Strictly finite-range potential for light and heavy nuclei

Strictly finite-range (SFR) potentials are exactly zero beyond their finite range. Single-particle energies and densities, as well as $S$-matrix pole trajectories, are studied in a few SFR potentials suited for the description of neutrons interacting with light and heavy nuclei. The SFR potentials considered are the standard cutoff Woods-Saxon (CWS) potentials and two potentials approaching zero smoothly: the SV potential introduced by Salamon and Vertse [Phys. Rev. C 77, 037302 (2008)] and the SS potential of Sahu and Sahu [Int. J. Mod. Phys. E 21, 1250067 (2012)]. The parameters of these latter potentials were set so that the potentials may be similar to the CWS shape. The range of the SV and SS potentials scales with the cube root of the mass number of the core like the nuclear radius itself. For light nuclei a single term of the SV potential (with a single parameter) is enough for a good description of the neutron-nucleus interaction. The trajectories are compared with a benchmark for which the starting points (belonging to potential depth zero) can be determined independently. Even the CWS potential is found to conform to this benchmark if the range is identified with the cutoff radius. For the CWS potentials some trajectories show irregular shapes, while for the SV and SS potentials all trajectories behave regularly.

Figure 1

Dependence of the mixing coefficient $c$ on the target mass number $A_T$ for two global parameter sets.

Figure 2

Radial shapes of Perey's WS and the SV ($c=0$) potentials and their derivatives for ${}^{18}\text{F}+n$. Derivatives appear in the spin-orbit terms in Eqs. (12) and (15).

Figure 3

Radial shapes of the neutron densities for the nucleus ${}^{18}$F in CWS and in SV potentials.

Figure 4

Radial shapes of the CWS and SS potentials with different $a_s$ values for ${}^{208}\text{Pb}+n$.

Figure 5

Best-fit SS shape to the CWS shape for ${}^{208}\text{Pb}+n$. WS parameters: $r_0=1.27$ fm, $a=0.7$ fm. SS parameters: ${\rho }_0=10.75$ fm, ${\rho }_1=8.94$ fm, $c=1.528$, $a_s=1.4$.

Figure 6

Dependence of the slope of the fitted line on the lower cut value of the node number $n_s$ for a potential in Eq. (24) with a range of $\mathcal{R}=10$ fm.

Figure 7

Convergence of the fitted $\sigma$ to the exact value (dotted line) obtained by using the lower cut value of the node number $n_s$ for a potential of Eq. (24).

Figure 8

Pole trajectories for a CWS potential with $R_{\max }=15$ fm for $l=0$ and $n=1,...,8$ for ${}^{18}$F. The solid circles denote the starting points of the trajectories with $V_0=0.005$ MeV.

Figure 9

The line is the linear function fitted to the $k_n^R$ values (dots) of the pole trajectories with node numbers $n=1,...,8$ for a CWS potential for ${}^{18}$F. These values correspond to the abscissas of the solid circles in Fig. 8. The fit results in a range $\mathcal{R}=14.67$ fm.

Figure 10

Dependence of the slope of the fitted line on the lower cut value of the node number $n_s$ ($n_s=1,\cdots ,n_u-1$ and $n_u=48$) for a CWS potential with $R_{\max }=10$ fm.

Figure 11

Pole trajectories for a SV potential with ${\rho }_0=5.3$ fm for $l=0$ and $n=1,...,5$ for ${}^{18}$F. The solid circles denote the $k_n$ points calculated with $V_0=0.005$ MeV.

Figure 12

Fit to the $k_n^R$ values of the solid circles in Fig. 11, with node numbers $n=1,...,5$ for a single-term SV potential for ${}^{18}$F. The range deduced from the slope $a_1$ is $\mathcal{R}=5.17$ fm.

Figure 13

Pole trajectories in SS potentials with different $a_s$ values. The value $a_s=1.0$ corresponds to the SV potential. The solid circles denote the $k_n$ values calculated with $V_0=0.005$ MeV.

Figure 14

Slope $a_1$ of the straight line fitted to the starting $k_n$ values ($n_s=1,\cdots ,n_u-1$ and $n_u=20$) for SS potentials of different $a_s$, with $a_s=1.0$ belonging to the SV potential; $A_1=\pi /11$ fm${}^{-1}$. Slopes belonging to $a_s=1.0$ and $a_s=1.6$ are hardly distinguishable in the given scale.