Three-cluster dynamics within an ab initio framework Universality in Few-Body Systems: Theoretical Challenges and New Directions INT 14-1, March 26, 2014 S. Quaglioni Collaborators: C. Romero-Redondo (TRIUMF) P. Navrátil (TRIUMF) G. Hupin (LLNL) LLNL-PRES-652341 This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC
Outline Introduction Microscopic three-cluster problem Formalism for the (A-2)+1+1 mass partition Applications to 6 He Conclusions Outlook Lawrence Livermore National Laboratory LLNL- PRES- 652341 2
Our goal is to develop a fundamental theory for the description of thermonuclear reactions and exotic nuclei Standard solar model Stellar Nucleosynthesis Exotic Nuclei Fusion Energy Generation What is the nature of the nuclear force? d+ 3 H à 4 He+n 3 H+ 3 H à 4 He+2n d+ 3 H à 4 He+n+γ Lawrence Livermore National Laboratory LLNL- PRES- 652341 3
Theory needed because fusion reactions are difficult or impossible to measure at astrophysical energies The nuclear fusion process operates mainly by tunneling through the Coulomb barrier 3 He+ 4 Heà 7 Be+γ Extremely low rates ( ) = S(E) σ E E exp " 2π Z Z % 1 2 $ e2 ' # 2mE & Projectiles and targets are not fully ionized 3 He+ 3 Heà 4 He+2p Electron screening can mask bare nuclear cross section Lawrence Livermore National Laboratory LLNL- PRES- 652341 4
Developing such a fundamental theory is extremely complicated and a longstanding goal of nuclear theory Ab initio many-body calculations: A (all active) point-like nucleons proton neutron quark Nuclear two- and three-body (NN+NNN) forces guided by Quantum Chromodynamics (QCD) A nucleons Unitary transformation to soften bare Hamiltonian: e.g., Similarity Renormalization Group (SRG) Efficient theoretical framework and! High Performance Computing (HPC)! SIERRA Lawrence Livermore National Laboratory LLNL- PRES- 652341 5
Our starting point is a method to describe static properties of light nuclei from first principles Ab initio no-core shell model (NCSM) approach N = N max +1 Helped to point out! the fundamental importance! of three-nucleon (NNN) forces! in structure calculations.! Energy spectrum of nuclear states (MeV) PRL 99, 042501 (2007) Lawrence Livermore National Laboratory LLNL- PRES- 652341 6
We extended this approach by adding the dynamics between nuclei with the resonating-group method (RGM) NCSM/RGM approach r ( A a) ( a) S-factor [MeV b] 35 30 25 20 15 10 5 Astrophysical S-factors! 3 H(d,n) 4 He d+ 3 H à 4 He+n BR51 AR52 CO52 AR54 He55 GA56 BA57 GO61 KO66 MC73 MA75 JA84 BR87 9d*+ 5d'* Ab initio NCSM wave functions of the nuclei NN interactions Pioneered ab initio calculations! of light-nuclei fusion reactions! 0 10 3 H(d,n) 4 100 1000 He astrophysical E kin [MeV] S-factor! PRL 108, 042503 (2012) S-factor [MeV b] 20 15 10 5 0 (b) d+ d+ 3 He 3 à p+ 4 He+p 4 Bo52 Kr87 Sch89 Ge99 Al01 Al01 Co05 0d*+0d'* 1d*+1d'* 3d*+3d'* 5d*+5d'* 7d*+5d'* 9d*+5d'* 10 100 1000 E kin [kev] Lawrence Livermore National Laboratory LLNL- PRES- 652341 7
We are now working to complete this picture Extended NCSM/RGM to include: 1) NNN force in reactions 2) States of the compound nucleus 3) Three-cluster states in the continuum States of Compound nucleus + r ( A a) + ( a 2 ) ( a) A a 23 ( ) y x ( a 3 ) This talk Lawrence Livermore National Laboratory LLNL- PRES- 652341 8
1) Importance of the NNN force in reactions G. Hupin, J. Langhammer, P. Navratil, S. Quaglioni, A. Calci and R. Roth, Phys. Rv. C 88, 054622 (2013) Elastic scattering of neutrons on 4 He! n+ 4 Heà n+ 4 He total cross section[b] 8 6 4 2 n- 4 He 6 billion NNN matrix elements! x NN+NNN NN (with ind. terms) 3N NN Expt. expt. E n =1.8 MeV 0 0 4 8 12 16 20 center-of-mass energy [MeV] This work sets the stage for a truly accurate prediction of! the d+ 3 Hà 4 He+n fusion from QCD-based NN+NNN forces! Lawrence Livermore National Laboratory LLNL- PRES- 652341 9
2) Importance of states of the compound system G. Hupin, S. Quaglioni, and P. Navratil, in progress angular differential cross section [mb/sr] 1000 100 d+ 4 Heà d+ 4 He with & without inclusion of 6 Li states! Expt. E d = 2.935 MeV d+ 4 He (d+ 4 He) + 6 Li 10 0 30 60 90 120 150 180 center-of-mass angle [deg] angular differential cross section [mb/sr] 1000 100 10 Expt. d+ 4 He (d+ 4 He) + 6 Li E d = 12.0 MeV 0 30 60 90 120 150 180 center-of-mass angle [deg] Six-body correlations important also for binding energy ( ~1 MeV)! Lawrence Livermore National Laboratory LLNL- PRES- 652341 10
3) We want to describe also systems for which the lowest threshold for particle decay is of the 3-body nature Exotic nuclei, (Borromean halos, dripline nuclei) 11 Li 6 He (= 4 He + n + n ) 6 Be (= α + p + p ) 11 Li (= 9 Li + n + n ) 14 Be (= 12 Be + n + n ) Constituents do not bind in pairs! 4 He n n 4 He-neutrons separation (fm) 208 Pb Probability density of 6 He g.s. Probability density of 6 He g.s. S. Quaglioni, C. Romero-Redondo, P. Navrátil, Phys. Rev. C 88, 034320 (2013) neutron s separation (fm) Lawrence Livermore National Laboratory LLNL- PRES- 652341 11
Microscopic three-cluster problem Starts from: Ψ (A) = d x d y G ν ( x, y) ν 3-body channels  ν Φ ν x y ( A a 23 ) ( a 2 ) η 1,23 η 23 ( a 3 ) (A a ψ 23 ) (a α1 ψ 2 ) (a α2 ψ 3 ) α3 δ ( x η 23 )δ ( y η 1,23 ) Projects (H E)Ψ (A) = 0 onto the channel basis: dx dy % & H v# v ( x #, y #, x, y) E N v# v ( x #, y #, x, y) ' ( G v ( x, y) = 0 ν Φ ν" x" y"  " ν HÂν Φ ν x y Φ " ν x" y"  " ν  ν Φ ν x y Hamiltonian kernel Norm or Overlap kernel Lawrence Livermore National Laboratory LLNL- PRES- 652341 12
This can be turned into a set of coupled-channels Schrödinger equations for the hyperradial motion Hyperspherical Harmonic (HH) functions form a natural basis: y ρ α x = ρ cosα y = ρ sinα x y, y x, x * Φ ν xy = φ x, y K K (α) Φ νkρ ϒ x y L (α η, ˆη 1,23, ˆη 23 ) δ(ρ ρ ) η ρ 5/2 5/2 ρ η K Then, with orthogonalization and projection over : dρ ρ 5 $ % N 1/2 H N 1/2 & ' ( ρ (, ρ) u (ρ) Kv = E u ( ρ () K ( v ( ν( ν ρ 5/2 ρ( 5/2 νk! φ x,! K! y ( α!)!n " 1/2 G# $ (x, y) = ρ 5/2 u ν νk (ρ) K φ K x y (α) Lawrence Livermore National Laboratory LLNL- PRES- 652341 13
These equations can be solved using R-matrix theory Internal region (ρ a) External region (ρ > a) 0 a ρ Expansion on a basis Bound state asymptotic behavior u Kv (ρ) = n c n Kν f n (ρ) u Kv (ρ) = C Kν kρ K K+2 (kρ) Scattering state asymptotic behavior u Kv (ρ) = A Kν $ % Η K (kρ) δ ν # ν δ K # K S vk, v # K# Η + K (kρ)& ' Lawrence Livermore National Laboratory LLNL- PRES- 652341 14
4 He+n+n within the NCSM/RGM S. Quaglioni, C. Romero-Redondo, P. Navratil, Phys. Rev. C 88, 034320 (2013) Accurate soft NN interaction: SRG-evolved chiral N 3 LO potential with Λ=1.5 fm -1 Fits NN data with high accuracy But: misses both chiral initial and SRG-induced NNN force Fortuitously: two effects mostly compensate each other for very light systems 4 He ab initio wave function obtained within the NCSM ( H A 2) ψ (A 2) β1 ( r 1, r 2,, r (A 2) A 2 ) = E β1 ψ (A 2) β1 ( r 1, r 2,, r A 2 ) N = N max Large expansions in A-body harmonic oscillator (HO) basis Preserves: 1) Pauli principle, and 2) translational invariance Can include NNN interactions 4 He binding energy close to experiment: 28.22 MeV (expt.: 28.3 MeV) Fully antisymmetric channel states: Â ν = (A 2)!2! A! # A 2 A A 2 & % 1 ˆPi,k + ˆPi,A 1 ˆPj,A ( $ % i=1 k=a 1 i< j=1 '( 1 ˆP A 1,A 2 Lawrence Livermore National Laboratory LLNL- PRES- 652341 15
The formalism is general for (A-2)+1+1 mass partitions Norm or overlap kernel (Pauli principle) y! x! # A 2 A A 2 & 1 ˆPi,k + ˆPi,A 1 ˆPj,A % ( 1 ˆP A 1,A $ i=1 k=a 1 i< j=1 ' ( ) x y N ν! ν! ( x, y!, x, y) = 1 # 1 1 2 $ ( )! x+ S 23! +! T 23 %# & $ 1 1 ( ) x+s 23 +T 23 2(A 2) R nx " " x ( x ")R ny " " y ( y " ) Φ " + n" x n" y n x n y (A 2)(A 3) 2 % & ( ) δ! * ν3 " n x " ν ν R nx " " x ( x ")R ny " " y ( y " ) Φ ν3 " n" x " n" x n" y n x n y ( ) δ x! x x! x ( ) δ y! y y! y n y P A 2,A Φ ν3 n x n y R nx x (x)r ny y (y) %' n y P A 3,A 1 P A 2,A Φ ν3 n x n y R nx x (x)r ny y (y)& (' - 2(A-2) + (A-2)(A-3)/2 SD ψ µ 1 (A 2) a + aψ ν 1 (A 2) SD SD ψ µ 1 (A 2) a + a + a aψ ν 1 (A 2) SD Lawrence Livermore National Laboratory LLNL- PRES- 652341 16
The formalism is general for (A-2)+1+1 mass partitions Norm or overlap kernel (Pauli principle) y! x! A 2 # A 2 A & %%1 P i,k + P i,a 1P j,a (( 1 P A 1,A $ i=1 k=a 1 ' i< j=1 ( ) x y Matrix element A ν2 1 over three-cluster states: n 4He - 2(A-2) SD Lawrence Livermore National Laboratory x, x y, y n + (A-2)(A-3)/2 ψ µ(a1 2) a + a ψν(a1 2) SD SD ψ µ( A1 2) a + a + a a ψν(a1 2) SD LLNL- PRES- 652341 17
The formalism is general for (A-2)+1+1 mass partitions Hamiltonian kernel (nucleon-nucleon-target potentials) y! x! # % $ A 2 A l=1 m=a 1 &# A 2 A A 2 & V lm +V A 1A ( 1 ˆPi,k + ˆPi,A 1 ˆPj,A % ( ' 1 ˆP A 1,A $ i=1 k=a 1 i< j=1 ' ( ) x y = V( x!)n! ( x, y!, x, y) + ν ν! (A 2) a + aψ ν 1 (A 2) (A 2) a + a + aaψ ν 1 (A 2) (A 2) a + a + a + a aaψ ν 1 (A 2) SD ψ µ 1 SD SD ψ µ 1 SD SD ψ µ 1 SD Lawrence Livermore National Laboratory LLNL- PRES- 652341 18
Part of the interaction kernel is localized only in x, x y! x! V A 1A ( 1 ˆP ) A 1,A x y R nx " " x ( x " )R (x) n" nx L x " x s 23 J 23 T 23 V n x L x s 23 J 23 T 23 x " n x n x 1 ( 1) x+s 23 +T ( 23 ) δ # δ( y# y) γ γ y# y Extended-size HO expansion N ext >> N max R ny y ( y ")R ny y (y) n y Lawrence Livermore National Laboratory LLNL- PRES- 652341 19
Results for 6 He ground state 6-body diagonalization vs 4 He(g.s)+n+n calculation χ ν (x, y) = 1 ρ 5/2 K u νk (ρ) φ x y K (α) NCSM 6-body diagonalization! NCSM/RGM 4 He(g.s.)+n+n! N tot = N 0 + N max Differences between NCSM 6-body and NCSM/RGM 4 He(g.s.)+n+n results due to core polarization Contrary to NCSM, NCSM/RGM wave function has appropriate asymptotic behavior Lawrence Livermore National Laboratory LLNL- PRES- 652341 20
Other convergence tests HH expansion Extended-size HO expansion χ ν (x, y) = 1 ρ 5/2 K max K u νk (ρ) φ x y K (α) V A 1A ( 1 ˆP A 1,A ) R ny y ( y #)R ny y (y) n y Lawrence Livermore National Laboratory LLNL- PRES- 652341 21
Probability density of 6 He ground state J π = 0 + ( x = y = S = 0) J π = 0 + ( x = y = S =1) Lawrence Livermore National Laboratory LLNL- PRES- 652341 22
Probability density of 6He ground state 4He-neutrons separation (fm) J π = 0+ ( x = y = S = 0) neutron s separation (fm) Lawrence Livermore National Laboratory LLNL- PRES- 652341 23
Results for 4 He(g.s.)+n+n continuum C. Romero-Redondo, S. Quaglioni, and P. Navratil, in progress Scattering phase shifts! Energy spectrum of states! new Γ = 2 dδ(e) de E=ER Lawrence Livermore National Laboratory LLNL- PRES- 652341 24
Convergence with respect to HO model space size Lawrence Livermore National Laboratory LLNL- PRES- 652341 25
Other convergence tests HH expansion χ ν (x, y) = 1 ρ 5/2 K max K u νk (ρ) φ x y K (α) Lawrence Livermore National Laboratory LLNL- PRES- 652341 26
Other convergence tests Extended-size HO expansion V A 1A 1 ˆP N ext ( A 1,A ) R ny y ( y #)R ny y (y) n y Sizable effects only when neutrons are in 1 S 0 partial wave (strong attraction) Lawrence Livermore National Laboratory LLNL- PRES- 652341 27
Conclusions We are building an efficient ab initio theory including the continuum NCSM eigenstates à short- to medium-range A-body structure NCSM/RGM cluster states à scattering physics of the system We map the many-body problem into a few-cluster problem The Pauli exclusion principle is treated exactly Inter-cluster interactions arise from underlying nuclear Hamiltonian First ab initio description of three-cluster dynamics 4 He+n+n bound and continuum states Good qualitative description of the low-lying spectrum of 6 He Lawrence Livermore National Laboratory LLNL- PRES- 652341 28
Outlook For a complete picture we need to: Run calculations with NNN forces (codes are ready) Introduce core excitations by coupling to NCSM A-body eigenstates Future applications of three-cluster formalism Calculations of radii, electric dipole transitions Other systems: 5 H ( 3 H+n+n), 11 Li (= 9 Li+n+n), 12 Be(= 10 Be+n+n) Ultimate goal: binary & ternary light-nucleus fusion reactions Transfer reactions: 3 H( 3 H,2n) 4 He Lawrence Livermore National Laboratory LLNL- PRES- 652341 29