Open problems at the N=Z line
P. Van Isacker, GANIL, France
S3 opportunity for N=Z
What’s so interesting about N=Z?
Some selected topics
SPIRAL2 week, Caen, November 2016
Nuclear regions covered by S3
What’s so interesting about N=Z?
Superallowed 0+ -> 0+ β decay and precision tests
of the standard model (unitarity CKM matrix).
Charge symmetry and charge independence of the
nuclear interaction (isospin symmetry).
Extra binding energy due to enhanced correlation
effects (e.g., Wigner energy).
Neutron-proton pairing.
D.D. Warner, M.A. Bentley & PVI, Nature Physics 2 (2006) 311
Superallowed 0+ -> 0+ β decay
Conservation of vector current (CVC) hypothesis:
2π 3! 7 ln 2 / me5c 4
ft1/2 =
= constant
2
2
GV M F
Requires corrections:
2π 3! 7 ln 2 / me5c 4
Ft1/2 ≡ ft1/2 (1+ δR" ) (1+ δNS − δC ) =
= constant
2
V
2GV (1+ Δ R )
Experiment gives t1/2, Q values & branching ratio.
(Nuclear structure) theory gives δNS and δC.
J.C. Hardy & I.S. Towner, Phys. Rev. C 91 (2015) 025501
Example:
74Rb
Superallowed 0+ -> 0+ β decay
SUPERALLOWED 0+ → 0+ NUCLEAR β DECAYS: A CRITICAL SURVEY WITH . . .
3090
74
PHYSICAL REVIEW
Rb
3070
34
ft (s)
Calculated
Ar
FIG. 4. Experimental
function of the charge on
Z. The bands represent th
Ft/[(1 + δR′ )(1 − δC + δN
distinguish those β em
nuclei have isospin Tz =
from those with Tz = 0 (l
22
Mg
38
Km
3050
34
Cl
42
10
C
14
46
V
O
26
Al
3030
0
54
Sc
10
50
Mn
Co
m
20
30
40
Z of daughter
J.C. Hardy & I.S. Towner, Phys. Rev. C 71 (2005) 055501
values. The method, which is illustrated in Fig. 4, is based
on the validity of the CVC hypothesis that the corrected
These new experiments can follow diff
to improve the precision on the nine super
J. C. HARDY
AND I. S.
TOWNER
Error
budget
Frac. Uncertainty (%)
have b
the ex
0.04
have t
is an
of a th
approa
compl
0.02
A=2
precisi
Of
the unc
the sta
`
resolu
V
Exp
R
R
C NS
Never
accept
2
FIG. 8. Uncertainty budget for |VJ.C.
| as& obtained
from
ud Hardy
I.S. Towner, Phys.
Rev. Csuperal91 (2015) 025501
sum is
+
+
lowed 0 → 0 β decay. The contributions are separated into four
|Vud |.
51Fe
51Mn
Experiment
Shell Model
6
fp-shell Model
Energy
(MeV)
Experiment
Isospin symmetry
25
21
CED
17
2J
3
13
9
0
5
100
0
-100
CED (keV)
The J=2 (0?) anomaly in 0f7/2
M.A. Bentley et al., Phys. Rev. C 92 (2015) 024310
Status of the anomaly
Bentley, Lenzi et al.: a comprehensive analysis of
mirror energy differences for A=42-54. Evidence for an nucleon-nucleon isovector term
with a strong J dependence.
The isospin symmetry breaking is not consistent
with that found in the bare nn interaction and
its J dependence differs from Coulomb.
The anomaly is not understood. This calls for
similar studies for higher A.
Wigner binding energy
Binding energy from Wigner supermultiplet theory:
2
( N − Z ) + 6δpairing ( N, Z ) + 8 N − Z + 8δN,Z π np
with δpairing(N,Z)=0, 1, 2 for even-even, odd-mass,
odd-odd, and πnp=1 for odd-odd.
Empirical expression for Wigner binding energy:
BW ( N, Z ) = −W ( A) N − Z − d ( A) δ N,Z π np
So, what’s the problem?
E. Wigner, Phys. Rev. 51 (1937) 106
Pairing
Pairing refers to the interaction between
particles in ‘time-reversed orbits’.
In nuclei: nucleons with orbital angular momenta
coupled to L=0.
Pairing is essential to understand the properties
of nuclei. Analogous to the superconducting
metallic state (BCS theory).
Pairing with neutrons and protons
For neutrons and protons two pairs and hence two
pairing interactions are possible:
1S
0
3S
1
isovector or spin singlet (S=0,T=1)
isoscalar or spin triplet (S=1,T=0)
Nucleon-nucleon interaction
ç
Nucleon-nucleon interaction
ç
Nucleon-nucleon interaction
ç
Neutron-proton correlations
The question is not whether T=0 interactions
between nucleons exist or whether they are
important. They do and they are.
The question is whether T=0 correlations exist,
similar to BCS-type ones for T=1.
Test from two-nucleon transfer reactions.
Related (but different) question: Are aligned T=0
neutron-proton pairs dominantly important?
Test from magnetic & electric moments of N=Z nuclei.
T=1 pair vibrations in Pb
A. Bohr & B.R. Mottelson, Volume 2, page 646
T=0 pair vibrations in Ca-Sc-Ti-V?
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Spin-aligned coupling scheme
B. Cederwall et al., Nature 469 (2011) 68
Moments of isomers in
94Ag
Quadrupole moment of 21+ isomer:
81949367824
2
3
Q ( b∞ 21) =
(eν + eπ ) ( ho ) ≈ 0.44 b
3489855625
Quadrupole moment of 7+ isomer:
30930277300923364
2
3
Q ( b [16 ] 7) =
(eν + eπ ) ( ho ) ≈ 0.64 b
627253477610841
∴ Measurement of quadrupole moments tests
spin-aligned scheme.
Conclusions
Superallowed 0+ -> 0+ β decay: mirror transitions
constrain the nuclear-structure corrections.
The J=2 (0?) anomaly remains unresolved and calls
for experiments at higher A.
A BCS-like phase for T=0 pairing? Test from
deuteron transfer.
Dominance of T=0 aligned pairs? Test from
magnetic and quadrupole moments.
T=1 pair vibrations
rm produces a spectrum
s.
any) collective correlations for T = 0 pairing. In fact,
this small lowering of the energy of the T = 0 state with
respect to the single-particle excitation, can be accounted
for by a residual np diagonal56
interaction, ∼ 20MeV/A,
rather than any collective effect.
Pair vibrations on
E
x
(MeV)
60
Pb isotopes
4.5
Zn
T=1
Isoscalar 58
Cu
Phonon
T=0
60Zn
T=0
3.0
58Zn
58
1.5
(1,0,0)
212
Aon spectrum for addition
b. The excitation energies
Pb isotopes derived from
60 Cu
-1
T=1
Ni
Ni
T=1
(0,0,1)
(0,1,0)
56
0
(0,1,1)
58
Cu
T=1
60Cu T=2
(1,0,1) T=2
(0,2,0)
T=1
208 210
Ni
hω + hω 0
1
hω0
60Ni T=2
(0,0,2)
Isovector
Phonon
2hω 1
hω 1
T=0
0
1
T
2
A.O. Macchiavelli et al., arXiv:nucl-th/9912041
Pair vibrations & rotations
Pairing hamiltonian for several shells. For two
shells and identical nucleons:
Ĥ ( Δε, g) = Δε ( n̂+ − n̂− ) − gV̂J=0,T =1
Phase transition occurs as a function of g/Δε:
For g<gcrit: pair vibrations (ex: Pb isotopes)
For g>gcrit: pair rotations (ex: Sn isotopes)
Two different coupling schemes:
$ U 1 ×U 1
+( )
−( )
&
SU + ( 2 ) × SU − ( 2 ) ⊃
&
SU ( 2 )
%
'
)⊃ U 1
()
)
(
Pair vibrations and rotations
Pair correlations result in a constructive interference of reaction amplitudes giving a
enhanced two-nucleon transfer.
σ ( Αgs→ Α+2gs)/σsp
Superfluids
∼ (Δ/G)2
∼ Ω2
Vibrations
~ (n +1 )
∼ Ω
Single
Particle
Ao
Closed shell
Closed shell
A.O. Macchiavelli
T=1 pair vibrations in Ca
T=1 pair vibrations in Ca-Sc-Ti-V?
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T=0 pair vibrations in Ca-Sc-Ti-V?
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T=0 aligned pairs in Ca-Sc-Ti-V?
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A new coupling scheme?
Our results reveal evidence for a spin-aligned,
isoscalar neutron–proton coupling scheme.
[T]his coupling scheme replaces normal
superfluidity (characterized by seniority
coupling) in the ground and low-lying excited
states of the heaviest N=Z nuclei.
B. Cederwall et al., Nature 469 (2011) 68
Spectrum of
94Ag
Expt
SM
!
21
18!!
21!
19
Energy #MeV$
16!
17!
5
14!!
15
12!!
13
7!
0
2! !
10!
11
4!!
6!
5!
9!
8!
3!
1!
7
B3
!
18
21!!
19!
16
17!
b3 !2"
18!!
21!
19!
16
17!
b3 !3"!
18
21!!
19!
16
17!
14!!
15
1!
4!!
3 !
12!
13
14!!
15
1!
4!!
3 !
12!
13
6!!
5 !
11!
10
6!!
5 !
11!
10
6!!
5 !
11!
10
9!!
8
7!
9!
8!
9!!
8
7!
14!!
15
1!
94
47 Ag 47
7!
4!!
3 !
12!
13
B-pair analysis of
1.0
0.5
0.0
94Ag
0 13 10 24 19 28 24 30 22 27 19 20 14 14 8
0 1 0 2 1 2 2 3 2 3 2 2 2 2 1
!J1 B3 ; J" 2
9
2
4
1
4
1
2
1
2
1
0
0
1
1
94
47Ag47
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 J
Magnetic dipole moments
For any state in a single-j shell
" 0.52 to 0.55 1 f
( 7/2 )
$
1
g (α J ) = ( gν + gπ ) ≈ #
2
$% 0.51 to 0.54 (1g9/2 )
The same result is obtained with b-IBM mapped
from a single-j shell model.
∴ Magnetic dipole moments test approximation
(A) but are insensitive to (B) and (C).
In 46V:
µ (3+ ) = 1.64(3) µ N
In 50Mn:
µ ( 5+ ) = 2.76(1) µ N
1
1
F.C. Charlwood et al., Phys. Lett. B 690 (2010) 346
Q moment of 21+ isomer in
94Ag
Shell model in pf5/2g9/2 space (M=21 -> dim=2):
+
Q
21
( 1 ) = 0.44 b
Shell model in 1g9/2 (J=21 -> dim=1):
196
2
Q ( 211+ ) =
e
+
e

(
ν
π ) ( ho ) ≈ 0.42 b
9
Expression in terms of b bosons:
81949367824
2
Q ( b 21) =
(eν + eπ ) ( ho ) ≈ 0.44 b
3489855625
3
∞
∴ Measurement of Q(21+) tests (A). Calculation
confirms (B+C).
A. Poves, private communication LSSM
Q moment of 7+ isomer in
94Ag
Shell model in pf5/2g9/2 space (M=7 -> dim=37327):
+
Q
7
( 1 ) = 0.62 b
Shell model in 1g9/2 (J=7 -> dim=84):
Q ( 71+ ) = 6.60 (eν + eπ ) ( ho )2 ≈ 0.60 b
Expression in terms of b bosons:
30930277300923364
2
Q ( b [16 ] 7) =
(eν + eπ ) ( ho ) ≈ 0.64 b
627253477610841
3
∴ Measurement of Q(7+) tests (A). Calculation
confirms (B+C).
A. Poves, private communication LSSM