Fusion and neutron transfer reactions with weakly bound nuclei within time-dependent and coupled channel approaches VIACHESLAV SAMARIN Flerov Laboratory of Nuclear Reactions, JINR, Dubna, Moscow Region, Russia samarin@jinr.ru

Outline The refinement of the initial condition for the nucleus-nucleus collision: the three body ground state and modified shell model of the 6He, 6Li, etc. Numerical solution of the time-dependent Schrödinger equation (TDSE): in the semi-classical model of the neutron rearrangement. Analysis of the time-dependent wave function with two-center shell model. Combination of the two-center channel coupling (CC) with TDSE in the model of transfer and fusion reactions. TDSE two- center shell model V.Samarin, NN2015, Catania

The ground state of the three body system 6Не (a+n+n) Fig. 1 The wave functions Y0 of the ground states of 6He nucleus were calculated by Feynman's continual integrals method in Euclidean time t=it [1,2]. The nucleon-nucleon interaction potentials similar to the M3Y potential were used (see Fig.1). The effective neutron-alpha-particle interaction potential included the centrifugal potential for the 1p shell. The probability densities are shown in Fig. 2 in Jacobi coordinates together with nucleons configuration. These results were used for the proposal of the new modified shell model and for the initial condition in the time-dependent calculations of reactions with 6He . 1. R.P.Feynman and A.R.Hibbs. Quantum Mechanics and Path Integrals. McGraw-Hill, New York. 1965 2. E.V.Shuryak. Sov. Phys. Usp. 27. 448 ( ) [UFN. 143. 309] Fig. 2 V.Samarin, NN2015, Catania

The modified shell model of 4He, 5He and 6He nuclei The modified mean nuclear potentials include effects of the self-consistency, pairing and the core polarization : for 4He for 5,6He Two variants of the model are: 1. E(1p3/2)=-Esep(2n)/2=-0.45 MeV 2. E(1p3/2)=-Esep(1n)=-1.8 MeV 4He 5He 6He The radial density for 4He is similar to the 4-body density. The radial density for 6He is similar to the 3-body density. V.Samarin, NN2015, Catania

Time-dependent Schrödinger equation (TDSE) The approximation of the independent external (valence) neutrons was used for the description of transfer reactions and the first capture stage of fusion reactions. The two-component spinor wave function of each neutron changes according to the time-dependent Schrödinger equation (TDSE) with the spin-orbital interaction [1] Light quantum particle (neutron) 3 r2(t) r1(t) [1] V. V. Samarin, EPJ Web Conf. 66, 03075 (2014) The potential energy of a neutron with a vector radius up to the contact with the colliding nuclei surfaces is the sum of the energies of its interaction with each nucleus Two heavy classical particles (cores) 1 and 2 Radii of nuclei centers r1(t), r2(t) are determined from equations of the classical mechanics for colliding nuclei. The operator of the spin–orbital interaction is TDSE approach may be very useful at energies near the Coulomb barrier, because it is the unperturbed method! Here, p is the momentum operator, s are Pauli matrices, l is the phenomenological dimensionless constant, l1 ~ 10, l2 ~ 40, c is the velocity of light and R0=1 fm V.Samarin, NN2015, Catania

The solution of the TDSE for the external neutron of 6He The change in the probability density and the rearrangement of the valence neutron of the 6He nucleus during a collision with the 197Au nucleus at an energy in the centre of mass system E = 19 MeV < VB, a scale factor is 1 fm, and radii of the circumferences equal to radii of the nuclei. The course of time corresponds to (a) − (d). The small typical grid spacing (~ 0.2 fm) of TDSE method leads to Correct calculations of the spatial structure of the external (valence) neutron wave function with the formation of two center (molecular) states [1, 2]. [1] V. V. Samarin, EPJ Web Conf. 66, 03075 (2014) [2] V. V. Samarin, EPJ Web Conf. 86, 00040 (2015) V.Samarin, NN2015, Catania

The visual presentations of TDSE solutions for collision 6Не +197Au: the rearrangement of the valence neutron of 6He Fig. 1 The change in the probability density of the valence neutron of the 6He nucleus during a collision with the 197Au nucleus in the models of the spherical Au nucleus (see Fig. 1) and in the models of the weakly deformed Au nucleus (see Fig. 2). An energy in the centre of mass system is E = 18 MeV < VB. Fig. 2 V.Samarin, NN2015, Catania

TDSE solution for collision 6Не +197Au: transfer and break up The change in the probability density of the valence neutron of the 6He nucleus during a collision with the 197Au in the models with E(1p3/2)=−Esep(1n)= =−1.8 MeV. An energy in the centre of mass system is E = 30 MeV > VB. Neutron transfer to the exited bound states of the discrete spectrum of Au and in the unbound states of the continuum spectrum. Au Potential well of 6He is transformed to potential well of 5He after neutron transfer from He to Au. 6He Break up of the 5He after neutron transfer to Au. 5He V.Samarin, NN2015, Catania

TDSE solution for collision 6Не +197Au: transfer and break up The change in the probability density of the valence neutron of the 6He nucleus during a collision with the 197Au in the models with E(1p3/2)=−Esep(1n)= =−1.8 MeV. An energy in the centre of mass system is E = 30 MeV > VB. n1 Neutron transfer to the exited bound states of the discrete spectrum of Au and in the unbound states of the continuum spectrum. n1 Potential well of 6He is transformed to potential well of 5He after neutron transfer from He to Au. Break up of the 5He after neutron transfer to Au. n2 5He V.Samarin, NN2015, Catania a

TDSE solution for collision 6Не +64Zn: transfer and break up The change in the probability density of the valence neutron of the 6He nucleus during a collision with the 64Zn in the models with E(1p3/2)=−Esep(1n)= =−1.8 MeV. An energy in the centre of mass system is E = 12 MeV > VB. Neutron transfer to the exited bound states of the discrete spectrum of Zn and in the unbound states of the continuum spectrum. Zn Potential well of 6He is transformed to potential well of 5He after neutron transfer from He to Zn. 6He Break up of the 5He after neutron transfer to Zn. 5He V.Samarin, NN2015, Catania

Neutron transfer in the reaction 6Не+ 64Zn TDSE probability w of the transfer of the external neutron of the 6Не nucleus (an energy of neutron separation E(1p3/2)=−1.8 MeV) as functions of the minimum distance Rmin between centers of nuclei 6Не, 64Zn for the energy in the center mass system E=12 MeV. Straight line represent the linear regression result. Transfer cross section was calculated by the integral Experimental radioactive 65Zn nucleus cross section (points) as a function of the center-of-mass energy for the 6He + 64Zn reaction [1]. The line correspond to the TDSE calculation. where b is the impact parameter, b0 is the minimum impact V. Scuderi et al., Phys. Rev. C. 84, 064604 (2011) parameter corresponding to the grazing collision when the surfaces of the nuclei approach the distance a = 0.5 fm, equal to the characteristic size (diffusivity) of the surface region of nuclei. V.Samarin, NN2015, Catania

Neutron levels in the shell model of the target nuclei, based the central potential of the Woods-Saxon type 197Au 64Zn A. Bohr, B. Mottelson. Nuclear structure V. 1: World Scientific Singapore. 1998, V.Samarin, NN2015, Catania

Probability densities of the TDSE solution for the central collision 6He+197Au and two-center wave functions Fa(R) W=1/2 2g9/2(Au) Two-center Shell Model 1p3/2(He) 3d5/2(Au) 1j15/2(Au) 1i11/2(Au) These two-center wave functions are overlapped greatly near the Coulomb barrier 4s1/2(Au) W=3/2 2g9/2(Au) is the projections of the total angular momentum to an internuclear axis 1p3/2(He) 3d5/2(Au) The two-center shell model is based on the Bessel series [1]. V.Samarin, NN2015, Catania 1j15/2(Au) 13 1. V. V. Samarin, Phys. Atom. Nucl. 78, 128 (2015)

Two-center Shell Model Two-center CC Combination of the two-center channel coupling (CC) with TDSE solution for 6He+197Au reaction TDSE Two-center Shell Model Two-center CC W=1/2 Neutron rearrangement and transfer Intermediate two center transitions 1p3/2(6He) 4s, 3d5/2, 3d3/2 (Au) are most important for W=1/2, Ecm<VB Two-center discrete energy levels and transitions in the models with E(1p3/2) =− Esep(1n)=−1.8 MeV The neutron transfer and fusion reactions are described by channel wave functions FaL(R) Ecm=18.5 МэВ plus discretesed continuum states L=0 U(R) V.Samarin, NN2015, Catania 14 The value of the coupling strength was determined by time-dependent two-center level populations.

Two-center Shell Model Two-center CC TDSE Combination of the two-center channel coupling (CC) with TDSE solution for 6He+197Au Two-center Shell Model Two-center CC TDSE Neutron rearrangement and transfer W=3/2 Intermediate two center transitions 1p3/2(6He) 3d5/2, 1j15/2 (Au) are most important for W=3/2, Ecm<VB The neutron transfer and fusion reactions are described by channel wave functions FaL(R) Two-center discrete energy levels and transitions in the models with E(1p3/2) =− Esep(1n)=−1.8 MeV Ecm=18.5 МэВ plus discretesed continuum states L=0 U(R) V.Samarin, NN2015, Catania 15 The value of the coupling strength was determined by time-dependent two-center level populations.

Reducted exact two-center CC equations are combined with TDSE There are three problems: the boundary conditions at R, the coupled matrix calculation and numerical solution of coupled equations [1]. The combination of the CC methods with TDSE method obviate this difficulties. Approximation of the isotropic coupling matrices Approximation of the reduction for coupling matrices. Ft is coupling strength for the compensation of the deletion some complicate expressions in exact equations. The value of the coupling strength Ft was determined by time-dependent two-center level populations [1]. . 1. V. V. Samarin, Phys. Atom. Nucl. 78, 128 (2015) V.Samarin, NN2015, Catania

The neutron transfer channels coupling equations are proposed: is the energy of the initial neutron state in the distant nucleus, is the reduced kinetic energy coupling matrix, is the wave function of valence neutron, is the energy of valence neutron, W is total angular momentum projection onto the axis connecting centers of colliding nuclei, is Schrödinger equation of two-centre shell model, is the two-centre potential, is the two-centre spin-orbital potential, is coupling strength for the compensation of the deletion some complicate expressions in exact equations of the perturbed stationary states method. V.Samarin, NN2015, Catania V. V. Samarin, Phys. Atom. Nucl. 78, 128 (2015)

Two-center levels and coupling matrix for 6He+197Au The coupling matrix elements: Two-center levels: The distant-dependent Q-values Q(R)>0 may lead to the enhancement of fusion cross section. V.Samarin, NN2015, Catania 18

Neutron transfer and fusion in the 6He+197Au reaction (b) Results of the cross section calculation for the formation of the 198Au (Fig. 1a) and fusion (Fig. 1b) in the 6He+197Au reaction [1, 2] agree satisfactorily with the experimental data near the barrier. Fig. 1. The excitation functions for the formation of the 198Au isotope (a) and fusion (b) in the reaction 6He+197Au. Experimental data (circles) is from [1, 2]. Theoretical curves were calculated within the coupled channel approach (solid lines) and the pure TDSE method (dashed line); VB is the Coulomb barrier. 1. Yu. E. Penionzhkevich et al., Eur. Phys. J. A 31, 185 (2007). 2. A. Kulko et al., J. Phys. G 34, 2297 (2007). V.Samarin, NN2015, Catania

Summary The traditional coupled channel (CC) approach was combined with the two-centre and time-dependent Schrödinger equations (TDSE). The value of the coupling strength was determined by time-dependent two-center level populations. Modified CC equations with neutron rearrangement coupling were proposed and solved. Distant-dependent Q-values Q(R)>0 were used near barrier. The satisfactory agreement between the experimental data and the calculation results is obtained for the neutron transfer cross section at the reaction 6He+64Zn and for the fusion and neutron transfer cross sections at the reaction 6He+197Au in the vicinity of Coulomb barrier. V.Samarin, NN2015, Catania

Thank you for attention! Dubna V.Samarin, NN2015, Catania