Near-barrier neutron transfer in reactions 3,6He + 45Sc and 3,6He + 197Au

Experimental cross sections for formation of 196,198Au isotopes in reactions 3,6He + 197Au and cross sections for formation of 44,46Sc isotopes in reactions 3,6He + 45Sc have been analyzed. To calculate neutron transfer probabilities and cross sections the time- dependent Schrödinger equation for external neutrons of 3He, 6He, 45Sc and 197Au nuclei has been solved numerically. It is shown that the contribution of fusion and subsequent evaporation is significant in the case of reactions 3,6He + 45Sc, whereas in the case of reactions 3,6He + 197Au, it is negligible. Fusion-evaporation was taken into account using NRV evaporation code. Results of calculations demonstrate overall satisfactory agreement with experimental data.


Introduction
Low-energy reactions with He isotopes ( 3 He and halo nuclei 6,8 He) attract interest due to the new possibilities of investigating the nuclear structure of light He projectiles and heavy target nuclei, e.g. [1]. We may explore target nuclei 45 Sc,197 Au by weakly bound neutrons of 6 He and the more strongly bound neutron of 3 He in reactions of neutron transfer from the projectile to the target as well as study the properties of external neutrons of 45 Sc,197 Au in the pick-up reaction 3 He+ 45 Sc, 3 He+ 197 Au. We may also test some theoretical models with both simple and complicated approximations by comparing them with experimental data.

Experiments
The experiments were performed with the extracted radioactive beams of 6 He obtained at the DRIBs accelerator complex, JINR, Dubna [2]. The beams of the 3 He nuclei were accelerated by the U-120M cyclotron of the Nuclear Physics Institute, the Academy of Sciences of the Czech Republic [3,4]. The target assemblies consisting of gold foils of different thicknesses were installed in the reaction chamber and irradiated by the beam of particles. For the reaction with 6 He the gold target assemblies were installed in the focal plane of the magnetic spectrometer MSP-144, where it was possible to obtain the lower energy beam with a good resolution. The maximum energy of the accelerated 6 He ions was ~10 MeV/nucleon, the intensity reached (2-5)·10 7 s -1 . The maximum energy of 3 He was 24.5 MeV, the beam intensity varied in the range (5-10)·10 11 s -1 . After each irradiation session induced γ-  196,198 Au isotopes was performed by their gamma-radiation energies and half-lives. The measurements of the yields were performed taking into account absolute intensities of gammatransitions and detector efficiency. The details of the experiment, data processing and results are described in [2 − 4].

Time-dependent model for neutron transfer
The experimental excitation functions for the formation of 196,198 Au isotopes in the reactions of 3,6 He with 197 Au and formation of 44,46 Sc isotopes in the reactions of 3,6 He with 45 Sc were analyzed and interpreted on the basis of the time-dependent Schrödinger equation [5].

Neutron states in 3,6 He
Information about the structure of loosely bound nuclei may be obtained by measuring their momentum distributions after breakup in a nuclear reaction. From the momentum distribution for the 6 Не nucleus it follows that it consists of the 4 He core and the two-neutron cluster, see e.g. [1,6]. There are several different approaches to the approximate analysis and solution of the three-body problem [7,8]. To determine the wave function of the ground state of the 6 He three-body system the Schrödinger equation may also be solved by Feynman's continual integrals method [9,10]. Within the simple approximation we use the shell model with the experimental value of the neutron separation energy 1.86 MeV and the root-mean-square charge radius 2.065 fm (the values were taken from [11]). The shell model mean field (Figure 1a) takes into account the properties of the ground state of the system of three particles 6 He (α + n + n), 3 He (p + p + n) and four particles 4 He (p + p + n+ n) calculated by Feynman's continual integrals method [12]. The central maximum of the mean field is the effect of the repulsive core of the nucleon-nucleon interaction. Results of calculations of the charge distributions for 3,4 He demonstrate overall satisfactory agreement with experimental data [13] ( Figure  1b).

The time-dependent Schrödinger equation (TDSE) and neutron wave functions
For theoretical description of neutron transfer during collisions of heavy atomic nuclei several semiclassical models are used [5,14,15]. They combine quantum description of internal one-particle and collective degrees of freedom with classical equations of motion of atomic nuclei, equation (1) r are the centers of nuclei with the masses 1 2 , m m and 12 ( ) V r is the potential energy of nuclear interaction [16]. We may assume that before contact of the surfaces of spherical nuclei with the radii R 1 , R 2 the potential energy of a neutron ( , ) W t r is equal to the sum of its shell model mean fields for both nuclei. The evolution of the components 1 2 , ψ ψ of the spinor wave function ( , ) t Ψ r for the neutron with the mass m during the collision of nuclei is determined by the equation (2) with the operator ˆL The evolution of the probability density for the external 3 2 1p shell neutrons of 6 He in the reaction 6 He + 197 Au was studied in Ref. [14]. In this case the neutron occupies free levels of the 197 Au nucleus. The neutron separation energies of 3 He and 197 Au are similar. In the case of pick-up to 197 Au the neutron occupies free levels only. The Pauli Exclusion Principle limits transfers to the unoccupied states. This principle was taken into account by two different approximations providing very similar results. Within the first, simple, approximation transfer to the occupied levels in the "frozen" shell structures of colliding nuclei is excluded. Within the second, more complicated, approximation the time-dependent few-body Slater determinant wave function is used [15]. In the case of pick-up to 3 He the neutron occupies the partially free 1s level of the 3 He nucleus and the mean field transforms from the 3 He shell model to the 4 He shell model.
The evolution of the probability density in the collisions of 3 He+ 45 Sc and 6 He+ 45 Sc is shown in Figures 2 and 3, respectively. After the transfer (stripping) from the 3,6 He nuclei (see Figure 4a) the neutron may occupy several initially vacant levels   Similarly, a weakly bound neutron of the 6 He nucleus may be transferred to several initially vacant top levels 9 2 2g , 11 2 1i , 5 2 3d , 15

Calculations of transfer cross sections
The total cross section of neutron transfer in the time-dependent model is calculated by equation (3) Here b is the impact parameter, b 0 is the minimum collision impact parameter which is equal to zero for the energy less than the Coulomb barrier. The function ( ) p b is the probability of neutron transfer during the collisions of nuclei without contact between their surfaces. Without taking into account the Pauli Exclusion Principle ( ) p b was determined by integrating the probability density over the volume after the collision (Figures 2 and 3). With the Pauli Exclusion Principle only stripping to initially vacant levels was taken into account. In the case of 6 He only stripping to discrete bound states of the target was taken into account.
The comparison of the experimental data and theoretical calculations for the reactions 45 Sc( 3 He, 2p) 46 Sc and 197 Au( 3 He, 2p) 198 Au is shown in Figures 5a and 5b, respectively. Symbols are the data from Refs [3,4].  [3,4] (filled squares) and Ref. [17] (empty squares), dashed curves are fusion-2p-evaporation calculations using NRV evaporation code [11], solid curves are neutron transfer calculations taking into account the Pauli Exclusion Principle, dash-dotted curve is sum of fusion-2p-evaporation and neutron transfer. Here and below arrows indicate barrier heights.
The comparison of the experimental data and theoretical calculations for the reactions 45 Sc( 3 He, 4 He) 44 Sc and 197 Au( 3 He, 4 He) 196 Au is shown in Figures 6a and 6b, respectively. For neutron pick-up the energy of the neutron in 197 Аu was chosen to coincide with the experimental neutron separation energy. As can be seen, in the reaction 3 He + 197 Аu neutrons are predominantly transferred from 3 He to 197 Аu. The comparison of the experimental data and theoretical calculations for the reaction 45 Sc( 6 He, 5 He) 46 Sc is shown in Figure 7. The comparison of the experimental data and theoretical calculations for the reaction 197 Au( 6 He, 5 He) 198 Au has been reported in [14].   46 Sc. Symbols are the experimental data from Ref. [18] (filled squares), dotted curve is total fusion cross section calculated using NRV fusion code [11], dashed curve is fusion-α n-evaporation calculation using NRV evaporation code [11], solid curve is neutron transfer calculation taking into account only stripping to discrete bound states, dash-dotted curve is sum of fusion-α n-evaporation and neutron transfer.
The model does not take into account the energy change associated with the Q-value of the neutron transfer. In the case of Q>0 it may lead to closer distances between nuclei and, thus, higher transfer probability resulting in higher transfer cross section. In the opposite case, Q<0, transfer probability is lower resulting in lower transfer cross section. This also implies the change of the structure of the projectile during the neutron transfer.
The effect of the Q-value and the quantum effects related to the motion of cores, e.g. tunneling, may be taken into account in the coupled channel approach [5,12,19] combined with the timedependent Schrödinger equation method.

Conclusion
The results of calculations within the time-dependent Schrödinger equation method demonstrate overall satisfactory agreement with experimental data. The method may also be applied for calculation of reactions with cluster nuclei [20].