Extrapolation from positive to negative energy of the Woods-Saxon parametrization of the n-208Pb mean field |
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Institution: | 1. Ho Chi Minh City University of Education, 280 An Duong Vuong, District 5, Ho Chi Minh City, Viet Nam;2. Institute of Nuclear Science and Technology, VINATOM, 179 Hoang Quoc Viet, Cau Giay, Hanoi, Viet Nam;3. University of Science, VNU-HCM, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Viet Nam;1. Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, Apdo. Postal 14-740, 07000 Ciudad de México, Mexico;2. Universidad de Pamplona, Km 1, vía salida a Bucaramanga, Campus Universitario, 543050, Pamplona, Colombia;3. Universidad Santiago de Cali, Campus Pampalinda, Calle 5 No. 6200, 760035, Santiago de Cali, Colombia;1. Belgorod State National Research University, Pobeda 85, Belgorod 308015, Russia;2. E.O. Paton Electric Welding Institute, NASU, Bozhenko 11, Kyiv 03650, Ukraine;3. N.N. Semenov Institute of Chemical Physics, RAS, Kosygina 4, Moscow 117977, Russia |
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Abstract: | The real part V(r); E) of the nucleon-nucleus mean field is assumed to have a Woods-Saxon shape, and accordingly to be fully specified by three quantities: the potential depth Uv(E), radius RV(E) and diffuseness av(E). At a given nucleon energy E these parameters can be determined from three different radial moments rq]v = (4π/A) ∝V(r; E)rq dr. This is useful because a dispersion relation approach has recently been developed for extrapolating rq]V(E) from positive to negative energy, using as inputs the radial moments of the real and imaginary parts of empirical optical-model potentials V(r; E) + iW(r; E). In the present work, the values of Uv(E), Rv(E) and av(E) are calculated in the case of neutrons in 208Pb in the energy domain −20 < E < 40 MeV from the values of rq]V(E) for q = 0.8, 2 and 4. It is found that both UV(E) and Rv(E) have a characteristic energy dependence. The energy dependence of the diffuseness aa(E) is less reliably predicted by the method. The radius RV(E) increases when E decreases from 40 to 5 MeV. This behaviour is in agreement with empirical evidence. In the energy domain −10 MeV < E < 0, RV(E) is predicted to decrease with decreasing energy. The energy dependence of the root mean square radius is similar to that of RV(E). The potential depth Uv slightly increases when E decreases from 40 to 15 MeV and slightly decreases between 10 and 5 MeV; it is consequently approximately constant in the energy domain 5 < E < 20 MeV, in keeping with empirical evidence. The depth Uv increases linearly with decreasing E in the domain −10 MeV < E < 0. These features are shown to persist when one modifies the detailed input of the calculation, namely the empirical values of rq]v(E) for E > 0 and the parametrization rq]w(E) of the energy dependence of the radial moments of the imaginary part of the empirical optical-model potentials. In the energy domain −10 MeV < E < 0, the calculated V(r; E) yields good agreement with the experimental single-particle energies; the model thus accurately predicts the shell-model potential (E < 0) from the extrapolation of the optical-model potential (E > 0). In the dispersion relation approach, the real part V(r; E) is the sum of a Hartree-Fock type contribution VHF(r; E) and of a dispersive contribution ΔV(r; E). The latter is due to the excitation of the 208Pb core. The dispersion relation approach enables the calculation of the radial moment rq]ΔV(E) from the parametrization rq]w(E): several schematic models are considered which yield algebraic expressions for rq]ΔV(E). The radial moments rq]HF(E) are approximated by linear functions of E. When in addition, it is assumed that VHF(r; E) has a Woods-Saxon radial shape, the energy dependence of its potential parameters (UHF, RHF, aHF) can be calculated. Furthermore, the values of ΔV(r; E) can then be derived. It turns out that ΔV(r; E) is peaked at the nuclear surface near the Fermi energy and acquires a Woods-Saxon type shape when the energy increases, in keeping with previous qualitative estimates. It is responsible for the peculiar energy dependence of RV(E) in the vicinity of the Fermi energy. |
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