Ab Initio CALCULATIONS OF NUCLEAR STRUCTURE (AND REACTIONS)

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1 GOALS Ab Initio CALCULATIONS OF NUCLEAR STRUCTURE (AND REACTIONS) 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, astrophysical reactions REQUIREMENTS Two-nucleon potentials that accurately describe elastic NN scattering data Consistent many-nucleon potentials and electroweak current operators Precise methods for solving the many-nucleon Schrödinger equation ISSUES TO DISCUSS Theoretical advantages and limitations of a given Hamiltonian or method What benchmarks to calculate for comparison of different methods What experiments to test and refine our models and methods

2 Recent Developments in Nuclear Quantum Monte Carlo Robert B. Wiringa Physics Division, Argonne National Laboratory WORK WITH Joe Carlson, Los Alamos Laura Marcucci, Pisa Ken Nollett, Argonne Muslema Pervin, Argonne Steve Pieper, Argonne Rocco Schiavilla, JLab & ODU WORK NOT POSSIBLE WITHOUT EXTENSIVE COMPUTER RESOURCES Argonne LCRC (Jazz), MCS Division (BlueGene/L), and ALCF (BlueGene/P) NERSC IBM SP s (Seaborg, Bassi) UNEDF SciDAC and INCITE programs Physics Division Work supported by U.S. Department of Energy, Office of Nuclear Physics

3 OUTLINE Hamiltonian Variational Monte Carlo Green s function Monte Carlo Binding energy results Nolen-Schiffer anomaly & 8 Be isospin-mixing Densities and radii Meson-exchange currents and magnetic moments M1, E, F, GT transitions NA scattering & astrophysical reactions Nucleon momentum distributions & A(e,e pn)

4 H = X i HAMILTONIAN K i + X i<j v ij + X i<j<k V ijk K i = K CI i Argonne v 18 (AV18) + K CSB i 4 [( 1 m p + 1 m n ) + ( 1 m p 1 m n )τ zi ] i v ij = v γ ij + vπ ij + v I ij + v S ij = P 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

5 Argonne v v c v τ v σ v στ v t v tτ V (MeV) r (fm) 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 and H binding energy

6 60 Argonne v 18 np Argonne v 18 pp Argonne v 18 nn Argonne v 18pq np Argonne v 18pq pp 10 3 S1 Argonne v 18 np Argonne v 18pq np SAID 7/06 np δ (deg) 0 Argonne v 18pq nn SAID 7/06 np δ (deg) S E lab (MeV) E lab (MeV) P0-10 Argonne v 18 np Argonne v 18pq np 0 SAID 7/06 np δ (deg) -10 Argonne v 18 np Argonne v 18 pp δ (deg) -0 Argonne v 18 nn -0 Argonne v 18pq np -30 Argonne v 18pq pp 1 P1-30 Argonne v 18pq nn SAID 7/06 np E lab (MeV) E lab (MeV)

7 THREE-NUCLEON POTENTIALS Urbana (UIX) V ijk = V πp ijk + V R ijk Illinois (IL,IL7) V ijk = V πp ijk + V πs ijk + V 3π R ijk + Vijk R + Vijk Rτ

8 Minimize expectation value of H VARIATIONAL MONTE CARLO E V = Ψ V H Ψ V Ψ V Ψ V E 0 Trial function (s-shell nuclei) Ψ V = 4 X Ψ J = i<j<k U T NI ijk 3 " 5 S Y i<j( U ij ) " # Y f c (r ij ) Φ A (JMTT 3 ) i<j # Ψ J Φ d (1100) = A p n ; Φ α (0000) = A p p n n U ij = X p=,6 u p (r ij )O p ij ; UTNI ijk = ǫv ijk ( r ij, r jk, r ki ) Functions f c (r ij ) and u p (r ij ) obtained from coupled differential equations with v ij.

9 Correlation functions f c φ p [f c ( 3 H)] f sp f c,u tτ ( H) f c,u tτ ( 3 H) f c,u tτ ( 4 He) f sp,u tτ ( 6 Li) φ p ( 6 Li) [f c ( 4 He)] u tτ r (fm)

10 Trial function (p-shell nuclei) 8 < Ψ J = A : Φ α (0000) 134 Y 4<l A Y i<j 4 f ss (r ij ) X Y β LS[n] LS[n] k 4<l A f sp (r kl ) Y 4<l<m A φ LS[n] p (R αl ) [Y m l 1 (Ω αl )] LML [χ l ( 1 m s)] SMS f pp (r lm ) JM [ν l( 1 t 3)] T T3 E 9 = ; 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) represented by vector with A ( A Z) spin-isospin components for each space configuration R = (r 1,r,...,r A ); Expectation values are given by summation over samples drawn from probability distribution W(R) = Ψ P (R) : Ψ V O Ψ V Ψ V Ψ V = X Ψ V (R)OΨ V (R) W(R) / X Ψ V (R)Ψ V (R) W(R) Ψ Ψ is a dot product and Ψ OΨ a sparse matrix operation.

11 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 using the short-time propagator accurate to order ( τ) 3 (V ijk term omitted for simplicity) " G αβ (R,R ) = e E0δτ G 0 (R,R ) α S Y # g ij (r ij,r ij) g 0,ij (r ij,r i<j ij ) β where the free many-body propagator is G 0 (R,R ) = R e K τ R =»r m π τ and g 0,ij and g ij are the free and exact two-body propagators g ij (r ij,r ij) = r ij e H ij τ r ij 3A» (R R ) exp τ/m

12 Mixed estimates 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 Propagator cannot contain spin or isospin-dependent p, L, or (L S) operators: G βα (R,R) has only v 8 v 18 v 8 computed perturbatively with extrapolation (small for AV18) Fermion sign problem limits maximum τ: G βα (R,R) brings in lower-energy boson solution Ψ V H Ψ(τ) projects back fermion solution. but statistical errors grow exponentially Constrained-path propagation, removes steps that have Ψ (τ,r)ψ(r) = 0 Possible systematic errors rduced by 10 0 unconstrained steps before evaluating observables.

13 GFMC propagation of three states in 6 Li Li gs E(τ) (MeV) α+d τ (MeV -1 )

14 Energy (MeV) H 1/+ 3 H 4 He AV / 7/ 6 1/ Li 3/ 7 Li 8 Li Be Li Argonne v 18 With Illinois- GFMC Calculations 7/ 5/ 1/ 3/ 9 Be 9/ 7/ + 1/ + 7/ 5/ + 3/ + 1/ 5/ 3/ , Be B IL Exp 1 C

15 Energy (MeV) He 6 He Li IL 7 He 3/ 5/ + 3/ 1/ + 5/ (5/) 1/ 3/ IL7 7 Li 5/ 5/ 7/ 1/ 3/ 5/ 5/ 7/ 1/ 3/ Argonne v 18 With Illinois V ijk GFMC Calculations October 007 Exp He Li Be He 9 Li 1/ 7/? 5/ 3/ 1/ 3/ 9 Be 10 He 3/ + 7/ + 5/ + 3/ 9/ 7/ + 5/ 1/ + 9/ + 7/ 5/ + 3/ 3/ + 3/ + 5/ + 1/ 5/ 1/ + 3/ 10 Li + (1 -, - ) , Be + 10 B

16 Nolen-Schiffer Anomaly Extract isovector [CSB (τ i + τ j ) z ] & isotensor [CD T ij ] energy components: E A,T (T z ) = X a n (A, T)Q n (T, T z ) n T Q 0 = 1 ; Q 1 = T z ; Q = 1 (3T z T ) Strong Type III CSB (constrained ±0% by nn scattering length) fixes isovector terms. Strong Type II CD (constrained by 1 S 0 pp and np scattering) overdoes isotensor; need P-wave NN scattering constraint? Isovector 3 H- 3 He 6 He- 6 Li * - 6 Be 7 He- 7 Li * - 7 Be * - 7 B 7 Li- 7 Be 8 He- 8 Li * - 8 Be * - 8 B * - 8 C 8 Li- 8 Be * - 8 B 9 Li- 9 Be * - 9 B * - 9 C 9 Be- 9 B 10 Be- 10 B * - 10 C Isotensor 6 He- 6 Li * - 6 Be 7 He- 7 Li * - 7 Be * - 7 B 8 He- 8 Li * C Coulomb E.M.+K.E. Total 8 Li- 8 Be * - 8 B 9 Li- 9 Be * - 9 B * - 9 C 10 Be- 10 B * - 10 C Ratio to experiment

17 GFMC isovector and isoscalar energy coefficients for AV18+IL7 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) 8 65(0) 757(0) 764 a 1 (6,1) () 14 54(1) 103() 1173 a 1 (7, 1 ) (3) 36 86(1) 159(4) 1641 a 1 (7, 3 ) (3) 9 5(1) 1348(4) 1373 a 1 (8,1) (4) 19 78(1) 1780(5) 1770 a 1 (8,) 164(4) 8 71(1) 176(4) 1659 a 1 (9, 1 ) (6) 4 55(1) 1786(7) 1851 a 1 (9, 3 ) (6) 19 90(1) 109(7) 104 a 1 (10,1) 5 13(7) 18 55(1) 50(8) 39 a (6,1) 167(0) (8) 95(9) 4 a (7, 3 ) 19(0) 7 38(5) 174(5) 175 a (8,1) 137(1) 4 10(8) 13(8) 145 a (8,) 144(0) 6 39(3) 189(3) 17 a (9, 3 ) 153(1) 7 38(8) 198(9) 176 a (10,1) 166(1) 1 119(18) 97(19) 41

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] 3 + ;1 ;1 3 P[431] 7 Li+n 3 + ;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 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) ;1 3 + ;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 i τ j ) z (σ i σ j ) L v(r) + (τ i τ j ) z (σ i σ j ) L w(r) Preliminary result: v γ µ n kev & w π (M n M p ) kev.

20 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).07(9) 8 He.79(3) 1.8() 1.94() 1.961(16) *Sick, PRC 77, 04130(R) (008) Wang, et al., PRL 93, (004) Mueller, et al., PRL 99, 5501 (007)

21 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 ρ pp (fm -3 ) r 1 (fm) RMS radii r pp r np r nn 4 He He He

22 FIG. 3: (Color online) Experimental charge radii of beryllium isotopes from isotope shift measurements ( ) compared with values from interaction cross section measurements ( ) and theoretical predictions: Greens-Function Monte-Carlo calculations (+) [0, 1], Fermionic Molecular Dynamics ( ) [], ab-initio No-Core Shell Model ( ) [1, 3, 4]. Nörtershäuser, et al., arxiv:

23 Nuclear Electromagnetic Currents Marcucci et al. (005) j = j (1) + j () (v) + + π π ρ,ω + j (3) (V π ) transverse Gauge invariant: q [ ] j (1) + j () (v) + j (3) (V π ) = [ ] T + v + V π, ρ ρ is the nuclear charge operator

24 MAGNETIC MOMENTS 3 3 H 7 Li µ H VMC(IA) GFMC(IA) 6 Li VMC(MEC) GFMC(MEC) EXPT 7 Be 8 Li 8 B 9 Be 3 He - Marcucci, Pervin, et al. (arxiv: )

25 M1, E, F, GT transitions 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 )/] F = P k τ k± ; GT = P k σ kτ k± Pervin, Pieper & Wiringa, PRC 76, (007) 6 He( + ;0 ;0) B(E) 6 Li(3 + ;0 ;0) B(E) 6 Li( + ;0 ;0) B(E) 6 Li( ;1 ;0) B(M1) 6 Li( + ;1 ;0) B(M1) 7 Li(1/ - 3/ - ) B(E) 7 Li(1/ - 3/ - ) B(M1) 7 Li(7/ - 3/ - ) B(E) 7 Be(1/ - 3/ - ) B(M1) CK NCSM VMC(IA) GFMC(IA) VMC(MEC) GFMC(MEC) Expt. 6 He( ) 6Li( ) log(ft) 7 Be(3/ - ) 7Li(3/ - ) log(ft) 7 Be(3/ - ) 7Li(1/ - ) log(ft) 7 Be(3/ - ) B.R Ratio to experiment

26 10 Be( + ) levels 10 Be( + ) levels at 3.37 and 5.96 MeV have same 1 D [44] spatial symmetry. Calculations give large but opposite sign quadrupole moments Q ±50 mb. Level ordering depends on H. Excitation energy (MeV) Argonne v 18 Without & with Illinois- 10 Be 3, + 9 Be+n Li+α GFMC calculations of B(E ) give: 10 Be( + ; Q < 0) = 7.9(3)e fm 4 10 Be( + ; Q > 0) =.1(3)e fm 4 compared to experimental compendium 10 Be( + 1 ;3.37) = 10.5(1.0)e fm 4 McCutchan & Lister doing ATLAS experiment to study transitions among these states preliminary result: 10 Be( + 1 ;3.37) : 7.9(1.0)e fm 4 suggesting first + state has Q < 0 Calculations for give: B(E ) = 6.9(1.0)e fm 4 (GFMC) B(M1 ) very small (VMC) Previous experiment found mostly M1-6 AV18 IL Exp 10 B

27 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 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 )

28 GFMC for 5 He as n+ 4 He Scattering States 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 R-Matrix Pole location E c.m. (MeV) Nollett, et al., PRL 99, 050 (007) E c.m. (MeV)

29 RADIATIVE CAPTURE REACTIONS σ(e cm ) = 8π 3 α v rel q X q/m Li LSJl h E LSJ l (q) + Ml LSJ (q) i He(α,γ) 7 Be 3 H(α,γ) 7 Li ( 4) 3 H(α, γ) 7 Li 3 He(α, γ) 7 Be Sources of 7 Li in big bang 7 Be key to solar ν e production S Factor (MeV b) C.M. Energy (MeV) Nollett, PRC 63, (001)

30 SINGLE-NUCLEON MOMENTUM DISTRIBUTIONS ρ στ (k) = Z dr 1 dr 1 dr dr A ψ JM J (r 1,r,...,r A ) e ik (r 1 r 1 ) P στ (1) ψ JMJ (r 1,r,...,r A ) 7 Li - AV18+UIX ρ n 10 1 ρ p ρ n ρ στ (k) ρ p ρ p - ρ p k (fm -1 )

31 TWO-NUCLEON & CLUSTER-CLUSTER DISTRIBUTIONS 4 He - AV18/UIX 4 He - AV18/UIX ρ 1 (k) ρ pp ρ np ρ(k) ρ tp ρ p ρ dd k (fm -1 ) k (fm -1 )

32 TWO-NUCLEON KNOCKOUT A(e,e pn) JLAB experiment for 1 C(e, e pn) measured back-to-back pp and np pairs Pairs with q rel = 3 fm 1 show np/pp ratio 10 0 Subedi et al., Science 30, 1476 (008) ρ NN (q,q=0) (fm 6 ) He 4 He 6 Li 8 Be q (fm -1 ) ρ NN (q,q=0) (fm 6 ) AV18+UIX 4 He AV4 4 He q (fm -1 ) VMC calculations for pairs with Q tot = 0 show this effect in A=3 8 nuclei Effect disappears when tensor correlations are turned off Shows importance of tensor correlations to > 3 fm 1 pp np Schiavilla, Wiringa, Pieper & Carlson, PRL 98, (007)

33 For Q tot > 0 (Q q), the minimum in pp distribution fills in: 0.6 ρ 1 (q,q) He Q=0.0 fm -1 Q=0.5 fm -1 Q=0.5 fm -1 Q=1.0 fm -1 Q=1.5 fm -1 R pp/pn He q (fm -1 ) He Q (fm -1 ) For q rel integrated over MeV/c, the ratio of pp to pn pairs R pp/pn compares well with preliminary analysis of CLAS data for 3 He(e,e pp)n Wiringa, Schiavilla, Pieper & Carlson, PRC 78, (008)

34 CONCLUSIONS ADVANTAGES/LIMITATIONS No basis required - very general trial functions with proper asymptotic structure can be used limited to configuration-space local potentials calculation requirements grow exponentially with A Most accurate method for 5 A 1 BENCHMARKS neutron drops 8 14 n for UNEDF 1 C ground state energy, + excited state, form factors 1 C( ) triple-alpha resonance EXPERIMENTS charge radii (both absolute and relative along isotope chains) B(M 1), B(E), B(GT), etc. for 6 A 1 with modern methods