Ab initio many-body calculations of nuclear scattering and reactions

Transcription

1 Canada s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Ab initio many-body calculations of nuclear scattering and reactions INT Workshop INT-3-53W Nuclear Reactions from Lattice QCD th March 03, Institute for Nuclear Theory Petr Navratil TRIUMF Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d un consortium d universités canadiennes, géré en co-entreprise à partir d une contribution administrée par le Conseil national de recherches Canada

2 Outline Chiral forces No-core shell model Including the continuum with the resonating group method NCSM/RGM NCSMC 7 He resonances 7 Be(p,γ 8 B capture 3 H(d,n 4 He fusion Outlook

3 Chiral Effective Field Theory First principles for Nuclear Physics: QCD Non-perturbative at low energies Lattice QCD in the future For now a good place to start: Inter-nucleon forces from chiral effective field theory Based on the symmetries of QCD Chiral symmetry of QCD (m u m d 0, spontaneously broken with pion as the Goldstone boson Degrees of freedom: nucleons + pions Systematic low-momentum expansion to a given order (Q/Λ χ Hierarchy Consistency Low energy constants (LEC Fitted to data Can be calculated by lattice QCD Λ χ ~ GeV : Chiral symmetry breaking scale

4 The ab initio no-core shell model (NCSM The NCSM is a technique for the solution of the A-nucleon bound-state problem Realistic nuclear Hamiltonian High-precision nucleon-nucleon potentials Three-nucleon interactions N = N max + Finite harmonic oscillator (HO basis A-nucleon HO basis states complete N max hω model space Effective interaction tailored to model-space truncation for NN(+NNN potentials Okubo-Lee-Suzuki unitary transformation Or a sequence of unitary transformations in momentum space: Similarity-Renormalization-Group (SRG evolved NN(+NNN potential A Ψ A = N max N=0 i A c Ni Φ Ni Convergence to exact solution with increasing N max for bound states. No coupling to continuum.

5 4 He from chiral EFT interactions: g.s. energy convergence 4 E [MeV] He bare (36 SRG (.0/8 N 3 LO (500 MeV NN + NNN Chiral N 3 LO NN plus N LO NNN potential Bare interaction (black line Strong short-range correlations Large basis needed SRG evolved effective interaction (red line Unitary transformation 8 ( H α =U α H U + α dh α dα = "# [ T, H α ], H α $ % α = λ N max A=3 binding energy and half life constraint c D =-0., c E =-0.05, Λ=500 MeV Two- plus three-body components, four-body omitted Softens the interaction Smaller basis sufficient

6 NCSM calculations of 6 He and 7 He g.s. energies - -0 E gs [MeV] He SRG-N 3 LO NN Λ=.0 fm - N max = N max = 4 N max = 6 N max = 8 N max =0 N max = extrap hω [MeV] E gs [MeV] NCSM Exp fit extrap 6 He SRG-N 3 LO NN Λ=.0 fm - hω=6 MeV 7 He 3/ N max 0 + ü N max convergence OK ü Extrapolation feasible E g.s. [MeV] 6 He: E gs =-9.5(5 MeV (Expt MeV 7 He: E gs =-8.7(5 MeV (Expt (30 MeV 7 He unbound (+0.430(3 MeV, width 0.8(5 MeV NCSM: no information about the width 4 He 6 He 7 He NCSM N max = NCSM extrap. -8.( -9.5(5-8.7(5 Expt He unbound

7 Include more many nucleon correlations NCSM/ A Ψ A = N max N=0 i A c Ni Φ Ni + + ( a µ ( a µ rµ r µ + ( a 3µ a µ + a µ + a 3µ = A using the Resonating Group Method (RGM ideas

8 ψ (A = c κ φ κ κ + Â ν φ ν ν ξκ { } ( ξν { } + Â µ φ µ ξµ µ + ( φ ν ({ ξ ν } g v( r v ({ } φ µ ( a κ = A φ κ ({ ξ µ } φ 3µ ({ ξ 3µ } G µ ( r µ, r µ φ ν ( a v r v ( a v a ν + a ν = A φ ν ( a µ φ µ ( a µ φ µ rµ r µ φ 3µ ( a 3µ a µ + a µ + a 3µ = A

9 ψ (A = c κ φ κ κ + Â ν φ ν ν ξκ { } ( ξν { } + Â µ φ µ ξµ µ + ( φ ν ({ ξ ν } g v( r v ({ } φ µ φ : antisymmetric cluster wave functions {ξ}: Translationally invariant internal coordinates (Jacobi relative coordinates These are known, they are an input ( a κ = A φ κ ({ ξ µ } φ 3µ ({ ξ 3µ } G µ ( r µ, r µ φ ν ( a v r v ( a v a ν + a ν = A φ ν ( a µ φ µ ( a µ φ µ rµ r µ φ 3µ ( a 3µ a µ + a µ + a 3µ = A

10 ψ (A = c κ φ κ κ + Â ν φ ν ν Α ν, Α µ : intercluster antisymmetrizers Antisymmetrize the wave function for exchanges of nucleons between clusters Example: ξκ { } ( ξν { } + Â µ φ µ ξµ µ + ( φ ν ({ ξ ν } g v( r v ({ } φ µ ( a κ = A φ κ ({ ξ µ } φ 3µ ({ ξ 3µ } G µ ( R µ, R µ φ ν ( a v a v = A, a v = Âν = A A ˆPiA i= r v ( a v a ν + a ν = A φ ν ( a µ φ µ ( a µ φ µ rµ r µ φ 3µ ( a 3µ a µ + a µ + a 3µ = A

11 ψ (A = c κ φ κ κ ξκ { } ( + g v ( r  ν φ ν ξν ν ({ } φ ν ( a κ = A φ κ ({ ξ ν } δ( r r v d r φ ν a ν + a ν = A r ( a v ( a v φ ν + G µ ( R, R  µ φ µ ξµ µ + ({ } φ µ c, g and G: discrete and continuous linear variational amplitudes Unknowns to be determined ({ ξ µ } φ 3µ ({ ξ 3µ } δ( R R µ δ( R R µ φ µ ( a µ ( a µ R d R dr φ µ R ( a 3µ a µ + a µ + a 3µ = A φ 3µ

12 ψ (A = c κ φ κ κ ξκ { } ( + g v ( r  ν φ ν ξν ν ({ } φ ν ( a κ = A φ κ ({ ξ ν } δ( r r v d r φ ν a ν + a ν = A r ( a v ( a v φ ν + G µ ( R, R  µ φ µ ξµ µ + ({ } φ µ Discrete and continuous set of basis functions Non-orthogonal Over-complete ({ ξ µ } φ 3µ ({ ξ 3µ } δ( R R µ δ( R R µ φ µ ( a µ ( a µ R d R dr φ µ R ( a 3µ φ 3µ a µ + a µ + a 3µ = A

13 ψ (A = c κ φ κ κ ξκ { } ( + g v ( r  ν φ ν ξν ν ({ } φ ν ({ ξ ν } δ( r r v d r φ ν a ν + a ν = A r ( a v ( a v φ ν + G µ ( R, R  µ φ µ ξµ µ + ({ } φ µ ({ ξ µ } φ 3µ ({ ξ 3µ } δ( R R µ δ( R R µ d R dr In practice: function space limited by using relatively simple forms of Ψ chosen according to physical intuition and energetical arguments Most common: expansion over binary-cluster basis

14 The ab initio NCSM/RGM in a snapshot Ansatz: eigenstates of H (A-a and H (a in the ab initio NCSM basis Many-body Schrödinger equation: ê realistic nuclear Hamiltonian Hamiltonian kernel Norm kernel 4

15 Consider the T = ½ case: 5 He ( 5 Li Five-nucleon cluster unbound; 4 He tightly bound, not easy to deform 4 He n H n d + p Energy 3 H MeV Satisfactory description of n- 4 He ( p- 4 He scattering at low excitation energies within single-channel approximation However, both n(p + 4 He and d + 3 H( 3 He channels needed to describe 3 H(d,n 4 He [ 3 He(d,p 4 He] fusion!

16 Unbound A=5 nuclei: 5 Heèn+ 4 He, 5 Lièp+ 4 He NCSM/RGM calculations 4 He SRG-N 3 LO NN potential with Λ=.0 fm - n A y (! c.m E n = 7 MeV d"/d#(! c.m NCSM/RGM Krupp et al He(n,n 4 He Differential cross section and analyzing MeV neutron energy Polarized neutron experiment at Karlsruhe 50 E n = 7 MeV ! c.m. NNN and 4 He polarization missing: Good agreement only for energies beyond low-lying 3/ - resonance

17 How about 7 He as n+ 6 He? He 0 SRG-N 3 LO Λ =.0 fm h- Ω = 6 MeV E x [MeV] ( +, -, h - Ω 4h - Ω 6h - Ω 8h - Ω 0h - Ω h - Ω Expt All 6 He excited states above + broad resonances or states in continuum Convergence of the NCSM/RGM n+ 6 He calculation slow with number of 6 He states Negative parity states also relevant Technically not feasible to include more than ~ 5 states 7

18 New developments: NCSM with continuum NCSM/ Ψ A J π T = c Ni ANiJ π T Ni

19 New developments: NCSM with continuum NCSM/ NCSM/RGM Ψ A J π T = c Ni ANiJ π T Ni The idea behind the NCSMC r J T A = X Z d r ( râ J T (A a,a r H = EN = N + (N HN = E NCSMC HNCSM + J T A = X c A J T + X h r Z h N HN d r X Z 0 c d r 0 N 0 ( r, r 0 0( r 0 ḡ = E ḡ! Â J T (A r c a,a

20 New developments: NCSM with continuum NCSM/ NCSM/RGM Ψ A J π T = c Ni ANiJ π T Ni The idea behind the NCSMC r J T A = X Z d r ( râ J T (A a,a r H = EN = N + (N HN = E NCSMC HNCSM + J T A = X c A J T + X h r Z h N HN d r X Z 0 c S. Baroni, P. N., and S. Quaglioni, PRL 0, 0505 (03; arxiv: [nucl-th] (submitted to PRC d r 0 N 0 ( r, r 0 0( r 0 ḡ = E ḡ! Â J T (A r c a,a

21 NCSMC formalism Start from ( HNCSM h h H ( c χ = E ( ḡ ḡ ( c χ NCSM sector: (H NCSM λλ = AλJ π T Ĥ Aλ J π T = ε J π T λ δ λλ NCSM/RGM sector: H νν (r, r = µµ dydy y y N νµ (r, yh µµ (y, y N µ ν (y,(,r

22 ψ J π T = ν g ν J πt (r r ( (st Y ( ˆr A a,a π Â ν A a α I π T a α I T (J π T δ(r r A a,a rr A a,a r dr J Φ π T νr (Jacobi channel basis Since we are using NCSM wave functions, it is convenient to introduce Jacobi channel states in the HO space J Φ π T νn ( (st Y ( ˆr A a,a π = A a α I π T a α I T (J π T R n (r A a,a The coordinate space channel states are given by J Φ π T νr = R n (r We used the closure properties of HO radial wave functions n J Φ π T νn δ(r r A a,a = R n (rr n (r A a,a r r A a,a n Target and projectile wave functions are both translational invariant NCSM eigenstates calculated in the Jacobi coordinate basis

23 J Φ π T ν r  J π T v  v Φ νr = ( A r ( a = ( a = ( A r J N πt δ( r r v v ( r,r = δ v v r r (A R n ( r R n (r Φ n n J π T ν n J ˆPA,A Φ π T νn Direct term: Treated exactly! (in the full space ν ν ( A ( A ( a = SD ψ µ (A a + aψ ν (A Exchange term: Obtained in the model space! (Many-body correction due to the exchange part of the intercluster antisymmetrizer SD δ(r r A a,a = R n (rr n (r A a,a r r A a,a n

24 Define SD channel states in which the eigenstates of the heaviest of the two clusters (target are described by a SD wave function: J Φ π T νn SD ( (st (a Y ˆRc.m. π = A a α I π T a α SD I T ( (J π T R n (a ( R c.m. (A a R c.m. O π A a α I (A a T ϕ 00 ( Rc.m. Vector proportional to the c.m. coordinate of the a nucleons Vector proportional to the c.m. coordinate of the A-a nucleons (a R c.m. R (A c.m. ξ 0 (A a (a ( ϕ 00 ( Rc.m. ϕ n ( Rc.m. = 00, n, n r r, NL, d= a n r r,nl O A a η A a ϕ nr ηa a r ( ϕ NL ξ0 η = ( (A aa r A a A A a,a ( (

25 More in detail: J Φ π T νn SD s = ˆĴ r ( s+ r+l+j r J r L J n r r,nl,j r 00, n, n, NL, r r d= a The spurious motion of the c.m. is mixed with the intrinsic motion A a π J rt r ϕ NL ( ξ 0 ( J π T Φ νr n r SD J Φ π T ν n Calculate these Matrix that can be inverted J Ô Φ π T νn SD = Φ n r r,n r r,j r J r π r T νr n r ˆˆ NL π J r Ô Φ r T νr n r Ĵr ( s+ s s r J r L J Interested in this Transformation is general: same for different A s or different a s s r J r L J 00, n, n r r, NL, d= a d = A a A a a Translational invariance preserved (exactly! also with SD channels 00, n, n r r, NL,

26 SD to Jacobi transformation is general and exact Can use powerful second quantization representation Matrix elements of translational invariant operators can be expressed in terms of matrix elements of density operators on the target eigenstates For example, for a =a = SD Φ ν n One-body density matrix elements J πt P A,A Φ νn J π T SD I s J j = A j j Kτ ŝ ŝ ĵ ĵ I s J j ˆK ˆ τ ( I + j +J ( T + +T I K I j J j T τ T T SD A α π I ( (Kτ π n j A α I T SD + T a nj a

27 Start from ( ( HNCSM h c = E h H χ NCSMC formalism ( ( ḡ ḡ ( c χ Coupling: ḡ λν (r = ν dr r AλJ π T Âν Φ J π T ν r N ν ν (r,r h λν (r = ν dr r AλJ π T ĤÂν ΦJ π T ν r N ν ν (r,r. Calculation of g from SD wave functions: g λνn = AλJ π T ÂνΦ J π T = = νn nl00, l 00nl, l (A nl00, l 00nl, l (A SD AλJ π T ÂνΦ J π T νn SD Ĵ ˆT j ( I +J+jŝĵ { I s l J j } SD AλJ π T a A α nlj I π T SD 7

28 NCSMC formalism Start from ( HNCSM h h H ( c χ = E ( ḡ ḡ ( c χ ( ( ( νrν r = δλλ ( ḡ λν (r ḡ λ ν(r δ νν δ(r r N λλ rr ( Orthogonalization: H = N ( HNCSM h h H N ( c χ = N + ( c χ Solve with generalized microscopic R-matrix Bloch operator ( ( (Ĥ + ˆL E = c χ ˆL c χ ˆL ν = ( δ(r a( d dr B ν r 8

29 Separation into internal and external regions at the channel radius a Internal region External region u c (r = A cn f n (r u c (r ~ v c δ ci I c (k c r U ci O c (k c r n [ ] 0 a r r c This is achieved through the Bloch operator: System of Bloch-Schrödinger equations: L c = d δ(r a µ c dr B c r T ˆ [ rel (r + L c + V Coul (r (E E c ]u c (r + dr'r'w cc' (r,r'u c' (r' Internal region: expansion on square-integrable basis External region: asymptotic form for large r c' u c (r = = L c u c (r n A cn f n (r [ ] u c (r ~ C c W (k c r or u c (r ~ v c δ ci I c (k c r U ci O c (k c r Scattering matrix Bound state Scattering state

30 After projection on the basis f n (r: [ C cn,c'n' (E E c δ cn,c'n' ]A c'n' = k c f / n L c I c δ ci U ci O c µ c v c c'n' f n ˆ T rel (r + L c + V Coul (r f n' δ cc' + f n W cc' (r,r' f n'. Solve for A cn. Match internal and external solutions at channel radius, a [ c aδ ci U c i O c (k c a ] = c k R c a c c I c (k µ c v c µ c v c I c (k c aδ ci U ci O c (k c a [ ] Lagrange basis associated with Lagrange mesh: { ax n [0,a]} g(xdx λ n g(x n In the process introduce R-matrix, projection of the Green s function operator on the channel-surface functions 0 0 a N n= f n (r f n' (rdr δ nn' R cc' = nn' µ c a f n (a[ C EI] cn,c'n' µ c' a f n' (a

31 3. Solve equation with respect to the scattering matrix U [ c aδ ci U c i O c (k c a ] = c k R c a c c I c (k µ c v c µ c v c I c (k c aδ ci U ci O c (k c a [ ] 4. You can demonstrate that the solution is given by: U = Z Z *, Z c c = ( k c a O c (k c aδ c c k c a R c c O c (k c a Scattering phase shifts are extracted from the scattering matrix elements U = exp(iδ δ

32 NCSM with continuum: 7 He 6 He+n δ [deg] P3/ δ [deg] P3/ n+ 6 He n+ 6 He E kin [MeV] E kin [MeV] NCSM/RGM with up to three 6 He states NCSMC with up to three 6 He states and four 7 He eigenstates More 7-nucleon correlations Fewer target states needed Expt. 3

33 NCSM with continuum: 7 He 6 He+n 50 0 P3/ 6 P5/ (a 50 0 P3/ 6 P5/ (b δ [deg] P/ 6 P3/ δ [deg] P/ 6 P3/ 0-30 S/ n+ 6 He(g.s S/ n+ 6 He(g.s E kin [MeV] E kin [MeV] NCSM/RGM with three 6 He states NCSMC with three 6 He states and ten 7 He eigenstates More 7-nucleon correlations Fewer target states needed Expt. 33

34 7 He: NCSMC vs. NCSM/RGM vs. NCSM J π experiment NCSMC NCSM/RGM NCSM E R Γ Ref. E R Γ E R Γ E R 3/ 0.430(3 0.8(5 [] / 3.35(0.99(7 [40] / 3.03(0 [] [5].0( 0.75(8 [5] [] A. H. Wuosmaa et al., Phys.Rev.C7, 0630(005. NCSMC and NSCM/RGM energies where phase shift derivative maximal NCSMC and NSCM/RGM widths from the derivatives of phase shifts Γ = δ(e kin / E kin Ekin =E R Experimental controversy: Existence of low-lying / - state not seen in these calculations Best agreement with the neutron pick-up and proton-removal reactions experiments [] 34

35 Solar p-p chain p-p chain 7 Solar neutrinos E ν < 5 MeV

36 Structure of the 8 B ground state NCSM/RGM p- 7 Be calculation five lowest 7 Be states: 3/ -, / -, 7/ -, 5/ -, 5/- Soft NN SRG-N 3 LO with Λ =.86 fm - 8 B + g.s. bound by 36 kev (Expt 37 kev Large P-wave 5/ - component 0. 0 g(r B( + g.s. SRG-N LO Λ=.86 fm - p+ 7 Be(3/ - +/ - +7/ - +5/ - +5/ - 3/ - s= l= 3/ - s= l= / - s= l= 7/ - s=3 l= 5/ - s= l= 5/ - s= l= r [fm] 5/ - state of 7 Be should be included in 7 Be(p,γ 8 B calculations 36

37 7 Be p p- 7 Be scattering NCSM/RGM calculation of p- 7 Be scattering 7 Be states 3/ -,/ -, 7/ -, 5/ -, 5/- Soft NN potential (SRG-N 3 LO with Λ =.86 fm - 8 B + g.s. bound by 36 kev (expt. bound by 37 kev New 0 +, +, and two + resonances predicted s = l = + clearly visible in (p,p cross sections PRC 8, (00 PLB 704 (0 379

38 7 Be p 7 Be(p,γ 8 B radiative capture NCSM/RGM calculation 7 Be states 3/ -,/ -, 7/ -, 5/ -, 5/- Soft NN potential (chiral SRG-N 3 LO with Λ =.86 fm - 8 B + g.s. bound by 36 kev (expt. 37 kev S(0 ~ 9.4(0.7 ev b Data evaluation: S(0=0.8(. ev b S 7 [ev b] (a 7 Be(p,γ 8 B Filippone Strieder Hammache Hass Baby Junghans Schuemann Davids Kikuchi De04 MN E NCSM/RGM E E kin [MeV] P.N., R. Roth, S. Quaglioni, Physics Letters B 704 (0 379

39 Ab initio calculation of the 3H(d,n4He fusion n 3H d NIF energy generation 4He ITER

40 d+ 3 H and n+ 4 He elastic scattering: phase shifts S3/ n+ 4 He, d*+ 3 H n+ 4 He no coupling P3/ P/ δ [deg] 30 0 d+ 3 H d+ 3 H S/ δ [deg] n+ 4 He n+ 4 He D3/ d+ 3 H, d*+ 3 H d+ 3 H no coupling S/ E kin [MeV] E kin [MeV] d+ 3 H elastic phase shifts: Resonance in the 4 S 3/ channel Repulsive behavior in the S / channel è Pauli principle d* deuteron pseudo state in 3 S - 3 D channel: deuteron polarization, virtual breakup n+ 4 He elastic phase shifts: d+ 3 H channels produces slight increase of the P phase shifts Appearance of resonance in the 3/ + D-wave, just above d- 3 H threshold The d- 3 H fusion takes place through a transition of d+ 3 H is S-wave to n+ 4 He in D-wave: Importance of the tensor force

41 3 H(d,n 4 He & 3 He(d,p 4 He fusion NCSM/RGM with SRG-N 3 LO NN potentials 35 0 S-factor [MeV b] H(d,n 4 He BR5 AR5 CO5 AR54 He55 GA56 BA57 GO6 KO66 MC73 MA75 JA84 BR87 d+9d*+5d'* SRG-N 3 LO Λ=.45 fm - S-factor [MeV b] (b d+ 3 He p+ 4 He Bo5 Kr87 Sch89 Ge99 Al0 Al0 Co05 0d* d*+d'* 3d*+3d'* 5d*+5d'* 7d*+5d'* 9d*+5d'* E kin [kev] E kin [kev] Potential to address unresolved fusion research related questions: 3 H(d,n 4 He fusion with polarized deuterium and/or tritium, 3 H(d,nγ 4 He bremsstrahlung, Electron screening at very low energies P.N., S. Quaglioni, PRL 08, (0

42 Conclusions and Outlook We developed a new unified approach to nuclear bound and unbound states Merging of the NCSM and the NCSM/RGM PRL 0, 0505 (03 We demonstrated its capabilities in calculations of 7 He resonances Successful NCSM/RGM applications to 7 Be(p,γ 8 B radiative capture 3 H(d,n 4 He and fusion PLB 704 (0 379 PRL 08, (0 Outlook: Inclusion of 3N interactions first results available for n- 4 He, p- 4 He Extension of the NCSMC formalism to composite projectiles (deuteron, 3 H, 3 He, 4 He Extension of the formalism to coupling of three-body clusters ( 6 He ~ 4 He+n+n

43 NCSMC and NCSM/RGM collaborators Sofia Quaglioni (LLNL Joachim Langhammer, Robert Roth (TU Darmstadt C. Romero-Redondo, F. Raimondi (TRIUMF G. Hupin, M. Kruse (LLNL S. Baroni (ULB W. Horiuchi (Hokkaido

44 Possible future benchmark: d- 3 H fusion Calculation of the 3 H(d,n 4 He and 3 He(d,p 4 He using the NCSMC formalism and chiral NN+3N forces is within the reach sensitive test of the chiral nuclear Hamiltonian complex reaction mechanism sensitive to the treatment of virtual excitations of the involved nuclei many-nucleon dynamics in the continuum Benchmark with lattice QCD ab initio calculations beneficial for both the standard nuclear calculations and the lattice QCD many-nucleon calculations Physics issue: Behavior of the d- 3 H and n- 4 He phase shifts at the resonance 44