N p pairing in n z nuclei
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the Wigner energy and the generalized. blocking phenomenon. cranking in isospace - response of t=0 pairing. against rotations in isospace. reality or fiction ?. n-p pairing in N=Z nuclei. Motivation & fingerprints ( basic concepts ): . W. Satuła University of Warsaw.

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n-p pairing in N=Z nuclei

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N p pairing in n z nuclei

  • theWigner energy and thegeneralized

blockingphenomenon

  • crankinginisospace - responseof t=0 pairing

againstrotationsinisospace

reality orfiction?

n-ppairinginN=Znuclei

Motivation & fingerprints (basicconcepts):

W. Satuła

University of Warsaw

  • symmetricnuclearmattercalculations

  • bindingenergies - mean-fieldcrisisaroundN~Zline

  • elementaryisobaricexciatationsinN~Znuclei

  • – a need for isosopinsymmetryrestoration

  • high-spin signatures of pn-pairing


Structure of nucleonic p airs

Neutron

-

Structure of nucleonic pairs

  • N=Z  nucleons start to occupy „identical” spatialorbitals

  • Nuclear interaction favoures L=0 coupling

  • Pair-structureisgoverned by thePauliprinciple:

    • Isovector(or1S0) pairs T=1, S=0

      - Isoscalar (deuteron-likeor3S1) pairsT=0, S=1

Proton

+

Tz+1 0 -1

Sz +1 0 -1


N p pairing in n z nuclei

3S1-3D1(coupled)pairing gap insymmetricNuclearMatterfromParis VNN

free-spacesp

spectrum

BHF sp

spectrum

(in-medium

corrections)

tensor-force

enhancement

From M. Baldo et al.

Phys. Rev. C52, 975 (1995)


N p pairing in n z nuclei

Gapsfromlocaleffectivepairinginteraction

DDDI usedinSkyrme-HFBcalculations

by Terasaki et al. NPA621 (1997) 706.

Isoscalarpairing

Tensor force

enhancement

Cut-off!!!

(otherwisedivergent!)

ro

DDDI:

E. Garrido et al. PRC60, 064312 (1999)

PRC63, 037304 (2001)


3 s 1 3 d 1 coupled pairing gap in symmetric nucl ear matter including relativistic corrections

3S1-3D1(coupled)pairing gap insymmetricNuclearMatterincludingrelativisticcorrections

Saturation

density

Includesrelativisticin-mediumcorrections

(splevelsfromDirac-Brueckner-HF)

O. Elgaroy, L.Engvik,

M.Hjorth-Jensen, E.Osnes,

Phys.Rev.C. 57 (1998) R1069


E mpirical nn interaction in n z

Empirical NN interactioninN~Z

  • T=1 channel:

    • J=0coupling dominates

  • T=0 channel:

    • J=1 and J=2j aresimilar

    • T=0 is, on theaverage, strongerthan T=1 by a factor of ~1.3

N.Anantaraman

& J.P. Schiffer

PL37B (1971) 229

Dufour & Zucker, Phys. Rev. C54, 1641 (1996)


N p pairing in n z nuclei

The model: deformedmean-field plus pairing:

0 0

Pairs p-n and p-ñ

Pairs:

~

Pairs ñ-n and p-p « usual » ; T=1

~

Pairs: p-ñ + n-p; T=1

~

Pairs: p-ñ – n-p ; T=0

Hamiltonian BCS:

N.Anantaraman

and J.P. Schiffer

PL37B (1971) 229


Comparison with delta force towards a local theory

Comparisonwithdelta-forcetowards a localtheory

M.Moinester, J.P. Schiffer,

W.P. Alford, PR179 (1969) 984


Bcs transformation

A.L.Goodman

Nucl. Phys. A186

(1972) 475

BCStransformation

BCS transformationtakesthefollowing form :

real

complex

wherethevariationalparametersare:

i 2

Densitymatrix(occupation) and thepairing tensor

Generalization:BCSHFB UiU &ViV matrices of dimension 4N


Bcs solution

BCS Solution

Energy(Routhian)

Variational equationinN=Z system (without Coulomb)

Occupationprobabilities; quasiparticleenergies:

Pairgaps:

Gap T=0 aã

Gap T=0 aa

n-ñ, p-p

T=1 n-p + p-ñ

~

~


T 0 t 1 no mixing

X

X

T=0/T=1 (no)mixing

X= /

48Ca

  • Incompletemixing?

  • T=1, Tz=+/-1 andTz=0

  • T=1, Tz=+/-1andT=0

W.S. &R.Wyss

PLB 393 (1997) 1


N p pairing in n z nuclei

Energy gain as a functionof T=0/T=1

pairing’smixing „x”

Energy gain:

DMass =E(T=0+1)- E(T=1)

Thomas-Fermi

X=1.1

X=1.2

X=1.3

X=1.4

X=

/

generalizedblockingeffect

n-excess

blocks pn-pairs

scattering

Wignerterm from

Myers & Swiatecki

neutrons

protons

Satuła & Wyss PLB393 (1997) 1


Wigner effect from self consistent skyrme hf

Wignereffectfromself-consistentSkyrme-HF

N=Z

Exp.

HFBCS T=1 Sph.

HFBCS T=1 Def.

(SIII)

  • Defficiency of conventionalself-consistentmodels:

    HF or HFBincluding standard

    T=1, |Tz|=1 ~ p-p & n-npairs:

  • (N-Z)2 ~ T2 termisOK!

  • no (orveryweak) |N-Z| ~ term

|N-Z|=2,4 (black)

A.S. Jensen, P.G.Hansen, B.Jonson,

Nucl.Phys. A431(1984) 393

o-o

e-e


N p pairing in n z nuclei

TheWignereffect

total

1

2

w / w

DE= asymT(T+x)

25

A=48

20

B (MeV)

15

48Cr

10

5

1.0

w

0

0.8

N-Z

-4

0

4

0?

1??

1.25??? exp. inN~Z

4 ???? Wigner SU(4)

0.6

0.4

24Mg

0.2

X=

0.0

0

1

2

3

4

5

6

7

Jmax


N p pairing in n z nuclei

Isobaricexcitations

inN~Z nuclei

2.0

1.5

1.0

0.5

47/A [MeV]

W(A) [MeV]

  • Thelowest:

  • T=0, T=1 & T=2 in e-e nuclei

  • T=0 & T=1 statesin o-o nuclei

GT=0

1.4

GT=1

  • The model needs to be extended to

  • includeisospinprojection isospincranking

0.6

strong T=0 pairing limit!

A

30

40

J.Janecke,Nucl. Phys. A73 (1965) 73

A. Macchiavelli et al.Phys. Rev.C61(2000) 041303(R)

P.Vogel,Nucl. Phys. A662 (2000) 148


N p pairing in n z nuclei

Theextremes.p.model:

4-fold degenerated

equidistant

s.p. spectrum

Energy:

Eigen-states (routhians) are

2-fold (Kramers) degene-

rated „stright lines”:

Crossings form simple

arithmetic serie:

„inertia” defined through

mean level spacing !!!


N p pairing in n z nuclei

T=2 states in e-e nuclei

20

14

28

1

2

DE=deT2

20

15

10

5

0

20

30

40

50

DET=2[MeV]

hWS+HT=1

+HT=0-wtx

T=2

hWS+HT=1

-wtx

iso-cranking

A

Iso-cranking gives excitation energy which goes like:

+ Epair

vacuum

mean level spaceing at the Fermi energy


N p pairing in n z nuclei

(iso)Coriolis

antipairingeffect

iso-MoI

Tx

1.5

1.0

48Cr

0.5

0

0

1

2

3

D/e = 0.001

3

6

e=1

iso-moment of inertia

2

1

D/e = 0.5;1.0;1.5

0

3

0.7

DT=0

2

D [MeV]

Tz

0.6

DT=1

0

iso-moment of inertia

1

1

0.5

2

3

0

0.4

4

0

1

2

3

hw [MeV]

0.3

hw


T 1 states in e e n z nuclei

T=1 statesine-e N=Znuclei

  • T=1 states: 2qp+ isocranking


Isocranking n z odd odd nuclei

odd-Tsequence

Isocranking N=Z odd-oddnuclei

T

de

5

de

de

2de

6de

4de

4

hw

even-Tsequence

3

de

2

de

1

de

0

hw

de

3de

5de

iso-signature

selection rule

Eeven-T = 1/2deTx2

Eodd-T = 1/2deTx2 - 1/2de


N p pairing in n z nuclei

T=0 vs T=1 statesino-o N=Znuclei

T=0

T=1

1.0

2qp

cranking

vacuum

0.5

DET=1 - DET=0 [MeV]

0.0

-0.5

exp

th

20

30

40

50

60

70

A


N p pairing in n z nuclei

Neutron-proton pairing collectivity

(a fit plus three easy steps)

(III)

ET=1 - ET=0 (even-even)

ET=1 - ET=0 (odd-odd)

(II)

  • Wigner energy linked to the n-p pairing collectivity

  • T=2 states in even-even nuclei obtained from isocranking

  • T=1 states in even-even nuclei obtained as 2qp excitations

  • T=1 states in odd-odd nuclei obtained from isocranking

  • T=0 states in odd-odd nuclei obtained as 2qp excitations

Fit of GT=0 /GT=1

ET=2 - ET=0 (even-even)

(I)

W. Satuła & R. Wyss Phys. Rev. Lett., 86, 4488(2001);

Phys. Rev. Lett., 87, 052504(2001)


N p pairing in n z nuclei

Schematic

isospin-isospin

interaction:

extreme sp model

even-even vacuum

de

H=hsp- wT+ kTT

l

2

de

de

de

de+k

3de

3(de+k)

hw

+ kT

E= (de+k)T2

1

1

1

E= (de+k)T2

2

2

2

seee.g. Bohr & Mottelson „NuclearStructure” vol. I

Neergard PLB572 (2003) 159

1

mean -

  • field

  • (Hartree)

HMF=hsp- (w - k T )T

iso-cranking with

isospin-dependent

frequency!!!

Hartree

Hartree-

-Fock


N p pairing in n z nuclei

Pairinginfastrotatingnuclei

Muller et al., Nucl. Phys. A383 (1982) 233

Resistance of nucleonic

paires against

fastrotation:


N p pairing in n z nuclei

48Cr ; HFB calculationsincluding T=0 & T=1 pairing

-1

d3/2 g9/2

4

4

[nf7/2pf7/2]

16+

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

  • Skyrmeinteractioninp-h

  • DDDI inp-p channel

  • fullyself-consistenttheory

  • no sphericalsymmetry

  • two-classes of solutions:

- T=0 dominatedat I=0

- T=1 dominatedat I=0

isoscalar

pairing

Non-collective (oblate)

rotation

no T=0 at

low spins

Collective (prolate)

rotation

T=1 collapses

(termination)

exp


N p pairing in n z nuclei

(1) 73Kr – manifestationof (dynamical) T=0 pairing?

3qp

2.5

30

2.0

3qp

1.5

25

1.0

20

0.5

15

0.0

-0.5

10

5

g

40

0.5

0.5

0.5

1.0

1.0

1.0

1.5

1.5

1.5

fp

R.Wyss, P.J. Davis, WS, R. Wadsworth

Conventional TRS calculations involving only T=1 pairing:

negative parity

negative parity

positive parity

Ix

(-,-)

(+,+)

73Kr

73Kr

(-,-)

Ew [MeV]

5qp

1qp

1qp

73Kr: Kelsall et al., Phys. Rev. C65 044331 (2005)

hw[MeV]

hw[MeV]

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

|3qp> = a+ng a+pg a+p(fp)|0>

<1qp|E2|3qp> ~ 0

(one-body operator)


N p pairing in n z nuclei

(2) 73Kr – manifestationof (dynamical) T=0 pairing?

1.0

n(fp)

p(fp)

p(fp)

0.5

pg9/2

pg9/2

ng9/2

0

n(fp)

p(fp)

p(fp)

30

ng9/2

pg9/2

pg9/2

25

20

15

10

5

0

1.4

0.4

0.8

1.0

1.2

1.6

0.2

0.6

What makes the 1qp and 3qp configurations alike?

Scattering of a T=0 np pair

TRS involving T=0 and T=1

pairing

in 73Kr

Dn

Dp

D [MeV]

DT=0

73Kr

n(fp)(-)

vacuum

1qp configuration

Ix

n(fp)

theory

ng9/2

exp

n(fp)

ng9/2

ng9/2(+)

pg9/2 p(fp)(-)

3qp configuration

hw [MeV]


N p pairing in n z nuclei

2.0

30

1.5

25

1.0

20

0.5

15

0.0

10

-0.5

5

0.5

1.0

1.5

0.5

0.5

1.0

1.0

1.5

1.5

(3) 73Kr – manifestationof (dynamical) T=0 pairing?

Conventional TRS calculations involving only T=1 pairing

in neighbouring nuclei:

negative parity

positive parity

all bands

Ix

(-,+)

75Rb

3qp

75Rb

(+,+)

Ew [MeV]

1qp

3qp

1qp

hw[MeV]

hw[MeV]

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)


N p pairing in n z nuclei

SUMMARY

Part of T=0 correlationsinN~Znucleiisdefinitely

beyond standard formulation of mean-field

(Wigner energy)

Adding T=0 pairinghelps but cannotsolvethe problem of

theWigner energy (symmetry energy) inN~Znuclei

whichseems to be beyondmean-field

Thereis no convincingarguments for coherency of

the T=0 phase

Theoreticaltreatment of T=1 statesin e-e nuclei and

T=0 states o-o nucleirequiresangularmomentum

and isospinprojections


N p pairing in n z nuclei

(**)

(*)

sn-1

aw

2as

sn-1

x

a

4asT(T+x);

x=aw/2as

0.153

1.33

0.239

6

1/2

8

0.125

1.27

0.213

11

2/3

14

0.107

1.24

0.196

38

1

47

0.106

1.26

0.196

31

0.95

39

Independent least-square fits of:

aw|N-Z|/Aa

the Wigner energy strength:

as(N-Z)2/Aa

the symmetry energy strength:

Głowacz, Satuła, Wyss, J. Phys. A19, 33 (2004)

very consistent with: Janecke, Nucl. Phys. (1965) 97

Fit includes N~Z nuclei with:

Z>10; 1<Tz<3

excluding odd-odd Tz=1 nuclei

-

-

-

(*)

See: Satuła et al. Phys. Lett. B407 (1997) 103

(**)

Based on double-difference formula:

J.-Y Zhang et al. Phys. Lett. B227 (1989) 1


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