symplectic amplitudes in shell model wave functions from e&m operators & electron...
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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). - PowerPoint PPT PresentationTRANSCRIPT
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)
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< 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
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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
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___
E E1 = 25MeV ,ε = 0___
E E1 = 30MeV ,ε =1___
E E1 = 18MeV ,ε = −1
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__
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’)
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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
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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
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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 +...
In p-shell (e,e’) data show thatC2(q) drops steadily with q
Calculation shows opposite
trend in both 6Li and 12C, but ….