It is well known that neutron rearrangement may play an important role in nuclear reactions. The aim of this work is the investigation of the reactions with light nuclei having different external neutron shells. The experiments on measurements of total cross sections were performed for reactions He + Si and Li + Si. The interesting results are the unusual wide enhancement of total cross section for Li + Si reaction as compared with Li + Si reactions. The similar weaker behavior was found for He + Si reaction as compared with He + Si reaction. The time-dependent quantum approach combined with the optical model was used for explanation of these effects. Based on this approach the observed local enhancements of total reaction cross sections for the studied reactions were explained by rearrangement of external weakly bound neutrons of projectile nuclei during the collision.
The experimental setup for the implementation of the transmission method using a multilayer telescope  is shown in Fig. 1. A system of silicon detectors E -E, (i = 0–4) (Si-telescope) was surrounded by the CsI(Tl) -spectrometer of complete geometry for registration of -rays and neutrons. The thin detectors E , E were used to identify the beam particles and determine the particle flux incident on the target. The position-sensitive detector E was used as a so-called active collimator  which determined the particle flux incident on the central region of the target E . The detectors E , E were used to analyze the products of reactions occurring in the material of the target E .
The experiment was performed on the accelerator U400M of the Flerov Laboratory of Nuclear Reactions (FLNR), Joint Institute for Nuclear Research (JINR). To obtain the secondary beam the fragmentation reaction of B beam with the energy E = 32 A MeV on the target Be was used. The secondary beam consisting of a mixture of particles He and Li was formed and purified by the magnetic system of the achromatic fragment separator ACCULINNA . The beam energy was varied by a fragment separator magnetic system, the choice of the thickness of the hydrogen-containing plates of CH absorbers in the range E 5 50 A MeV without significant loss of intensity of the beam of particles. Identification was carried out by energy losses of particles in E , E detectors of the telescope and the time of flight. In order to reduce the energy uncertainty, detectors of different thickness (100, 380, or 500 microns) were used in the experiment depending on the beam energy. Detectors of γ-spectrometer recorded γ-quanta and neutrons in coincidence with the start signal from the detector E . The number of events of the reaction was determined from the analysis of energy losses in natural Si-target as well as the analysis of gamma and neutron radiation detected by the spectrometer.
The results of measurements of total cross sections for reactions Li + Si and He + Si are presented in Fig. 2.
The cross section for the reaction for the He nucleus exceeds the cross section with the He nucleus in the entire energy range, which may be explained by the large size of the He nucleus. The measurements showed that the dependence on energy of the total cross section for the reaction Li + Si has a broad maximum. Enhancement of the cross section for Li nuclei compared to Li is observed in the energy range 10 30 A MeV.
The analysis of these effects using the microscopic complex folding potential in Ref.  as well as within the optical model in Ref.  did not provide satisfactory explanation of the observed features in the behavior of the energy dependence of the total cross section. In this study, the potentials of the optical model were modified to take into account the dynamic rearrangement of two external neutrons of projectile nuclei He and Li. The obtained results are in good agreement with the experimental data (Fig. 2).
For theoretical description of neutron rearrangement during collisions of atomic nuclei we used the time-dependent Schrödinger equation (TDSE) approach [12 14] for the external neutrons combined with the classical equations of motion of atomic nuclei. The evolution of the components of the spinor wave function for the neutron with the mass m during the collision of nuclei is determined by Eq. (1) with the operator of the spin-orbit interaction
(1) The centers of nuclei with the masses move along classical trajectories. We may assume that before contact of the surfaces of spherical nuclei with the radii R , R the potential energy of a neutron is equal to the sum of its interaction energies with both nuclei. The initial conditions for the wave functions were obtained based on the shell model calculations with the parameters providing neutron separation energies close to the experimental values.
Examples of the evolution of the probability density of the external neutrons of Li nucleus when colliding with the nucleus Si at different energies were given in Ref. . During a slow (adiabatic) relative motion of the colliding nuclei the external neutrons (dineutron cluster) of Li nucleus are penetrating the Si nucleus and populating the slowly changing two-center (“molecular”) states, the probability density for which fills a large part of the volume of the target nucleus. During the rapid (diabatic) relative motion the probability density of neutrons does not have time to fill all the target nucleus and its change is more local. After the separation of the nuclei the wave packet in the surface region of the target nucleus remains spreading and rotating with large angular momentum. At intermediate velocities there is a transition from the adiabatic regime to the diabatic one.
The qualitative character of the rearrangement of external neutrons during the approach of nuclei depends on the ratio of the average velocity of the external neutron and the relative velocity of the nuclei in the process of collision. The average kinetic energy of weakly bound neutrons in the nuclei He and Li may be approximately calculated within the shell model. Using estimation , where is the energy of the projectile nucleus with the mass , is the atomic mass unit, we obtain the ratio of velocities
(2) At low energies, when , , during the flight of the projectile nucleus close to the target nucleus the weakly bound neutrons may, relatively speaking, make many turns around the cores of both nuclei. In the extremely diabatic case (at intermediate energies), when , , the neutron may not be able to move to the target nucleus during the time of flight. The value of the parameter may be used to estimate the degree of adiabaticity of the collision.
The real part of the potential for nuclei with “frozen” neutrons was supplemented with the diabatic correction arising from an increase in neutron density between the surfaces of the nuclei as they approach
(3) with the function
(4) where is the mean field for neutrons in the target nucleus, , is the probability density of the external neutrons of the projectile nucleus, is the same density calculated in the absence of interaction of these neutrons with the target nucleus, is the region between the surfaces of the nuclei,
(5) with the variable parameters 10 MeV determining the position of the transition region and 2 MeV determining its width. The diabatic correction reduces the height and shifts to the right the position of the Coulomb barrier
(6) For the imaginary part of the potential we used the approximation with the exponential dependence
(7) and the radius , increasing according to the shift of the barrier position
(8) where b = 1 fm, k = 2, = 5.8 fm for the reaction Li + Si. In the case of reactions with nuclei He, Li for the real and the imaginary parts of the nuclear potential the Woods Saxon form was used
For collisions Li + Si the parameters of the real part of the potential , , were obtained by fitting the angular distributions of the elastic scattering. The results of calculation of the total cross sections for reactions Li + Si thus obtained (Fig. 2) are also in good agreement with the experimental data [1, 6, 8].
In this paper we presented experimental results of a direct measurement of the total cross sections for the reactions He + Si and Li + Si in the beam energy range 5 50 A MeV. The enhancements of the total cross sections for reactions He + Si and Li + Si have been observed. The theoretical analysis of the possible causes of these effects in the collisions of nuclei He and Li with Si nuclei was performed including the influence of external neutrons of weakly bound projectile nuclei. The time-dependent model proposed in the paper shows that the rearrangement of external weakly bound neutrons of nuclei He and Li during the collision changes the real and the imaginary parts of the interaction potential, which may cause a local enhancement in the total reaction cross section. This enhancement is most noticeable in the range of energies where the relative velocity of the nuclei is close in magnitude to the average velocity of external neutrons of the studied light weakly bound nuclei.
We express our deep gratitude to the team of researchers of the ACCULINNA experimental facility (FLNR, JINR) as well as to the team of the U400M accelerator (FLNR, JINR) for maintenance of experiments. The work was supported by Russian Science Foundation (RSF), grant No. 17-12-01170.