Resonances and collisional properties of neutron-rich helium isotopes in the adiabatic hyperspherical representation

TL;DR

This work employs the adiabatic hyperspherical representation to solve the N-body Schrödinger equation for neutron-rich helium isotopes, focusing on $^{6}$He and $^{7}$He within a simplified $^{4}$He core–neutron and core–neutron–neutron framework. By combining angular-momentum–dependent two-body interactions, a spin-dependent neutron–neutron force, and a spin-dependent three-body force, the authors reproduce the $^{6}$He ground state and the $2^{+}$ resonance, and predict the $^{7}$He $3/2^{-}$ resonance, along with elastic, inelastic, and four-body continuum processes. They compute a quadrupole transition between $^{6}$He $0^{+}$ and $2^{+}$ states and analyze phase shifts, cross sections, and recombination rates, comparing with experimental data and No-Core Shell Model results. The results validate the utility of a constrained few-body Hamiltonian in capturing essential features of light halo nuclei and lay groundwork for extending the approach to the more complex $^{8}$He system and related five-body dynamics.

Abstract

This work treats few-body systems consisting of neutrons interacting with a $^{4}{\mathrm{He}}$ nucleus. The adiabatic hyperspherical representation is utilized to solve the $N$-body Schr$\ddot{\mathrm{o}}$dinger equation for the three- and four-body systems, treating both $^{6}{\mathrm{He}}$ and $^{7}{\mathrm{He}}$ nuclei. A simplified central potential model for the $^{4}{\mathrm{He}}-n$ interaction is used in conjunction with a spin-dependent three-body interaction to reproduce $^{6}{\mathrm{He}}$ bound-state and resonance properties as well as properties for the $^{8}{\mathrm{He}}$ nucleus in its ground-state. With this Hamiltonian, the adiabatic hyperspherical representation is used to compute bound and scattering states for both $^{6}{\mathrm{He}}$ and $^{7}{\mathrm{He}}$ nuclei. For the $^{6}{\mathrm{He}}$ system, the electric quadrupole transition between the $0^{+}$ and $2^{+}$ state is investigated. For the $^{7}{\mathrm{He}}$ system, $^{6}{\mathrm{He}}+n$ elastic scattering is investigated along with the four-body recombination process $^{4}{\mathrm{He}}+n+n+n\rightarrow$$^{6}{\mathrm{He}}+n$ and breakup process $^{6}{\mathrm{He}}+n\rightarrow$$^{4}{\mathrm{He}}+n+n+n$.

Figures (9)

• Figure 1: In (a) the $\prescript{4}{}{\mathrm{He}}-n$ phase shifts in the energy range of $0<E<30~\mathrm{MeV}$. The $s$--wave interaction is taken from PhysRevLett.79.2411 and the interactions for $l>0$, along with the spin--orbit interaction, are taken from PhysRevC.61.024318. In (b) the $\prescript{4}{}{\mathrm{He}}-n$ phase shifts in the energy range of $0<E<30~\mathrm{MeV}$ for the simplified model, without a spin--orbit interaction. The $s$--wave and $d$--wave interactions are taken from PhysRevLett.79.2411 and the $p$--wave interaction is taken from PhysRevC.61.024318.
• Figure 2: The spin singlet (left panel) and triplet (right panel) three--body interaction potentials for the $\prescript{4}{}{\mathrm{He}}-nn$ subsystem. These potentials are plotted as a function of the magnitude of three--body Jacobi vectors $\vec{\rho}_{1}$ and $\vec{\rho}_{2}$. The magnitude of $\vec{\rho}_{1}$ is proportional to the $nn$ inter-particle distance and $\vec{\rho}_{2}$ is the vector connecting the center of mass of the $nn$ subsystem to the $\prescript{4}{}{\mathrm{He}}$ nucleus. The interaction parameters for the singlet interaction were tuned to reproduce the $\prescript{6}{}{\mathrm{He}}(0^+)$ ground state energy and the $\prescript{6}{}{\mathrm{He}}(2^+)$ resonance. The parameters for the triplet interaction were tuned to reproduce the $\prescript{8}{}{\mathrm{He}}(0^+)$ ground state energy and rms matter radius.
• Figure 3: The lowest few adiabatic hyperspherical potential curves for the $\prescript{4}{}{\mathrm{He}}-nn$ system are displayed in the $(L^{\pi},S)J^{\pi}=(0^{+},0)0^{+}$ symmetry in (a) and $(L^{\pi},S)J^{\pi}=(2^{+},0)2^{+}$ in (b). The solid curves are the adiabatic potentials $U_{\nu}(\rho)$ and the dashed curves include the second--derivative non--adiabatic coupling term. The visible peaks in the second and third potentials in (b) are due to the avoided crossing clearly seen near $\rho=17~\mathrm{fm}$.
• Figure 4: The sum of the eigenphase shifts (see inset) and their energy derivatives (see main figure) are shown for the $\prescript{6}{}{\mathrm{He}}(2^{+})$ system over an energy range from $0.6~\mathrm{MeV}$ to $1.4~\mathrm{MeV}$. Each curve represents a calculation performed by solving Eq. \ref{['eq:coupled_channel_eqs']} where 1, 3, 6, and 8 channels are included. There is a clear peak in the energy derivative at $0.80~\mathrm{MeV}$ with a corresponding increase in phase of $\approx2$ radians, indicating the presence of the $2^{+}$ resonance state of $\prescript{6}{}{\mathrm{He}}$.
• Figure 5: The square of the quadrupole transition between the $\prescript{6}{}{\mathrm{He}}$$0^{+}$ ground state and the $\prescript{6}{}{\mathrm{He}}$$2^{+}$ continuum states at some final state energy $E$. The peak in the transition matrix element corresponds to the resonance position of the $2^{+}$ state and the FWHM is on the order of the width of the resonance. The inset shows a comparison with a previous hyperspherical calculation PhysRevLett.79.2411.