Halo nuclei ${}^6\mathrm{He}$ and ${}^8\mathrm{He}$ with the Coulomb-Sturmian basis

The rapid Gaussian falloff of the oscillator functions at large radius makes them poorly suited for the description of the asymptotic properties of the nuclear wave function, a problem which becomes particularly acute for halo nuclei. We consider an alternative basis for ab initio no-core configuration interaction (NCCI) calculations, built from Coulomb-Sturmian radial functions, allowing for realistic exponential falloff. NCCI calculations are carried out for the neutron halo nuclei ${}^{6,8}\mathrm{He}$, as well as for the baseline case ${}^4\mathrm{He}$, with the JISP16 nucleon-nucleon interaction. Estimates are made for the root-mean-square radii of the proton and matter distributions.

Figure 1

The calculated wave function obtained when a problem with exponential asymptotics—here, the Woods-Saxon problem is taken for illustration—is solved in a finite basis of oscillator functions. The radial probability density $r^2{\left|\phi \left(r\right)\right|}^2$ is shown on a logarithmic scale, so that exponential asymptotics would appear as a straight line. The Woods-Saxon (WS) and harmonic oscillator (HO) potentials are shown in the inset. (Solutions are for the Woods-Saxon $1s_{1/2}$ function, with potential parameters appropriate to neutrons in ${}^{16}\mathrm{O}$ [16], with maximal basis radial quantum numbers $n$ as indicated.)

Figure 2

Radial functions (a) $R_{nl}(b;r)$ of the harmonic oscillator basis and (b) $S_{nl}(b_l;r)$ of the Coulomb-Sturmian basis, with $b_l$ given by the node-matching prescription (6). These functions are shown arranged according to the harmonic oscillator principal quantum number $N\equiv 2n+l$ (see text) and are labeled by $l$. The dotted curves show the same functions dilated outward by a factor of $\sqrt{2}\approx 1.414$, corresponding to a factor of 2 reduction in $\hslash \Omega$.

Figure 3

The calculated ${}^4\mathrm{He}$ ground-state energy (top) and rms proton radius $r_p$ (bottom), using the conventional oscillator (left) and Coulomb-Sturmian (right) bases. These are shown as functions of the basis $\hslash \Omega$ parameter, for $N_{\max }=4$ to 16 (as labeled) and for transformation cutoffs $N_{\mathrm{cut}}=9$, 11, and 13 (Coulomb-Sturmian basis only, indicated by dashing, curves nearly indistinguishable). The converged values obtained with the JISP16 interaction are indicated by dashed horizontal lines. The spreads in radius values over this $\hslash \Omega$ range, at the highest $N_{\max }$, are indicated by vertical bars (bottom).

Figure 4

The calculated ${}^6\mathrm{He}$ ground-state energy (top) and rms proton radius $r_p$ and matter radius $r_m$ (bottom), using the conventional oscillator basis (left), Coulomb-Sturmian basis (center), and proton-neutron asymmetric Coulomb-Sturmian basis with $\beta =1.414$ (right). These are shown as functions of the basis $\hslash \Omega$ parameter, for $N_{\max }=6$ to 16 (as labeled). Exponentially extrapolated energies from the present calculations are indicated by open symbols, the best extrapolated energy from Ref. [47] is indicated by the dashed horizontal line, and the ${}^4\mathrm{He}+2n$ breakup threshold obtained with JISP16 [47] is marked by the solid horizontal line (top panels). The spreads in radii over this $\hslash \Omega$ range, at the highest $N_{\max }$, are indicated by vertical bars (bottom panels).

Figure 5

The calculated ${}^8\mathrm{He}$ ground-state energy (top) and rms proton radius $r_p$ and matter radius $r_m$ (bottom), using the conventional oscillator basis (left), Coulomb-Sturmian basis (center), and proton-neutron asymmetric Coulomb-Sturmian basis with $\beta =1.414$ (right). These are shown as functions of the basis $\hslash \Omega$ parameter, for $N_{\max }=4$ to 14 (as labeled). Exponentially extrapolated energies from the present calculations are indicated by open symbols, the best extrapolated energy from Ref. [47] is indicated by the dashed horizontal line, and the ${}^6\mathrm{He}+2n$ breakup threshold obtained with JISP16 [47] is marked by the solid horizontal line (top panels). The spreads in radii over this $\hslash \Omega$ range, at the highest $N_{\max }$, are indicated by vertical bars (bottom panels).

Figure 6

The ${}^4\mathrm{He}$ ground-state rms proton radius $r_p$, as estimated from the crossover point (see text), calculated for the harmonic oscillator and Coulomb-Sturmian bases (as indicated in the legend). The relative deviations from the converged value may be read from the right-hand axis.

Figure 7

The (a) ${}^6\mathrm{He}$ and (b) ${}^8\mathrm{He}$ ground-state rms proton radius $r_p$ (lower curves) and matter radius $r_m$ (upper curves), as estimated from the crossover point (see text), calculated for the harmonic oscillator basis, the Coulomb-Sturmian basis, and the proton-neutron asymmetric Coulomb-Sturmian basis with $\beta =1.414$ (as indicated in the legend).

Figure 8

The ${}^4\mathrm{He}$ proton radius and ${}^{6,8}\mathrm{He}$ proton and matter radii, as estimated from the crossover point, for the harmonic oscillator basis, the Coulomb-Sturmian basis, and the proton-neutron asymmetric Coulomb-Sturmian basis with $\beta =1.414$. For each of these bases, the extracted radii are shown for the highest three successive $N_{\max }$ values ($12\le N_{\max }\le 16$ for ${}^{4,6}\mathrm{He}$ or $10\le N_{\max }\le 14$ for ${}^8\mathrm{He}$), in some cases visually indistinguishable, with the largest symbol indicating the highest $N_{\max }$ value. Experimental values or ranges are shown as horizontal bands.