1. Role of neutron transfer and deformation effect in capture process at sub-barrier energies
V.V. Sargsyan, G.G. Adamian, N.V.Antonenko
International Workshop: Nuclear Reactions on Nucleons and Nuclei Messina - October 24-26, 2017

2. Outline Quantum Diffusion Approach: Formalism
Role of deformations of colliding nuclei
Role of neutron pair transfer
Summary

3. The quantum diffusion approach based on the following assumptions:
QD Model
The quantum diffusion approach based on the following assumptions:
The capture (fusion) can be treated in term of a single collective variable: the relative distance R between the colliding nuclei.
The internal excitations (for example low-lying collective modes such as dynamical quadropole and octupole excitations of the target and projectile, single particle excitations etc. ) can be presented as an environment.
Collective motion is effectively coupled with internal excitations through the environment.

4. QD Model The full Hamiltonian of the system: The internal subsystem
Sargsyan, EPJ. A 45 (2010)
Sargsyan, EPJ. A 47 (2011)
Sargsyan, PRC 77 (2008)
The full Hamiltonian of the system:
The collective subsystem
(inverted harmonic oscillator)
The internal subsystem
(set of harmonic oscillators)
Coupling between the subsystems (linear coupling)

5. Potential Nucleus-nucleus interaction potential:
Adamian, Int. J. Mod. Phys E 5 (1996).
Nucleus-nucleus interaction potential:
Double-folding formalism used for nuclear part:
Nucleus-nucleus potential:
density - dependent effective nucleon-nucleon interaction
Woods-Saxon parameterization for nucleus density
Coulomb interaction:

6. Capture cross-section
The capture cross-section
Sum of partial capture cross-sections
--- the reduced de Broglie wavelength
--- the partial capture probability at fixed energy and angular momenta
The capture is calculated by integration of the propagator:

7. Capture probability The expression for the propagator:
Dadonov, Tr. Fiz. Inst. Akad. Nauk SSSR 167 (1986).
The expression for the capture probability:
The first and second moments are influenced by the internal excitations!

8. Realistic nucleus-nucleus potential  inverted oscillator
The real interaction between the nuclei can be approximated by the inverted oscillator.
The frequency of oscillator is found from the condition of equality of classical action

9. Effect of orientation
The lowest Coulomb barrier
The highest Coulomb barrier
At fixed bombarding energy the capture occurs above or below the Coulomb barrier depending on mutual orientations of colliding nuclei !

10. Effect of orientation

11. Calculated results
Raman, At. Data Nucl. Data Tables 78 (2001)
The experimental data at energies ~10 MeV below and ~25 MeV above the barrier is well described !
Sargsyan et. al., PRC 85, (2012)

12. Calculated results
Raman, At. Data Nucl. Data Tables 78 (2001)
The experimental data at energies ~10 MeV below and ~25 MeV above the barrier is well described !
dashed curves represent the calculations by Wong’s formula

13. Calculated results
Raman, At. Data Nucl. Data Tables 78 (2001)
The experimental data at energies ~10 MeV below and ~25 MeV above the barrier is well described !
dashed curve represents the calculation by Wong’s formula

14. Role of neutron transfer
Reactions with 2 neutron transfer

15. Role of neutron transfer
Why the influence of the neutron transfer is strong in some reactions, but is weak in others ?

16. Role of neutron transfer
The importance of 2n transfer channel in capture.
Enhancement or suppression ?
Model assumptions
Sub-barrier capture depends on two-neutron transfer with positive Q-value.
Before the crossing of Coulomb barrier, 2-neutron transfer occurs and lead to population of first 2+ state in recipient nucleus (donor nucleus remains in ground state).
Because after two-neutron transfer, the mass numbers, the deformation parameters of interacting nuclei, the height and shape of the Coulomb barrier are changed.

17. Reactions with 2n transfer
40Ca(β2=0) + 96Zr(β2=0.08)
42Ca(β2=0.247) + 94Zr(β2=0.09)
18O(β2=0.1) + 74Ge(β2=0.283)
16O(β2=0) + 76Ge(β2=0.262)

18. Reactions with two neutron transfer

19. Reactions with 2n transfer
40Ca(β2=0) U(β2=0.286)
42Ca(β2=0.247) U(β2=0.282)
32S(β2=0.312) Th(β2=0.261)
34S(β2=0.252) Th(β2=0.244)
The data of 40Ca + 238U is reproduced taking into account the 2n transfer !
At low energies the excitation function of 32S + 232Th reaction is underestimated !

20. Reactions with 2n transfer
40Ar(β2=0.25) U(β2=0.286)
42Ar(β2=0.275) U(β2=0.282)
34S(β2=0.11) U(β2=0.286)
36S(β2=0.168) U(β2=0.282)

21. Enhancement or suppression ?
If deformation of the system decreases due to neutron transfer, capture cross section becomes smaller!

22. Predictions
50Ti(β2=0) U(β2=0.286)
52Ti(β2=0.233) U(β2=0.282)
50Ti(β2=0) Pu(β2=0.293)
52Ti(β2=0.233) U(β2=0.292)
At energies ~10 MeV below and above the barrier the capture is practically the same !

23. Predictions
50Ti(β2=0) Cm(β2=0.297)
52Ti(β2=0.233) U(β2=0.298)
50Ti(β2=0) + 250Cf(β2=0.299) Ti(β2=0.233) + 248Cf(β2=0.31)
51V(β2=0.26) + 249Bk(β2=0.297) V(β2=0.265) + 247Bk(β2=0.299)
At energies ~10 MeV below and above the barrier the capture is practically the same !

24. Summary
The quantum diffusion approach is applied to study the capture process in the reactions at sub-barrier energies. The behavior of experimental excitation function above and below the Coulomb barrier is well described.
The static quadrupole deformations of interacting nuclei play an important role in capture process at sub-barrier energies.
The change of capture cross section after neutron transfer occurs due to change of deformations of nuclei. The neutron transfer is an indirect effect of quadrupole deformation.
Neutron transfer can enhance, suppress or weakly influence the capture cross section.

25. C. J. Lin & H. Q. Zhang reactions

26. The equation for the collective momentum contains dissipative kern and random force:
One can assumes some spectra for the environment and replace the summation over the integral:
--- relaxation time for the internal subsystem

27. The analytical expressions for the first and second moments in case of linear coupling
Functions determine the dynamic of the first and second moments
--- are the roots of the following
equation

28. Initial conditions for two regimes of interaction1. 2.
rex
rin
Rint
Ec.m. > U(Rint) -- relative motion is coupled with other degrees of freedom
Ec.m. < U(Rint) -- almost free motion
Nuclear forces start to act at Rint=Rb+1.1 fm, where the nucleon density of colliding nuclei reaches 10% of saturation density.