1 / 30

Resonances and strength functions of few-body systems

Resonances and strength functions of few-body systems. Y. Suzuki (Niigata, RIKEN). Outline I. Hoyle resonance in 12 C A narrow, near threshold state playing a key role for producing 12 C element in stars through triple-α reactions Coulomb 3-body problem

wscalzo
Download Presentation

Resonances and strength functions of few-body systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Resonancesandstrength functions of few-body systems Y. Suzuki (Niigata, RIKEN) Outline I. Hoyleresonance in 12C A narrow, near threshold state playing a key role for producing 12C element in stars through triple-α reactions Coulomb 3-body problem Adiabatic hyperspherical method + Complex absorbing potential (CAP) H. Suno, Y. S., P. Descouvemont, PRC91(2015), PRC94(2016) II. Resonances in A=4 nuclei, 4H, 4He, 4Li, studied through strength functions The first excited state of a magic nucleus 4He is not of negative parity but a 0+ resonance (What structure?) Other broad negative-parity resonances Correlated Gaussian bases + Complex scaling method (CSM) W. Horiuchi, Y. S., PRC78 (2008) W. Horiuchi, Y. S., K. Arai, PRC85 (2012) W. Horiuchi, Y. S., PRC87(2013), FBS54(2013) Resonance workshop July3-9, 2017, Bled, Slovenia

  2. I. Hoyle resonance The synthesis of 12C is essential to 12C-based life Its process is sequential via a narrow resonance of 8Be (0+): ER=0.092 MeV above 2α threshold, ΓR=6.8 eV Hoyle predicted a 0+ state around 7.7 MeV (1952) to explain the abundance of 12C. Later it was experimentally found: ER=0.38 MeV above 3α threshold, ΓR=8.5 eV ThisHoyle’stheoryledtothe establishment of the nucleosynthesis in stars

  3. Difficulty in calculating Hoyle resonance width Precise prediction of its width requires calculations of the barrier for 3-α penetration, which is challenging because (1)3 charged-particles interacting via long-range Coulomb forces always cause couplings even at large distances (2)No asymptotic wave functions of 3 charged-particles are known (3)2-α subsystem (8Be) has a sharp resonance that causes successive avoided crossings with 3-α continuum ? γ

  4. Basic idea of Hyperspherical method Extension of spherical coordinates (r, θ, φ) to N-particle system Number of degrees of freedom = 3(N-1) excluding c.m. motion N=3: hyperadius R of length dimension 5 dimensionless coordinates (hyperangles) Ω (system’s size) (system’s shape) Ω (α,β,γ) Euler angles for rotation (θ,φ)angles to determine triangular shape R is a convenient quantity to describe a large-scale change of the system, i.e., R-dependence of the energy is important to determine the dynamics

  5. Kinetic energy: T=TR+TΩ (separated into hyperradial and hyperangle parts) Schrödinger eq. for rescaled wave function in HS coordinates Λ2:squared grand angular momentum No differentiation wrt R is present in then 2nd and 3rd terms Channel wave function Φνand adiabatic HS potential Uνare defined by solving the eigenvalue problem where R is just a parameter (ν=1,2,…) To solve the eigenvalue problem accurately is crucially important Λ2 does not commute with V especially the Coulomb potential ΛΛ is expanded in terms of basis spline functions

  6. Adiabatic HS potentials for Jπ=0+ Uν modified Ali-Bodmer a-type αα potential 3α potential to fit the energies of Hoyle resonance and 2+ bound state R [fm] For R ≦140 fm (Rrms60fm), the lowest ν=1 curve is dominated by α+8Be(0+) structure Beyond that, 3-body continuum curves supersede the 2-body continuum Appearance of successive avoided crossings

  7. Focusing on the potential barrier top region R [fm] Barrier top: The centrifugal term does not decrease with increasing R The combined contribution of the centrifugal and Coulomb terms decrease centrifugal + Coulomb terms ∝ (A+BR)/R2 [A,B] = 0 The nuclear potential term is flat due to 8Be(0+) resonance structure |

  8. Complex absorbing potential CAP converts outgoing waves into exponentially decreasing functions, allows for the use of L2 basis functions CAP is nonzero only in the asymptotic region (large R), and is designed to be transmission-free and to minimize reflection Uν(R) → Uν(R) – iWν(R) C=2.62206 D.E.Manolopoulos, JCP 117 (2002) T.P.Grozdanov et al., JCP 126 (2007) Energies E of hyperradial eq. vs CAP parameters Bound state: EB is unchanged Resonance state: ER – iΓR/2 is independent of CAP parameters Continuum state: E depends on CAP parameters (de Broglie wavelength corresponding to the lowest scattering energy Emin)

  9. Hoyle resonance parameters vs CAP parameters Stable enough to calculate precise resonance parameters!

  10. By replacing ε with R-dependent CAPW(R), Calculation of E2 strength function Triple-α reaction rate is determined by the E2 transition strength from 3α 0+ continuum to the first excited 2+ state of 12C Using detailed balance, we relate its strength to E2 strength function f for the inverse process initiall state←operator←Green'soperator←operator←initial state Final continuum sum is taken by the closure relation and reduces to Green’s operator (resolvent) outgoing waves are made exponentially damp, i.e., discretized

  11. Triple-α reaction rate: α + α + α → 12C(2+) + γ HHR: N.B. Nguyen et al., PRC 87 (2013) Imaginary time: T. Akahori et al., PRC 92 (2015) CDCC: K. Ogata et al., PTP 122 (2009) Faddeev: S. Ishikawa, PRC 87 (2013) Dotter, B. Paxton, Astro. Astrophys.507 (2009) CDCC results contradict observations

  12. II. Resonances inA=4nuclei (in MeV) IsobardiagramforA=4nuclei All levels but the ground state of 4He are unbound Many of them are based on R-matrix analysis

  13. Physics motivation: 4He: The first excited state is a near threshold 0+ resonance that has the same quantum number as the ground state The level sequence of the seven negative-parity resonances 4H and 4Li: Theoretical prediction of broad resonances with T(isospin)=1 (Experimental access to these resonances is hard) Both bound and unbound states of these four-body systems are described with Correlated Gaussians (CG) K.Varga, Y.S., PRC52 (1995) Y.S., K. Varga, Lecture notes in physics 54 (1998)

  14. Structure calculations with realistic NN interactions strong short-range repulsion ΔL=2 and ΔS=2 mixings due to tensor force To cope with the needed correlations we use Correlated Gaussians ux=u1x1+u2x2+u3x3 CG: orbital, spin, isospin parts are explicitly included Parameters of CG: (4-body system) (u1, u2, u3) sets of (L1, L2, S) We superpose many basis functions, each contains many parameters Basis selection: Stochastic variational method

  15. ± Energy levels of 4He threshold ………. CG basis with realistic force reproduces the spectrum The first excited 0+ state (Γ=0.5 MeV) has cluster structure of (3H+p)+(3He+n) JπT Spectroscopic amplitude of the first excited 0+ state to 3He+n channel effective force w/o tensor force

  16. : E1 operator Strengthfunction many-body resolvent Continuum discretization with CSM Moiseyev, Phys. Rep. 302 (1998) damp at large r Eigenvalue problem of complex-scaled Hamiltonian can be solved with L2 basis within certain θ (complex-scaled resolvent) λ: label of eigenvalues μ: z component of multipoles

  17. act one-body operator on the ground state: To construct CG bases for the Gren’s operator, we prepared them not for minimizing energy but for representing different excitation mechanism as well as decay channels (indicated red bold lines) To make sure, we also check the sum rule

  18. Photoabsorption of 4He CSM, θ=17° Experimental data are in discrepancy near the threshold Electric dipole excitation: T=1, Jπ=1- states at 23.64 MeV and 25.96 MeV CG calculation in reasonable agreement

  19. Scaling angle θ dependence + Check of CSM calculation Check of CSM by MRM Detailed balance Photoabsorption cross section can be calculated from the inverse process, radiative capture cross section, which can be calculated by standard reaction theory (microscopic R-matrix theory) A+γ B+C By comparing both results CSM is tested

  20. Isoscalar (IS) Isovector (IV0) Charge-exc.(IV±) (4H, 4Li) A=4 resonances and spin-dipole excitations of 4He All states in 4H and 4Li (2-,1-,0-) are broad resonances They could be reached from the 4He g.s.by spin-dipole (SD) operator SD operator λ=0,1,2(multipolarity)

  21. Spin-dipole excitations of 4He SD resonance is narrower than E1 resonance

  22. Determining resonance parameters from strength functions The first term of the numerator gives the Lorentz distribution The second term contributes to the background distribution Resonance energy and width are determined by the peak position and its half-falloff width of the strength function

  23. Resonance parameters of negative-parity levels of 4He E(θ): Eigenvalues of H(θ) stable against θ: CSM S(E): Peaks of strength function B.A.: Bound-state approximation Both E(θ) and S(E) givefair results In case E(θ) does not work, S(E) may help (MeV) M

  24. Resonance parameters of negative-parity levels of 4H and 4Li

  25. Negative-parity levels of A=4 nuclei Cal. Left E(θ): Stable eigenvalues of H(θ): CSM Middle S(E): Peaks of strength function Right B.A.: Bound-state approximation

  26. Summary Hoyle resonance is studied in adiabatic hyperspherical approach to 3 α particles With CAP its resonance parameter is reliably calculated for the first time, which makes it possible to compute the triple-α reaction rates down to 0.01 GK A=4 resonances are described reasonably well in complex scaling method using Correlated Gaussians: they span both configurations that can reach from the ground state and that are needed to represent the decaying channels Calculation of partial widthremains to be explored Very recent paper concerning a tetra neutron （4n） resonance K. Kisamori et al, PRL116(2016) E. Hiyama, R. Lazauskas, J. Carbonell, M. Kamimura, PRC93 (2016)

  27. Cf. Hyperspherical harmonics expansion popular in nuclear physics Hyperspherical harmonics (HH) is an eigenfunction of Λ2 with eigenvalue K(K+4), where φ is expressed in terms of Jacobi polynomials The advantage is that the channel wave function is expanded in known HH The convergence wrt K, l1, l2 is reasonably fast for short-ranged interactions, especially for small R Since Λ2does not commute with the Coulomb potentials, the couplings among HH’s due to Coulomb interactions persist even at large R

  28. 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 Grid points and weights (Rl, ωl) Gauss-Lobatto quadrature Basis functions Coefficients c are determined from linear equation Convergence wrt channel numbers and grid points are checked

  29. Properties of 12C states [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)

  30. Charge-exchange reaction Almost no data available for comparing to theoretical SD strength functions Spin-nonflip parts E1 Spin-flip parts SD Nakayama et al., PRC76 (2007) 4He 4H

More Related