Abstract: While large-scale shell-model calculations may be useful, perhaps even essential, for reproducing experimental data, insight into the physical underpinnings of many-body phenomena requires a deeper appreciation of the underlying symmetries or near-symmetries of a system. Our program is focused on standard as well as novel algebraic approaches to nuclear structure … many-body shell-model methods within a group theoretical framework. J. P. Draayer, J. G. Hirsch, Feng Pan, A. I. Gueorguieva C. Bahri, G. Popa, G. Stoitcheva V. G. Gueorguiev, K. D. Sviracheva, K. Drumev H. Ganev, H. Matevosyan, H. Grigoryan Principals Students

- End of the Day - Q Space P Space xh  yh  postulate: x(~2) + y(~12) = ~12 … symplectic …

- Ongoing Research Projects - Pseudo-SU(3) calculations for heavy nuclei E2 / M1 (scissors) modes in deformed nuclei Algebraic solution for non-degenerate pairing Non-linear modes (surface solitons) of a drop Boson and Fermion representations of sp(4) Mixed-mode (oblique) shell-model methods SU(3) calculations for lower fp-shell nuclei good more WORKINGWORKING

Castaños / Draayer / Hirsch Dirk Rompf (PhD-1998) Thomas Beuschel (PhD-1999) Carlos Vargas (PhD-2000) Gabriela Popa (PhD-2001) Study shell-model dynamics against the background of an extremely elegant algebraic/geometric picture Rot(3)  SU(3) Macroscopic Microscopic Reproduce experimental M1 spectra including observed fragmentation and clustering in a microscopic model Ann. Phys. 180 (1987) 290 Z. Phys. A 354 (1996) 359 PRC 57 (1998) 1233; PRC 61 (2000) ; 62 (2000)

… For the record... Enhanced M1 strengths first predicted in 1978 by Lo Iudice and Palumbo within the framework of the so- called phenomenological two-rotor model (TRM) … [N. Lo Iudice and F. Palumbo, PRL 41 (1978) 1532] Enhanced M1 strengths first observed in 1984 in 156 Gd … [D. Bohle, A. Richter, W. Steffen, A. E. L. Dieperink, N. Lo Iudice, F. Palumbo, and O. Scholten, PL 137 B (1984) 27 & U.E.P. Berg, C. Basing, J. Drexler, R. D. Heil, U. Kneisl, W. Naatz, R. Ratzek, S. Schennach, R, Stock, T. Weber, B. Fischer, H. Hollick, and D. Kollewe, PL 149 B (1984) 59] Discussed within the context of the pseudo-SU(3) model … [O. Castaños, J. P. Draayer and Y. Leschber, Ann. Phys. 180 (1987) 290] Considered within the framework of the interacting boson (proton-neutron) model (IBM-2) … [K. Heyde and C. De Coster, PRC 44 (1991) 2262]

Typical M1 Spectrum Clustering Fragmentation M1 GS s1+s 2 MeV 4 MeV * * Adiabatic representation mixing...

Basis States: Pseudo-SU(3) (spherical  cartesian)  n[f]  (   (  )   particle number permutation symmetry multiplicity label (  ) deformation [SU(3)] K-band [SU(3)] total orbital total spin total a.m. a.m. proj.  n[f]  (   SMs  Q 0 -> z-deformation xy - orbital a.m. xy - a.m. proj. total spin spin proj. cartesian system (Slater Determinants) spherical system transform between these yields a bit representation of basis states …

Tricks of the Trade (config geometry  algebraic shape) Invariants  Invariants Rot(3) SU(3) Tr(Q 2 )  C 2 Tr(Q 3 )  C 3    ~    +  +  +  +   =  tan   + 

          +      +      +   -   +   +     +     +   -     +   -    +   -     m,l   +  -2m+l   +  +m-2l) k Direct Product Coupling Coupling proton and neutron irreps to total (coupled) SU(3): TWIST SCISSORS SCISSORS + TWIST … Orientation of the  - system is quantized with the multiplicity denoted by k = k(m,l) … multiplicity GROUND STATE

Hamiltonian: Symmetry Preserving SU(3) preserving Hamiltonian: H = c 1 (Q   Q  + Q   Q ) + c 2 Q   Q + c 3 L 2 + c 4 K 2 + c 5 (L  2 + L 2 ) Using the identities... Q   Q  =  C 2  - 3 L  2 with  or Q   Q  =  Q Q - Q   Q  - Q   Q ]/2 L = L  + L   = L  - L the Hamiltonian becomes... H = H rot + H int with rotor & interactions terms … H rot = a L 2 + b K 2 H int = c 2 + d C 2 + e (C 2  + C 2 )

H int = c 2 + dC 2 + e(C 2  + C 2      kinetic potential System Dynamics H rot = a L 2 + b K 2  quantum rotor It can be shown that: H int = c   2 + d     + c  -  d  -  -  where:  -    so that: E int = h    (n  + 1/2) + h   -  (n  -  + 1/2) where: n  = m, n  - = l Scissors Mode Twist Mode …requires triaxial shape distributions …always present, axial and triaxial Recall: dynamics driven by interplay between the interaction and statistics

M1 Operator M1 = (3/4) 1/2  N    g orbit  L   g spin  S   g orbit (proton)  g orbit (neutron)  g spin (proton)  g spin (neutron)  Actual Values

Scissors and Twist Mode Examples Axial + Triaxial + Triaxial

Example: M1 Modes in Gd Particle distributions: total normalunique   Gd (10,4)  156 Gd (18,0) 158 Gd (18,0) 160 Gd (18,4) Valence space: U(   )  U(  ) total normal unique   = 32  n  = 20 (pf-shell)  u  = 12 (h 11/2 )  = 44  n = 30 (sdg-shell)  u = 14 ( i 13/2 ) ~ ~ ~ ~ ~

Hamiltonian: Realistic (Version 1) H = H 0 Oscillator … mixed irrep shell-model basis with 5 proton and 5 neutron irreps plus all products having S = 0 and J  8 Symmetry Preserving Symmetry Breaking + d   l 2  Proton single-particle l 2 + d  l 2 Neutron single-particle l 2 + g  H p  Proton Pairing + g H p Neutron Pairing - a 2 C 2 Q Q symmetric + a 3 C 3 Q (QxQ) asymmetric + b K 2 K-Band splitting + c L 2 Rotational bands

Results for Gd Nuclei

160 Gd: Excitation Spectrum

160 Gd: M1 Strength Distribution

Fragmentation of M1 Strength in 160 Gd

Results for Dy Nuclei

M1 Sumrule vs E2 Strength

Triaxial Case: 196 Pt Sumrule ~ 1.5  2 or about one-half that of the strongly deformed case

Analytic Results Castaños ‘87 Rompf ‘96 Beuschel ‘98 Castaños ‘87 Rompf ‘97 Note: t  s … (  )  (  )

Odd-A Case: 163 Dy Recent Results: Beuschel/Hirsch

Hamiltonian: Realistic (Version 2) H = H 0 Oscillator … mixed irrep shell-model basis with ~5 proton and ~5 neutron irreps & all products having S = 0 and J  8 Symmetry Preserving Symmetry Breaking + d   l 2  Proton single-particle l 2 + d  l 2 Neutron single-particle l 2 + g  H p  Proton Pairing + g H p Neutron Pairing - (  Q   Q Q Q symmetric + a 3 C 3 Q (QxQ) asymmetric + b K 2 K-Band splitting + c L 2 Rotational bands + a sym  C 2 Intrinsic symmetry FIXED FINE TUNE

Hamiltonian Parameters FIXED FINE TUNE - (  Q   Q + d   l 2  + d  l 2 + g  H p  + g H p + a 3 C 3 + b K 2 + c L 2 + a sym  C 2

158 Gd

160 Gd Exp Th Energy [MeV] 160 Gd

Some Observations Fragmentation can be interpreted as effect of non-collective residual interactions Gross structure reproduced by adding a new “twist” to the “scissors” system Triaxial proton-neutron configuration gives a physical interpretation of multiplicity Analytic results for scissors as well as twist modes and combination now available Future research with S=1 mode for odd-A as well as even-even nuclei underway Scissors Twist Scissors mode (proton-neutron) oscillation always exists in deformed systems Extra “twist” mode if triaxial proton or neutron distributions are included

Conclusions - Part 1: E2 / M1 modes... E X C I T I N G (simple) P H Y S I C S ! E X C I T I N G (simple) P H Y S I C S ! Scissors mode always present … Twist mode(s) requires triaxiality … Fragmentation via symmetry mixing... B(E2) values good (effective charge) … Near perfect excitation spectra …

Mixed-Mode SMC … a new shell-model code (SMC) that integrates the best shell model methods available: m-scheme spherical shell-model SU(3) symmetry based shell-model Developers Vesselin Gueorguiev Jerry Draayer Erich Ormand Calvin Johnson (H - Eg) 

The Challenge... Nuclei display unique and distinguishable characteristics: Single-particle Featues Pairing Correlations Deformation/Rotations Closed Shell Closed Shell SingleSingle SingleSingle Rotations SU(3) Pairing SU Q (2) Pairing SU Q (2).

Usual Shell-Model Approach Select a model space based on a simple approximation (e.g. Nilsson scheme) for the selection of basis states. Diagonalize a realistic Hamiltonian in the model space to obtain the the energy spectrum and eigenstates. Evaluate electromagnetic transition strengths (E2, M1, etc.) and compare the results with experimental data. Dual Basis “Mixed Mode” Scheme Select a dual (non-orthonormal) basis (e.g. pairing plus quadrupole scheme) for the interaction Hamiltonian. Diagonalize a realistic Hamiltonian in the model space to obtain the the energy spectrum and eigenstates. Evaluate electromagnetic transition strengths (E2, M1, etc.) and compare the results with experimental data.

Hamiltonian Matrix SU(3) standard jj-coupling shell-model scheme single-particle ___________ pairing driven o v e r l a p overlapoverlap

Shell-Model Hamiltonian  Spherical shell-model basis states are eigenstates of the one-body part of the Hamiltonian - single-particle states.  The two-body part of the Hamiltonian H is dominated by the quadrupole-quadrupole interaction Q·Q ~ C 2 of SU(3).  SU(3) basis states - collective states - are eigenstates of H for degenerate single particle energies  and a pure Q·Q interaction.

Eigenvalue Problem in an Oblique Basis  Spherical basis states e i  SU(3) basis states E   Overlap matrix  The eigenvalue problem

Current Evaluation Steps Matrix elements ( H and g )  i and m- scheme SM basis (spherical) SU(3) basis (cylindrical) H and g g=UU T ( Cholesky ) (U -1 ) T HU -1 (U  )=E (U  ) Eigenstates ( Lanczos ) Expectation values and matrix elements of operators and O and g |E>

We use the Wildenthal USD interaction and denote the spherical basis by SM(#) where # is the number of nucleons outside the d 5/2 shell, the SU(3) basis consists of the leading irrep (8,4) and the next to the leading irrep, (9,2). Example of an Oblique Basis Calculation: 24 Mg Visualizing the SU(3) space with respect to the SM space using the naturally induced basis in the SU(3) space. SM(4) & SU3+ SU(3) basis space SM space SU(3) basis space SM(2) & SU % Dimension (m-scheme) FullSM(4)SM(2)SM(1)SM(0)GT100SU3+ (8,4) & (9,2) SU3 (8,4) Model Space

3.3 MeV (0.5%) 4.2 MeV(54%) Better Dimensional Convergence!

Energy ( MeV) SM(0) (8,4) (9,2) More irreps FULL SM(4) SM(2) SM(1) Level Structure

Oblique Basis Spectral Results

Oblique basis calculations have the levels in the correct order! Spherical basis needs 64.2% of the total space - SM(4)! SU(3) basis needs 0.4% of the total space - (8,4)&(9,2)! Oblique basis 1.6% of the total space for SM(1)+(8,4)! SM(0)+(8,4) is almost right… 0.2%

Oblique basis calculations have better overlap with the exact states! Using 10% of the total space in oblique basis SM(2)+(8,4)+(9,2) as good as using 64% of the total space in spherical basis SM(4)

Mixed mode (oblique basis) calculations lead to:  better dimensional convergence  correct level order of the low-lying states  significant overlap with the exact states Conclusions - Part 2: Mixed modes...

Further Considerations #1 Traditional configuration shell-model scheme(s) versus SU(3) and symplectic shell-model approaches  Common “bit arithmetic” (Slater determinants)  Both use harmonic oscillator basis functions …  Translation invariance okay for both approaches  Both use effective reduced matrix element logic  One single-particle the other quadrupole driven Complementary … ???

Further Considerations #2 4h  0h  2h  Configuration Shell-Model (multi-h  horizontal slices Symplectic Shell-Model (multi-h  vertical slices 6h  symplectic slice translation invariance okay for both SU(3) limit (0h 

Further Considerations #3 … mixed-mode shell-model … Q Space P Space xh  yh  postulate: x(~2) + y(~12) = ~12 … symplectic …

Abstract: … consider standard as well as novel algebraic approaches to nuclear structure, including use of the Bethe ansatz and quantum groups, that are being used to explore special features of nuclei: pairing correlations, quadrupole collectivity, scissors modes, etc. In each case the underlying physics is linked to symmetries of the system and their group theoretical representation. E2 / M1 (scissors) modes in deformed nuclei Mixed-mode shell-model methods feasible Possible configuration + symplectic extension ?