Neutron-proton correlations, UHK, Hong Kong, July 2015

Aligned neutron-proton pairs�in N=Z nuclei�P. Van Isacker, GANIL, France�

Motivation �Shell-model analysis�A model with high-spin bosons�Experimental tests�

Spin-aligned T=0 np pairs�Motivation: A simple description of the N=Z nuclei

98In, 96Cd, 94Ag, 92Pd, 90Rh.�Starting point: Shell-model interpretation in terms

of spin-aligned T=0 np pairs (Blomqvist).�Experiments have been proposed and carried out

at GANIL (Cederwall, de France, Wadsworth…).�[Also, analysis of N=Z nuclei in the 1f7/2 shell

(42Sc, 44Ti, 46V, 48Cr, 50Mn, 52Fe, 54Co).] �

Plus ça change (1)�

Plus ça change (2)�

Nuclear belly dancer�

B. Cederwall et al., Nature 469 (2011) 68 �

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 �

Approximations�Hypothesis: N=Z nuclei can be described in the

(spherical) shell model, in an appropriate model space and with an appropriate interaction.�

Approximations: �(A) Truncate shell model to a single high-j shell.�(B) Truncate single-j shell space to one written in

terms of aligned-spin B (J=9) pairs.�(C) Replace aligned-spin B pairs by b bosons.�

Shell-model interaction: 1g9/2 �

SLGT0

T!1

T!0

0 1 2 3 4 5 6 7 8 9

"2

"1

0

angular momentum J

#!J,T"!MeV

"

E.J.D. Serduke et al., Nucl. Phys. A 256 (1976) 45�H. Herndl and B.A. Brown, Nucl. Phys. A 627 (1997) 35 �

Pair analysis in the shell model�Define different types of nucleon pairs: ����Calculate overlap with shell-model wave functions

with the nucleon-pair shell model in an isospin-invariant formulation.�

BJT+ = aj1/2

+ × aj1/2+( )

JT( )

S+ : J = 0, T =1; D+ : J = 2, T =1; B+ : J = 9, T = 0.

J.-Q. Chen, Nucl. Phys. A 562 (1993) 218; 626 (1997) 686�G.J. Fu et al., Phys. Rev. C 87 (2013) 044310 �

B-pair analysis of 96Cd�

Cd964848

!J1 B2; J" 23 6 7 7 7 5 3 2 11 1 1 1 1 1 1 1 1

0 2 4 6 8 10 12 14 16 J0.0

0.5

1.0

Spectrum of 96Cd�

Cd964848

Expt SM B2 b2 !2" b2 !3"

0! 0!

2!

4!

6!8!

10!12!16!14!

0!

2!

4!

6!

8!

10!16!12!14!

0!

2!

4!

6!

8!

10!16!12!14!

0

5Energy#MeV

$

B-pair analysis of 94Ag �

Ag944747

!J1 B3; J" 20 13 10 24 19 28 24 30 22 27 19 20 14 14 8 9 4 4 2 2 0 10 1 0 2 1 2 2 3 2 3 2 2 2 2 1 2 1 1 1 1 0 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 J0.0

0.5

1.0

Spectrum of 94Ag �

Ag944747

Expt SM B3 b3 !2" b3 !3"

7!

21!

7!1!3!8!9!5!6!4!11!10!2!13!12!15!14!

17!16!19!21!18!

7!8!9!

10!11!5!6!

13!12!3!4!1!15!14!

17!16!19!21!

18!

7!8!9!

10!11!5!6!

13!12!3!4!1!15!14!

17!16!19!21!18!

7!8!9!

10!11!5!6!

13!12!3!4!1!15!14!

17!16!19!21!

18!

0

5Energy#MeV

$

Mapping to bosons�Transform to a much simpler problem in terms of

interacting bosons: B+ -> b+ �

Boson energies and boson-boson interactions are derived from the shell model (i.e., not fitted).�

���

T. Otsuka et al., Nucl. Phys. A 309 (1978) 1 �L.D. Skouras et al., Nucl. Phys. A 516 (1990) 255 �

Magnetic dipole moments�For any state in a single-j shell����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: �In 50Mn: �

g αJ( ) = 12gν + gπ( ) ≈

0.52 to 0.55µN 1 f7/2( )0.51 to 0.54 µN 1g9/2( )

"#$

%$

µ 31+( ) =1.64(3) µNµ 51+( ) = 2.76(1) µN

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): ��Shell model in 1g9/2 (J=21 -> dim=1): ���Expression in terms of b bosons: � ∴ Measurement of Q(21+) tests (A). Calculation

confirms (B+C).�

Q 211+( ) = 196

9eν + eπ( ) ho( )2 ≈ 0.42 b

Q b∞3 21( ) = 81949367824

3489855625eν + eπ( ) ho( )2 ≈ 0.44 b

Q 211+( ) = 0.44 b

A. Poves, private communication LSSM�

Q moment of 7+ isomer in 94Ag �Shell model in pf5/2g9/2 space (M=7 -> dim=37327): ��Shell model in 1g9/2 (J=7 -> dim=84): ��Expression in terms of b bosons: � ∴ Measurement of Q(7+) tests (A). Calculation

confirms (B+C).�

Q 71+( ) = 0.62 b

Q 71+( ) = 6.60 eν + eπ( ) ho( )2 ≈ 0.60 b

Q b3 16[ ]7( ) = 30930277300923364627253477610841

eν + eπ( ) ho( )2 ≈ 0.64 b

A. Poves, private communication LSSM�

A discussion between physicists�What is the correct coupling scheme for these

nuclei? Seniority or aligned pairs?�Physicist 1: The 7+ state in 94Ag can be written as���

Physicist 2: Not at all, this 7+ state should be written as�

��7+ = B2 4[ ] B;J = 7

7+ = B2 16[ ] B;J = 7

(Anti)-symmetric amnesia�

⌅ b2 �16⇥ b; J � 7 ⇤

(Anti)-symmetric amnesia�

⌅ b2 �16⇥ b; J � 7 ⇤

(Anti)-symmetric amnesia�

⌅ b2 �16⇥ b; J � 7 ⇤

(Anti)-symmetric amnesia�

⌅ b2 �16⇥ b; J � 7 ⇤

(Anti)-symmetric amnesia�

⌅ b2 �4⇥ b; J � 7 ⇤⌅ b2 �16⇥ b; J � 7 ⇤

(Anti)-symmetric amnesia�

⌅ b2 �4⇥ b; J � 7 ⇤⌅ b2 �16⇥ b; J � 7 ⇤

(Anti)-symmetric amnesia�

⌅ b2 �4⇥ b; J � 7 ⇤⌅ b2 �16⇥ b; J � 7 ⇤

(Anti)-symmetric amnesia�

⌅ b2 �4⇥ b; J � 7 ⇤⌅ b2 �16⇥ b; J � 7 ⇤

(Anti)-symmetric amnesia�

⌅ b2 �4⇥ b; J � 7 ⇤⌅ b2 �16⇥ b; J � 7 ⇤

Fermion pairs versus bosons�In the 1g9/2 nucleon-pair shell model: ����In the b-IBM: ���∴ The B pair behaves as a boson. �

b2 4[ ] b;J = 7 b2 16[ ] b;J = 7 =70122008733503

≈ 0.896

B2 4[ ] B;J = 7 B2 16[ ] B;J = 7

=112919600563049280139849953265085321

≈ 0.899

Conclusions�(A) Truncation of the shell model to a single-j

shell. (?)�(B) Truncation of the single-j shell space to one

written in terms of aligned-spin B (J=7 or 9) pairs. (✓)�

(C) Replacement of aligned-spin B pairs by b bosons. (✔)�

S. Zerguine & P. Van Isacker, Phys. Rev. C 83 (2011) 064314�P. Van Isacker, Int. J. Mod. Phys. E 22 (2013) 1330028 �

Shell-model interaction: 1f7/2 �

Sc422121

T!1

T!0

0 1 2 3 4 5 6 7

"3

"2

"1

0

angular momentum J

# JTf!MeV

"

Shell-model interaction: 1f7/2 �

Co542727

T!1

T!0

0 1 2 3 4 5 6 7

"3

"2

"1

0

angular momentum J

# JTf!MeV

"

B-pair analysis of 44Ti and 52Fe�

Ti442222

Fe522626

!J1 B2; J" 23 4 5 5 3 2 11 1 1 1 1 1 1

0 2 4 6 8 10 12 J0.0

0.5

1.0

B-pair analysis of 46V and 50Mn �

V46 2323

Mn502525

!J1 B3; J" 20 7 4 12 8 12 9 11 6 8 4 4 2 2 0 10 1 0 2 1 2 2 2 1 2 1 1 1 1 0 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 J0.0

0.5

1.0

B-pair analysis of 48Cr�

Cr482424

!J1 B4; J" 26 13 18 17 14 9 5 2 12 3 4 4 4 3 2 1 1

0 2 4 6 8 10 12 14 16 J0.0

0.5

1.0

Structure of 96Cd�

Spectrum of 44Ti�

Ti442222

Expt SM B2 b2 !2" b2 !3"

0!

2!

4!

6!

8!

10!12!

0!

2!

4!

6!

8!

10!12!

0!2!

4!

6!

8!10!12!

0!2!

4!

6!

8!10!12!

0

5

10

Energy#MeV

$

Spectrum of 52Fe �

Fe522626

Expt SM B2 b2 !2" b2 !3"

0!2!

4!

6!

8!

10!

0!

2!

4!

6!

8!12!10!

0!2!

4!

6!

8!12!10!

0!2!

4!

6!

8!12!10!

0

5

10

Energy#MeV

$

Spectrum of 46V �

V46 2323

Expt SM B3 b3 !2" b3 !3"

3!1!4!5!6!7!9!8!

11!10!

13!

12!15!

5!1!3!6!7!4!9!2!8!

11!10!

13!

12!15!

5!6!7!3!9!8!4!1!11!10!

13!

12!15!

5!6!7!3!9!8!4!1!

11!10!

13!12!15!

5!6!7!3!9!8!4!1!11!10!

13!

12!15!

0

5

10

Energy#MeV

$

Spectrum of 50Mn �

Mn502525

Expt SM B3 b3 !2" b3 !3"

5!# 6! $1!7!3!2!# 8! $# 9! $# 11! $# 13! $# 15! $

5!1!3!6!7!4!9!8!2!11!10!

13!

12!15!

5!6!7!3!9!8!4!1!11!10!

13!

12!15!

5!6!7!

3!9!8!4!1!11!10!

13!

12!15!

5!6!7!3!9!8!4!1!11!10!

13!

12!15!

0

5

10

Energy#MeV

$

Spectrum of 48Cr�

Cr482424

Expt SM B4 b4 !2" b4 !3"

0!2!

4!

6!

8!

10!

12!

0!

2!4!

6!

8!

10!

12!

0!

2!4!

6!

8!

10!

12!

0!

2!4!

6!

8!

10!

12!

0!2!4!

6!

8!

10!

12!

0

5

10

Energy#MeV

$

Spectrum of 92Pd�

Pd924646

Expt SM B4 b4 !2" b4 !3"

0!

2!

4!

6!

0!

2!

4!

6!

8!

10!

12!14!16!

0!

2!4!

0!2!4!

6!

8!

10!

12!14!

16!

0!2!4!

6!

8!

10!

12!14!16!

0

5Energy#MeV

$

Q moment of 5+ state in 50Mn �Experimental value: �Large-scale shell model: �Numerical result in the 1f7/2 shell: ��Expression in terms of b bosons: ��Parameter-free test: �� �

Q 51+( ) = 0.80(12) bQ 51+( ) = 0.58 b

Q 51+( ) = 4.2 eν + eπ( ) ho( )2

Q b3 12[ ]5( ) = 649485150241eν + eπ( ) ho( )2 ≈ 4.3 eν + eπ( ) ho( )2

Q 51+; 46V( )

Q 71+; 42Sc( )

≈Q b3 12[ ]5( )

Q b( )=216495150241

≈1.44

F.C. Charlwood et al., Phys. Lett. B 690 (2010) 346 �

E2 properties in b-IBM�The E2 operator in the shell model: ���The E2 operator in terms of b bosons: ��The mapping implies the relation: ����

€

ˆ T µF E2( ) = eν ri

2Y2µ θ i,φi( )i∈ν∑ + eπ ri

2Y2µ θ i,φi( )i∈π∑

€

ˆ T µB E2( ) = eb b+ × ˜ b ( )

µ

2( )

€

g9 / 2( )2;9+ ˆ T F E2( ) g9 / 2( )2

;9+ = b ˆ T B E2( ) b

⇒ eb =553π ho( )2

×266187

eν + eπ( )

B(E2) values in 96Cd�

B(E2) values in 96Cd�A simple consequence of the aligned-pair

assumption: ����

€

B E2;J + 2→J( ) = eb220 2J +1( )

9 9 2J + 2 J 9# $ %

& ' (

2

B(E2) values in 94Ag �

B(E2) values in 94Ag �

B(E2) values in 92Pd�

Energy of 21+ isomer in 94Ag �How many independent states of three b-bosons

can couple to angular momentum 21?����Answer: d (3,9,21)=2 one of which is spurious.�After elimination of the spurious state, the

energy of the physical 21+ state is ��

€

d υ,,J( ) =i2π

z2J +1 −1( ) z2υ +2−1 −1( ) zυ +k −1( )k=1

2−2∏

zυ +J +2 zk+1 −1( )k=1

2−2∏z =1

∫

€

E 21+( ) = 3εb +685120155

ν12b +

1548821545

ν14b +

1212882624805

ν16b

P. Van Isacker, Phys. Scr. T150 (2012) 014042 �

Energy of 21+ isomer in 94Ag �In turn, we know the boson matrix elements in

terms of the shell-model matrix elements: �

€

ν12b =

121869355

ν3 +63423138710

ν4 +2995763050

ν5 +10988153350

ν6

+11483372358070

ν7 +1523131525

ν8 +10893535925

ν9

ν14b =

8688515

ν5 +19531310

ν6 +4625157902

ν7 +19771310

ν8 +221122270

ν9

ν16b =

817ν7 + 3ν8 +

917ν9

Energy of 21+ isomer in 94Ag �Therefore, we know the energy of the 21+ isomer

in 94Ag in terms of the shell-model interaction matrix elements: �

€

Eb 21+( ) =22134

3707825 ν3 +

11525497415650

ν4 +13477519535740387250

ν5

+86061497494857250750

ν6 +354940047213214690483150

ν7

+1561553973220784125

ν8 +154111070943753330125

ν9

Energy of 21+ isomer in 94Ag �The 21+ state is unique in the 1g9/2 shell model.

Its energy is therefore known analytically: ���Comparison tests the reliability of the mapping: �

€

E f 21+( ) =2165

ν5 +2110

ν6 +645442

ν7 +6910

ν8 +717170

ν9

€

E f 21+( ) ≈ 0.323 ν5 +2.1 ν6 +1.459 ν7 +6.9 ν8 + 4.218 ν9

Eb 21+( ) ≈ 0.006 ν3 +0.155 ν4 +0.235 ν5 +1.772 ν6

+1.653 ν7 + 7.073 ν8 + 4.106 ν9

Conservation of n, J and T�A unique n-particle shell-model state with angular

momentum J and isospin T has energy��The coefficients aλ satisfy �

€

E f jnJT( ) = aλvλ

λ

∑

€

aλλ =0

2 j

∑ =n n −1( )2

,

λ λ +1( )aλλ =0

2 j

∑ = J J +1( ) + j j +1( ) × n n − 2( )

2aλλ =0even

2 j

∑ = T T +1( ) +34n n − 2( )