Constraints on energies of ${}^{10}\mathrm{He}\left(0^{+}\right)$ and ${}^9\mathrm{He}(1/2^{+})$

I have used the relationship between computed energies in ${}^{10}\mathrm{He}$ and single-particle energies in ${}^9\mathrm{He}$ to provide limits on the $s_{1/2}$ energy. The absence of any bound states in ${}^{10}\mathrm{He}$ requires $E_{\mathrm{s}}>1\mathrm{MeV}$, contradicting all the experiments that have reported an $s$ state near threshold. The present analysis supports the view that the variation of ${}^{10}\mathrm{He}$ “ground-state” (g.s.) energies determined in various reactions is caused by the presence of two overlapping $0^{+}$ resonances. Results of the two simplest reactions—proton knockout and ($t,p$)—have been used to extract the g.s. and excited $0^{+}$ energies as a function of the mixing parameter $b^2$ between the $p$-shell and the ${\left(sd\right)}^2$ basis states.

Figure 1

Experimental energies [4, 5, 6, 7, 8, 9, 10, 11, 12] of ${1/2}^{+}$ (closed) and ${5/2}^{+}$ (open) resonances in ${}^9\mathrm{He}$. Labels for various reactions are the same as in Table 1.

Figure 2

Experimental energies [13, 14, 15, 16, 17, 18, 19] of “g.s.” resonance in ${}^{10}\mathrm{He}$. Labels for various reactions are the same as in Table 2.

Figure 3

Experimental widths of “g.s.” resonance in ${}^{10}\mathrm{He}$. Labels for various reactions are the same as in Table 2.

Figure 4

Relationship between the assumed energy of ${1/2}^{+}$ resonance in ${}^9\mathrm{He}$ and the computed energy of the ${\left(sd\right)}^2$ $0^{+}$ state in ${}^{10}\mathrm{He}$ for $E_{\mathrm{d}}=4.20$ (solid line) and 4.93 MeV (dashed line).

Figure 5

In ${}^{10}\mathrm{He}$, plotted vs $E_{\mathrm{s}}$ in ${}^9\mathrm{He}$ are the energies of the $p$-shell $0^{+}$ state (Ref. [20]) (wide solid line), the ${\left(sd\right)}^2$ $0^{+}$ state computed herein (thin solid curve), and the resulting energies of the g.s. (short-dashed curve) and excited $0^{+}$ state (long-dashed curve), assuming a mixing matrix element of $V=1.05\mathrm{MeV}$ (see text).

Figure 6

Plotted vs $b^2$ [the amount of ${\left(sd\right)}^2$ in ${}^{10}\mathrm{He}$(g.s.)] are the energies of the g.s. (lower solid curve) and excited $0^{+}$ state (upper dashed curve) that are required to fit results of energy measurements in proton knockout from ${}^{11}\mathrm{Li}$ and in the reaction ${}^8\mathrm{He}$($t,p$)—assuming the experiments measure the centroid of two overlapping $0^{+}$ resonances. The upper solid curve is the resulting energy of the $p$-shell basis state, and the lower dashed curve is the energy of the ${\left(sd\right)}^2$ $0^{+}$ basis state. This fit makes no assumption about the mixing, but only that a two-state model suffices.

Figure 7

By combining the dependence of $E{\left(sd\right)}^2$ on $b^2$ from Fig. 6 with the dependence of $E_{\mathrm{s}}$ on $E{\left(sd\right)}^2$ from Fig. 4, this plot displays vs $b^2$ the ${1/2}^{+}$ energy in ${}^9\mathrm{He}$ that will fit the ${}^{10}\mathrm{He}$ “g.s.” energies measured in $p$ knockout and ($t,p$).