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)

16O

()=(2,0)

()=(4,0), (0,2)

0h

2h

4h

Vh

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

2h

4h

Vh

•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) 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)…

Vh

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

Rcm 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 18O; 4.45 MeV 1-

()=(1,0)

()=(2,1)

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(λ ,μ ) = (1,0) : − 1/6 (p → s)+ 5 /6(p → d) FL(e,e' ) = 8 / 3 y1/ 2 (1− 1

2 y)e−y

(λ ,μ ) = (2,1) : − 5 /6 (p → s) − 1/6(p → d) FL(e,e' ) = 2 /15 y3 / 2e−y

y = (bq /q)2

()=(2,1)

18O(')

GDR:

Low-lying:

Multi-shell calculationspotentially

plagued with lack ofself-consistency

Need some constraint onh=2 =(2,0) monopole

interactions

Amplitude of Symplectic terms determined by h=2,=(2,0) matrix elements

€

ph (n+2hω) | T +V | (nhω)( λ ,μ )=( 2,0)L= 0

€

T ~ − V

T =p2

2m~

1b2

€

< 0hω | H | 0hω > < 0hω | H | 2hω > ... ... ...

< 2hω | H | 0hω > < 2hω | H | 2hω >... ... ...

< n −2hω | H | nhω > < nhω | H | nhω >

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

If no constraints introduced, the amplitude and even the signof the symplectic terms vary

with the oscillator parameter

h=2, =(2,0)L=0:

Simple CaseJ=0, T=0 (0+2)h states in 16O

Basis: closed shell and 2h [f]=[4444] ()= (4,2), (2,0)

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p−2 (sd)2 p−2 (sd)2 p−2 (sd)2 p−1(pf ) s−1(sd)

(02)x(40) (02)x(02) (10)x(21) (01)x(30) (00)x(20)

| (4,2) > 1

| GMR > −.143 0.045 −.13 .874 .443

| (2,0) >1

.402 .635 −.66

Diagonalize (0+2)h space:no h=2 interaction

h=2 interaction onMK interaction b=1.7 fm

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E(MeV ) 0.0 9.5 12.4 13.7

%(2,0)1p1h 0.0 1.6 55.5 43.5

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E(MeV ) −9.9 10.8 13.7 14.7

%(2,0)1p1h 3.1 18.7 48.5 29.0

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 nucleus2. G.S. energy perturbed very far from 0h position3. 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)

€

< ph(λ ,μ ) = (2,0) |T +V | nhω >= 0

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)

Either by choosing a suitable oscillator parameter, or invoking by hand

E1 strength and electric polarizability (E1.E1) of 16ODetermined by the h=2 ()=(2,0) interaction

nh(n+2)hx nh

(n+1)h

0+

1-E1 (1,0)

V(2,0)= VMK(2,0)), a parameter

02+

0+gs

E1.E1

€

αE1

if = 20

f

+ | D |1n

− 1n

− | D | 0i

+

ΔEn

n

∑

Dial h=2 =(2,0)L=0, S=0 interaction strength

Somewhat analogous to dialing oscillator parameter

Under closureTwo-photon-decay + α

E1.E1 transforms as (2,0)E1 (1,0)

E1 strength, two-photon decay, and polarizability with h=2, ()=(2,0) interaction

(x10-3 fm3)

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α gs α if

0 −43.2 700

1 −8.9 508

−1 −83.6 921

exp −17± 4 585

€

___

E E1 = 25MeV ,ε = 0___

E E1 = 30MeV ,ε =1___

E E1 = 18MeV ,ε = −1

€

__

EE1=B(E1:0+→1n

− )•Exn∑

B(E1:0+→1n− )

n∑

Similar Sensitivity seen for M1 Strength

€

M1 Strength 1+T =1 → g.s. (μn

2

J π ε =1 ε = 0

1+ (16.22MeV ) 0.01 0.001

1+ (17.14MeV ) 0.028 0.006

1+ (18.80MeV ) 0.008 0.0004

Main effect from changes in theSU(4) symmetries introduced in g.s.

Symplectic amplitudesin abinitio NCSM

Very large model spaces achieved ~20h for at beginning of p-shell ~10hfor at the end of p-shell

Examine symplectic and hmonopole amplitudes through predicted C0 and C2 (e,e’) form factors

Basic (e,e’)Form Factors in HO basisDonnelly +Haxton, Millener, Ellis+Hayes, Escher+ Draayer

Elastic C0 (e,e’) Inelastic 0+-2+ C2 (e,e’)

€

hω = 0 (λ ,μ ) = (0,0)

F0s−0s

(q 2 ) = exp(−y)

F0 p−0 p

(q 2 ) = 3y(1− 23 y)exp(−y)

Δhω = 2 (λ ,μ ) = (2,0)

F0s−1s

(q 2 ) = 2 / 3yexp(−y)

F0 p−1p

(q 2 ) = 10 / 3(1−2 5y)exp(−y)

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hω = 0 (λ ,μ ) = (1,1)

F0 p−0 p

(q 2 ) = − 8 /15yexp(−y)

Δhω = 2 (λ ,μ ) = (2,0)

FGQR

(q 2 ) = 24 /15y(1−1 3y)exp(−y)

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y = (bq /2)2

Sign and Magnitude of cross-shell Amplitudes determine rate of convergence

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F(q 2 ) =1−< r 2 > q 2

6+o(q 4 )

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

q=0, determined by total charge

q>0, as <r2> increase <=> (e,e’) form factor pulled in in q

NCSM for 6Li ground state

C0 Form Factor for 12C shows similar effect

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F(q 2 ) =1−< r 2 > q 2

6+o(q 4 )

C2 Form Factors in 6Li

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B(Cλ ) = f −2 Z 2

4π(2λ +1)!!

q λ

⎛

⎝ ⎜

⎞

⎠ ⎟Fλ

2

€

C2(q) ≡ B(C2)1/ 2 = A + By+Cy2 +...

C2 transitions in r-space

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B(E2) Values for 6Li (e2 fm 4 )

b( fm) hw(MeV ) 0hw 2hw 4hw 6hw 8hw 10hw Expt.

1.94 11MeV 6.84 8.14 8.93 9.89 10.73 11.63 21.8(4.8)

1.79 13MeV 4.91 6.25 7.03 8.16 9.14 10.22

12C6Li

GMR and GQR Strengths Strongly Affected

Simple (0+2)hcalcs. in 12C

b=1.18 fm is just below the value of b needed to changethe sign of

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ph (0 +2hω) | T +V | (0hω) (λ ,μ )=(2,0)L=0

Seek Physical Truncation of Model Space

SU(3) & Sp(3,R) very promising

€

ph (n+2hω) | T +V | (nhω) (λ ,μ )=(2,0)L=0

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

€

B(Cλ ) = f −2 Z 2

4π(2λ +1)!!

q λ

⎛

⎝ ⎜

⎞

⎠ ⎟Fλ

2

€

C2(q) ≡ B(C2)1/ 2 = A + By+Cy2 +...

In p-shell (e,e’) data show thatC2(q) drops steadily with q

Calculation shows opposite

trend in both 6Li and 12C, but ….