samarin@jinr.ru

1. Microscopic time-dependent analysis of neutrons transfers at low-energy nuclear reactions with spherical and deformed nuclei V.V. Samarin Joint Institute for Nuclear Research, Dubna samarin@jinr.ru The aim of report is application of based quantum mechanics equations for neutron transfers description.

2. Motivation • Neutron transfers in the low energy nuclear reactions allow us to obtain new isotopes of atomic nuclei with increased neutron content. • The probability of neutron transfer is highest during so-called grazing nuclear collisions. In this case the distances between the surfaces of the atomic nuclei do not exceed the range of the action of nuclear forces (1–2 fm). • The most probable transition is the one between the nuclei of the external, most weakly bound neutrons. • A new possibility for theoretical study of this reactions is provided by numerical solution for the non-stationary Schrodinger equation for external neutrons [1]. • In this study the spin-orbital interaction and Pauli's exclusion principle were taken into consideration for spherical and deformed nuclei. 1. V.Samarin, V. Zagrebaev, Walter Greiner.Phys. Rev., C 75, 035809 (2007) .

3. 6Не +197Au 6Не +208Pb 18O +48Са 48Са+238U Time-dependent Schrödinger equation with spin-orbital interaction and Pauli's exclusion principle Spherical and deformed nuclei shell model for external nucleons We studied four nuclear reactions (three from them more detail) and used two methods: More detail and less detail

4. Time-dependent Schrödinger equation with spin-orbital interaction Time-dependent Schrödinger equation with spin-orbital interaction R0=1 fm in Cartesian coordinates is numerically solved by difference method [1-3] for external neutrons of spherical nuclei at their grazing collisions with energies near to a Coulomb barrier. 1. Samarin V. V., Samarin K. V. // Bull. Russ. Acad. Sci. Phys. 2010. V. 74. P. 567. 2. Samarin V. V., Samarin K. V. // Bull. Russ. Acad. Sci. Phys. 2011. V. 75. P. 964. 3. Samarin V. V., Samarin K. V. // Bull. Russ. Acad. Sci. Phys. 2012. V. 76. P. 450.

5. Deformed nuclei shell model Schrödinger equation for the nucleon energy levels and wave functions at arbitrary axial-symmetrical field with spin-orbit interactionare calculated by a numerical solution of a Schrödinger equation for an arbitrary axial-symmetrical field with spin-orbit interactions, basedon decomposing on Bessel functions and difference scheme along internuclear axis. Samarin V.V. Phys. Atom. Nucl., 2010, 73, p. 1416.

6. 6Не +208Pb, Еcm=18MeV, frontal collision Time-dependent Schrödinger equation solution is the way of visual simulation of neutron transfer for reactions with halo nuclei 6Не. At first, for frontal and grazing collisions with energy below Coulomb barrier you can see, that 6Не neutron wave function of 1p3/2 state is arranged between projectile and target nuclei. There is stable space structure, like to 3d state of Au or Pb, with zero angular momentums projection to internuclear axis. 6Не +197Au, Еcm=18MeV, grazing collision

7. 6Не +208Pb, Еcm=20MeV, frontal collision At second, for frontal and grazing collisions with energy in vicinity of Coulomb barrier you can see, that 6Не neutron wave function of 1p3/2 state is arrangedbetween projectile and target nuclei with some space structure too. Е=20MeV 6Не +197Au, Еcm=21MeV, grazing collision

8. At next, for grazing collision with energy above of Coulomb barrier you can see, that 6Не neutron wave function of 1p3/2 state is arranged between projectile and target nuclei with stable space structure like to rotated 3d state of Au. 6Не +197Au, Еcm=30MeV, grazing collision

9. At last, for grazing collision with energy more above of Coulomb barrier you can see, that 6Не neutron wave function of 1p3/2 state is arranged between projectile and target nuclei with some space structure like to rotated 3d state of Au too. 6Не +197Au, Еcm=60MeV, grazing collision

10. If distances between nuclear centers and surfaces is decreasing, potential barrier between two potential walls is decreasing too ( Fig. 1), transfer probability is increasing ( Fig. 2). This reason lead to cross section of neutron transfer at 6He+197Au, which satisfactorily agrees with experimental data (Fig. 3). Spacing interval between nuclear surfaces Fig. 1. p Fig. 3. Fig. 2. Probabilities p of the transfer of the external neutron of the 6Не as functions of the minimum distance s between the surfaces of 6Не, 197Au nuclei for the energies in the centerof mass systemnearbarrier energies from 18 to 22 MeV ( ), 30 MeV ( ) and 60 MeV ( ).

11. We may illustrate semiclassically dominant neutron transfer for neutron orbits with zero angular momentums projection to internuclear axis, which touching each other. Neutron energy and momentum for classical orbit in projectile equal approximately to neutron energy and momentum for classical orbit in target. In this case angular momentums relatively to centers of nuclei will be proportional to radii of nuclei (Fig. 1). 6Не 197Au R2 R1 1p3/2 For reaction6Не +197Au, infrontal collision dominant transfer is: 1p3/2(6Не) 3d5/2,3/2(197Au). Calculated by time-dependent Schrödinger equation occupied states probabilities in Au for stripping neutron with full momentum projection W are shown on Fig. 2. 3d5/2,3/2 Fig. 1 Fig. 2

12. Pauli's exclusion principlelimit transfers to occupied states for reactions 18O+48Ca. Preliminary: unlimited transfers Begin of collision 1d5/2 1d5/2 18O 48Ca End of collision Neutron states in spherical shellmodel

13. Pauli's exclusion principlelimit transfers to occupied states for reaction: 48Ca+248U Preliminary: unlimited transfers Begin of transfer 48Ca 238U 2g9/2 End of collision 48Ca 238U Neutron states in spherical shellmodel Neutron states in deformed shellmodel are studied in next slides Begin of collision Begin of transfer 3d5/2 1f7/2 End of collision End of collision 2g9/2 48Ca 238U 48Ca 238U

14. Pauli's exclusion principle were taken into consideration by:1. exception with transfer to occupied states in “frozen” nuclei shell structures (simple approximation). 2. time dependent many body wave function (M=2, 3) (more correct approximation). Fig. 1. Probabilities of neutrons pick-up (solid curves) and stripping (dashed curves)for 48Са at reactions 48Са + 18O (curves 1) и48Са + 238U (curves 2) as functionon minimum value of internuclear distance At reaction 48Ca+238U probabilities of neutrons stripping and pick-up are commensurable. Fig. 2. Probabilities of neutrons pick-up (solid curves) and stripping(dashed curves) for 48Са at reactions 48Са + 238U asfunction on minimum value of internuclear distance

15. Visual simulation of neutron transfers at reaction 18O+48Ca. Two external neutrons: 1d5/2 from 18O and 1f7/2 from 48Ca with moment projection W=1/2, 3/2 are took into account Visual simulation of neutron transfers at reaction 48Ca+238U Three externalneutrons: 1f7/2 from 48Ca and 2g9/2, 1i11/2 from 238U with moment projection W=1/2, 3/2 are took into account

16. Nucleons transfers in low-energy nuclear reactions40,48Ca+238U with deformed nucleus 238U b a a) Some upper energy levels for neutron states with module Wof total angular momentum projectionto symmetry axis at hypothetic nuclei 238U with quadrupole deformation b2 and octupoledeformation b4 = b2/2 (a) and at real nucleus 238U with b2 = 0.215, b4 = 0.095 (b), dashed lines correspond unoccupied levels

17. Neutron pick-up at reaction 48Ca+238U Visual simulation of neutron pick-up at reaction 40Ca+238U during frontal collision at energy in the center of mass system E=192 MeV. In the beginning external neutron of 238U is in initial state 1j15/2 with angular momentum projections on symmetry axis W=5/2. Angle between symmetry axes of deformed nucleus 238U and initial velocity of 40Ca nucleus equal 45o.

18. Neutrons pick-up in low-energy nuclear reactions40Ca+238U with deformed nucleus 238U a b a) Probability density of the external neutron of 238U for initial state 1j15/2 with angular momentum projections on symmetry axis W=5/2 during frontal collision with the 40Ca at energy in the center of mass system E=192 MeV. Angle between symmetry axes of deformed nucleus 238U and initial velocity of 40Ca nucleus equal 45o. b) The probabilities of neutron pick-up at reaction 40Са+238U as a function of minimum distance s between nuclear surfaces. Angles between symmetry axes of deformed nucleus 238U and initial velocity of 40Ca nuclei equal 45o (solid line) and 90o (dashed line).

19. Neutron stripping at reaction 48Ca+238U Visual simulation of neutron stripping at reaction 48Ca+238U during frontal collision at energy in the center of mass system E=175 MeV. In the beginning external neutrons of 48Ca is in initial state 1f7/2. Angle between symmetry axes of deformed nucleus 238U and initial velocity of 48Ca nucleus equal 0o.

20. Neutron stripping at reaction 48Ca+238U Visual simulation of neutron stripping at reaction 48Ca+238U during frontal collision at energy in the center of mass system E=192 MeV. In the beginning external neutrons of 48Ca is in initial state 1f7/2. Angle between symmetry axes of deformed nucleus 238U and initial velocity of 48Ca nucleus equal 45o.

21. Neutrons stripping in low-energy nuclear reactions48Ca+238U with deformed nucleus 238U a b a) Probability density of the external neutrons of 1f7/2 shell of 48Ca during a frontal collision with the 238U at energy in the center of mass system E=192 MeV. Angle between symmetry axes of deformed nucleus 238U and initial velocity of 40Ca nucleus equal 45o. b) The probabilities of neutron strippingat reaction 48Са+238U (a) as a function of minimum distance s between nuclear surfaces. Angles between symmetry axes of deformed nucleus 238U and initial velocity of Ca nuclei equal 45o (solid line), 90o (dashed line) and 0 (dotted line).

22. Conclusion • A new possibility for theoretical study of this reactions is provided by numerical solution for the non-stationary Schrödinger equation for external neutrons. • In this study the spin-orbital interaction and Pauli's exclusion principle were taken into consideration. Time-dependent Schrödinger equation is numerically solved by difference method for external neutrons of spherical nuclei 6He, 18O, 48Са and deformed nucleus 238U at their grazing collisions with energies near to a Coulomb barrier. • The probabilities of transfer of neutrons at reactions 6He+197Аu, 18O+48Ca, 40,48Ca+238U are determined as function on minimum internuclear distances. • The calculation results of cross section for formation of the 198Au isotope in the 6Не+197Au reaction agree satisfactorily with the experimental data in vicinity of the Coulomb barrier. • At reactions 6He+197Аu, 18O+48Ca, neutrons are predominantly transferred from a smaller nucleus to the greater nucleus. At reaction 48Ca+238U probabilities of neutrons stripping and pick-up are commensurable. • Nonstationary quantum approach applied in this work may be used for internal nucleons too. It may be useful for nucleons transfer experimental data analysis.

23. Thank youforattantion! Dubna