Pairing in the continuum: The quadrupole response of the Borromean nucleus ${}^6\mathrm{He}$

The ground state (g.s.) and low-lying continuum states of ${}^6\mathrm{He}$ are found within a shell model scheme, in a basis of two-particle states built out of continuum $p$ states of the unbound ${}^5\mathrm{He}$ nucleus, using a simple pairing contact-delta interaction. This accounts for the Borromean character of the bound ground state, revealing its composition. We investigate the quadrupole response of the system and we put our calculations into perspective with the latest experimental results. The calculated quadrupole strength distribution reproduces the narrow $2^{+}$ resonance, while a second wider peak is found at about 3.9 MeV above the g.s. energy.

Figure 1

Left: experimental energy levels (resonances) in ${}^5\mathrm{He}$. Center: unperturbed energies of two-particle states built upon the scheme on the left. Right: experimental energy levels (bound ground state and resonances) of ${}^6\mathrm{He}$. Shades of gray and pink indicate widths. The experimental energies are from Ref. [18] (in black) and Ref. [19] (in red). Parentheses indicate uncertain spin-parity assignment.

Figure 2

Calculated square of ${}^5\mathrm{He}$ continuum wave functions (top: $p_{1/2}$; bottom: $p_{3/2}$) as a function of radial variable and continuum energy. Notice the different scales in the color schemes.

Figure 3

(a) Eigenspectrum of the interacting two-particle case for $J=0$ for increasing basis dimensions, $N$. (b) The coefficient of the $\delta$-contact matrix, $G$, has been adjusted each time to reproduce the g.s. energy. The actual strength of the pairing interaction, $g$, is obtained by correcting with the energy spacing $\Delta E$, and it is practically a constant.

Figure 4

Ground state wave function ($S=0$) for $N=100$ as a function of the coordinates of the two neutrons and corresponding contour plot (upper part). Decomposition of the g.s. into the $J=0$ basis (lower part) as a function of an arbitrary basis state label: the basis is divided in two blocks, ${10}^4$ ${[p_{1/2}\times p_{1/2}]}^{\left(0\right)}$ components and then ${10}^4$ ${[p_{3/2}\times p_{3/2}]}^{\left(0\right)}$ components. The ordering in each block is established by the sequential energies of each pair of continuum single-particle states, i.e., $(E_{C_1},E_{C_2})=(0.1,0.1)$, (0.1, 0.2),…, (0.1, 10.0), (0.2, 0.1), (0.2, 0.2),…, (10.0, 10.0).

Figure 5

Quadrupole strength distribution with respect to the breakup threshold. The total strength (black) is split into the contribution of the ${\left(p_{3/2}\right)}^2$ and $\left(p_{3/2}{p}_{1/2}\right)$ components, dashed (blue) and dot-dashed (red) respectively. The inset shows the full curve for the total strength.