Nonresonant Density of States Enhancement at Low Energies for Three or Four Neutrons

The low energy systems of three or four neutrons are treated within the adiabatic hyperspherical framework, yielding an understanding of the low energy quantum states in terms of an adiabatic potential energy curve. The dominant low energy potential curve for each system, computed here using widely accepted nucleon-nucleon interactions with and without the inclusion of a three-nucleon force, shows no sign of a low energy resonance. However, both systems exhibit a low energy enhancement of the density of states, or of the Wigner–Smith time delay, which derives from long-range universal physics analogous to the Efimov effect. That enhancement could be relevant to understanding the low energy excess of correlated four-neutron ejection events observed experimentally in a nuclear reaction by Kisamori et al. [Phys. Rev. Lett. 116, 052501 (2016)].

Figure 1

(a) Hyperspherical potential curve for the most attractive channel ($0^{+}$ for 4n, ${\frac{3}{2}}^{-}$ for 3n in the inset) for both the 4n and 3n systems. Comparison of the lowest $0^{+}$ 4n adiabatic hyperspherical potential energy curves computed with the HH method (blue dash–dotted curve, which shows the best calculation for the AV18 Hamiltonian with Kmax $=140$ and is not accurately converged at hyperradii beyond approximately 20 fm). The lowest solid magenta 4n (and 3n in the inset) potential energy curve is computed using the CGHS method applied to the AV8’ Hamiltonian. The open circles are the adiabatic potentials calculated using a simple NN Gaussian potential (see text). The lower dashed gray curves in both the main figure and the inset are the expected long-range ${\rho }^{-2}$ potentials at unitarity for this symmetry of the 4n and 3n systems, i.e., in the infinite scattering length limit (see text and Table 2). The upper dashed gray curves are the corresponding potentials for noninteracting neutrons. Clearly there is no local minimum and no local maximum of the type that is always associated with a quasibound resonance. (b) Plot of the function $C(\rho )\equiv (\rho /a)[{\rho }^2{u}_0(\rho )2\mu /{\hslash }^2-l_{\mathrm{eff}}(l_{\mathrm{eff}}+1)]$ for the 3n case. According to Eq. (3), we should obtain $C(\rho \to \infty )=C$, where $C$ is the coefficient listed in Table 2. We observe the slow convergence for large $\rho$ of the adiabatic potentials calculated using the HH basis. However, it has to be noted that where the convergence is achieved, the functions $C(\rho )$ obtained for the different interactions used in this work almost collapse onto a single curve. Noticeably, this happens already for fairly small values of $\rho$, showing that the adiabatic potentials are already universal at moderate values of the hyperradius. In fact, the limit $C(\rho )=C$ is reached only for $\rho >500\mathrm{fm}$.

Figure 2

Rescaled Wigner–Smith time delays $2\sqrt{E}d\delta /dE$ for the AV8’ interaction (solid curves) and the simple Gaussian interaction (open circles). The 4n results are the higher curve, the 3n results the lower. These are computed in the lowest adiabatic hyperspherical potential energy curve for the 4n $0^{+}$ symmetry, and for the 3n ${\frac{3}{2}}^{-}$ symmetry. These show no local maximum that would be expected for a low energy resonance. Note the $E^{-1/2}$ dependence of the Wigner–Smith time delay (or density of states) in the zero energy limit, a consequence of the ${\rho }^{-3}$ term in the long-range potentials for the 3n and 4n systems. The inset shows the elastic scattering phaseshift versus the square root of the energy. Both cases show the proportionality to $\sqrt{E}$ dependence that holds in the zero energy limit, a consequence of the ${\rho }^{-3}$ long-range potential energy term. The open circles that lie almost exactly on top of these curves are computed using the lowest 3n and 4n hyperspherical potential curves based on a simple two-body Gaussian potential interaction (see text).