Transitions, Overlaps. Spectroscopic Factors

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Καλλίστρατος Νικολαΐδης
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1 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)

2 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

3 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

4 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 ) 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 ) 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)

5 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.

6 -20 Energy (MeV) He 1/2 α+n 3/2 α+2n 5 He He Li α+d 5/2 1/2 3/2 6 He+n 7 He Argonne v 18 + UIX VMC Trial & GFMC Calculations 15 July 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 Li 7 Li+n 8 Li+n 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 Be 3/2

7 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 IA IA+MEC Amsterdam Saskatoon Stanford 10 7 C0, C2 M q (fm 1 ) 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 IA IA+MEC Saskatoon Mainz 10 5 C2, C4 M q (fm 1 ) q (fm 1 ) Wiringa & Schiavilla, Phys.Rev.Lett. 81, 4317 (1998)

8 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 n=p 6 Li ( ) 7 Li ( 3 / 2 1 / 2 ) ρ E2 t 3 (fm 1 ) n p 7 Li ( 3 / 2 7 / 2 ) 7 Li ( 3 / 2 5 / 2 ) r (fm) B(E2 ) Experiment VMC p p n 6 Li( ) 21.8± ± ±0.4 7 Li( ) 7.59± ± ±0.3 7 Li( ) 15.5± ± ±0.5 7 Li( ) 4.1± ± ±0.2

9 Input to DWIA analysis of pion scattering: 6 Li(π,π) 6 Li 6 Li(π,π ) 6 Li Li(1 + ) 240 MeV π 6Li(3 + ) 240 MeV π dσ/dω (mb/sr) Li(1 + ) 180 MeV π 6 Li(3 + ) 180 MeV π Li(1 + ) 100 MeV π 6 Li(3 + ) 100 MeV π θ (deg) 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)

10 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 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 θ (deg) 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, (2001)

11 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 (x 10) ρ(p m ) [(MeV/c) -3 ] /2 - -> 0 + 3/2 - -> 2 + VMC: S=0.39 VMC: S=0.27 MFT: S=0.42(4) MFT: S=0.16(2) p m [MeV/c] Lapikás, Wesseling, & Wiringa, Phys.Rev.Lett. 82, 4404 (1999)

12 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

13 < 6 He(J + ) + p(p j ) 7 Li(3/2 - ) > AV18/UIX (VMC) Spectroscopic Factor CK VMC Expt* 7 6 Li(p) - He(p) He(0 )+p(p3/2) (4) 6 + He(2 )+p(p3/2) (2) He(2 )+p(p1/2) He + p total (5) N(k) (fm -1 ) * Lapikas et al., PRL 82, 4404 (1999) k (fm -1 )

14 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 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)

15 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-) / He(0+) / (1) 6 He(2+) / (1)+0.26(1) Total 1.16(1) 7 Li(3/2-) / He(0+) / (1) 6 He(2+) / (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

16 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-) AV (2) 1.47/ He(0+) -67.5(1) 1.31/ (1) 6 He(2+) -67.5(1) 1.31/ (1)+0.24(1) Total 1.10(2) 7 Li(3/2-) AV4-42.5(1) 2.32/ He(0+) -31.0(1) 1.83/ (2) 6 He(2+) -29.0(1) 1.86/ (1)+0.19(1) Total 0.82(4) 7 Li(3/2-) AV18/UIX -33.1(1) 2.32/ He(0+) -24.2(1) 1.96/ (1) 6 He(2+) -22.3(1) 1.98/ (1)+0.13(1) Total 0.66(2)

17 < 6 Li(J + ) + n(p j ) 7 Li(3/2 - ) > AV18/UIX (VMC) Spectroscopic Factor CK VMC 2[ 7 Li(n) - 6 Li(n)] Li(1 + ;0)+n(pj ) Li(3 + ;0)+n(pj ) N(k) (fm -1 ) Li(0 ;1)+n(pj ) Li(2 ;0)+n(pj ) Li(2 ;1)+n(pj ) Li + n total k (fm -1 )

18 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

19 Cluster-cluster overlaps A αd A αt 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 Be(0 + ) 8 Be(2 + ) 8 Be(4 + ) r (fm)

20 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,...).

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