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Band Structure of nuclei in Deformed Hartree-Fock and Angular Momentum Projection theory
C. R. PraharajC. R. PraharajInstitute of PhysicsInstitute of Physics
Bhubaneswar . IndiaBhubaneswar . India
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Outline of talk Motivation Formalism
HF calculation Angular momentum projection Band mixing
Results and Discussion Configurations and band structure
• Normal deformed bands• Large K bands• “Chiral” bands• Superdeformed bands
B(E2) & B(M1) Qs, Qt & µN
Conclusion
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Motivation
After coming of Heavy Ion accelerators and new generation of detector arrays:
Extend upto very high spins Many bands are known
Need theoretical frame work to correlate and understand these starting from a fundamental principle :-
Nucleon-nucleon interaction (Strong) Shell model (both closed and away from
that)
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Some interesting phenomena in nuclear spectroscopy are
Regular and irregular spectra Known upto very high spins
Band crossings Inter-band transitions Signature effects
Large K bands. K selection violation Identical bands Staggering in B(E2) and B(M1)
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Hartree-Fock (HF) method
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RESIDUAL INTERACTIONS
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See Ripka in Adv Nucl Phys vol 2
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Superdeformed HF solutions in 84Zr
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66Zn :
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Angular momentum intensities in (a) K=0+ and RAL band (b,c).
Bandcrossing andneutron RAL in 198Hg CRP and Khadkikar PRL 50,
1254 (1983)
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Model Space in Rare-earth Region Orbits for protons are 3s1/2, 2d3/2, 2d5/2, 1g7/2, 1h9/2, 1h11/2
and orbits for neutrons are 3p1/2, 3p3/2, 2f5/2, 2f7/2, 1h9/2, 1i13/2.
132Sn is considered as spherical inert core.
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The nucleon orbits ( ) are approximately two fold degenerate.
HF orbits for 171Lu
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Angular Momentum (J) Projection From the HF single particle orbits, HF
intrinsic and particle-hole intrinsic states are constructed by J projection.
Intrinsic state does not have a unique J quantum number.
By angular momentum projection from intrinsic states the spectra and other spectroscopic properties are obtained.
Angular momentum projection operator is
Where =
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Band-mixing In general, two states and projected
from two intrinsic configurations are not orthogonal to each other even if and are orthogonal.
Thus, whenever necessary, we do band-mixing using the following equation to get better results.
Results for Lu, Re, Nd and K Isomeric bands are given below:
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Regular and irregular bands
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169Lu
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169Lu
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171Lu 173Lu
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171Lu3qp bands
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171Lu
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HF orbits for 150Nd
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Multi-quasi particle Large K bands
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Oblate K Isomers in 146Ba. The ground band is Prolate. Not shown is a
band crossing due to i13/2 neutrons (prolate).
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“Chiral” bands
The deformed HF orbits are calculated with a spherical closed shell core with Z=N=28
PHF studies of Chiral and Superdeformed bands See Ph.D. Thesis of Zashmir Naik
(Institute of Physics, Bhubaneswar) (unpublished)
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Superdeformed bands
Superdeformed band structure of 104Pd is studied Model space and interaction are same as 122CS caseWe have performed calculation in two different ways
● 1st one is by occupying prolate deriving orbits and un-occupying oblate deriving orbits● 2nd one is by constraint HF calculation
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105Ag
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B(E2), B(M1) and B(M1)/B(E2)
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Conclusions Deformed HF and J projection – a general manybody method No phenomenological assumptions. Reliable. Good predictive power Superdeformed bands. K isomers. Chiral bands. Bandcrossing. Signature effects. Need for effective interactions in large model spaces.
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Band crossing due to intruder orbits are predicted
B(E2) and B(M1) values are predicted
Trends of the B(M1)/B(E2) ratios are predicted.
“Chiral” band structure are explained
Superdeformed bands are studied
Other spectroscopic properties (spectra, quadrupolemoments, magnetic moments, etc.,) are also quite well reproduced as compared to experimental results.
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References For HFReferences For HF
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References For J ProjectionReferences For J Projection
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References For Expt.References For Expt.
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QS and B(E2)
Effective charges proton :1.7e neutron: 0.7e
Core polarization are account by considering Effective charges.
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µ and B(M1) Magnetic moment µ is given by
B(M1) value for γ-transition from initial state αJ1 to final state βJ2 is
g-factors (in µΝ)gsgl
½ × (-3.826)0Neutron½ × 5.5861proton
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Superdeformed and Chiral Bands in Z = 50 region
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Shape Coexistence in 52Cr
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Conclusions We are able to explain both regular and irregular
rotational bands.
The doublet structure and departure from regular rotational behavior of the low-K bands are explained.
We could relate staggering in spectrum with angular momentum carried by large-j nucleon.
Structure of one, three, five quasi-particles bands for odd A and zero, two, four quasi-particle bands for even A are explained.
Def HF and J Projection formalism (J Phys G 14, 843 (1988)):
COLLABORATORS● Zashmir Naik (TIFR)● A.K. Rath (Sambalpur Univ) ● S.B. Khadkikar (PRL)● S.P. Pandya (PRL)● Experimentalists and Colleagues
from Universities and Institutions.