International Workshop on Shapesand Dynamics of Atomic Nuclei:

Contemporary Aspects October 8-10, 2015 at Bulgarian

Academy of Sciences, Sofia, Bulgaria

Algebraic Models for Structure of Heavy N=Z Nuclei: IBM-4 Results for

Gamow-Teller and α-transfer Strengths

V.K.B. KotaTheoretical Physics Division

Physical Research Laboratory Ahmedabad 380009, India

Thanks to R. Sahu (Berhampur, India) and P.C. Srivastava (Roorkee, India)

• Introduction to N=Z nuclei

•IBM-4:

•GT strengths for

•α-transfer strengths involving N=Z nuclei

•Conclusions

Plan of the Talk

UsdST(36) ⊃ SOsdST (36) ⊃ SOsST(6) ⊕ SOdST(30) limit

4 2 4 22 2 2 2 1 2 1

n nn n n nX Y+ ++ + +→

1. INTRODUCTION

23-09-2015 4SDANCA15 (V.K.B. Kota)

N=Z nuclei start from deuteron and stop at 100Sn

Light N=Z nuclei: 12C , 16O , 24Mg , ------

Hoyle state at 7.654 MeVand Hoyle band Tetrahedral

symmetry

Rod like structure athigh excitation

Isospin and SU(4) symmetry are important

N=Z nuclei with A>60 are heavy N=Z nuclei - they start from 62Ga

N=Z odd-odd nuclei – new insights into pn (or T=0) superfluidity

For N=Z nuclei, protons and neutrons occupy same orbits

Isoscalar (T=0) and isovector (T=1) pairing

SU(4) degenerate T=0 and T=1 states in o-o N=Z nuclei23-09-2015 5SDANCA15 (V.K.B. Kota)

δVnp

restorationof SU(4)symmetry In odd-oddN=Z nuclei ?

Competition between Wigner energy, symmetry energy and isovector pairing energy

Doublebinding energydifference

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Pair is defined to have

In LST coupling with two-particle L=0:isovector pairs : (ST)=(01) isoscalar pairs : (ST)=(10)

SO(8) algebraic model

†Dµ

†Pµ

† † † † † †( ) -(1- ) ( ) - (1 ) ( )pH x x P P x D Dµ µ µ µµ µ

= +∑ ∑

(8) (6) (3) (3)(8) [ (5) (3)] (3)(8) [ (5) (3)] (3)

ST S T

S T

T S

SO SO SO SOSO SO SO SOSO SO SO SO

⊃ ⊃ ⊕⊃ ⊃ ⊗⊃ ⊃ ⊗

U(4Ω) ⊃ [U(Ω) ⊃ ⊃ ---] ⊗ [SU(4) ⊃ SUS(2) ⊗ SUT(2)]

Wigner’s SU(4)Iso vibrationalIso rotational

SO(Ω)SU(3)

⊃Sp(2Ω) ⊃[SO(Ω) ⊃----]⊗SU(2)⊗SU(2)

I

IIIII

I

II,III VKBK and CA, Nucl. Phys. A764, 181 (2006)23-09-2015 7SDANCA15 (V.K.B. Kota)

Kalin Drumev

0 :x =1:x =1:x = −

A hermitization procedure for the SO(8) Hamiltonian +Dyson mapping gives interacting s-boson U(6) model withthe bosons carrying (ST) = (10) ⊕ (01) degrees of freedom.

By adding d-bosons gives the full IBM-4 with U(36) SGA This will include deformation effects

A group chain of IBM-4 that is proved to useful for heavy N=Z nuclei is [ VKBK, Ann. Phys. (N.Y.) 280, 1 (2000)]:

Usd:ST(36) ⊃ SOsd:ST (36) ⊃ SOs:ST(6) ⊕ SOd:ST(30) ⊃ …⊃ SOL(3)⊗ SOS(3)⊗ SOT(3)

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2. IBM-4 for N=Z nuclei: UsdST(36) ⊃ SOsdST (36) ⊃ SOsST(6) ⊕ SOdST(30) limit

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IBM-4: s and d bosons each with (ST)=(10)+(01)

U(36)

Usd(6) ⊗UST(6)

UsdS(18) ⊕UsdT(18)

UsST(6) ⊕UdST(30)

SOsdST(36)

allow for Wigner SU(4)

Soft , shape mixing etc., like SO(6) of IBM-1expected to be good for heavy N=Z nuclei

SOsdST(36)⊃SOsST(6) ⊕SOdST(30) limit is used in the past andin the present work

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Note: SO(6) ∼ SU(4)

We have not only good Wigner SU(4) in the total space but also in the s and d boson spaces seperately

ω, ωs and ωd are seniority quantum numbers, also ω = N for low-lying states

gs: for o-o N=Z SOST(6) irrep [1] giving (ST)=(10) and (01) for e-e N=Z SOST(6) irrep [0] giving (ST)=(00)

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Energy formula will have many terms

Group theoryis more involved

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Known results in the exactsymmetry limit:

(1) B(E2; L → L-2) exhibit a ∆L=4 staggering

(2) Formulas for B(E2)’s involving low-lying statesare derived

(3) Some of the observed properties of low-lyinglevels in o-o N=Z nucleus 74Rb are described including aligned spin 1in the lowest T=0 band just as in CSM

(4) Predictions for deuterontransfer strengths are available

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2 : :ˆ ˆ( (6)) ( / ) ( / )mix s S T s S dH C SO n nα β α γ α= + +

, , ( ) , , , 0,( ) ; , 2,...,0 or 1

s s d

s

N ST N N STN N

ω ω ω ωω

= = =

= −

Simple Hamiltonian and a basis for incorporating the competition between T=1 and T=0 pairing

, , , 0,

( )

, , ( ) (6,30) , , , , 0, ( )s d

s d

s d

N Ns n n s d s d

n nN ST C N n n STω ω ωω ω ω= == =∑

, ,( ),

( )

, , ( ) (3,3) , , , ( ) ; s s

sS sT

sS sT

n STs s n n s sS sT s sS sT

n nn ST C n n n ST n n nωω = = +∑

, ,( ), , ( ) , , ( )s

s

N STsN ST N STα

ωω

α ω=∑VKBK, J. Math. Phys. 38, 6639 (1997)

mixing term

Significance of the β/α termSingle boson energies:

( 0) / 5 / , ( 1) / 5, ( 0,1) / 5 /

s

s

d

TTT

ε α β αε αε α γ α

= = += == = +

For a N boson system with N odd β/α<0 gives T=0 gsβ/α>0 gives T=1 gsthey will be degenerate for β/α=0.Thus β/α term generates competition between T=0 and T=1pairing correlations.

f(T=0) exhibits odd-even staggering in number of T=0 pairs in the gs of N=Z nuclei just as in the shell model and the staggering is maximum for | β/α |<2.

VKBK, Prog. Theo. Phys. 118 (2007) 893.23-09-2015 15SDANCA15 (V.K.B. Kota)

e-e to o-oN→N+1

†STs

Results close to those obtained using SO(8)model

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operator:

3. GT strengths

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4 2 4 22 2 2 2 1 2 1 ( ) (01) (10); is odd for bothn n

n n n nX Y ST N+ ++ + +→ ⇔ = →

0, 1, 1† †10 01 01 10 ,

( )s t

L S TGT effAT g s s s s s

µ µ

= = = = +

For β+ decay we have µt = +1 generator of SO(6) in s boson space

2( ) , , ( ) (10) , , ( ) (01) ;GT gs

f f i iM GT Y S T T X S Tα α= = =

, ,( ), ,( )

1/2

2

, ,( ), ,( )

( )

[ ][ ] [11](2 1)(2 1) ( (6))

( )( ) (11)

( ) ( 2)

gsf fi i

s s

s

s

gsf fi i

s s

s

N S TN S TGT effA

ssf f s

f fi i

N S TN S TeffA s

T g s

S T C SOS TS T

g s

ααω ω

ω

ω

ααω ω

ω

ωω

ω

=

× + +

= +

∑

∑

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(a) GT strength increases as N increases and also as |β/α| increases

(b)-(d) for e-e and o-o involved, the |β/α| need not be same

β/α -ve means T=0 pairing is stronger +ve means T=1 pairing stronger

-- these explain the results

Depending on β/α it is possible that gs to excited state strength can be larger than gs to gs

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The GT strengths calculated are all only with L=0– thus they are only spin excited

For 62Ge → 62Ga, data is now available in PRL 113, 092501 (2014)

low-lying 1+ states have more complex structure [SU(4) broken]

in IBM-4, the L and S need not be real (but T must be)

Therefore we need terms such as

Also we need to include ωd ≠ 0 states in the basis and also perhapssome more SU(4) breaking terms (SU(3) Casimir?)

2, 1; 1, 1 2, 1; 1, 1† † † †01 10 10 01 10 01 01 10,

L S J T L S J Td s s d d s s d

= = = = = = = = + +

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4. α-transfer strengths

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† † † †10 10 01 01T s s s sα κ = +

2( : : 2) : 2, , ( ) : , , ( )gs gsS A N B N B N ST T A N STα αβ δ→ + = +

, , , 0 2, 2, , 0, ,( ) 2, ,( ), 2,

( ),

: 2, , ( ) : , , ( ) (2 1)(2 1)

(6,30) (6,30)

2, , ( ) , , ( )

gs gss d s d

s s s d s d

s d s

gs gs

N N N NN ST N STn n n n

n n

s s s s

B N ST T A N ST S T

C C

n ST T n ST

α

ω ω ω ωδ βω ω

ω

α

β δ κ

ω ω

= + + =++

+ = + +

×

× +

∑

( 2)( 6)s s s sn nω ω− + + +

For β/α=0:2 ( 6 )( 30 )( 34 )( 2 )(2 1)(2 1)

4( 17)( 18)s s s sN N N NS S T

N Nαω ω ω ωκ

+ + + − + + + −= + + + +

generator of SU(1,1)complimentary to SOsST(6)

Note that ωs = 0 for e-e and 1 for o-o23-09-2015 22SDANCA15 (V.K.B. Kota)

using Tα

Variation in the transfer strength is quite small as β/α changes. There is a small peak at β/α =0

The structures seen in the figure are same as those obtained in the fermionic SO(8) model

Strengths have ∼ N2 scaling

possible to consider ( ) † † † †

10 10 01 01( ) † †

10 10

'( ) ''( )

' ( )

eff

d

T s s s s

T s sα

α

κ κ

κ

= +

=

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( )

, , , 0, ,( ) 2, ,( ),

( ), ,

2, 2, , 0 , , , 22, , 2,

: 2, , ( ) : , , ( )

' (2 1)(2 1)

(6,30)

(6,30) (3,3)

igs gss d

i f s ds si fs d s s

f is d s s s

s d sS sT sS sT

sS sT s

gs d gs

N NN ST N STn n

n n

N N n S T nn n n n n n

n n n

B N ST T A N ST

S T

C

C C C

α

ω ωδ βω ω

ω ω

ω ω ω

β δ

κ=+

+ + = ++ +

+ =

+ =

+ +

×

×

∑

∑

, , , (3,3)

( 2)( 3)

fs S T

sS sSn S n S

ω

× − + + +

note that here is allowedi fs sω ω≠

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for odd-odd N=Z nucleiN is odd, gs → gs

using the operator Tα(ST)=(01) or (10)

using the operator(ST)=(10)

results shifted by 20

( )dTα

using the operator(ST)=(01)

( )dTα

**general structure for e-e and o-o is same

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5. Conclusions

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Formulation and first numerical results for GT strengths and α-transfer strengths are presented

These are in addition to deuteron transfer reported before

Employed a simple basis within IBM-4 and a one parameter mixing Hamiltonian

Many results are close to those given by the SO(8) model

Experimental tests ??

possible extensions of the formulation [inclusion of ωd ≠ 0 states, SU(6) ⊃ SU(3) term in H, more general transition operators]