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 , 2016 1 1

2. γ Triple-alpha reactions ? Prime importance for synthesizing
12C due to helium burning γ ?
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 2 2

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) 3 3

4. Physics problem Energy-averaged triple-α reaction rate
E2 photoabsorption (photodissociation) cross section
for
E: 3α kinetic 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 4 4

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

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) 6

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
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
The Coulomb potential term is most repulsive 8 8

9. Hyperspherical harmonics method commonly used in nuclear physics
ρ1,
ρ2 space-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 Λ2 does 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 10 10

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) 12 12

13. Calculation of E2 strength function
Final-state sum is replaced by Green’s operator
By replacing
with R-dependent CAP
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 13

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

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= MeV,
suggesting a transition from the
sequential to direct processes
Hyperspherical Harmonics R-matrix propagation calculation neglects the Coulomb couplings at large R 15

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 16

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)
β is 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-3α components

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) 21 21