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J. P. Draayer, J. G. Hirsch, Feng Pan, A. I. Gueorguieva C. Bahri, G. Popa, G. Stoitcheva

Principals Students. Group Theory Methods (Nuclear Shell Model). 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.

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J. P. Draayer, J. G. Hirsch, Feng Pan, A. I. Gueorguieva C. Bahri, G. Popa, G. Stoitcheva

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  1. Principals Students Group Theory Methods (Nuclear Shell Model) 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 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.

  2. Q Space … symplectic … yhw xhw P Space Q Space - End of the Day - postulate: x(~2) + y(~12) = ~12

  3. good WORKING more - 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

  4. E2 / M1 (scissors) modes in deformed nuclei ... 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 Castaños / Draayer / Hirsch Dirk Rompf (PhD-1998) Thomas Beuschel (PhD-1999) Carlos Vargas (PhD-2000) Gabriela Popa (PhD-2001) Ann. Phys. 180 (1987) 290 Z. Phys. A 354 (1996) 359 PRC 57 (1998) 1233; 1703 ... PRC 61 (2000) 031301; 62 (2000) 064313 ...

  5. … 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 156Gd … [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]

  6. 4 MeV M1 2 MeV 0+ 1+s GS Typical M1 Spectrum Clustering * Fragmentation *Adiabatic representation mixing ...

  7. a.m. proj. spin proj. total a.m. total spin spherical system total spin xy - a.m. proj. total orbital xy - orbital a.m. K-band [SU(3)] Q0 -> z-deformation deformation [SU(3)] multiplicity label (l m) cartesian system (Slater Determinants) permutation symmetry particle number transform between these yields a bit representation of basis states … Basis States: Pseudo-SU(3) (spherical  cartesian) |n[f]a(l m() |n[f]a(l meLMLSMs

  8. Invariants Invariants Rot(3) SU(3) Tr(Q2)  C2 Tr(Q3)  C3 Tricks of the Trade(config geometry  algebraic shape) b2~ l2+ lm + m + 3(l + m+1) g =tan-1 [3 m / (2 l + m+3)]

  9. Direct Product Coupling Coupling proton and neutron irreps to total (coupled) SU(3): (lp, mp)  (ln, mn)  (lp+ln , mp+mn) + (lp+ln- 2, mp +mn+ 1) + (lp+ln+1, mp+mn- 2) + (lp+ln- 1, mp+mn- 1)2 + ... m,l  (lp+ln-2m+l,mp+mn+m-2l)k GROUND STATE SCISSORS TWIST SCISSORS + TWIST … multiplicity … Orientation of the p-n system is quantized with the multiplicity denoted by k = k(m,l)

  10. Hamiltonian: Symmetry Preserving SU(3) preserving Hamiltonian: H = c1(Qp •Qp + Qn •Qn) + c2Qp •Qn + c3 L2 + c4 K2 + c5 (Lp2 + Ln2) Using the identities ... Qs •Qs = 4 C2s - 3 Ls2 with s = p or n Qp •Qn= [Q • Q - Qp •Qp - Qn •Qn]/2 L = Lp + Ln l =Lp - Ln the Hamiltonian becomes ... H = Hrot + Hint with rotor & interactions terms … Hrot = a L2 + b K2 Hint = c l2 + d C2 + e (C2p + C2n)

  11. System Dynamics Hrot = a L2 + b K2 quantum rotor Recall: dynamics driven by interplay between the interaction and statistics Hint = c l2 + dC2 + e(C2p + C2n)  kinetic potential …always present, axial and triaxial It can be shown that: Hint = cq lq2+ dqq2 + cf-lf-2+ df-f-2 where: f- = fp - fn so that: Eint = hwq(nq+ 1/2) + hwf- (nf-+ 1/2) where: nq = m, nf- = l Scissors Mode Twist Mode …requires triaxial shape distributions

  12. M1 Operator gorbit(proton) = 1 gorbit(neutron) = 0 gspin(proton) = 5.5857 gspin(neutron) = -3.8263 M1 = (3/4)1/2mNSs (gorbit (s) Ls + gspin (s) Ss ) Actual Values

  13. Scissors and Twist Mode Examples Triaxial + Triaxial Axial + Triaxial

  14. Example: M1 Modes in 156-160Gd Valence space: U(Wp) U(Wn) total normal unique Wp = 32 Wnp = 20 (pf-shell) Wup = 12 (h11/2) Wn = 44 Wnn = 30 (sdg-shell) Wun= 14 ( i13/2 ) ~ ~ ~ ~ ~ Particle distributions: total normal unique (l,m) p:156-160Gd 14 8 6 (10,4) n: 156Gd 10 6 4 (18,0) 158Gd 12 6 6 (18,0) 160Gd 14 8 6 (18,4)

  15. Symmetry Preserving Symmetry Breaking Hamiltonian: Realistic (Version 1) H = H0Oscillator -a2 C2Q•Q symmetric + a3 C3Q•(QxQ) asymmetric + b K2K-Band splitting + c L2Rotational bands + dp l2pProton single-particle l2 + dn l2nNeutron single-particle l2 + gp HppProton Pairing + gn HpnNeutron Pairing … mixed irrep shell-model basis with 5 proton and 5 neutron irreps plus all products having S = 0 and J £ 8

  16. Results for 156-160 Gd Nuclei

  17. 160Gd: Excitation Spectrum

  18. 160Gd: M1 Strength Distribution

  19. Fragmentation of M1 Strength in 160Gd

  20. Results for 160-164 Dy Nuclei

  21. M1 Sumrule vs E2 Strength

  22. Triaxial Case: 196 Pt Sumrule ~ 1.5 m2 or about one-half that of the strongly deformed case

  23. Castaños ‘87 Castaños ‘87 Rompf ‘96 Rompf ‘97 Beuschel ‘98 Analytic Results Note: t  s … (l,m) (m,l)

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

  25. Symmetry Preserving Symmetry Breaking Hamiltonian: Realistic (Version 2) H = H0Oscillator - (c/2) Q•QQ•Q symmetric + a3 C3Q•(QxQ) asymmetric + b K2K-Band splitting + c L2Rotational bands +asymDC2 Intrinsic symmetry FINE TUNE + dp l2pProton single-particle l2 + dn l2nNeutron single-particle l2 + gp HppProton Pairing + gn HpnNeutron Pairing … mixed irrep shell-model basis with ~5 proton and ~5 neutron irreps & all products having S = 0 and J £ 8 FIXED

  26. Hamiltonian Parameters FIXED FINE TUNE - (c/2) Q•Q + dp l2p + dn l2n + gp Hpp + gn Hpn + a3 C3 + b K2 + c L2 + asymDC2

  27. 158Gd

  28. 2 + 8 + 4 + 7 1.5 + 6 + 2 + 5 + 0 + 1 + 4 + 3 1 Energy [MeV] + + 2 8 160 Gd + 6 0.5 + 4 + 2 0 + 0 Exp Th Exp Th Exp Th -0.5 160Gd

  29. Some Observations • Scissors mode (proton-neutron) oscillation always exists in deformed systems • Extra “twist” mode if triaxial proton or neutron distributions are included • 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

  30. Conclusions - Part 1: E2 / M1 modes ... • Near perfect excitation spectra … • B(E2) values good (effective charge) … • Scissors mode always present … • Twist mode(s) requires triaxiality … • Fragmentation via symmetry mixing ... EXCITING (simple) PHYSICS!

  31. Mixed mode (oblique) shell-model methods ... • Mixed-Mode SMC… anewshell-model code (SMC) that integrates the best shell model methods available: • m-schemespherical shell-model • SU(3)symmetry based shell-model • ----- Developers ----- • Vesselin Gueorguiev • Jerry Draayer • Erich Ormand • Calvin Johnson (H - Eg)Y Oblique SMC

  32. . Pairing Pairing S i n g l e S i n g l e SUQ(2) SUQ(2) Closed Shell Closed Shell Rotations SU(3) The Challenge ... Nuclei display unique and distinguishable characteristics: • Single-particle Featues • Pairing Correlations • Deformation/Rotations

  33. 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.

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

  35. 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 ~ C2 of SU(3). • SU(3) basis states - collective states - are eigenstates of H for degenerate single particle energies  and a pure Q·Q interaction.

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

  37. H and g Current Evaluation Steps g=UUT (Cholesky) (U-1)THU-1 (U)=E (U) Eigenstates (Lanczos) Matrix elements ( H and g) O and g |E> SM basis (spherical) SU(3) basis (cylindrical) Expectation values and matrix elements of operators <O> and <E1| O |E2> m-scheme iand <j1j2J'T'|V|j3j4J''T''>

  38. Model Space SU3 (8,4) SU3+ (8,4) & (9,2) GT100 SM(0) SM(1) SM(2) SM(4) Full Dimension (m-scheme) 23 128 500 29 449 2829 18290 28503 SU(3) basis space % 0.08 0.45 1.75 0.10 1.57 9.92 64.17 100 SU(3) basis space SM space SM(2) & SU3+ SM space SM(4) & SU3+ Example of an Oblique Basis Calculation: 24Mg We use the Wildenthal USD interaction and denote the spherical basis by SM(#) where #is thenumber of nucleons outside the d5/2 shell, the SU(3) basis consists of the leading irrep (8,4) and the next to the leading irrep, (9,2). Visualizing the SU(3) space with respect to the SM space using the naturally induced basis in the SU(3) space.

  39. 3.3 MeV (0.5%) 4.2 MeV(54%) Better Dimensional Convergence!

  40. Level Structure -70 -75 -80 SM(0) SM(1) -85 4 (8,4) 3 4 (8,4) 2 SM(2) Energy ( MeV) (9,2) More -90 2 irreps 0 SM(4) FULL -95

  41. Oblique Basis Spectral Results

  42. Oblique basis calculations have the levels in thecorrect 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 basis1.6% of the total space for SM(1)+(8,4)! • SM(0)+(8,4) is almost right… 0.2%

  43. Overlaps With The Exact Eigenvectors For 24Mg 120 100 80 Main Contribution % 60 40 20 5 3 4 3 5 8 4 4 0 1 2 3 4 5 6 57.77 53.02 39.78 42.50 42.99 35.92 SM(2) 63.02 63.77 71.49 59.46 70.15 54.14 SU(3)+ 91.58 90.95 87.72 89.06 87.35 82.23 SM(2)+SU(3)+ 93.25 92.81 89.98 92.47 91.10 88.33 SM(4) 98.57 98.73 97.92 98.41 98.55 96.59 SM(4)+SU(3)+ Eigenvectors

  44. 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)

  45. Conclusions - Part 2: Mixed modes ... Mixed mode (oblique basis) calculations lead to: • better dimensional convergence • correct level orderof the low-lying states • significant overlapwith the exact states

  46. 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 … ???

  47. 6hw symplectic slice Configuration Shell-Model (multi-hw Symplectic Shell-Model (multi-hw SU(3) limit 0hw (0hw 4hw 2hw vertical slices horizontal slices Further Considerations #2 translation invariance okay for both

  48. Q Space … symplectic … yhw xhw P Space Q Space Further Considerations #3 … mixed-mode shell-model … postulate: x(~2) + y(~12) = ~12

  49. E2 / M1 (scissors) modes in deformed nuclei • Mixed-mode shell-model methods feasible • Possible configuration + symplectic extension ? Group Theory Methods (Nuclear Shell Model) 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. The End

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