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 (  )=(0,0) 16 O (  )=(2,0) (  )=(4,0), (0,2) 0h0h 2h2h 4h4h V  h  Algebraic model provides understanding of the underlying many-body physics, including collectivity Physical means to truncate the basis Straightforward to eliminate spurious states

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. {(    ) i } (    )=(2,0)x(    ) + ( i,  i ) (    )=(2,0)x(    ) + ( j,  j ) 0h0h 2h2h 4h4h V  h  Applied to numerical multi-h  shell model calculations by diagonalizing Hamiltonian in SU(3) basis (up to 5h  in 16 O); Up to ~ 30h  in Sp(3,r) basis D.,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. +(4,2)+(2,1)… V  h 

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 R cm transforms as  =(1,0) Most physics operators transform simply under SU(3) Electromagnetic transitions Giant resonances Electron scattering form factors Weak interactions Pion Scattering …

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

Multi-shell calculations potentially plagued with lack of self-consistency Need some constraint on  h  =2  =(2,0) monopole interactions

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

Simple Case J=0, T=0 (0+2)h  states in 16 O 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

Problem noted in many multi-h  shell calculations 1.Radial (monopole) excitations appear at low-energies though these excitations determined by compressibility of nucleus 2.G.S. energy perturbed very far from 0h  position 3.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) S.S.M. Wong, Phys. Lett 20,188, (1966) P.J. Ellis, L. Zamick, Ann Phys (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) Either by choosing a suitable oscillator parameter, or invoking by hand

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

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

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

Symplectic amplitudes in 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

Basic (e,e’)Form Factors in HO basis Donnelly +Haxton, Millener, Ellis+Hayes, Escher+ Draayer Elastic C0 (e,e’) Inelastic C2 (e,e’)

Sign and Magnitude of cross-shell  Amplitudes determine rate of convergence in-shell contributions to always adds constructively cross-shell contributions determined by (  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 q=0, determined by total charge q>0, as increase (e,e’) form factor pulled in in q

NCSM for 6 Li ground state

C0 Form Factor for 12 C shows similar effect

C2 Form Factors in 6 Li

C2 transitions in r-space 12 C 6 Li

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

Seek Physical Truncation of Model Space SU(3) & Sp(3,R) very promising   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 problematic

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 6 Li and 12 C, but ….