Extensive systematizations of theoretical and experimental nuclear densities and of optical potential strengths extracted from heavy-ion elastic scattering data analyses at low and intermediate energies are presented. The energy dependence of the nuclear potential is accounted for within a model based on the nonlocal nature of the interaction. The systematics indicates that the heavy-ion nuclear potential can be described in a simple global way through a double-folding shape, which basically depends only on the density of nucleons of the partners in the collision. The possibility of extracting information about the nucleon-nucleon interaction from the heavy-ion potential is investigated.

A parameter-free nonlocal double-folding-inspired interaction is proposed for the nucleus-nucleus systems. Excellent reproductions of elastic scattering differential cross section data were obtained for several systems over a wide range of bombarding energies. Our results should be of value in the description of the scattering of other many-body systems. [S0031-9007(97)05269-1] PACS numbers: 21.30. Fe, 25.70.Bc The mean field interaction between complex quantum many-body systems (nucleus-nucleus, cluster-cluster, etc.) is still an open question in current physics research. The study of this matter is a fundamental step in the understanding of many-body dynamics. In the nucleus-nucleus case, significant progress has been achieved concerning this question during the last decade [1], as a consequence of the measurement of accurate and extensive elastic scattering data at intermediate energies. Nuclear rainbow scattering, first observed in a systems [2][3][4] and later in light heavy ions [5][6][7], probes the nucleus-nucleus potential not only at the surface region but also at smaller distances, and ambiguities in the real part of the potentials have been removed. The resulting phenomenological interactions have significant dependence upon the bombarding energies. Some theoretical models have been developed to account for this energy dependence through realistic mean field potentials. Nowadays, the most successful models seem to be those based on the DDM3Y interaction [8][9][10] which is an improvement of the originally energyindependent double-folding potential [11]. But, in order to fit the data, the density-and energy-dependent DDM3Y potential needs a renormalization factor which besides being system dependent [1,12] is still slightly energy dependent [1].In this Letter we show, by an extensive description of elastic scattering data using an optical integro-differential equation, that the dependence on the bombarding energy of the real bare potential is mostly due to the intrinsically nonlocal nature of the effective one-body interaction. The real bare potential (by bare we mean the average, mean field, interaction with no coupled channels effects) is constructed using the folding model. It contains no adjustable parameters and is energy independent. The absorptive part is taken to be a three parameters WoodsSaxon interaction. We also supply a simple approach to obtain the local-equivalent energy-dependent potential.Before we set the stage for the analysis of elastic scattering data, we first describe our theoretical model.When dealing with nonlocal interactions, one is required to solve the integro-differential equationwhere, on physical grounds [13], the kernel function is taken to be symmetric: U͑ R, R 0 ͒ U͑ R 0 , R͒. We take for U͑ R, R 0 ͒ the following form motivated by the physics problem at handIn our analysis, the Coulomb interaction, V C ͑R͒, was obtained using an expression for the double sharp cutoff folded potential [14] and the local energy-dependent imaginary potential, W ͑R, E͒, was taken t...

Nonlocal interactions are an intrinsically quantum phenomenon. In this work we point out that, in the context of heavy ions, such interactions can be studied through the refractive elastic scattering of these systems at intermediate energies. We show that most of the observed energy dependence of the local equivalent bare potential arises from the exchange nonlocality. The nonlocality parameter extracted from the data was found to be very close to the one obtained from folding models. The effective mass of the colliding, heavy-ion, system was found to be close to the nucleon effective mass in nuclear matter. [S0031-9007(97)02958-X] PACS numbers: 25.70. Bc, 21.30.Fe, 21.65. + f, Of fundamental importance in nuclear physics are the effects arising from the Fermi nature of the nucleons. When calculating interaction potentials between nuclei, these effects translate into a nonlocality. This Pauli nonlocality has been discussed in the context of the nucleon-nucleus scattering [1,2]. A fully microscopic calculation of the nucleus-nucleus interaction is quite complicated and one relies here on procedures such as the resonating-group method [3]. Other methods rely on relating the nucleusnucleus nonlocality to that of the nucleon-nucleus one using folding procedure [4]. However, the prediction of Jackson and Johnson [4], namely, the nonlocality range in the nucleus-nucleus systems, is smaller than that in the nucleon-nucleus one by about the inverse of the reduced mass in the former, was never really subjected to tests. In this Letter we supply such a test through a careful analysis of the elastic scattering of the systems 12 C 1 12 C and 16 O 1 12 C at intermediate energies.Before we set the stage for our analysis of exchange effects in the ion-ion interaction, we first say a few words about this interaction. The effective, one-body interaction that determines the elastic scattering between two nuclei can be written in a schematic way asThe first term, V bare ͑r, r 0 ͒, is usually called the bare interaction. It represents the ground state expectation value of the interaction operator, which contains as basic input the average effective nucleon-nucleon force (G matrix). The nonlocality here is solely due to the Pauli exclusion principle and in what follows we refer to it as the Pauli nonlocality. The second term contains the contribution arising from virtual transitions to intermediate states i (inelastic channels, transfer channels, etc). The corresponding nonlocality arises almost entirely from the polarizations that ensue in the heavy-ion system owing to the propagation in the intermediate channels. This is exemplified by the channel Green's function G ͑1͒ i ͑r, r 0 ; E͒, which contains an explicit energy dependence. This latter contribution is called the Feshbach term and thus we refer to its nonlocality as the Feshbach nonlocality. When confronting theory with experiment one usually relies on a one-body optical model with a local potential. This brings into light immediately the issue of extracting from Eq. (1) a l...

Self-similar potentials generalize the concept of shape-invariance which was originally introduced to explore exactly-solvable potentials in quantum mechanics. In this article it is shown that previously introduced algebraic approach to the latter can be generalized to the former. The infinite Lie algebras introduced in this context are shown to be closely related to the q-algebras. The associated coherent states are investigated.

O objetivo deste estudo é analisar as características relativas às vitimizações sexuais intrafamiliares cometidas contra grupos etários categorizados segundo o critério de idade estabelecido pelo Estatuto da Criança e do Adolescente,atendidos no Centro de Referência da Criança e do Adolescente e nos Conselhos Tutelares de Ribeirão Preto, São Paulo, no período de 1995 a 2000. Identificou-se 234 agressões cometidas por 217 agressores contra 210 famílias e 226 vítimas. Foram vitimizadas 131 crianças (48,7%) e 95 adolescentes (41,2%) com predomínio do sexo feminino. Crianças com idade entre dez anos e um mês e 12 anos incompletos foram as mais atingidas (19,5%) e nos adolescentes, entre 12 e 14 anos completos (17,3%). A maioria das vítimas reside com famílias que possuem três (19,9%) ou quatro filhos (17,7%), sendo os primogênitos os mais agredidos (33,6%). Agressores únicos vitimizaram em sua maioria apenas uma vítima (86,7%). Pais (34,2%) e padrastos (30,3%) foram os que mais agrediram, com os primeiros vitimizando mais crianças (19,7%) e os segundos, adolescentes (17,1%).