Su3 Symmetry in Atomic Nuclei

Preface
Contents
About the Author
1 Introduction
	References
2 SU(3) Algebra in Nuclei: Preliminaries
	2.1 Introduction
	2.2 SU(3) supsetSO(3) supsetSO(2) Algebra: Quadrupole Operator
	2.3 SU(3) supset[SU(2) supsetSO(2)] otimesU(1) Algebra
	2.4 SU(3) Irreps (λµ)
		2.4.1 Young Tableaux
		2.4.2 Kronecker Products of SU(3) Irreps
		2.4.3 Dimension of (λµ) Irreps
		2.4.4 Leading SU(3) Irrep in U((η+1)(η+2)/2)
	2.5 SU(3) Quadratic and Cubic Casimir Operators
	2.6 SU(3) supsetSO(3) supsetSO(2) States: K Label
	2.7 SU(3) supsetSU(2) otimesU(1) States
	2.8 Preliminary Applications of SU(3) Symmetry
		2.8.1 SU(3) in Shell Model
		2.8.2 SU(3) in Interacting Boson Model
	2.9 SU(3) in Particle Physics
	2.10 Summary
	References
3 SU(3) Wigner–Racah Algebra I
	3.1 Introduction
	3.2 SU(3) Irreps for Many-Particle Systems
		3.2.1 Plethysm Method
		3.2.2 Recursion Method
		3.2.3 Difference Method
		3.2.4 Method for Obtaining a Few Lower SU(3) Irreps
	3.3 SU(3) Wigner and Racah Coefficients
		3.3.1 SU(3) supsetSU(2) otimesU(1) Reduced Wigner Coefficients
		3.3.2 SU(3) supsetSO(3) Reduced Wigner Coefficients
		3.3.3 SU(3) Racah or U- and Z- Coefficients
	3.4 Building Up Principle and General Comments
	3.5 Summary
	References
4 SU(3) Wigner–Racah Algebra II
	4.1 Introduction
	4.2 SU(3) Tensorial Decomposition and Wigner–Eckart Theorem
		4.2.1 Examples from sd and sdgIBM
		4.2.2 Shell Model Two-Body Interactions
		4.2.3 Analytical Results for Electric Quadrupole Transition Strengths
	4.3 SU(3) Fractional Parentage Coefficients
		4.3.1 Construction of SU(3) Intrinsic States: IBM Examples
		4.3.2 SU(3) Intrinsic States: Fermion Examples
		4.3.3 Triple Barred SU(3) Reduced Matrix Elements
	4.4 9-SU(3) Coefficients
	4.5 SU(3) D-Functions
	4.6 Summary
	References
5 SU(3) supsetSO(3) Integrity Basis Operators
	5.1 Introduction
	5.2 X3 and X4 Operators and Their Matrix Elements
	5.3 Shape Parameters and (λµ) Irreps Correspondence
	5.4 Integrity Basis Hamiltonian and Asymmetric Rotor
	5.5 K Quantum Number from X3 and X4 Operators
	5.6 Extension to KJ Quantum Number
	5.7 Summary
	References
6 SU(3) in Shell Model Based Approaches and Their Applications
	6.1 Introduction
	6.2 Pseudo SU(3) Model with Pseudo-spin
		6.2.1 Mapping of Operators to Pseudo-(tildeell tildes) Basis
		6.2.2 Basic Results from Pseudo-spin and Pseudo-Nilsson Orbits
		6.2.3 Spectroscopy with Pseudo-SU(3) Symmetry
	6.3 Proxy-SU(3) Model
		6.3.1 Prolate Dominance over Oblate Shape
		6.3.2 Results for the Deformation Parameters (β, γ) and B(E2)\'s
	6.4 Sp(6,R) Model with SU(3) Subalgebra
		6.4.1 SU(3) Limit of Sp(6,R)
	6.5 Fermion Dynamical Symmetry Model with SU(3) Limit
		6.5.1 i- Active and k- Active Schemes
		6.5.2 Fermion Dynamical Symmetry Model
	6.6 Summary
	References
7 SU(3) in Interacting Boson Models
	7.1 Introduction
	7.2 SU(3) in sdgIBM
		7.2.1 SUsd(3) times1g Limit
		7.2.2 SUsdg(3) Limit
		7.2.3 ΔL=4 Staggering in sdgIBM
	7.3 SU(3) in sdpfIBM
		7.3.1 Introduction
		7.3.2 Dynamical Symmetries of sdpfIBM and the SU(3) Limit
		7.3.3 Analytical Results for E1 Transitions in SUsd(3) oplusSUpf(3) Limit
	7.4 SU(3) in Proton–Neutron IBM (IBM-2)
		7.4.1 SU(3) in pn-sdIBM
		7.4.2 SU(3) in pn-sdgIBM
	7.5 SU(3) in IBM-3 and IBM-4 Models
		7.5.1 SU(3) Limit of IBM-3
		7.5.2 SU(3) Limit of IBM-4
	7.6 Summary
	References
8 SU(3) in Interacting Boson–Fermion Models
	8.1 Introduction
	8.2 SU(3) timesj Limit of IBFM for Odd-A Nuclei
	8.3 SUBF(3) Limit of IBFM for Odd-A Nuclei
		8.3.1 Nilsson Correspondence I
		8.3.2 Nilsson Correspondence II
		8.3.3 Application to E2 Transition Strengths: 187Os Example
		8.3.4 Application to M1 Transition Strengths: 185Re Example
		8.3.5 Single Nucleon Transfer: 185W Example
	8.4 SU(3) in IBFFM for Odd–Odd Nuclei
		8.4.1 SUBF(3) otimesU(2j+1) Limit: 186Re Example
		8.4.2 SUBFF(3) Limit : 190Ir Example
	8.5 SU(3) in IBF2M for 2 Quasi-particle Excitations
	8.6 SU(3) in sdgIBFM-2 and M1 Distributions
	8.7 Summary
	References
9 Extended Applications of SU(3)
	9.1 Introduction
	9.2 Phase Transitions with SU(3)
		9.2.1 U(5) to SU(3) Transition
		9.2.2 Example of an Analytically Solvable QPT
		9.2.3 Critical Point X(5) Symmetry for U(5) rightarrowSU(3) Transition
	9.3 Partial SU(3) Dynamical Symmetry
	9.4 SU(3) for Removal of Spurious c.m. States
	9.5 SU(3) for Clustering in Nuclei
		9.5.1 Nuclear Vibron Model
		9.5.2 Semi-microscopic Algebraic Cluster Model with SU(3)
	9.6 SU(3) in No-Core Shell Model
		9.6.1 Symmetry Adopted SU(3) Based No-Core Shell Model (SA-NCSM)
		9.6.2 No-Core Symplectic Shell Model (NCSpM)
	9.7 Summary
	References
10 Statistical Nuclear Physics with SU(3)
	10.1 Introduction
	10.2 Preliminaries of Statistical Spectroscopy
		10.2.1 Averages, Traces, State Densities, and Partial Densities
		10.2.2 General Principles of Trace Propagation
	10.3 SU(3) Energy Centroids and Goodness of SU(3) Symmetry
		10.3.1 (2s1d) Shell Model Example
		10.3.2 Sp(6,R) supsetSU(3) Example
	10.4 Application of SU(3) Energy Centroids: Regularities with Random Interactions
		10.4.1 Regular Structures from Random Interactions: sdpfIBM Example
		10.4.2 Regular Structures from Random Interactions: sdIBM-T Example
	10.5 Partition Functions and Level Density Enhancement in Deformed Nuclei with SU(3)
	10.6 Statistical Group Theory for SU(3) Multiplicities
	10.7 Example of a Random Matrix Ensemble with SU(3) Symmetry
		10.7.1 Definition of EGUE(2)-SU(3) Ensemble
		10.7.2 Basic Formulation for Analytical Treatment of EGUE(2)-SU(3)
		10.7.3 Results for Lower Order Moments of One- and Two-Point Functions
	10.8 Summary
	References
11 Multiple SU(3) Algebras in Interacting Boson Model and Shell Model
	11.1 Introduction
	11.2 Four SU(3) Algebras in sdgIBM: Results  for Quadrupole Properties
		11.2.1 Structure of Intrinsic States
		11.2.2 Large-N Limit Results for Quadrupole Moments  and B(E2)\'s
	11.3 Eight SU(3) Algebras in sdgiIBM: Results  for Quadrupole Properties
	11.4 Multiple SU(3) Algebras in Shell Model
		11.4.1 (sdg)6p,2n Example
		11.4.2 (sdgi)6p Example
	11.5 Summary
	References
12 Summary and Future Outlook
	References
Appendix A Angular Momentum Algebra
Appendix B Elements of U(n) Lie Algebra and Its Subalgebras
Appendix C Asymptotic Nilsson Wavefunctions
Appendix D Correspondence Between SUBF2(3) Irreps  and 2 q.p. Nilsson Configurations for η= 3 Shell
Appendix E Bivariate Moments, Cumulants, and Edgeworth Expansion
Appendix  References