Energy dependence of the total cross section for the ${}^{11}\mathrm{Li}+{}^{28}\mathrm{Si}$ reaction

In this paper the results of measurements of the total cross sections for the ${}^{11}\mathrm{Li}+{}^{28}\mathrm{Si}$ reaction in the beam energy range $7A$–$25A$ MeV are presented. The experimental cross sections were obtained by registration of the prompt gamma and neutron radiation accompanying the interaction of ${}^{11}\mathrm{Li}$ nuclei with ${}^{28}\mathrm{Si}$ nuclei. The total cross sections for the ${}^{11}\mathrm{Li}+{}^{28}\mathrm{Si}$ reaction are calculated based on a numerical solution of the time-dependent Schrödinger equation for the external weakly bound neutrons of the projectile nucleus ${}^{11}\mathrm{Li}$. The calculated total reaction cross sections are in good agreement with the experimental data.

Figure 1

Three-dimensional scheme of the experimental setup (a) and its section with the dimensions (b): (1) polyethylene plates for reducing the energy of the beam particles; (2) scintillation detector AC1; (3) position-sensitive Si detector $\mathrm{\Delta }E$ (16X:16Y); (4) start $\mathrm{\Delta }{E}_0$ Si detector; (5) scintillation detector AC2; (6) target $\mathrm{\Delta }{E}_{\mathrm{T}}$ Si PIN detector; (7) CsI(Tl) scintillation $\gamma$ spectrometer; (8) window for the output of the beam (stainless steel foil).

Figure 2

(a) Two-dimensional spectrum $\mathrm{\Delta }{E}_0{T}_{\mathrm{TOF}}$ used for identification of secondary beam particles consisting of ${}^8\mathrm{He}$ and ${}^{11}\mathrm{Li}$; the ionization losses of particles in the $\mathrm{\Delta }{E}_0$ detector are along the ordinate axis; the time of flight of the beam particles at the distance of the time-of-flight base $L_{\mathrm{TOF}}=8.5\mathrm{m}$ is along the abscissa. (b) The two-dimensional spectrum $T_{\mathrm{AC}1}{T}_{\mathrm{AC}2}$ used to select the events of ${}^8\mathrm{He}$ and ${}^{11}\mathrm{Li}$ particles incident on the central spot of the target. The times $T_{\mathrm{AC}1}$ and $T_{\mathrm{AC}2}$ of arrival of the signals from the detectors AC1 and AC2 relative to the start time from the $\mathrm{\Delta }{E}_0$ detector are plotted along the abscissa and the ordinate axes, respectively. Events marked by points inside the contours on the spectra $\mathrm{\Delta }{E}_0{T}_{\mathrm{TOF}}$ and $T_{\mathrm{AC}1}{T}_{\mathrm{AC}2}$ form a set of events corresponding the beam particles incident on the central spot of the target.

Figure 3

(a) Detection efficiency of a group of $M$ photons by the spectrometer (spectrometer response). Experimental results for gamma energy 1332 keV (filled squares) and geant4 [13] calculations for gamma energies 400 keV (triangles), 1332 keV (empty squares), and 5000 keV (circles). (b) The relative frequencies $w_M\left(k\right)$ of the number of triggered spectrometer detectors in the registration of $M$ photons.

Figure 4

Experimental (symbols) and smoothed by splines (curves) energy dependence of the total cross sections for the reactions ${}^9\mathrm{Li}+{}^{28}\mathrm{Si}$ (a): filled circles (Sobolev 2017 [3]), filled triangles (Penionzhkevich 2017 [2]), empty triangles (Warner 1996 [5]); (b) ${}^6\mathrm{He}+{}^{28}\mathrm{Si}$: empty circles (Warner 1996 [5]), filled triangles (Penionzhkevich 2017 [2]), filled circles (Sobolev 2005 [14], Ugryumov 2005 [15]); the results of this work: empty stars are the lower boundaries of the total reaction cross sections ${\tilde{\sigma }}_{\mathrm{R}}$, filled stars are the total reaction cross sections ${\sigma }_{\mathrm{R}}$ for correction $\eta ={\eta }_0=0.85$ (a) and $\eta =0.78\pm 0.04$ (b). The dashed lines (b) are the upper and lower boundaries of the confidence interval for the total reaction cross section.

Figure 5

The dependence of the correction $\eta$ on the separation energy of one (empty circles) and two (filled circles) external neutrons for the ${}^9\mathrm{Li}$ and ${}^6\mathrm{He}$ nuclei. Solid line is the result of linear interpolation.

Figure 6

Experimental energy dependence of the total cross sections for the reaction ${}^{11}\mathrm{Li}+{}^{28}\mathrm{Si}$: filled (red) squares are the results of this work. The remaining data for ${}^{11}\mathrm{Li}$: filled diamonds (Warner 1996 [5]), empty circle (Villari 1991 [16]), and empty diamonds (Li Chen 2007 [17]). Data for ${}^9\mathrm{Li}+{}^{28}\mathrm{Si}$: filled circles (Sobolev 2017 [3]), filled triangles (Penionzhkevich 2017 [2]), and empty triangles (Warner 1996 [5]).

Figure 7

Schemes of neutron levels for the nuclei ${}^7\mathrm{Li}$ (a), ${}^9\mathrm{Li}$ (b), and ${}^{11}\mathrm{Li}$ (c) in the shell model of the deformed nucleus and in the shell model of the spherical nucleus ${}^{11}\mathrm{Li}$ (d).

Figure 8

(a) The total cross section for the ${}^9\mathrm{Li}+{}^{28}\mathrm{Si}$ reaction. Filled circles are the experimental data from Fig. 4 (Sobolev 2017 [3], Penionzhkevich 2017 [2], Warner 1996 [5]); the curve is the result of calculations in the optical model with the energy-dependent optical potential [2]. (b) Probabilities $P_{\mathrm{core}}(b,E)={\tilde{P}}_{\mathrm{core}}(kb,E)$ depending on the impact parameter $b$ for energies $2.5A$ MeV (solid line), $5A$ MeV (dashed line), $12.3A$ MeV (dash-dotted line), and $50.6A$ MeV (dotted line).

Figure 9

An example of the evolution of the probability density for external neutrons of the ${}^{11}\mathrm{Li}$ nucleus in the collision with the ${}^{28}\mathrm{Si}$ nucleus at energy $E=12.6A$ MeV. The location of panels (a)–(d) corresponds to time.

Figure 10

The probabilities of neutron transfer to the unoccupied bound states of the discrete spectrum in the ${}^{28}\mathrm{Si}$ nucleus (a) and the probabilities of transfer to the states of the continuous spectrum calculated by formula (28) with $C=1, \mathrm{\Delta }{R}_1=3$ fm, and $\mathrm{\Delta }{R}_2=10$ fm (b) for energies $2.5A$ MeV (solid line), $6.3A$ MeV (dash-dotted line), $12.6A$ MeV (dashed line), $25.3A$ MeV (dotted line), and $50.6A$ MeV (dash-dot-dotted line). Arrows indicate the position of the Coulomb barrier.

Figure 11

The total cross section for the ${}^{11}\mathrm{Li}+{}^{28}\mathrm{Si}$ reaction. Symbols are experimental data. Filled (red) squares are the results of this work, filled diamonds (Warner 1996 [5]), empty circle (Villari 1991 [16]), and empty diamonds (Li Chen 2007 [17]). Curves are the results of calculations for the values of the adjustable parameter $C=2$ (solid curve) and $C=1$ (dashed curve) with the probability $P_{\mathrm{c}}$ of transition to the states of the continuous spectrum calculated by formula (28).