Triple-α reactions at low temperatures

1. Triple-α reactions at low temperatures Y. Suzuki (Niigata, RIKEN) Motivation: Triple-alpha reaction process for 12C synthesis (no way to measure) sequential or direct capture process discrepancies in low-temperature theoretical rates Plan: Solving 3-charged particle Schrödinger equation involving continuum adiabatic hyperspherical formalism complex absorbing potential (CAP) comparison with NACRE in collaboration with H. Suno (RIKEN) and P. Descouvemont (ULB) submitted to PRC ECT* workshop ‘Three-body systems in reactions with rare isotopes’ Oct. 03-07, 2016

2. γ Triple-alpha reactions Prime importance for synthesizing 12C due to helium burning ? Sequential process via the narrow resonances of 8Be (0+) and 12C Hoyle state NACRE rate: based on the sequential process with Breit-Wigner resonance formula At high temperatures (above 0.1 GK) the sequential process dominates

3. Direct process α + α + α → 12C(2+) + γ expected to dominate below 0.1 GK Huge discrepancies at low temperatures due to the apparent difficulty in treating continuum of 3 charged particles NACRE: C. Angulo et al., NPA 656 (1999), CDCC: K. Ogata et al., PTP 122 (2009) HHR: N.B. Nguyen et al., PRC 87 (2013), Faddeev: S. Ishikawa, PRC 87 (2013) Imag. Time: T. Akahori et al., PRC 92 (2015) A. Dotter, B. Paxton, Astro. Astrophys.507 (2009)

4. Physics problem Energy-averaged triple-α reaction rate E2 photoabsorption (photodissociation) cross section for E: 3αkinetic energy Q = －2.836 MeV This formula can describe both direct and sequential processes We use adiabatic hyperspherical approach that makes it possible to describe bound, narrow resonance, and continuum states

5. Rescaled wave function Schrödinger eq. in hyperspherical coordinates Coordinates in body-fixedframe (size of the system) Hyperradius Hyperangles Ω (α,β,γ) Euler angles for rotation (θ,φ)angles to specify triangular shape θ=0 (equilateral) π/2(colinear) Λ2:squared grand angular momentum No differentiation wrt R in 2-4th terms

6. Adiabatic hyperspherical potentials Channel wave function for a fixed R is defined by ΛΛ (ν = 1, 2, …) Channel wave function is expanded in D-functions as well as in terms of basis spline functions for Boson symmetry → boundary conditions at φ=0, π/3 For details, H. Suno et al., PRA65 (2002), PRC91 (2015)

7. Adiabatic hyperspherical potentials for Jπ=0+ modified Ali-Bodmer a-type αα potential 3α potential to fit the energies of Hoyle resonance and 2+ bound state For R ≦140 fm, the lowest curve displays α+8Be(0+) nature • Appearance of successive avoided crossings with 3α continua

8. Focusing on the barrier top region Barrier top: The Coulomb potential term is most repulsive The centrifugal term does not decrease with increasing R The combined contribution of the centrifugal and Coulomb terms decrease The nuclear potential term is flat due to 8Be(0+) resonance structure

9. Hyperspherical harmonics method commonly used in nuclear physics ρ1, ρ2space-fixed coordinates Hyperspherical coordinates Hyperspherical harmonics (HH) The advantage of HH: an eigenfunction of Λ2 with eigenvalue K(K+4) Channel wave function is expanded in HH. The convergence must be checked by increasing K, l1, l2. The convergence is reasonably fast for short-ranged interactions. Since Λ2does not commute with the Coulomb potentials, the Coulomb couplings between HH’s persist even at large R

10. Solving hyperradial eq. with FEM-DVR basis Basis functions in Finite-Element-Methods-Discrete Variable Representation T.N.Rescigno et al., PRA62 (2000), H. Suno, JCP 134 (2011) Divide the integration range (R1=Rmin, Rmax) by L grid points The grid points and weights (Rl, ωl) are generated by dividing the range with a set of N grid points, and further subdividing each interval with M-th order Gauss-Lobatto quadrature L=(M-1)(N-1)+1 FEM-DVR basis functions cf. Lagrange mesh meth

11. Hyperradial coupled-channels equation • Coefficients c are determined from linear equation hyperradial kinetic energy matrix elements Hamiltonian matrix, becomes symmetric block diagonal

12. Properties of 12C states The narrow width of Hoyle resonance is calculated with TF-CAP. [40] E.Garrido et al., PRC91 (2015) [41] M. Chernykh et al., PRL98 (2007) [42] T. Neff et al., J. Phys. Conf. Ser. 569 (2014)

13. By replacing with R-dependent CAP Calculation of E2 strength function Final-state sum is replaced by Green’s operator outgoing waves are made exponentially damp, i.e., discretized This is equivalent to replacing Calculation of G(E+) reduces to the matrix inversion of H-iW, depending on the grid points No need to determine c for all the final-states

14. Hoyle resonance H. Suno, Y.S., P. Descouvemont, PRC91 (2015) Transmission-free (and minimum reflection) CAP D.E.Manolopoulos, JCP 117 (2002) T.P.Grozdanov et al., JCP 126 (2007) Absorption length = de Broglie wave length for Emin

15. Comparison of σγ Convergence test Comparison with other calculations No need to normalize to NACRE A sharp peak at Hoyle resonance A kink around E=0.13-0.14 MeV, suggesting a transition from the sequential to direct processes Hyperspherical Harmonics R-matrix propagation calculation neglects the Coulomb couplings at large R

16. Triple-alpha reaction rate At T < 0.08GK, our rate is by far smaller than CDCC and HHR, and one or two orders of magnitude larger than NACRE, Faddeev, Imaginary time At T > 0.2GK, all the rates agree within one order of magnitude Our rate is found to be larger by a factor of 2.66 than NACRE at T > 0.1GK

17. Reason for the discrepancy from NACRE Focusing on the peak of σγ at Hoyle resonance The Breit-Wigner resonance formula is numerically confirmed to work excellently The peak height ~ The peak width ~ The rate near Hoyle resonance region is proportional to Our B(E2) is 2.52 times larger than the experimental value

18. Adjusted rates Reason for adjustment: Use of better αα potentials (l-dependence is here ignored) αα potential is originally non-local Y. S. et al., PLB 659 (2008) 12C states should contain non-3α components According to FMD, P3α(21+) = 0.67, P3α(02+) = 0.85 M.Chernykh et al., PRL98 (2007) βis determined to reproduce B(E2)

19. Summary The triple-α reaction process at low temperatures has been studied. The TF-CAP has enabled us to compute the E2 strength function as well as the very narrow width of Hoyle resonance. Our reaction rate is up to 3 orders of magnitude larger than NACRE at T = 0.01 GK, while the adjusted rate reproduces it at T > 0.1 GK. Further challenges: use of better (nonlocal) αα potential explicit inclusion of non-3αcomponents

20. Energy surface of boson symmetry Two-dimensional contour plot of the potential energy surface of αα potentials at a fixed R = 10 fm as a function of θ and φ The solid lines show the lowest contour line here, the dashed and other lines correspond to an increase by every 2 MeV

21. Total reaction and charge-changing cross sections lead to determining nuclear radii !? Y. Suzuki (Niigata, RIKEN) in collaboration with W. Horiuchi (Hokkaido), I. Tanihata (RCNP), and others References for total reaction cross sections and skin thickness charge-changing cross sections W. Horiuchi, Y.S.,T. Inakura, PRC89 (2014) W. Horiuchi, S. Hatakeyama, S. Ebata, Y.S., PRC93(2016) Y.S., W. Horiuchi et al., PRC94 (2016)