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Symplectic Amplitudes in Shell Model Wave Functions from E&M operators & Electron Scattering

Symplectic Amplitudes in Shell Model Wave Functions from E&M operators & Electron Scattering. Symplectic Group provided natural extension of Elliot’s SU(3) model to multi-shells Draayer, Rosensteel, Rowe, and colleagues. (  )=(4,0), (0,2). 4 h . 2 h . (  )=(2,0).

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Symplectic Amplitudes in Shell Model Wave Functions from E&M operators & Electron Scattering

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  1. Symplectic Amplitudes in Shell Model Wave Functions fromE&M operators &Electron Scattering

  2. Symplectic Group provided natural extension of Elliot’s SU(3) model to multi-shellsDraayer, Rosensteel, Rowe, and colleagues ()=(4,0), (0,2) 4h 2h ()=(2,0) Vh ()=(0,0) 0h 16O • Algebraic model provides • understanding of the underlying many-body physics, including collectivity • Physical means to truncate the basis • Straightforward to eliminate spurious states

  3. SU(3) & Sp(3,R) used in multi-h numerical shell model calculations as a very physical truncation scheme D.,R.,R., Ellis, England, Harvey, Millener, Strottman,Towner,Vergados, Hayes, et al. ()=(2,0)x() + (j,j) 4h Vh 2h ()=(2,0)x() + (i,i) +(4,2)+(2,1)… Vh {()i} 0h • Applied to numerical multi-h shell model calculations by diagonalizing Hamiltonian in SU(3) basis (up to 5h in 16O); Up to ~ 30h in Sp(3,r) basisD.,R.,R., Ellis, England, Harvey, Millener, Strottman,Towner,Vergados, Hayes, et al. • Today: Abinitio No-core shell model (multi-h) Barrett, Navratil,Vary, et al.

  4. Several Advantage to SU(3) & SP(3,R) classification of states and operators • Truncation of basis by SU(3) repns. is physical • Straight forward to eliminate spurious states Rcmtransforms as =(1,0) • Most physics operators transform simply under SU(3) Electromagnetic transitions Giant resonances Electron scattering form factors Weak interactions Pion Scattering …

  5. Example: Electron and Pion Scattering in 18O; 4.45 MeV 1- ()=(1,0) ()=(2,1) ()=(2,1) 18O(') GDR: Low-lying:

  6. Multi-shell calculationspotentiallyplagued with lack ofself-consistencyNeed some constraint onh=2 =(2,0) monopole interactions

  7. Amplitude of Symplectic terms determined by h=2,=(2,0) matrix elements h=2, =(2,0)L=0: If no constraints introduced, the amplitude and even the sign of the symplectic terms vary with the oscillator parameter

  8. Simple CaseJ=0, T=0 (0+2)h states in 16O Basis: closed shell and 2h [f]=[4444] ()= (4,2), (2,0) Diagonalize (0+2)h space: no h=2 interaction h=2 interaction on MK interaction b=1.7 fm Vary b, get very different answer for 2hw 1p1h amplitudes

  9. Problem noted in many multi-h shell calculations Radial (monopole) excitations appear at low-energies though these excitations determined by compressibility of nucleus G.S. energy perturbed very far from 0h position GMR and GQR strong functions of oscillator parameter Solutions proposed: 1. Use weak coupling scheme (diagonalize each h first, then urn on cross-shell interactions)(Ellis + England) 2. Introduce Hartree-Fock-like condition (Arima) Either by choosing a suitable oscillator parameter, or invoking by hand S.S.M. Wong, Phys. Lett 20,188, (1966) P.J. Ellis, L. Zamick, Ann Phys. 55 61 (1969) A. Feassler, et al. N.P. A330, 333 (1979) D.J. Millener, et al. AIP, 163, 402 (1988) A.C. Hayes, et al, PRC 41, 1727 (1990) W. Haxton, C. Johnson, PRL 65, 1325 (1990) J.P. Blaizot, Phys. Rep. 64, 1 (1980) M.W. Kirson, N.P.A 257, 58 (1976) T. Hoshino, W. Sagawa, A.Arima, N.P. A 481, 458 (1988)

  10. E1 strength and electric polarizability (E1.E1) of 16ODetermined by the h=2 ()=(2,0) interaction 02+ E1.E1 0+gs (n+1)h 1- Under closure Two-photon-decay +  E1.E1 transforms as (2,0) E1 (1,0) E1 (1,0) 0+ nh(n+2)hx nh Dial h=2 =(2,0)L=0, S=0 interaction strength V(2,0)= VMK(2,0)),  a parameter Somewhat analogous to dialing oscillator parameter

  11. E1 strength, two-photon decay, and polarizability with h=2, ()=(2,0) interaction (x10-3 fm3)

  12. Similar Sensitivity seen for M1 Strength Main effect from changes in the SU(4) symmetries introduced in g.s.

  13. Symplectic amplitudesin abinitio NCSM Very large model spaces achieved ~20h for at beginning of p-shell ~10hfor at the end of p-shell Examine symplectic and hmonopole amplitudes through predicted C0 and C2 (e,e’) form factors

  14. Basic (e,e’)Form Factors in HO basisDonnelly +Haxton, Millener, Ellis+Hayes, Escher+ Draayer Inelastic 0+-2+ C2 (e,e’) Elastic C0 (e,e’)

  15. Sign and Magnitude of cross-shell  Amplitudes determine rate of convergence q=0, determined by total charge q>0, as <r2> increase <=> (e,e’) form factor pulled in in q in-shell contributions to <r2> always adds constructively cross-shell contributions determined by <nh|T+V|n+2h>( • In symplectic model cross-shell constructive, building up collectivity • In numerical diagonalizations, cross-shell difficult to determine • - strongly effect by oscillator parameter • - need Hartree-Fock-like constraint

  16. NCSM for 6Li ground state

  17. C0 Form Factor for 12C shows similar effect

  18. C2 Form Factors in 6Li

  19. C2 transitions in r-space 12C 6Li

  20. GMR and GQR Strengths Strongly Affected Simple (0+2)hcalcs. in 12C b=1.18 fm is just below the value of b needed to change the sign of

  21. Seek Physical Truncation of Model Space SU(3) & Sp(3,R) very promising  problematic • Drive g.s. energy down • Shift GMR, GQR energies dramatically • Can lead to unphysical symplectic terms in wave fns., (including wrong sign) Need to introduce a constraint (Hartree-Fock-like) Should improve convergence

  22. C2 (e,e’) Matrix elements In p-shell (e,e’) data show that C2(q) drops steadily with q Calculation shows opposite trend in both 6Li and 12C, but ….

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