Emergent universality in the two-neutron halo structure of ${}^{22}\mathbf{C}$

The structure of the two-neutron halo ${}^{22}\mathrm{C}$ is investigated by means of a renormalized zero-range three-body model, with interactions in the $s$-wave channel, and a finite-range model with two- and three-body forces provided by the hyperspherical adiabatic expansion method. In both models the halo wave function in configuration space is obtained by using as inputs the two-body scattering lengths and the two-neutron separation energy. The halo-matter density is computed for ${}^{22}\mathrm{C}$ with different three-body forces and low-energy parameters, with two-neutron separation energy within the range $50\text{keV}\le S_{2n}\le 1000$ keV. The halo-neutron density depends weakly on the neutron-${}^{20}\mathrm{C}$ scattering length as long as its absolute value is larger than the neutron-neutron one. The halo-neutron density is then analyzed by means of the root-mean-square radius, the probability density, and also the geometry, taking into account the angle between the two Jacobi coordinates. The results of finite-range and zero-range two-neutron-core models are compared. The effects in the halo structure of short-range and long-range three-body forces are studied, and the emergent universal behavior of the halo-neutron density and its geometry is pointed out.

Figure 1

Two lowest angular eigenvalue functions, ${\lambda }_n\left(\rho \right)$, for different values of $K_{\max }$ in the expansion in Eq. (4). The thick (red) solid line corresponds to $K_{\max }=1600$, which is the value used in the calculations.

Figure 2

Computed values of $S_{2n}$ (a) and $\langle {\rho }^2\rangle$ (b) for the two states in Table 2 with ${\ell }_x={\ell }_y=0$ and $V_{3b}^P=0$ (black circles) and $V_{3b}^P=3.1$ MeV (red squares). In the last case the results for $S_{2n}$ have been multiplied by 40 and the ones of $\langle {\rho }^2\rangle$ have been divided by 15.

Figure 3

Two-neutron separation energy $S_{2n}$ as a function of $r_n$. The circles and the squares are the calculations including the ${\ell }_x,{\ell }_y\le 8$ components and the ${\ell }_x={\ell }_y=0$ components, respectively. Panel (a): the solid curves are the ones shown in Fig. 22 of Ref. [15]. The triangles and the stars are the calculations with only $s$ waves but replacing the three-body force in Eq. (11) by a Gaussian and an exponential shape (12) (with ${\rho }_0=5\mathrm{fm})$, respectively. Panel (b): the same as in panel (a) but for ${\rho }_0=1$ fm for the range of the three-body force. The point with error bars expresses the estimated ${}^{22}\mathrm{C}$ neutron halo radius obtained in Ref. [17].

Figure 4

Definition of the polar angles ${\theta }_x^{\prime }$ used for the two possible sets of Jacobi coordinates [panels (a) and (b)] in ${}^{22}\mathrm{C}$.

Figure 5

Halo-neutron density ${\eta }_n\left(r_n\right)$ in ${}^{22}\mathrm{C}$ with the power-dependent three-body potential (11) in panel (a) and the Gaussian three-body potential (12) in panel (b), compared to the zero-range model (represented by the black line with circles in both frames). Two values of $S_{2n}$ are used for the calculations: 250 keV (dashed curves) and 1000 keV (solid curves). The energy of ${}^{21}\mathrm{C}$ is fixed to $E_{nc}=0$. Two different ranges have been considered for the three-body force: ${\rho }_0=1$ fm (blue curves) and ${\rho }_0=5$ fm (red curves).

Figure 6

Black curves: The lowest effective adiabatic potential $V\left(\rho \right)$ [as given in Eq. (6)] for ${}^{22}\mathrm{C}$ when $E_{nc}=0$ and ${\ell }_x={\ell }_y=0$. Solid curves: The lowest effective potential plus the three-body force producing a ${}^{22}\mathrm{C}$ state with $S_{2n}=1000$ keV. Dashed curves: The lowest effective potential plus the three-body force producing a ${}^{22}\mathrm{C}$ state with $S_{2n}=250$ keV. Two different ranges have been considered for the three-body force, ${\rho }_0=1$ fm (blue curves), and ${\rho }_0=5$ fm (red curves). In panel (a) the three-body force corresponds to the power dependence in Eq. (11), and in panel (b) the three-body force has the Gaussian shape given in Eq. (12).

Figure 7

Contour plot of the density $F(r_x^{\perp },r_x^{\left|\right|})$ defined in Eq. (22) for the polar angle ${\theta }_x^{\prime }$ defined as in Fig. 4. Panels (a), (b), (c), and (d) correspond to a ${}^{22}\mathrm{C}$ two-neutron separation energy of 1000, 500, 250, and 50 keV, respectively.