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SU(3) realization of the Pairing-plus- Quadrupole Model in One or More Oscillator Shells. K. P. Drumev. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria. SU(3) Realization of the PPQ Model in One or More Oscillator Shells Debrecen, 2012. - PowerPoint PPT Presentation
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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.
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Results: Pure Pairing Spectrum
highdegeneracy
Potential to describecomplicated structures
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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), …
∩ ∩∩
∩
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Results: Wave Function Contents – scenario 2
G
hω4p in ds+fp shell
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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+
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Beta shape parameter – scenario 2
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+
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Beta shape parameter – scenario 2
G
hω
J = 0+ J = 0+J = 0+
J = 1/2+
J = 1/2+
J = 1/2+
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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+
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Gamma shape parameter – scenario 2
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+
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Gamma shape parameter – scenario 2
G
hω
J = 0+J = 0+
J = 1/2+ J = 1/2+
J = 0+
J = 1/2+
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Uncertainty of the beta shape parameter for 3p and 4p
Scenario 1 Scenario 2
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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)
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Results: Pure Pairing Spectrum for proton-neutron systems: 2p+2n Isovector (T=1) pairing Total pairing (T=0 + T=1)
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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 )
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Gamma shape parameter Scenario 1 Scenario 2
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Wave function: Effects of Gπν ≠ Gππ(and Gνν)
Scenario 1 Scenario 2
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Beta: Effects of Gπν ≠ Gππ (and Gνν)
Scenario 1 Scenario 2
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Gamma: Effects of Gπν ≠ Gππ (and Gνν)
Scenario 1 Scenario 2
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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.
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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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
SU(3) Realization of the PPQ Model in One or More Oscillator ShellsDebrecen, 2012
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Thank you !
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