Pairing effects on vorticity of incident neutron currents at quasiparticle resonance energies in $n\text{−}A$ elastic scattering

In this study, we analyzed how the incident neutron current is affected by the pairing effect in neutron-nucleus scattering described within the framework of Hartree-Fock-Bogoliubov theory by performing numerical calculations in terms of current, vorticity, and circulation of the incident neutron current. We found that the pairing effect on the incident neutron flux is completely different between particle-type and hole-type quasiparticle resonances. In the case of h-type quasiparticle resonance, the pairing acts to prevent the neutron flux from entering the nucleus, reducing circulation. In the case of p-type quasiparticle resonance, pairing acts to reduce circulation at energies lower than the resonance energy, but, at energies higher than the resonance energy, the effect of pairing on the neutron flux is reversed and, conversely, circulation is increased. These properties are consistent with the approximate expression for the vorticity characteristics at p-type and h-type resonances, implying that the vorticity appearing near p-type resonances due to the pair-correlation effect is inversely proportional to the resonance width (proportional to the resonance lifetime). In contrast, at h-type resonances, the pair-correlation effect has the effect of canceling out the vortex created by MF scattering and is independent of the resonance width.

Figure 1

The neutron currents (a) $\mathit{j}$, (c) ${\mathit{j}}_0$, and (e) ${\mathit{j}}_{\mathrm{pair}}$ at $\epsilon =3.94$ MeV (an energy corresponding to the h-type $d_{3/2}$ resonance) with $\lambda =-1.0$ MeV and $\langle \mathrm{\Delta }\rangle =3.0$ MeV. The panels (b), (d), and (f) are the vortices corresponding to the currents (a), (c), and (e), respectively. The arrows representing current are the same length. The strength (absolute value) of the current, which is normalized by $k(=\sqrt{{\frac{2m}{{\hslash }^2}(\lambda +E)}^{\phantom{\int }}})$ (the absolute value of the plane wave) is represented by color. The unit of the vorticity is ${\mathrm{fm}}^{-2}$. The black solid semicircle shows the size of the target nucleus given by $r_0{A}^{\frac{1}{3}}=3.66$ fm.

Figure 2

The contour plot of $\mathrm{\Gamma }$ as a function of $\epsilon$ and $\langle \mathrm{\Delta }\rangle$. The dotted curves represent the real part of the $S$-matrix poles for the quasiparticle resonances which were shown in Ref. [20].

Figure 3

The circulation obtained by integrating the area of the region $0\le x\le 10$ fm, $-10\le z\le 10$ fm (the area as shown in Fig. 1), or by line integrating this region in a counterclockwise direction. $\mathrm{\Gamma }$ (red solid), ${\mathrm{\Gamma }}_0$ (black dash), ${\mathrm{\Gamma }}_{\mathrm{pair}}$ (blue dot), and $\delta \mathrm{\Gamma }$ (purple dot-dash) are the circulations which are corresponding to $\mathit{\omega }$, ${\mathit{\omega }}_0$, ${\mathit{\omega }}_{\mathrm{pair}}$, and $\delta \mathit{\omega }$, respectively. The lower panel shows an enlargement of the region $0\le \epsilon \le 2$ MeV to make it easier to see the circulation for the p-type resonances.

Figure 4

The current ${\mathit{j}}_{\mathrm{pair}}$ and vorticity ${\mathit{\omega }}_{\mathrm{pair}}$ at $\epsilon =1.45-1.16\times {10}^{-2}$, 1.45, and $1.45+1.16\times {10}^{-2}$ MeV; $\epsilon =1.45$ MeV is the energy at which ${\mathrm{\Gamma }}_{\mathrm{pair}}$ becomes zero (found near the real part of the $S$-matrix pole of the p-type $f_{7/2}$ resonance), and $\pm 1.16\times {10}^{-2}$ MeV is the imaginary part of the $S$-matrix pole (half-width of the resonance).

Figure 5

The pairing dependence of the circulations $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}_{\mathrm{pair}}$ around the energy region for the p-type $p_{3/2}$ resonance. The solid blue, red, and purple curves represent the circulation with $\langle \mathrm{\Delta }\rangle =2.0$, 3.0, and 4.0 MeV, respectively. The dotted curves show the circulation calculated every 0.2 MeV.