1. n-p pairing in N=Z nuclei
reality or fiction?
W. Satuła
University of Warsaw
Motivation & fingerprints (basic concepts):
symmetric nuclear matter calculations
binding energies - mean-field crisis around N~Z line
the Wigner energy and the generalized
blocking phenomenon
elementary isobaric exciatations in N~Z nuclei
– a need for isosopin symmetry restoration
cranking in isospace - response of t=0 pairing
against rotations in isospace
high-spin signatures of pn-pairing

2. Structure of nucleonic pairs
N=Z  nucleons start to occupy „identical” spatial orbitals
Nuclear interaction favoures L=0 coupling
Pair-structure is governed by the Pauli principle:
Isovector (or 1S0) pairs T=1, S=0
- Isoscalar (deuteron-like or 3S1) pairs T=0, S=1
Neutron
Proton + Tz - Sz

3. 3S1-3D1 (coupled) pairing gap in symmetric Nuclear Matter from Paris VNN
free-space sp
spectrum
BHF sp
spectrum
(in-medium
corrections)
tensor-force
enhancement
From M. Baldo et al.
Phys. Rev. C52, 975 (1995)

4. Gaps from local effective pairing interaction
DDDI:
E. Garrido et al. PRC60, (1999)
PRC63, (2001)
DDDI used in Skyrme-HFB calculations
by Terasaki et al. NPA621 (1997) 706.
Isoscalar pairing
Tensor force
enhancement ro
Cut-off!!!
(otherwise divergent!)

5. 3S1-3D1 (coupled) pairing gap in symmetric Nuclear Matter including relativistic corrections
Saturation
density
O. Elgaroy, L.Engvik,
M.Hjorth-Jensen, E.Osnes,
Phys.Rev.C. 57 (1998) R1069
Includes relativistic in-medium corrections
(sp levels from Dirac-Brueckner-HF)

6. Empirical NN interaction in N~Z
T=1 channel:
J=0 coupling dominates
T=0 channel:
J=1 and J=2j are similar
T=0 is, on the average, stronger than T=1 by a factor of ~1.3
N.Anantaraman
& J.P. Schiffer
PL37B (1971) 229
Dufour & Zucker, Phys. Rev. C54, 1641 (1996)

7. The model: deformed mean-field plus pairing:
Pairs p-n and p-ñ
Pairs: ~
Pairs ñ-n and p-p « usual » ; T=1 ~
Pairs: p-ñ + n-p; T=1
0 0 ~
Pairs: p-ñ – n-p ; T=0
Hamiltonian BCS:
N.Anantaraman
and J.P. Schiffer
PL37B (1971) 229

8. Comparison with delta-force towards a local theory
M.Moinester, J.P. Schiffer,
W.P. Alford, PR179 (1969) 984

9. BCS transformation BCS transformation takes the following form : real
A.L.Goodman
Nucl. Phys. A186
(1972) 475
BCS transformation takes the following form :
real
complex
where the variational parameters are:
i 2
Density matrix (occupation) and the pairing tensor
Generalization: BCSHFB UiU & ViV matrices of dimension 4N

10. BCS Solution Energy (Routhian)
Variational equation in N=Z system (without Coulomb)
Occupation probabilities ; quasiparticle energies:
Pair gaps:
 n-ñ, p-p
T=1 n-p + p-ñ ~
Gap T=0 aã
Gap T=0 aa ~

11. T=0/T=1 (no)mixing 48Ca X= / Incomplete mixing? T=1, Tz=+/-1 andTz=0
W.S. & R.Wyss
PLB 393 (1997) 1

12. Energy gain as a function of T=0/T=1
pairing’s mixing „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=
generalized blocking effect
n-excess
blocks pn-pairs
scattering
Wigner term from
Myers & Swiatecki
protons
neutrons
Satuła & Wyss PLB393 (1997) 1

13. Wigner effect from self-consistent Skyrme-HF
N=Z
Exp.
HFBCS T=1 Sph.
HFBCS T=1 Def.
(SIII)
Defficiency of conventional self-consistent models:
HF or HFB including standard
T=1, |Tz|=1 ~ p-p & n-n pairs:
(N-Z)2 ~ T2 term is OK!
no (or very weak) |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

14. DE= asymT(T+x) The Wigner effect 1 2 w / w X= Jmax 0? 1??25
A=48 20
B (MeV) 15
48Cr 10
1.0 5
DE= asymT(T+x) 2 1 w
w / w
total
0.8 -4 4
N-Z
0.6
0.4 0?
1??
1.25??? exp. in N~Z
4 ???? Wigner SU(4)
0.2
24Mg X=
0.0 1 2 3 4 5 6 7
Jmax

15. Isobaric excitations in N~Z nuclei 0.5 1.0 1.5 2.0 47/A [MeV]
W(A) [MeV] 30 40
0.6
1.4
GT=0
GT=1 A
The lowest:
T=0, T=1 & T=2 in e-e nuclei
T=0 & T=1 states in o-o nuclei
The model needs to be extended to
include isospin projection  isospin cranking
strong T=0 pairing limit!
J.Janecke, Nucl. Phys. A73 (1965) 73
A. Macchiavelli et al. Phys. Rev. C61 (2000) (R)
P.Vogel, Nucl. Phys. A662 (2000) 148

16. „inertia” defined through
The extreme s.p.model:
4-fold degenerated
equidistant
s.p. spectrum
Eigen-states (routhians) are
2-fold (Kramers) degene-
rated „stright lines”:
Crossings form simple
arithmetic serie:  
Energy:
„inertia” defined through
mean level spacing !!!

17. 5 10 15 20 DET=2 [MeV] T=2 iso-cranking 20 30 40 50 A DE= deT2 1 214
T=2 states in e-e nuclei 5 10 15 20 20 28
DET=2 [MeV]
hWS+HT=1
+HT=0-wtx
T=2
hWS+HT=1
-wtx
iso-cranking 20 30 40 50 A
Iso-cranking gives excitation energy which goes like:
DE= deT2 1 2
+ Epair
vacuum
mean level spaceing at the Fermi energy

18. (iso)Coriolis antipairing effect iso-MoI D [MeV] hw [MeV] 48Cr 6 Tx 3
0.5
1.0
1.5
iso-moment of inertia
D/e = 0.001
D [MeV] 1 2 3
hw [MeV]
DT=0
DT=1
48Cr 6 Tx
e=1
D/e = 0.5;1.0;1.5
0.7
0.6
iso-moment of inertia Tz 1 2 3 4
0.5
0.4
0.3 1 2 3 hw

19. T=1 states in e-e N=Z nuclei
T=1 states: 2qp + isocranking

20. Isocranking N=Z odd-odd nuclei
odd-T sequence T de 5 de de
2de
4de
6de 4 hw
even-T sequence 3 de 2 de de 1 hw de
3de
5de
iso-signature
selection rule
Eeven-T = 1/2deTx2
Eodd-T = 1/2deTx2 - 1/2de

21. T=0 vs T=1 states in o-o N=Z nuclei
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

22. 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, (2001)

23. see e.g. Bohr & Mottelson „Nuclear Structure” vol. I
Schematic
isospin-isospin
interaction:
extreme sp model de
3de hw
de+k
3(de+k) l
even-even vacuum
see e.g. Bohr & Mottelson „Nuclear Structure” vol. I
Neergard PLB572 (2003) 159 1
H=hsp- wT+ kTT 2
mean -
field
(Hartree)
HMF =hsp- (w - k T )T
iso-cranking with
isospin-dependent
frequency!!!
E= (de+k)T2 1 2
Hartree
E= (de+k)T2
+ kT 1 2 1 2
Hartree-
-Fock

24. Pairing in fast rotating nuclei
Muller et al., Nucl. Phys. A383 (1982) 233
Resistance of nucleonic
paires against
fast rotation:

25. 48Cr ; HFB calculations including T=0 & T=1 pairing
J. Terasaki, R. Wyss, and P.H. Heenen PLB437, 1 (1998)
Skyrme interaction in p-h
DDDI in p-p channel
fully self-consistent theory
no spherical symmetry
two-classes of solutions:
d3/2 g9/2 -1
- T=0 dominated at I=0
- T=1 dominated at I=0
isoscalar
pairing
Non-collective (oblate)
rotation
no T=0 at
low spins
Collective (prolate)
rotation
T=1 collapses
[nf7/2 pf7/2] 4
16+
(termination)
exp

26. (1) 73Kr – manifestation of (dynamical) T=0 pairing?
R.Wyss, P.J. Davis, WS, R. Wadsworth
Conventional TRS calculations involving only T=1 pairing:
positive parity
negative parity
negative parity Ix
73Kr
(+,+)
3qp
(-,-)
73Kr
-0.5
0.0
0.5
1.0
1.5
2.0
2.5 5 10 15 20 25 30
(-,-)
Ew [MeV]
5qp
1qp
1qp
73Kr: Kelsall et al., Phys. Rev. C (2005)
0.5
1.0
1.5
0.5
1.0
1.5
0.5
1.0
1.5
hw [MeV]
hw [MeV] g 40 fp
|1qp> = a+n(fp)|0>
|3qp> = a+ng a+pg a+p(fp)|0>
<1qp|E2|3qp> ~ 0
(one-body operator)

27. 73Kr (2) 73Kr – manifestation of (dynamical) T=0 pairing?
What makes the 1qp and 3qp configurations alike?
Scattering of a T=0 np pair 5 10 15 20 25 30
0.4
0.8
1.2
1.6
0.2
0.6
1.0
1.4
hw [MeV] Ix
73Kr
theory
exp
DT=0 Dp Dn
0.5
D [MeV]
TRS involving T=0 and T=1
pairing
in 73Kr
n(fp)
ng9/2
p(fp)
pg9/2
n(fp)(-)
vacuum
1qp configuration
n(fp)
p(fp)
pg9/2
ng9/2
n(fp)
ng9/2
ng9/2(+)
pg9/2 p(fp)(-)
3qp configuration

28. (3) 73Kr – manifestation of (dynamical) T=0 pairing?
Conventional TRS calculations involving only T=1 pairing
in neighbouring nuclei:
all bands
positive parity
negative parity Ix
(-,+)
75Rb
75Rb
3qp 5 10 15 20 25 30
-0.5
0.0
0.5
1.0
1.5
2.0
(+,+)
Ew [MeV]
1qp
3qp
1qp
0.5
1.0
1.5
0.5
1.0
1.5
0.5
1.0
1.5
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)

29. SUMMARY Part of T=0 correlations in N~Z nuclei is definitely
beyond standard formulation of mean-field
(Wigner energy)
Adding T=0 pairing helps but cannot solve the problem of
the Wigner energy (symmetry energy) in N~Z nuclei
which seems to be beyond mean-field
There is no convincing arguments for coherency of
the T=0 phase
Theoretical treatment of T=1 states in e-e nuclei and
T=0 states o-o nuclei requires angular momentum
and isospin projections

30. Independent least-square fits of:
the Wigner energy strength:
aw|N-Z|/Aa
the symmetry energy strength:
as(N-Z)2/Aa
4asT(T+x);
x=aw/2as a aw
2as
sn-1 x
(*)
(**)
0.95 39
0.196 31
0.106
1.26
1/2 8
0.239 6
0.153
1.33
2/3 14
0.213 11
0.125
1.27 1 47
0.196 38
0.107
1.24
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