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Influence of the neutron-pair transfer on fusion V. V. Sargsyan*, G. G. Adamian, N. V. Antonenko In collaboration with W. Scheid, H. Q. Zhang, D. Lacroix, G. Scamps ECT*, Trento, Italy *Joint Institute For Nuclear Research

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Outline I.Quantum Diffusion Approach: Formalism II.Role of deformations of colliding nuclei III.Role of neutron pair transfer IV.Summary

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The assumptions of the QD approach The quantum diffusion approach based on the following assumptions: 1.The capture (fusion) can be treated in term of a single collective variable: the relative distance R between the colliding nuclei. 2.The internal excitations (for example low-lying collective modes such as dynamical quadropole and octupole excitations of the target and projectile, one particle excitations etc. ) can be presented as an environment. 3.Collective motion is effectively coupled with internal excitations through the environment.

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The formalism of quantum-diffusion approach The full Hamiltonian of the system:

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The formalism of quantum-diffusion approach The full Hamiltonian of the system: The collective subsystem (inverted harmonic oscillator)

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The formalism of quantum-diffusion approach The full Hamiltonian of the system: The collective subsystem (inverted harmonic oscillator) The internal subsystem (set of harmonic oscillators)

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The formalism of quantum-diffusion approach The full Hamiltonian of the system: The collective subsystem (inverted harmonic oscillator) The internal subsystem (set of harmonic oscillators) Coupling between the subsystems (linearity is assumed) Adamian et al., PRE71,016121(2005) Sargsyan et al., PRC77,024607(2008)

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The random force and dissipative kern The equation for the collective momentum contains dissipative kern and random force: --- Random force --- Dissipative Kern

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The random force and dissipative kern 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: --- Random force --- Dissipative Kern --- relaxation time for the internal subsystem

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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

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Nucleus-nucleus interaction potential: Double-folding formalism used for nuclear part: Nucleus-nucleus potential: 1.density - dependent effective nucleon-nucleon interaction 2.Woods-Saxon parameterization for nucleus density Adamian et al., Int. J. Mod. Phys E 5, 191 (1996).

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

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The capture cross section The capture cross-section is a sum of partial capture cross-sections G (R 0,P 0 ) t=0 (R, P) The partial capture probability obtained by integrating the propagator G from the initial state (R 0,P 0 ) at time t=0 to the finale state (R, P) at time t: --- the reduced de Broglie wavelength --- the partial capture probability at fixed energy and angular momenta

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Propagator for the inverted oscillator For the inverted oscillator the propagator has the form: The expression for the capture probability --- the mean value of the collective coordinate and momentum --- the variances and Dadonov, Man’ko, Tr. Fiz. Inst. Akad. Nauk SSSR 167, 7 (1986).

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Initial conditions for two regimes of interaction 1. 2. r ex r in R int E c.m. > U(R int ) -- relative motion is coupled with other degrees of freedom E c.m. < U(R int ) -- almost free motion Nuclear forces start to act at R int =R b +1.1 fm, where the nucleon density of colliding nuclei reaches 10% of saturation density.

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16 O+ 208 Pb reaction Reactions with spherical nuclear are good test for the verification of the approach. Using these reaction we fixed the parameters used in calculation. The change of the slope of the excitation function means, that at sub-barrier energies the diffusion becomes a dominant component.

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Reactions with spherical nuclei Reactions with the spherical nuclei more clearly shows the behavior of the excitation function.

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The features of quantum diffusion approach 1.The coupling with respect to the relative coordinate results in a random force and a dissipation kernel. t 2.The integral term in the equations of motion means that the system is non-Markovian and has a “memory” of the motion over the trajectory preceding the instant t. 3.Predictive power. Sargsyan et. al., EPJ A45, 125 (2010) Sargsyan et. al., PRA 83, 062117 (2011) Sargsyan et. al., PRA 84, 032117 (2011) Our approach takes into account the fluctuation and dissipation effects in the collisions of heavy ions which model the coupling with various channels.

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Role of deformation of nuclei in capture process At fixed bombarding energy the capture occurs above or below the Coulomb barrier depending on mutual orientations of colliding nuclei ! The lowest Coulomb barrier The highest Coulomb barrier

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Reactions with deformed nuclei The effect depends on the charges and deformations of the colliding nuclei. The used averaging procedure seems to work correct. Sargsyan et. al., PRC 85, 037602 (2012)

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Role of neutron transfer Neutrons are insensitive to the Coulomb barrier and, therefore, their transfer starts at larger separations before the projectile is captured by the target nucleus. It is generally thought that the sub-barrier capture (fusion) cross section increases because of the neutron transfer. The present experimental data (for example 60 Ni + 100 Mo system, Scarlassara et. al, EPJ Web Of Conf. 17, 05002 (2011) ) specify in complexity of the role of neutron transfer in the capture (fusion) process and provide a useful benchmark for theoretical models. Why the influence of the neutron transfer is strong in some reactions, but is weak in others ?

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Large enhancement of the excitation function for the 40 Ca+ 96 Zr reaction with respect to the 40 Ca+ 90 Zr reaction!

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The 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, and, respectively, the height and shape of the Coulomb barrier are changed.

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Sargsyan et. al., PRC 84, 064614 (2011) Sargsyan et. al., PRC 85, 024616 (2012)

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Sargsyan et. al., PRC 84, 064614 (2011) Sargsyan et. al., PRC 85, 024616 (2012)

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Reactions with two neutron transfer

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Pair transfer ? Reactions with Q 1n 0 Good agreement between calculations and experimental data is an argument of pair transfer By describing sub- barrier capture, we demonstrate indirectly strong spatial 2-neutron correlation and nuclear surface enhancement of neutron pairing Indication for Surface character of pairing interaction ?

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Enhancement or suppression ? 2n-transfer can also suppress capture If deformation of the system decreases due to neutron transfer, capture cross section becomes smaller

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Summary The quantum diffusion approach is applied to study the capture process in the reactions with spherical and deformed nuclei at sub-barrier energies. The available experimental data at energies above and below the Coulomb barrier are well described. Change of capture cross section after neutron transfer occurs due to change of deformations of nuclei. The neutron transfer is indirect effect of quadrupole deformation. Neutron transfer can enhance or suppress or weakly influence the capture cross section.

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Reactions with weakly bound projectiles There are no systematic trends of breakup in reactions studied! For some system with larger (smaller) Z T breakup is smaller (larger).? The break-up probability:

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Reactions with weakly bound projectiles There are no systematic trends of breakup in reactions studied! For some system with larger (smaller) Z T breakup is smaller (larger).? The break-up probability:

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Friction depending on the relative distance of colliding nuclei Frictions is a result of the overlapping of the nuclear densities. For the light systems, the coupling parameter should depend on the relative distance between the colliding nuclei and, as a result the friction becomes coordinate-dependent. Comparing the results, obtained with the analytic expressions (constant friction) for the equations of motions with the numerical one (coordinate- dependent), one can assume that the linear coupling limit is suitable for the heavy systems and not very deep sub-barrier energies.

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Calculations with constant and R- dependent friction

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