1. 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

2. 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

3. 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 E.V.Shuryak. Sov. Phys. Usp ( ) [UFN ]
Fig. 2
V.Samarin, NN2015, Catania

4. 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

5. 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, (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

6. 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, (2014)
[2] V. V. Samarin, EPJ Web Conf. 86, (2015)
V.Samarin, NN2015, Catania

7. 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

8. 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

9. 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

10. 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

11. 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,
(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

12. 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

13. 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)

14. 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.

15. 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.

16. 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

17. 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)

18. 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

19. 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

20. 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

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