Open Shell Effects in a Microscopic Optical Potential for Elastic Scattering of Exotic Helium Isotopes (2013)

Microscopic optical potentials taking into account the open shell structure of the halo nuclei 6 He and 8 He are derived and calculated for the first time for the elastic scattering of these nuclei from a polarized proton target.It is demonstrated that the inclusion of open shell effects leads to new terms in the optical potential.Terms associated with the scalar single particle density matrix in spin space are the same as the terms obtained for closed shell nuclei.It is shown that the single particle density matrices corresponding to vectors in spin space vanish in the case of closed shell nuclei.For the open shell nuclei 6 He and 8 He, the vector in spin space can couple to spin-dependent pieces of the nucleon-nucleon interaction.This leads to additional non-zero terms in the optical potential, which are derived for the first time.In the case of 6 He and 8 He, one additional central term and one additional spin-orbit term have been identified.The effect of the additional non-zero terms on the differential cross section and analyzing power is found to be small for the scattering energies between 71 and 300 MeV per nucleon.As the projectile kinetic energy increases, the differences between results calculated with the closed shell and modified optical potentials diminish.The nuclei 6 He and 8 He are described by harmonic oscillator wave functions, and the ground states are assumed to have 1s and 1p shells occupied.The higher levels of harmonic oscillator basis are excluded from consideration.Under this assumption, the harmonic oscillator wave functions do not describe the extended exponential distribution of matter specific to halo nuclei.In order to simulate 3.4 Diagonal densities of 6 He and 8 He as functions of momentum transfer q.Red dashed lines represent diagonal densities in the s-shell; black solid lines correspond to the diagonal densities in the p-shell; blue dash-dotted lines depict total diagonal densities.Panel [a] depicts diagonal densities for 6 He, while panel [b] represents diagonal densities of 8 He. . . . . . . . . . . . . . . . . .3.5 The matrix elements p (q, P) as function of vectors q and P. The angle between the vectors q and P is equal to 90 .The contour lines represent density values in the units of fm 3 .The density matrix elements of 6 He and 8 He are shown in panels [a] and [b] respectively. . . . . . . . . . . . . . . . . . . . . . . . . . .3.6 The diagonal matrix element p (q) as function of momentum transfer q.Black solid line corresponds to 6 He, blue dashed line represents 8 He. . . . . . . . . .4.1 The Wolfenstein amplitudes A, C and M for neutron-proton scattering at 71 MeV laboratory kinetic energy based on the CD-Bonn (red solid lines) and Idaho N3LO (blue dashed lines) potentials.The left panels represent the real parts, while the right panels depict the imaginary parts of the amplitudes.The data points are the amplitudes extracted from the GW-DAC current analysis [5].4.2 The Wolfenstein amplitudes G+H, G and H for neutron-proton scattering at 71 MeV laboratory kinetic energy based on the CD-Bonn (red solid lines) and Idaho N3LO (blue dashed lines) potentials.The left panels represent the real parts, while the right panels depict the imaginary parts of the amplitudes.The data points are the amplitudes extracted from the GW-DAC current analysis [5].4.3 The imaginary and real parts of the off-shell Wolfenstein amplitudes A calculated with the CD-Bonn potential are portrayed in panels [a] and [b] respectively.The off-shell Wolfenstein amplitude A calculated with the Idaho N3LO potential is depicted by panels [c] and [d], the imaginary and real parts respectively.The dashed line represents the on-shell condition.The projectile kinetic energy in the laboratory frame is 71MeV per nucleon.The angle between vectors q and K NN is 90 . . . . . . . . . . . . . . . . . . . . . . . . .4.4 The imaginary and real parts of the off-shell Wolfenstein amplitudes C calculated with the CD-Bonn potential are portrayed in panels [a] and [b] respectively.The off-shell Wolfenstein amplitude C calculated with the Idaho N3LO potential is depicted by panels [c] and [d], the imaginary and real parts respectively.The dashed line represents the on-shell condition.The projectile kinetic energy in the laboratory frame is 71MeV per nucleon.The angle between vectors q and K NN is 90 . . . . . . . . . . . . . . . . . . . . . . . . .4.5 The imaginary and real parts of the off-shell Wolfenstein amplitude M calculated with the CD-Bonn potential are portrayed in panels [a] and [b] respectively.The off-shell Wolfenstein amplitude M calculated with the Idaho N3LO potential is depicted by panels [c] and [d], the imaginary and real parts respectively.The dashed line represents the on-shell condition.The projectile kinetic energy in the laboratory frame is 71MeV per nucleon.The angle between vectors q and K NN is 90 . . . . . . . . . . . . . . . . . . . . . . . . .4.6 The imaginary and real parts of the off-shell Wolfenstein amplitudes G+H calculated with the CD-Bonn potential are portrayed in panels [a] and [b] respectively.The off-shell Wolfenstein amplitudes G+H calculated with the Idaho N3LO potential are depicted by panels [c] and [d], the imaginary and real parts respectively.The dashed line represents the on-shell condition.The projectile kinetic energy in the laboratory frame is 71MeV per nucleon.The angle between vectors q and K NN is 90 . . . . . . . . . . . . . . . . . . . . . .4.7 The imaginary and real parts of the off-shell Wolfenstein amplitudes G-H calculated with the CD-Bonn potential are portrayed in panels [a] and [b] respectively.The off-shell Wolfenstein amplitudes G-H calculated with the Idaho N3LO potential are depicted by panels [c] and [d], the imaginary and real parts respectively.The dashed line represents the on-shell condition.The projectile kinetic energy in the laboratory frame is 71MeV per nucleon.The angle between vectors q and K NN is 90 . . . . . . . . . . . . . . . . . . . . .5.1 The imaginary and real parts of the central term U A (q, K) of the closed shell optical potential calculated for 6 He are depicted in panels [a] and [b] respectively.Panels [c] and [d] represent the imaginary and real parts of the additional central term U C A (q, K).The projectile kinetic energy is 71 MeV per nucleon.The angle between the vectors q and K is 90 .Contour lines represent the optical potential in the units of MeV fm 3 .The black dashed line represents the on-shell condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2 The imaginary and real parts of the spin-orbit term U C (q, K) of the closed shell optical potential calculated for 6 He are depicted in panels [a] and [b] respectively.Panels [c] and [d] represent the imaginary and real parts of the additional spin-orbit term U M C (q, K).The projectile kinetic energy is 71 MeV per nucleon.The angle between the vectors q and K is 90 .Contour lines represent the optical potential in the units of MeV fm 3 .The black dashed line represents the on-shell condition. . . . . . . . . . . . . . . . . . . . . . . . .5.3 Same as Fig. 5.1, but for projectile kinetic energy 200 MeV per nucleon. . . .5.4 Same as Fig. 5.2, but for projectile kinetic energy 200 MeV per nucleon. . . .5.5 The angular distributions of the differential cross section, d d , and analyzing power A y as functions of the momentum transfer q for elastic scattering of 6 He are shown in the top and bottom panels respectively.The projectile kinetic energy is 71 MeV per nucleon.The calculations are performed with an optical potential based on the CD-Bonn potential.Red dotted lines represent observables for the closed shell microscopic optical potential.Blue dashed lines correspond to the microscopic optical potential with the additional central term.Green dash-dotted lines are for the modified optical potential with the additional spin-orbit term.Black solid lines depict the optical potentials with the additional central and spin-orbit terms.The data are taken from Refs.[6, 7].5.6 Same as Fig. 5.5, but for projectile kinetic energy 100 MeV per nucleon. . . .5.7 Same as Fig. 5.5, but for projectile kinetic energy 200 MeV per nucleon. . . .