K. P. Drumev

K. P. Drumev

Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria

SU(3) realization of the Pairing-plus-Quadrupole Model

in One or More Oscillator Shells

• Motivation

• SU(3) Realization of Pairing-plus-Quadrupole Model

- full-space results for 20Ne in the ds shell - full-space results for 2, 3 and 4 particles in the ds+fp shell - full-space results with 2 protons and 2 neutrons in the

ds+fp shell

• Extended (pseudo-) SU(3) shell model - application to upper-fp (f5/2,p3/2,p1/2) + g9/2 shell model space - 64Ge and

68Se

• Conclusion

Outline

SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012

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Motivation for the study of N~Z systems

• Interesting area for research (P+QQ compete, nucleosynthesis – rp-process nuclei , interesting N ~ Z effects – isoscalar pairing)

• Full-space microscopic calculations in two (upper-fp+gds) shells – beyond current capabilities (max ~109 basis states)

• Ab-initio no-core techniques – applicable for light nuclei only

• A challenge - not many realistic interactions available in the pf5/2g9/2 model space (none in the fp-gds space?)

• Add the pair scattering and the isoscalar pairing part in the interaction. Classification of states in SO(8) pn-pairing model – not fully resolved.


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A. Bohr, B. R. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31 (1959) No. 11L. S. Kisslinger and R. A. Sorensen, Mat. Fys. Medd. Dan. Vid. Selsk. 32 (1960) No. 9K. Kumar and M. Baranger, Nucl. Phys. 62 (1965) 113Bahri, J. Escher, J. P. Draayer, Nucl. Phys. A592 (1995) 171 (SU(3) basis in 1 shell only )M. Hasegawa, K. Kaneko, T. Mizusaki, J. Zhang - tens of articles published in1998 – 2011 period

H = Hpairing + HQQ

SU(3) (β,γ) shape parameters ~ (λ,μ) labels

Elliott`s model

Pairing-plus-Quadrupole Model

SU(3): Microscopic theory since the SU(3) group generators – Lμ and Qμ (μ=1,2,3) are given in terms of individual nucleon coordinate and momentum variables


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Four Model Spaces

FOUR SPACES (uniquespaces explicitly included)

…mixed with …

πU

πN

νU

νN

πU

πN

νU

νN

πU

πN

νU

νN

πU

πN

νU

νN

πU

πN

νU

νN ν ν ν ν

+

+ +

Shell U

π ππ π

π νπ ν π π + ν νπ π + ν ν

XAZ N

Shell N


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Quadrupole-Quadrupole Model ≡ Extended SU(3) Shell Model

• Inter-shell (N and U) coupling of irreps

• Well-defined particle number and total angular momentum

U(2Ω) {U(Ω) SU(3) } x SU(2) U(2Ω) {U(Ω) SU(3) } x SU(2) SfN ),,(][

JMSkLSS UNUN ;,),(}),(}{;),(}{{

Basis States

Eigenstates:

jijjiji LSJMaa }){(... jj

∩∩SU(3) Realization of the PPQ Model in One or More Oscillator Shells

Debrecen, 2012 6

Extended Pairing-plus-Quadrupole Hamiltonian

H =

..,, chHG UNpsc

UN

Upair

UNpair

NUpair

UNpair

N HGHGHGHG

QQ.2

1

ππ pair-scattering

UUpair

UUNNpair

NN HGHG ,,,,

ππ and νν pairing

πν pairing

..chHG NUpsc

NU πν pair-scattering

..,, chHG UNpsc

UN νν pair-scattering

mixes configurations with different distributions of particles over the shells

mixes configurations with a specific distribution of particles over the shells

SU(3) symmetry preserving interaction

NEW TERMS

in the model!

single-particle energies)( ........Ups

Nps

Ups

Nps HHHHh


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Results for 20Ne in the ds shell

( C. E. Vargas, J. G. Hirsch and J. P. Draayer, Nucl. Phys A690 , 409 (2001)

Calculations in ds+fp shellsScenario 1 f7/2 is an intruder level (belongs to the lower shell - ds)

Systems:2, 3 and 4 particles of the same kind in the ds+fp shells2p+2n in the ds+fp shells

Scenario 2 f7/2 NOT an intruder level (belongs to the upper shell - fp)

full-space calculation pairing strength G = 0.05 MeV, 0.2 MeV (mild to medium) single-particle strength hω = 5, 10, 20 MeV (small to considerable) quadrupole-quadrupole strength χ = 0 ,…, 0.3 MeV

ds

hω

0

f7/2

fp

ds

hω

0

f7/2

fp


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Pairing and Pair-Scattering Operator

ds shell

ds and fp shells

fp shell

S+S-

For pairing η = η’

For pair scattering η ≠ η’

strength ≡ P


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Results: Pure Pairing Spectrum

highdegeneracy

Potential to describecomplicated structures


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∩

Results: Wave Function Contents – scenario 1

G

hω

4p in ds+fp shell

[ NN , NU ] ( λ , μ ) [ 4 , 0 ] (4,2), (0,4), (3,1), … [ 2 , 2 ] (10,0), (8,1), (6,2), …[ 0 , 4 ] (8,2), (7,1), (4,4), …

∩ ∩∩

∩


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Results: Wave Function Contents – scenario 2

G

hω4p in ds+fp shell


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Beta shape parameter – scenario 1

G

hωk = (5/9π)1/2A<r2><r2> r.m.s. radius A mass number

J = 1/2+

J = 0+

J = 1/2+J = 1/2+

J = 1/2+ J = 1/2+J = 1/2+

J = 0+J = 0+

J = 0+

J = 0+J = 0+


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G

hω

J = 0+ J = 0+

J = 0+

J = 0+

J = 0+ J = 0+

J = 1/2+

J = 1/2+

J = 1/2+

J = 1/2+

J = 1/2+ J = 1/2+


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G

hω

J = 0+ J = 0+J = 0+

J = 1/2+

J = 1/2+

J = 1/2+


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Gamma shape parameter – scenario 1

G

hω

J = 0+J = 0+

J = 0+

J = 0+J = 0+ J = 0+

J = 1/2+J = 1/2+

J = 1/2+

J = 1/2+

J = 1/2+ J = 1/2+


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G

hω

J = 0+

J = 0+

J = 0+

J = 0+

J = 0+J = 0+

J = 1/2+

J = 1/2+J = 1/2+

J = 1/2+ J = 1/2+ J = 1/2+


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G

hω

J = 0+J = 0+

J = 1/2+ J = 1/2+

J = 0+

J = 1/2+


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Uncertainty of the beta shape parameter for 3p and 4p

Scenario 1 Scenario 2


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Quadrupole collectivity for 3p and 4pScenario 1

Scenario 2

J = 1/2+

J = 1/2+

J = 1/2+

J = 1/2+ J = 1/2+

J = 1/2+

J = 0+

J = 0+

J = 0+

J = 0+

J = 0+ J = 0+

Quadr. Coll. = < C2(λ,μ) >/C2,ref(λref,μref)


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Results: Pure Pairing Spectrum for proton-neutron systems: 2p+2n Isovector (T=1) pairing Total pairing (T=0 + T=1)


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Beta shape parameter Scenario 1 Scenario 2

( ( K. P. Drumev, A. I. Georgieva and J. P. Draayer, J. Phys: Conf. Ser., 356 , 012015 (2012) - hw = 0 case only )


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Gamma shape parameter Scenario 1 Scenario 2


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Wave function: Effects of Gπν ≠ Gππ(and Gνν)



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Beta: Effects of Gπν ≠ Gππ (and Gνν)



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Gamma: Effects of Gπν ≠ Gππ (and Gνν)



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Extended (Pseudo-) SU(3) Shell Model (SUMMARY)

K. P. Drumev – Towards an Extended Microscopic Theory for Upper-fp-Shell Nuclei,Ph.D. Dissertation, Louisiana State University, USA, 2008

Microscopic theory since the SU(3) group generators – Lμ and Qμ (μ=1,2,3)are given in terms of individual nucleon coordinate and momentum variables

Related to the Bohr-Mottelson model upper-fp (f5/2p)

pseudo-ds(ds)

SU(3) symmetry brokenby the s.p. terms in theHamiltonian

f7/2

f5/2

SU(3) symmetry isreasonably good

p1/2

p3/2d3/2

d5/2

f7/2

g9/2

INERT CORE

s1/2pseudospin

transformation

~ ~

~ ~

~Usp

Nsp

Usp

Nsp HHHH

Hext SU(3)=

+GHpairing – χ/2 HQQ +aKJ

2 +bJ2


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[Interactions provided by P. Van Isacker, see e.g.: E. Caurier, F. Nowacki, A. Poves, & J. Retamosa, Phys. Rev. Lett. 77, 1954 (1996)]

up to 50-60% dominance of the leading irreps !

C2 1

4 Q.Q 3 ˆ L 2

Pseudo-SU(3) Symmetry in 64Ge and 68Se


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Conclusions

• Calculations for the systems 2p(n), 3p(n), 4p(n) and 2p+2nwere perfomed• Effects of the quadrupole, pairing and the single-particleterms of the Hamiltonian were studied, two scenarios for theposition of the intruder level were considered • Results suggest that the two scenarios lead to a very distinctbehavior of the wave functions, shape parameters and thequadrupole collectivity for the ground states of all the systems

While the pairing interaction mostly softens the effects, the strength of the s.p. energies drives the main (rapid) changes in the behavior of the systems.


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• Application of the extended SU(3) model to additional upper-fp + gds shell nuclei (Br and Kr isotopes of particular interest) (Challenges: need other realistic interactions in the pf5/2g9/2 ( JUN45?* Honma et al. PRC 80, 064323 (2009) ) or pf5/2gds model space, huge model spaces in full-space calculations)

• Application of the theory to heavier deformed (rare-earth / actinide) nuclei- Origin and multiplicity of 0+ states- B(E2) & B(M1) transition strengths, clusterization effects

- Double beta decay - Study of nuclear reactions

• Role of truncations [e.g., () & S] in the symmetry-adapted basis

• Search for new and improved interactions (parameter optimization)

• Evolution of key parameters from the theory of effective interactions

Future Work


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Thank you !

Documents

K. P. Drumev