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)

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

T-, N-, Z-, projected HFB

| T,Tz,N,Z sinddT ,TT ()e iTy d

0

2

d0

2

e i(N ˆ N ) i(Z ˆ Z )

0

(uk vkPk ) | 0

k,1

vk

T,Tz,N,Z | H | T,Tz,N,Z T,Tz,N,Z | T,Tz,N,Z

0, uk2 uk

2 1

Simplified version: projected BCS

uk

vk

1

21

ek

ek 2 2

1/ 2

, 2 vk2 ( , )

Z, 1

N, 1k

E(1, 1) T,Tz,N,Z | H | T,Tz,N,Z T,Tz,N,Z | T,Tz,N,Z

minimum

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• ?