Transitions, Overlaps and Spectroscopic Factors A. Arriaga J. Carlson D. Kurath L. Lapikas T.-S. H. Lee L. E. Marcucci K. M. Nollett V. R. Pandharipande S. C. Pieper R. Schiavilla K. Varga R. B. Wiringa Pieper & Wiringa, Ann.Rev.Nucl.Part.Sci. 51, 53 (2002)
Hamiltonian H = K i + v ij + V ijk i i<j i<j<k K i = K CI i + K CSB i h2 4 [( 1 m p + 1 m n ) + ( 1 m p 1 m n )τ zi ] 2 i Argonne v 18 (AV18) v ij = v γ ij + vπ ij + v R ij = p v p (r ij )O p ij O p ij = [1, σ i σ j, S ij, L S, L 2, L 2 (σ i σ j ), (L S) 2 ] [1, τ i τ j ] + [1, σ i σ j, S ij, L S] [T ij, (τ iz + τ jz )] Urbana IX (UIX) V ijk = V 2πP ijk + V R ijk V 2πP ijk = A 2π {Xij, π Xik}τ π j τ k + i 4 [Xπ ij, Xik]τ π i τ j τ k cyc
Variational Monte Carlo E V = Ψ V H Ψ V Ψ V Ψ V E 0 Trial function Ψ V = [ 1 + U T NI ijk ] Ψ P Ψ P = [ S i<j i<j<k (1 + U ij ) ] Ψ J U ij = p=2,6 u p (r ij )O p ij ; U T NI ijk = ɛv ijk ( r ij, r jk, r ki ) s-shell nuclei [ ] Ψ J = f c (r ij ) Φ A (JMT T 3 ) i<j Φ α (0000) = A p p n n Functions f c (r ij ) and u p (r ij ) are obtained numerically from solution of coupled differential equations containing v ij
p-shell nuclei Ψ J = A i<j 4 4<l<m A f ss (r ij ) LS[n] Φ A (LS[n]JMT T 3 ) 1234 56...A = Φ α(0000) 1234 (β LS[n] k 4<l A f LS[n] sp (r kl ) ) } fpp LS[n] (r lm ) Φ A (LS[n]JMT T 3 ) 1234 56...A 4<l A φ LS[n] p (R αl ) { [Y m l 1 (Ω αl )] LML [χ l ( 1 2 m s)] SMS }JM [ν l( 1 2 t 3)] T T3 Permutation symmetry A [n] L (T, S) 6 [2] 0, 2 (1, 0), (0, 1) [11] 1 (1, 1), (0, 0) 7 [3] 1, 3 (1/2, 1/2) [21] 1, 2 (3/2, 1/2), (1/2, 3/2), (1/2, 1/2) [111] 0 (3/2, 3/2), (1/2, 1/2) 8 [4] 0, 2, 4 (0, 0) [31] 1, 2, 3 (1, 1), (1, 0), (0, 1) [22] 0, 2 (2, 0), (1, 1), (0, 2), (0, 0) [211] 1 (2, 1), (1, 2), (1, 1), (1, 0), (0, 1)
Diagonalization in β LS[n] basis to produce energy spectra E(Jx π ) and orthogonal excited states Ψ V (Jx π ) Expectation values Ψ V (R) is represented by a vector with 2 A ( A Z) spin-isospin components for each space configuration R = (r 1, r 2,..., r A ); Expectation values are given by summation over samples drawn from probability distribution W (R) = Ψ P (R) 2 : Ψ V O Ψ V Ψ V Ψ V = Ψ V (R)OΨ V (R) Ψ V / (R)Ψ V (R) W (R) W (R) Ψ Ψ is a dot product and Ψ OΨ a sparse matrix operation. Transition matrix elements Generate W from either initial or final state: Ψ f T Ψ i Ψ f Ψ i = Ψ f T Ψ i W / Ψf 2 W Ψi 2 W T may be a non-square matrix.
-20 Energy (MeV) -30-40 -50 0 + 4 He 1/2 α+n 3/2 α+2n 5 He 1 + 2 + 0 + 1 + 2 + 3 + 6 He 6 1 + Li α+d 5/2 1/2 3/2 6 He+n 7 He Argonne v 18 + UIX VMC Trial & GFMC Calculations 15 July 2004 6 He+2n 5/2 5/2 7/2 3/2 5/2 5/2 7/2 7 Li α+t 3/2 1/2 8 He 1 + 2 + 0 + 4 + 1 + 3 + 0 + 1 + 8 2 + Li 7 Li+n 8 Li+n 3 + 2 + 1 + 4 + 2 + 9 Li 7/2 3/2 5/2 1/2 3/2 8 Be+n 1/2 5/2-60 Ψ T GFMC α+α 8 Be 0 + 9 Be 3/2
APPLICATIONS F 2 T (q) = 1 F 2 L (q) = 1 2J i + 1 J=1 2J i + 1 J=0 J f T El J Electromagnetic form factors J f T J Coul (q) J i 2 (q) J i 2 + J f T Mag (q) J J i 2 6 Li(e,e) 6 Li 6 Li(e,e) 6 Li F L (q 2 ) IA IA+MEC Stanford (q 2 ) F T 2 10 4 10 5 10 6 IA IA+MEC Amsterdam Saskatoon Stanford 10 7 C0, C2 M1 10 4 0 1 2 3 4 q (fm 1 ) 10 8 0 1 2 3 4 q (fm 1 ) 6 Li(e,e ) 6 Li* (3 +,T=0) 6 Li(e,e ) 6 Li* (0 +,T=1) F L 2 (q 2) 10 4 IA IA+MEC Mainz Saskatoon Stanford (q 2 ) F T 2 10 4 10 5 IA IA+MEC Saskatoon Mainz 10 5 C2, C4 M1 10 6 10 6 0 1 2 3 4 q (fm 1 ) 10 7 0 1 2 3 4 q (fm 1 ) Wiringa & Schiavilla, Phys.Rev.Lett. 81, 4317 (1998)
APPLICATIONS Pion scattering Compute quadrupole transition densities for p and n: ρ t 2J f + 1 Ψ Jf M f 3 i δ(r r i )r2 E2 (r) = i Y 2 M (ˆr i ) 1+2t 3 τ(i) 2 J f M f J i 2M i M Ψ Ji M i 0.00 0.02 0.00 0.02 0.04 n=p 6 Li (0 + 3 + ) 7 Li ( 3 / 2 1 / 2 ) ρ E2 t 3 (fm 1 ) 0.00 0.02 0.04 0.06 0.00 0.02 0.04 n p 7 Li ( 3 / 2 7 / 2 ) 7 Li ( 3 / 2 5 / 2 ) 0 1 2 3 4 5 6 r (fm) B(E2 ) Experiment VMC p p n 6 Li(0 + 3 + ) 21.8±4.8 21.1±0.4 21.1±0.4 7 Li( 3 2 1 2 ) 7.59±0.1 5.7±0.1 16.5±0.3 7 Li( 3 2 7 2 ) 15.5±0.8 13.2±0.2 34.6±0.5 7 Li( 3 2 5 2 ) 4.1±2.0 2.3±0.1 5.4±0.2
Input to DWIA analysis of pion scattering: 6 Li(π,π) 6 Li 6 Li(π,π ) 6 Li 10 3 10 3 10 2 10 1 6Li(1 + ) 240 MeV π 6Li(3 + ) 240 MeV π 10 2 10 1 dσ/dω (mb/sr) 10 2 10 1 6 Li(1 + ) 180 MeV π 6 Li(3 + ) 180 MeV π 10 2 10 1 6 Li(1 + ) 100 MeV π 6 Li(3 + ) 100 MeV π 10 1 10 1 0 30 60 90 120 150 θ (deg) 0 30 60 90 120 150 180 10 1 Conventional shell-model prediction required enhancement factor E p = E n = 2.5 to fit data; no such factor needed with Ψ V. Lee & Kurath, Phys.Rev.C 21, 293 (1980)
π + (π ) scattering sensitive to p (n) transition strength: 7 Li(π,π ) 7 Li * @ 164 MeV π + 7 Li(5/2 ) p + n p only π 7 Li(5/2 ) p + n p only 10 4 10 4 dσ/dω (mb/sr) π + 7 Li(7/2 ) p + n p only π 7 Li(7/2 ) p + n p only π + 7 Li(1/2 ) p + n p only π 7 Li(1/2 ) p + n p only 0 30 60 90 120 150 θ (deg) 0 30 60 90 120 150 180 10 3 Conventional shell-model required different enhancement factors E p = 2.5, E n = 1.75 to fit data; again none needed with Ψ V. Lee & Wiringa, Phys.Rev.C 63, 014006 (2001)
APPLICATIONS 7 Li(e, e p) 6 He reaction Coulomb DWIA analysis using VMC overlap in momentum distribution: ρ(p m ) = e ipm r 6 He a(r) 7 Li dr 2 10-6 10-7 (x 10) ρ(p m ) [(MeV/c) -3 ] 10-8 10-9 10-10 10-11 3/2 - -> 0 + 3/2 - -> 2 + VMC: S=0.39 VMC: S=0.27 MFT: S=0.42(4) MFT: S=0.16(2) 10-12 -100 0 100 200 300 p m [MeV/c] Lapikás, Wesseling, & Wiringa, Phys.Rev.Lett. 82, 4404 (1999)
Cluster overlaps & spectroscopic factors Two-cluster overlap A ab (J a, J b, J, r ab ) = AΨ a (J a )Ψ b (J b ), r ab Ψ(J) = LM L,SM S LM L, SM S JM J J a M a, J b M b SM S R L (r ab )Y LML (ˆr ab ) Radial functions R L (r ab ) are evaluated in VMC calculation Momentum distribution N ab (k) = Ãab(J a, J b, J, k) 2 computed from Fourier transform of A ab Spectroscopic factor S ab (J a, J b, J) = = A 2 ab (J a, J b, J, r) d 3 r 1 (2π) 3 N ab (k) d 3 k
< 6 He(J + ) + p(p j ) 7 Li(3/2 - ) > 10 3 10 2 10 1 AV18/UIX (VMC) Spectroscopic Factor CK VMC Expt* 7 6 Li(p) - He(p) 1.00 1.00 6 + He(0 )+p(p3/2) 0.59 0.39 0.42(4) 6 + He(2 )+p(p3/2) 0.22 0.14 6 + 0.16(2) He(2 )+p(p1/2) 0.18 0.13 6 He + p total 0.99 0.66 0.58(5) N(k) (fm -1 ) 10-1 10-2 10-3 10-4 0 1 2 3 4 5 * Lapikas et al., PRL 82, 4404 (1999) k (fm -1 )
Sum rules for p-shell spectroscopic factors Fixed Center For single-nucleon pickup or knockout reactions J,j while in stripping reactions, J,j 2J A + 1 S(J A 1, j, J A ) N p 2J A 1 + 1 S(J A 1, j, J A ) N h where N p (N h) is the number of p-shell particles (holes) Translationally Invariant For Harmonic Oscillator wave functions S T I = A A 1 S F C assuming same HO parameter in s- and p-shells; similar modifcation to sum rules (known since 1970s)
Tests for 6 He + p 7 Li HO wave functions Ψ s (r i ) = exp[ α s (r i R A ) 2 ] Ψ p (r i ) = (r i R A )exp[ α p (r i R A ) 2 ] choose α s to give correct 4 He rms radius r p = r n = 1.47 fm A Z α s α p r p /r n S T I p 3/2 + p 1/2 7 Li(3/2-) 0.26 0.26 1.76/1.85 6 He(0+) 0.26 0.26 1.55/1.83 0.64(1) 6 He(2+) 0.26 0.26 1.55/1.83 0.26(1)+0.26(1) Total 1.16(1) 7 Li(3/2-) 0.26 0.11 2.22/2.47 6 He(0+) 0.26 0.11 1.60/2.44 0.59(1) 6 He(2+) 0.26 0.11 1.60/2.44 0.22(1)+0.22(1) Total 1.04(1) Woods-Saxon wave functions Single WS cannot produce both A=4,7 radii; TI/FC = 1.09 Double WS can produce both A=4,7 radii; TI/FC = 0.99
Fully correlated Ψ V and different Hamiltonians A Z H E(VMC) r p /r n S T I p 3/2 + p 1/2 7 Li(3/2-) AV1-101.5(2) 1.47/1.56 6 He(0+) -67.5(1) 1.31/1.64 0.62(1) 6 He(2+) -67.5(1) 1.31/1.64 0.24(1)+0.24(1) Total 1.10(2) 7 Li(3/2-) AV4-42.5(1) 2.32/2.48 6 He(0+) -31.0(1) 1.83/2.72 0.44(2) 6 He(2+) -29.0(1) 1.86/2.91 0.19(1)+0.19(1) Total 0.82(4) 7 Li(3/2-) AV18/UIX -33.1(1) 2.32/2.47 6 He(0+) -24.2(1) 1.96/2.95 0.39(1) 6 He(2+) -22.3(1) 1.98/3.19 0.14(1)+0.13(1) Total 0.66(2)
< 6 Li(J + ) + n(p j ) 7 Li(3/2 - ) > 10 3 10 2 AV18/UIX (VMC) Spectroscopic Factor CK VMC 2[ 7 Li(n) - 6 Li(n)] 2.00 2.00 6 Li(1 + ;0)+n(pj ) 0.72 0.68 6 Li(3 + ;0)+n(pj ) 0.55 0.41 N(k) (fm -1 ) 10 1 10-1 6 + Li(0 ;1)+n(pj ) 0.30 0.20 6 + Li(2 ;0)+n(pj ) 0.16 0.13 6 + Li(2 ;1)+n(pj ) 0.20 0.14 6 Li + n total 1.93 1.56 10-2 10-3 10-4 0 1 2 3 4 5 k (fm -1 )
Results Single-nucleon spectroscopic factors for A = 5 10 for AV18/UIX compared to N p and N h J + j J Pickup (J =gs) Stripping (J=gs) 4 He+n 5 He 6.07 / 6 = 101% 5 He+n 6 He 1.90 / 2 = 95% 5 He+p 6 Li 0.83 / 1 = 83% 6 He+n 7 He 4.25 / 4 = 106% 6 He+p 7 Li 0.71 / 1 = 71% 4.79 / 6 = 80% 6 Li +n 7 Li 1.57 / 2 = 79% 4.59 / 5 = 92% 7 He+n 8 He 3.83 / 4 = 96% 7 He+p 8 Li 0.86 / 1 = 86% 7 Li +n 8 Li 2.72 / 3 = 91% 3.66 / 4 = 91% 7 Li +p 8 Be 4.29 / 5 = 86% 8 He+n 9 He 2.04 / 2 = 102% 8 He+p 9 Li 0.73 / 1 = 73% 4.11 / 5.6 = 74% 8 Li +n 9 Li 3.46 / 4 = 87% 2.94 / 3 = 98% 8 Li +p 9 Be 1.53 / 2 = 77% 3.62 / 4.2 = 87% 8 Be+n 9 Be 2.50 / 3 = 83% 9 Be+p 10 B 2.07 / 3 = 69% 2.73 / 3.6 = 76% Stable A He+n unstable A+1 He no quenching Stable A He unstable A 1 He+n almost no quenching Pickup A Z much less bound A 1 Z-1+p lots of quenching Pickup A Z somewhat less bound A 1 Z+n moderate quenching
Cluster-cluster overlaps A αd A αt 0.6 0.4 0.2 0-0.2 0.3 0.2 0.1 0-0.1-0.2 6 + Li(1 ) L=0 6 + Li(1 ) L=2 (x10) 6 + Li(3 ) L=2 6 + Li(3 ) L=4 (x10) 7 - Li(3/2 ) 7 - Li(7/2 ) A αα 0.4 0.3 0.2 0.1 0-0.1-0.2 8 Be(0 + ) 8 Be(2 + ) 8 Be(4 + ) -0.3 0 2 4 6 8 r (fm)
Future prospects Paper on A = 5 10 VMC survey in preparation; will include CK factors for rare isotopes from Kurath. Incorporate A ab (J, j, J, r) overlaps into ptolemy code for direct reactions (Pieper). Utilize A ab (J, j, J, r) overlaps in coupled-channels reaction codes (Nunes). Explore relation between R-matrix reduced widths and spectroscopic factors (Schiffer & Nollett). Astrophysically interesting transitions such as 3 He(α, γ) 7 Li, 7 Be(ε) 7 Li, being studied (Marcucci, Nollett, Schiavilla,...).