Nuclear quantum Monte Carlo

Transcription

1 Nuclear quantum Monte Carlo Robert B. Wiringa, Physics Division, Argonne National Laboratory Ivan Brida, Los Alamos Joseph Carlson, Los Alamos Kenneth M. Nollett, Ohio Saori Pastore, South Carolina Steven C. Pieper, Argonne Rocco Schiavilla, JLab & ODU WORK NOT POSSIBLE WITHOUT EXTENSIVE COMPUTER RESOURCES Argonne Laboratory Computing Resource Center (Jazz & Fusion) Argonne Math. & Comp. Science Division (BlueGene/L & SiCortex) Argonne Leadership Computing Facility (Intrepid & Mira) Physics Division Work supported by U.S. Department of Energy, Office of Nuclear Physics

2 GOALS Ab Initio CALCULATIONS OF LIGHT NUCLEI Understand nuclei at the level of elementary interactions between individual nucleons, including Binding energies, excitation spectra, relative stability Densities, electromagnetic moments, transition amplitudes, cluster-cluster overlaps Low-energy NA &AA scattering, asymptotic normalizations, astrophysical reactions REQUIREMENTS Two-nucleon potentials that accurately describe elastic NN scattering data Consistent three-nucleon potentials and electroweak current operators Accurate methods for solving the many-nucleon Schrödinger equation RESULTS Quantum Monte Carlo methods can evaluate realistic Hamiltonians accurate to 1 % About 100 states calculated for A 1 nuclei in good agreement with experiment Applications to elastic & ineleastic e, π scattering, (e, e p), (d,p) reactions, etc. Electromagnetic moments, M1, E, F, GT transitions calculated 5 He = nα scattering and 3 A 9 ANCs and widths

3 NUCLEAR HAMILTONIAN H = X i K i + X i<j v ij + X i<j<k V ijk K i = 4 [( 1 m p + 1 m n ) + ( 1 m p 1 m n )τ iz ] i Argonne v 18 Wiringa, Stoks, & Schiavilla, PRC 51, 38 (1995) v ij = v γ ij + vπ ij + v I ij + v S ij = P v p (r ij )O p ij v γ ij : pp, pn & nn electromagnetic terms v π ij [Y π (r ij )σ i σ j + T π (r ij )S ij ] τ i τ j v I ij = P p Ip T π(r ij )O p ij v S ij = P p [P p + Q p r + R p r ]W(r)O p ij O p ij = [1, σ i σ j, S ij,l S,L,L (σ i σ j ), (L S) ] + [1, σ i σ j, S ij,l S,L,L (σ i σ j ), (L S) ] τ i τ j + [1, σ i σ j, S ij,l S] T ij + [1, σ i σ j, S ij,l S] (τ i + τ j ) z S ij = 3σ i ˆr ij σ j ˆr ij σ i σ j T ij = 3τ iz τ jz τ i τ j

4 60 40 Argonne v 18 np Argonne v 18 pp Argonne v 18 nn S1 Argonne v 18 np SAID 7/06 np δ (deg) 0 0 SAID 7/06 np δ (deg) S E lab (MeV) E lab (MeV) δ (deg) Argonne v 18 np Argonne v 18 pp Argonne v 18 nn SAID 7/06 np 3 P E lab (MeV) δ (deg) P1 Argonne v 18 np SAID 7/06 np E lab (MeV) Argonne v 18 fits Nijmegen PWA93 data base of 1787 pp & 514 np observables for E lab 350 MeV with χ /datum = 1.1 plus nn scattering length & H binding energy

5 Argonne v 18 V (MeV) c τ σ στ t tτ ls lsτ r (fm) Uses 4 I p, P p, Q p, R p parameters [ constrained so that v t (r=0) = 0 & v p t r r=0 = 0 ] plus f πnn coupling strength & one cutoff parameter in Y π (r), T π (r).

6 THREE-NUCLEON POTENTIALS Urbana V ijk = V πp ijk + V R ijk Carlson, Pandharipande, & Wiringa, NP A401, 59 (1983) Illinois V ijk = V πp ijk + V πs ijk + V 3π R ijk + V R ijk Pieper, Pandharipande, Wiringa, & Carlson, PRC 64, (001) Illinois-7 has 4 strength parameters fit to 3 energy levels in A 10 nuclei. In light nuclei we find (thanks to large cancellation between K & v ij ): V ijk (0.0 to 0.07) v ij (0.15 to 0.5) H We expect V ijkl 0.05 V ijk (0.01 to 0.03) H 1 MeV in 1 C.

7 Minimize expectation value of H VARIATIONAL MONTE CARLO E V = Ψ V H Ψ V Ψ V Ψ V E 0 using MeT R opolis Monte Carlo and trial function 3 Ψ V = 4S Y ( U ij + X " # Y U ijk ) 5 f c (r ij ) Φ A (JMTT 3 ) i<j k i,j i<j single-particle Φ A (JMTT 3 ) is fully antisymmetric and translationally invariant central pair correlations f c (r) keep nucleons at favorable pair separation pair correlation operators U ij = P p u p(r ij )O p ij reflect influence of v ij triple correlation operator U ijk added when V ijk is present multiple J π states constructed and diagonalized for p-shell nuclei ability to construct clusterized or asymptotically correct trial functions Ψ V are spin-isospin vectors in 3A dimensions with A `A Z components Lomnitz-Adler, Pandharipande, & Smith, NP A361, 399 (1981) Wiringa, PRC 43, 1585 (1991)

8 SCALING OF VMC CALCULATION TIME WITH NUCLEUS Scales with # particles (6A w.f. calculations for kinetic energy) # pairs (operations to construct w.f.) spin isospin (size of w.f. vector): A Pairs Spin Isospin Q (/ 8 Be) 4 He He Li Li Be Be Be B C C , O , Ca n n

9 GREEN S FUNCTION MONTE CARLO Projects out lowest energy state from variational trial function Ψ(τ) = exp[ (H E 0 )τ]ψ V = X n exp[ (E n E 0 )τ]a n ψ n Ψ(τ ) = a 0 ψ 0 Evaluation of Ψ(τ) done stochastically in small time steps τ Z Ψ(R n, τ) = G(R n,r n 1 ) G(R 1,R 0 )Ψ V (R 0 )dr n 1 dr 0 Mixed estimates used for expectation values O(τ) = Ψ(τ) O Ψ(τ) Ψ(τ) Ψ(τ) O(τ) Mixed = Ψ V O Ψ(τ) Ψ V Ψ(τ) O(τ) Mixed + [ O(τ) Mixed O V ] ; H(τ) Mixed = Ψ(τ/) H Ψ(τ/) Ψ(τ/) Ψ(τ/) E 0 Cannot propagate p, L, or (L S) operators use H = AV8 + Ṽijk Fermion sign problem would limit maximum τ, but... Constrained-path propagation removes steps that have Ψ (τ,r)ψ V (R) = 0 Multiple excited states of same J π stay orthogonal Pudliner, Pandharipande, Carlson, Pieper, & Wiringa, PRC 56, 170 (1997) Wiringa, Pieper, Carlson, & Pandharipande, PRC 6, (000)

10 EXAMPLES OF GFMC PROPAGATION E (MeV) He E(τ) (MeV) Li = g.s α+d τ (MeV 1 ) τ (MeV -1 ) Curve has P i a iexp( E i τ) with E i = 1480, 340 & 0. MeV (0. MeV is first 4 He excitation) Ψ V has small amounts of 1.5 GeV contamination g.s. ( ) & stable after τ = 0. MeV 1 (a broad resonance) never stable decaying to separated α & d E(τ=0.) is best GFMC estimate of resonance energy

11 GFMC FOR SECOND EXCITED STATES OF SAME J π The Ψ V are constructed by non-orthogonal basis diagonalization in p-shell wave functions. Example: 7 Li(5/-) has 4 symmetry possibilities: F[43], 4 P[41], 4 D[41], D[41] Ψ V ( nd 5 ) Ψ V (1 st 5 ) = 0, but Ψ GFMC ( nd 5 ) Ψ V (1 st 5 ) need not be zero. Will e (H Ẽ0)τ Ψ V ( nd 5 ) Ψ GFMC (1 st 5 )? Can use Ψ GFMC (i) H Ψ GFMC (j) and Ψ GFMC (i) Ψ GFMC (j) to rediagonalize Li(5/ - ) AV18 + Illinois- GFMC Propagation 4 th 5/ - 3 rd 5/ - nd 5/ - 1 st 5/ E(τ) (MeV) -5 Ψ i Ψ Ψ Ψ 1 Ψ 3 Ψ 1 Ψ 4 Ψ τ (MeV -1 ) τ (MeV -1 ) Pieper, Wiringa, & Carlson, Phys. Rev. C 70, (004)

12 Energy (MeV) He 6 He 6 Li 7 Li 7/ 5/ 5/ 7/ 1/ 3/ 8 He 8 Li Argonne v 18 with Illinois-7 GFMC Calculations Be IL7: 4 parameters fit to 3 states 600 kev rms error, 51 states ~60 isobaric analogs also computed Li 5/ 1/ 3/ AV18 9 Be 7/ 5/ 7/ 7/ 3/ 3/ 5/ 1/ 5/ 1/ 3/ AV18 +IL7 Expt. 10 Be 3, B C

13 Excitation energy (MeV) Argonne v 18 with Illinois-7 GFMC Calculations Including IL7 gives correct s.-o. splitting & 10 B g.s. 6 He 6 Li 7 Li 7/ 5/ 5/ 7/ 1/ 3/ 8 Li Be Be 7/ 5/ 7/ 7/ 3/ 3/ 5/ 1/ 5/ 1/ 3/ AV18 10 Be AV18 +IL7 Expt. 3, B 4 +

14 1 st AND nd (HOYLE) STATES IN 1 C Constructing the Jastrow part of the trial wave function is major effort: There are 5 LS-basis J= states in 1 C in the 0P shell: 1 S[444], 3 P[4431], 1 S[44], 5 D[44], 3 P[433] All can be constructed by projections from a closed (p3/) 8 shell (Carlson) Dominant 3-α symmetry is easily constructed with one α in the 0S shell and two αs in the 0P shell (Pandharipande) Additional components generated by promoting one whole α to the 1S-0D shell, and also promoting pairs, e.g., 0P 0D and 0P 1S Total of 11 Jastrow components (some with considerable overlap) to be diagonalized Challenge in GFMC propagation is keeping the nd state orthogonal to the ground state UNEDF SciDAC grant to develop general-purpose load-balancing library (ADLB) to run under MPI on 3,768 nodes with OpenMP for 4 cores/node INCITE grant of Argonne s IBM BlueGene/P time used for calculations

15 1 st AND nd (HOYLE) STATES IN 1 C PRELIMINARY Convergence as a function of imaginary time (τ) 1 C states AV18+IL7 H 16 Jun C states AV18+IL7 RMS r(p) 16 Jun E(τ) (MeV) g.s.; 6-state corr 18 g.s.; O: 4+5 state nd ; corr 4+ G nd ; corr 6+5 P 3 4 He r p 1/ (fm) g.s.; 6-state corr 18 g.s.; O: 4+5 state nd ; corr 4+ G nd ; corr 6+5 P C τ (MeV -1 ) τ (MeV -1 ) g.s. energy nd E VMC GFMC Expt. VMC GFMC Expt. AV () 73.(5) 10.0(3) 7.9(6) AV18+IL7 65.7() 93.3(4) () 10.4(5) 7.65

16 1 st AND nd (HOYLE) STATES IN 1 C PRELIMINARY 1 C(1 st & nd ) AV18+IL7 - ρ p - 16 Jun C(1 st & nd ) AV18+IL7 - ρ p - 16 Jun g.s., Corr O, Ψ V g.s., Corr O, GFMC 10-1 ρ p (fm -3 ) nd, P, Ψ V nd, P, GFMC Experiment ρ p r (fm -1 ) g.s., Corr O, Ψ V g.s., Corr O, GFMC C 10-5 nd, P, Ψ V nd, P, GFMC Experiment r (fm) r (fm) Central density dip for ground state may be interpreted as three α s in a triangle Central density peak for Hoyle state may be evidence for a linear configuration of three α s Lusk, Pieper, & Butler, SciDAC Review Spring 010

17 NOLEN-SCHIFFER ANOMALY Nuclear forces are mostly charge-independent [CI 1,τ i τ j ], but have small charge-dependent [CD T ij ] and charge-symmetry-breaking [CSB (τ i + τ j ) z ] components, while electromagnetic forces are a mix of CI, CD, & CSB terms. Evidence for strong charge-independence-breaking (CIB) comes from the energy differences of isobaric multiplets: E A,T (T z ) = P n T a n(a, T)Q n (T, T z ) For example, Q 0 = 1 ; Q 1 = T z ; Q = 1 (3T z T ) a 1 (3, 1 ) = E(3 He) E( 3 H) a (6,1) = 1 3 [E(6 Be) E( 6 Li ) + E( 6 He)] The Nolen-Schiffer anomaly is the difference not explained by Coulomb force; strong CIB and other electromagnetic terms in Argonne v 18 explain the remainder (shown in kev): a n (A, T) K CSB v C1 (pp) v γ,r v CSB +v CD Total Expt. a 1 (3, 1 ) (0) 7 65(0) 753(0) 764 a 1 (7, 1 ) (3) 34 85(1) 1599(4) 1645 a (6,1) 166(0) (3) 89(4) 4 a (8,1) 137(1) 4 10(8) 13(8) 145 Wiringa, AIP CP 118, 1 (009)

18 Isospin-mixing in 8 Be Experimental energies of states E a = 16.66(3) MeV Γ α a = 108.1(5) kev E b = 16.9(3) MeV Γ α b = 74.0(4) kev D[431] 3 P[431] ;1 ;1 3 P[431] 7 Li+n ;0+1 ;0 ;1 ;0+1 ;1 7 + Be+p ;1 8 B Isospin mixing of ;1 and ;0* states due to isovector interaction H 01 : ;1 8 Li 1 G[44] 4 + ;0 Ψ a = βψ γψ 1 ; Ψ b = γψ 0 βψ 1 decay through T = 0 component only Γ α a/γ α b = β /γ β = 0.77 ; γ = 0.64 Energy (MeV) -48 E a,b = H 0 H 11 s H00 «H 11 ± + (H 01) H 00 = () MeV H 11 = 16.80() MeV H 01 = 145(3) kev D[44] 1 S[44] ;0 8 Be ;0 α+α

19 GFMC mixed estimates for off-diagonal matrix elements Ψ f (τ) O Ψ i (τ) p Ψf (τ) Ψ f (τ) p Ψ i (τ) Ψ i (τ) O(τ) M i + O(τ) Mf O V, where O V = Ψ f V O Ψi V q Ψ fv ΨfV p Ψ iv ΨiV, O(τ) Mi = Ψf V O Ψi (τ) Ψ i V Ψi (τ) O(τ) Mf = Ψf (τ) O Ψ i V Ψ f (τ) Ψ f V s Ψ i V Ψi V Ψ f V Ψf V, s Ψ f V Ψf V Ψ i V Ψi V,

20 Isospin-mixing matrix elements in kev H 01 K CSB V CSB V γ (Coul) (MM) ;1 ;0 GFMC 115(3) 3.1() 1.3(6) 90.3(6) 78.3(5) 1.0() Barker 145(3) 67 ;1 ;0 GFMC 10(4).9() 18.(6) 80.3(30) 79.5(30) 0.8() Barker 10(1) 54 ;1 ;0 GFMC 90(3).5() 14.8(6) 73.1(1) 60.9(1) 1.() Barker 6(15) 3 ;1 1 ;0 GFMC 6() 0.4() 1.3(4) 4.4(1) Barker, Nucl.Phys. 83, 418 (1966) Coulomb terms are about half of H 01, but magnetic moment and strong Type III CSB are relatively more important than in Nolen-Schiffer anomaly; still missing 0% of strength. Strong Type IV CSB will also contribute (probably best nuclear structure place to look): V CSB IV = (τ 1 τ ) z (σ 1 σ ) L v(r) + (τ 1 τ ) z (σ 1 σ ) L w(r) Preliminary result: VIV CSB few kev.

21 SINGLE-NUCLEON DENSITIES ρ p,n (r) = X i Ψ δ(r r i ) 1 ± τ i Ψ He 6 He - Proton He 6 He - Proton 6 He - Neutron 6 He - Neutron Density (fm -3 ) He - Proton 8 He - Neutron Density r (fm -1 ) He - Proton 8 He - Neutron r (fm) r (fm) RMS radii r n r p r c Expt 4 He 1.45(1) 1.45(1) 1.67(1) 1.681(4)* 6 He.86(6) 1.9(4).06(4).060(8) 8 He.79(3) 1.8() 1.94() 1.959(16) *Sick, PRC 77, 04130(R) (008) Wang, et al., PRL 93, (004) Mueller, et al., PRL 99, 5501 (007) Brodeur, et al., PRL 108, (01)

22 TWO-NUCLEON DENSITIES ρ pp (r) = X i<j Ψ δ(r r i r j ) τ i τ j Ψ He 6 He - Proton 8 He - Proton RMS radii r pp r np r nn 4 He ρ pp (fm -3 ) He He r 1 (fm)

23 INTRINSIC DENSITY OF 8 BE 8 Be w.f.: 4 He core + 4 p-shell nucleons + pair corr. M. C. ρ(r): random walk in Ψ(r 1, r,, r 8 ) & periodically for each set (r 1, r,, r 8 ) Lab ρ(r): bin r 1, r,, r 8 Intrinsic ρ(r): find eigenvectors of moment of inertia matrix: 0 1 x i x i y i x i z i M = X i y i x i y i y i z i C A, rotate to them, and bin r 1, r,, r 8. 6 z i x i z i y i z i Be( ) Be(4 + ) z (fm) 0 z (fm) Laboratory Intrinsic r = (x y ) 1/ (fm) Laboratory Intrinsic r = (x y ) 1/ (fm) Wiringa, Pieper, Carlson, & Pandharipande, Phys. Rev. C 6, (000)

24 TWO-NUCLEON HALO DENSITIES ρ nn (r) = X i<j Ψ(J π, T, T z = +1) δ(r r i r j )τ + i τ + j Ψ(J π, T, T z = 1) He( ) 8 10 Be( ) 8 10 Be( ) r core-nn (fm) 6 4 r core-nn (fm) He( ) Be( 10 Be( 1 ) ) r core-nn (fm) 6 4 r core-nn (fm) r nn (fm) r nn (fm) r nn (fm)

25 NUCLEON MOMENTUM DISTRIBUTIONS Probability of finding a nucleon in a nucleus with momentum k in a given spin-isospin state: Z ρ στ (k) = dr 1dr 1 dr dr A ψ A (r 1,r,...,r A ) e ik (r 1 r 1 ) P στ ψ A (r 1,r,...,r A ) Useful input for electron scattering studies Universal character of high-momentum tails from np tensor interaction ρ(k) (fm 3 ) 10 B C 8 Be Li He H k (fm -1 ) VMC tabulations for A = 1 available at:

26 M1, E, F, GT transitions NO EFFECTIVE CHARGES! E = e P k 1 ˆr k Y (ˆr k ) ( τ kz ) M1 = µ N Pk [(L k + g p S k )( τ kz )/ + g n S k (1 τ kz )/] 6 He( ;0 ;0) B(E) 6 Li( ;0 ;0) B(E) 6 Li( ;1 ;0) B(M1) 7 Li(1/ - 3/ - ) B(E) 7 Li(1/ - 3/ - ) B(M1) F = P k τ k± ; GT = P k σ kτ k± 7 Li(7/ - 3/ - ) B(E) Pervin, Pieper & Wiringa, PRC 76, (007) Marcucci, Pervin, et al., PRC 78, (008) 7 Be(1/ - 3/ - ) B(M1) 6 He( ) 6 Li( ) B(GT) Cohen-Kurath NCSM GFMC(IA) GFMC(MEC) 7 Be(3/ - ) 7 Li(3/ - ) B(GT) 7 Be(3/ - ) 7 Li(1/ - ) B(GT) 7 Be(3/ - ) Branching ratio Experiment Ratio to experiment

27 A 10 MAGNETIC MOMENTS W/ χeft EXCHANGE CURRENTS Hybrid calculations using AV18+IL7 wave functions and χeft exchange currents developed in: Pastore, Schiavilla, & Goity, PRC 78, (008) ; Pastore, et al., PRC 80, (009) 4 3 p 3 H 7 Li 9 Li 9 B µ (µ N ) H GFMC(IA) GFMC(FULL) EXPT n 3 He 6 Li 6 Li* 7 Be 8 Li 8 B 9 Be 9 C 10 B 10 B* -3

28 M1 TRANSITIONS W/ χeft 9 Be( 5 / - 3 / - ) B(M1) dominant contribution is from OPE five LECs at N3LO d V and d V 1 are fixed assuming resonance saturation d S and c S are fit to experimental µ d and µ S ( 3 H/ 3 He) c V is fit to experimental µ V ( 3 H/ 3 He) Λ = 600 MeV 8 B( ) B(M1) 8 Li( ) B(M1) 8 B( ) B(M1) 8 Li( ) B(M1) 7 Be( 1 / - 3 / - ) B(M1) 7 Li( 1 / - 3 / - ) B(M1) 6 Li( ) B(M1) Pastore, Pieper, Schiavilla & Wiringa PRC 87, (013) EXPT GFMC(IA) GFMC(MEC) Ratio to experiment

29 Energy (MeV) D[431] 3 P[431] TRANSITIONS IN/TO ;1 ; ;1 ; Be+p P[431] ;0 ;1 7 Li+n ;1 ;0+1 8 B ;1 8 Li 1 G[44] 4 + ;0 8 Be 8 Be presents new challenges in transition calculations E transitions between rotational band states which have large widths M 1 transitions involving isospin-mixed states GT transitions that are not super-allowed and go to a broad final state -5 1 D[44] ;0 J π ; T GFMC Expt -56.3() () (1) S[44] ;0 8 Be α+α () (15) ;16.8(3) (3) ;16.8(3) (3) 16.9 ;17.4() ;18.0(3)

30 E TRANSITIONS IN 8 Be New experiment at Tata Institute, Mumbai for 4 + transition Experimental AND theoretical challenge: 4 + and states are wide and breakup into two αs GFMC calculation is extrapolated back to τ = 0.1 MeV 1 ; predicts B(E) = 7.(15) Experiment detects α+α+γ in coincidence for range of beam energies Assuming Breit-Wigner shape, simple analysis gives B(E) = 1.3(3) r p (fm) E (MeV) E (e fm ) (a) (b) (4 + ) (c) ( ) τ (MeV -1 ) Datar, Chakrabarty, Kumar, Nanal, Pastore, Wiringa, et al., arxiv:

31 M1 TRANSITIONS IN 8 Be BETWEEN ISOSPIN-MIXED STATES We calculate between states of pure isospin: matrix element IA MEC TOT ; 1 M1 ;0.9(1) 0.6(1).91(1) ; 1 M1 ;1 0.14(0) 0.04(1) 0.18(1) ; 0 M1 ;0 0.17(0) 0.0(0) 0.19(0) ; 0 M1 ;1.60(1) 0.9(1).89(1) Then have to combine them using the physical states: to get the final results: = 0.77 ; ; = 0.4 ; ; = 0.64 ; ; = 0.97 ; ; 1 B(M 1) IA TOT Expt ().54(3).65(5) (1) 0.46(1) 0.30(7) (1) 0.6(1) 1.88(46) ().01().89(33) We evaluate the isospin-mixing matrix elements H 01 to make sure we have the correct relative signs of our wave functions.

32 APPLICATIONS TO LIGHT-ION REACTIONS The availability of radioactive-ion beams has renewed interest in reactions like (d,p) in inverse kinematics 10 (a) H( 7 Li, 3 He) 6 He (0 + ) We have helped analyze a number of RIB experiments 1 such as d( 8 Li,p) 9 Li (ATLAS) & d( 9 Li,t) 8 Li (TRIUMF) (b) H( 7 Li,t) 6 Li (1 + ) PTOLEMY DWBA calculations for transfer 10 (d,p) vertex from AV18 (d,t), ( 8 Li, 9 Li), etc. vertices computed as A-body overlaps using VMC Ψ V (A-1) a Ψ V (A) Norm is spectroscopic factor dσ/dω (mb/sr) 1 5 (c) (d) H( Li,t) Li (0 ) H( Li, He) He (3/ ) Absolute prediction for dσ/dω Good predictions of n-knockout from 10 Be and 10 C (NSCL) 1 (e) H( 8 Li, 3 He) 7 He (5/ - ) Macfarlane & Pieper, PTOLEMY, ANL-76-11, Rev. 1 (1978) Wuosmaa et al., PRL 94, 0850 (005) +... Kanungo et al., PLB 660, 6 (008) Grinyer et al., PRL 106, 1650 (011) θ CM (deg)

33 ONE-NUCLEON OVERLAPS IN VMC/GFMC For antisymmetric and translationally invariant parent Ψ A (α) and daughter Ψ A 1 (γ) wave functions, with α [J π A, T A, T za ], γ [J π A 1,T A 1, T za 1 ], and single-nucleon quantum numbers ν [l,s, j, t,t z ], the translationally invariant overlap function is: R(α,γ, ν; r) = fl A fiˆψa 1(γ) Y(ν)(ˆr δ(r r ) J A,T A ) r ΨA(α) where Y(ν)(ˆr ) = [Y l (ˆr ) χ s ] j χ t and Ψ A 1 (γ) = 1, Ψ A (α) = 1. The corresponding spectroscopic factor is the norm of the overlap: Z S(α, γ,ν) = R(α, γ, ν; r) r dr Overlap functions R satisfy a one-body Schrödinger equation with appropriate source terms. Asymptotically, at r, these source terms contain core-valence Coulomb interaction at most, and hence for parent states below core-valence separation thresholds: R(α, γ,ν; r) r C(α, γ,ν) W η,l+1/(kr), r where W η,l+1/ (kr) is a Whitakker function with k = µb/, B is the separation energy, and C(α,γ, ν) is the asymptotic normalization coefficient or ANC.

34 GFMC evaluation of R is by extrapolation requiring two mixed estimates minus the VMC result: R(α, γ, ν; r; τ) R(α, γ, ν; r; τ) MA + R(α,γ, ν; r; τ) MA 1 R(α, γ, ν; r) V, where M A denotes a mixed estimate where parent Ψ A (α; τ) has been propagated in GFMC and M A 1 is a mixed estimate where daughter Ψ A 1 (γ; τ) has been propagated. 0. R [fm -3/ ] VMC mixed 6 He mixed 7 Li GFMC fit R [fm -3/ ] VMC mixed 6 He mixed 7 Li GFMC fit W/r r [fm] r [fm] Imaginary time evolution of overlaps in the p 3/ channel of the overlap 6 He + p 7 Li Brida, Pieper, & Wiringa, PRC 84, (011)

35 GFMC FOR SCATTERING STATES GFMC treats nuclei as particle-stable system should be good for energies of narrow resonances Need better treatment for locations and widths of wide states and for capture reactions METHOD Pick a logarithmic derivative, χ, at some large boundary radius (R B 9 fm) GFMC propagation, using method of images to preserve χ at R, finds E(R B, χ) Phase shift, δ(e), is function of R B, χ, E Repeat for a number of χ until δ(e) is mapped out need E accurate to 1/3% Example for 5 He( 1 ) -3 "bound state" Log-deriv = fm -1 Bound-state boundary condition does not give stable energy; Decaying to n+ 4 He threshold Scattering boundary condition produces stable energy. E(τ) (MeV) τ (MeV -1 )

36 5 He AS n+ 4 He SCATTERING Black curves: Hale phase shifts from R-matrix analysis up to J = 9 of data AV18 with no V ijk underbinds 5 He(3/ ) & overbinds 5 He(1/ ) AV18+UIX improves 5 He(1/ ) but still too small spin-orbit splitting AV18+IL reproduces locations and widths of both P -wave resonances δ LJ (degrees) AV18 AV18+UIX AV18+IL R-Matrix σ LJ (b) AV18+IL Experiment Pole location E c.m. (MeV) E c.m. (MeV) Nollett, Pieper, Wiringa, Carlson, & Hale, PRL 99, 050 (007)

37 CONCLUSIONS We have demonstrated that realistic nuclear Hamiltonians and accurate QMC calculations can reproduce many properties of light nuclei: Argonne v ij + Illinois V ijk gives rms binding-energy errors < 0.6 MeV for A = 3 1 Successfully predict/reproduce densities, radii, moments, & transition matrix elements Can obtain energies and widths of low-energy nucleon-nucleus scattering states There are many more exciting challenges in the structure and reactions of A 1 nuclei, which we want to tackle in the next few years, such as: 1 C excited states and transitions; ν- 1 C scattering Single- & double-intruder states in 9,10,11 Be, 10,11 B; 11 Li More electroweak transitions in A 1 Charge-independence breaking in 10 C(β + ) 10 B Parity-violating n-α scattering: 5 He( 1 ) HP V 5 He( ) Cluster-cluster overlaps, SFs, ANCs, for (A-)d A, (A-4)α A Astrophysical reactions such as 3 He(α,γ) 7 Be For larger nuclei A > 1 some possibilities are: exascale computing for 16 O ( 1000 more expensive than 1 C) cluster GFMC (cluster VMC for 16 O done in 1990s) AFDMC (auxiliary field diffusion Monte Carlo) or hybrid GFMC-AFDMC Gandolfi, Pederiva, Fantoni, & Schmidt ( 79), PRL 99, 0507 (007)

38 Energy (MeV) He 6 He 6 Li 7 Li 7/ 5/ 5/ 7/ 1/ 3/ 8 He Argonne v 18 with Illinois-7 GFMC Calculations 8 Li Be AV18 +IL7 9 Li 5/ 1/ 3/ 9 Be Expt. 7/ 5/ 7/ 7/ 3/ 3/ 5/ 1/ 5/ 1/ 3/ 3, Be + 10 B 1 C