:

Isovector Pairing Hamiltonian

Generate all configurations by lifting pp, nn, pn pairs and diagonalize.6 or 7 levels around the Fermi level -> dimension ~ 10000 Few cases with 8 levels -> no significant change if G is scaled.

H (k, )k, ˆ N k, G ˆ P k,

ˆ P k,

ˆ P k, 1 ˆ p k

ˆ p k ˆ P k,1

ˆ n k ˆ n

k

ˆ P k,0

1

2ˆ n k ˆ p

k ˆ p k

ˆ n k

[H,r T ]0 : Isospin is conserved.

Solve the pairing problem by diagonalization:-Isospin is good-No problems with instabilities of the pair field

Why X=1? Strong pairing limit Spontaneous breaking of isorotational symmetry

hmf (k, )k, ˆ N k

ˆ P k, ˆ P k,

k,

quasiparticles mixed from proton and neutron particles and holes

ˆ P k,

k

Pair field is a vector, which spontaneously breaks the isorotational symmetry.

Since [Tx ,Ty ] iTz (SU2) strong breaking generates an isorotational band

E(T)T(T 1)

"strong" means >> level distance

Frauendorf SG, Sheikh JACranked shell model and isospin symmetry near N=Z NUCLEAR PHYSICS A 645, 509 (1999)

http://wos22.isiknowledge.com.lib-proxy.nd.edu/?SID=A69dP32CDjAfCkKFaph&Func=Abstract&doc=2/7

02

ˆ

0

ˆ

np

ppnn

np

ppnn

y

z

N Z p n : E(r ) E()

All directions of the pair field are equivalent.

Isorotations (strong symmetry breaking)Bayman, Bes, Broglia, PRL 23 (1969) 1299 ( 2 particle transfer)Frauendorf, Sheikh, NPA 645, 509 (1999) Frauendorf, Sheikh, Physica Scripta T88, 162 (2000)

intrinsic state : | Described by common

quasi proton and quasi neutron excitations without an pn - pair field

isorotational state : DTz

T

0( ,,0) |

isorotational bands : E(T,Tz) hmf Tz T(T 1)

2Spectra of deformed N Z nuclei organanize into

spatio - iso - rotational bands.

Afanasjev AV, Frauendorf S, PRC 71, 064318 (2005)Afanasjev AV, Frauendorf S, NPA 746, 575C (2004 )Kelsall NS, Svensson CE, Fischer S, et al. EURO. PHYS. J. A 20, 131 (2004)….

level spacing dominates

pair fielddominates 1 3

T

spherical sphericaldeformed

Wigner X with AutoTAC Deformations

• Not perfect, but promising.• Two problems :

44≤A≤58 too strong scatter74≤A≤88 Xc~1 Xe~4

• Why?• Calculated deformations • not good enough

smallmediumlarge

Rotational response

Optimize the deformation

• Nilsson calculated• Woods Saxon calculated•Folded Yukawa calculated•Experimental (BE2(2->0)•Experimental yrast energies

“adopted deformations”

Adjusted deformations

2 1

2E(N 1,Z 1) 2E(N,Z) E(N 1,Z 1) , N,Z even

Tz 12

(N Z)

• Isovector proton neutron pairing with the strength fixed by isospin conservation gives the correct X•Mean field treatment (HFB) is insufficient – violates isospin conservation•In devising approximations beyond mean field it is decisive to incorporate restoration of isospin

Isovector and isoscalar pairing

H (k, )k, ˆ N k, GV

ˆ P k, ˆ P k,

,k

GSˆ Q k ˆ Q k

k

ˆ P k, 1 ˆ p k

ˆ p k ˆ P k,1

ˆ n k ˆ n

k

ˆ P k,0 1

2ˆ n k ˆ p

k ˆ p k

ˆ n k

ˆ Q k

1

2ˆ n k ˆ p

k ˆ p k

ˆ n k

1

4E(N 1,Z 1) 2E(N,Z) E(N 1,Z 1) , N Z even

Indication for weak isoscalar pairing correlations?

• Isoscalar pairing attenuates the staggering between the even-even and odd-odd N=Z nuclei: some indication from experiment•Small isoscalar pair correlation would only slightly increase the X values: within the tolerance range of the isovector scenario•What is GS/GV ?

Implementation into mean field approaches

•8 levels around the Fermi level is not enough-> dimensions explode->approximations.•Iso-cranking approximation•HFB + RPA•HFB + SCRPA•T-,N-,Z- projected HFB•BCS-truncation

Iso-cranking Frauendorf, Sheikh, NPA 645, 509 (1999)

For spatial rotations of well deformed nuclei do HFB with:

H 'H Jx, with constraint | Jx | >= I(I 1)

E | H | >I(I 1)

2

In analogy do HBF with:

H 'H Tz ˆ A ,

with constraints , | Tz |, >= T(T 1), , | ˆ A |, >= A

or equivalentely :

H 'H 1ˆ N 1

ˆ Z

with constraints , | ˆ N |, >= N , , | ˆ Z |, >= Z

T(T 1) T 1/2 and N Z

Problem: It works only for a sufficiently strong pair field.

HFB+Lipkin-Nogami may mend the problem.

HFB+QRPA

H (k, )k, ˆ N k, G ˆ P k,

ˆ P k,

2

r T

r T

ˆ P k, 1 ˆ p k

ˆ p k ˆ P k,1

ˆ n k ˆ n

k

ˆ P k,0

1

2ˆ n k ˆ p

k ˆ p k

ˆ n k

[H,r T ]0 : Isospin is conserved.

K. Neergard PLB 537, 287 (2002); PLB 572, 159 (2003); PRC 80, 044313 (2009)

T

TdT

dE )(0

ji

ji eeTTEETETE1

220 )(

2)0()(

2)1(

)(

TT

TE

Equidistant levels

Iso-cranking

HFB+QRPA unreliable near the critical G. We are close by.

SCQRPA is not worked out for full isovector pairing. Hung & Dang RIKEN working on it.

nn pairing

Only nn pairing

In BCS state a certain configuration has the weight : w fk with fk uk if no pair

vk if pair

k

All configuarations with w taken.

BCS

BCS

• Generalization to full isovector pairing OK• Not implemented yet• ?

I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA

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