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## Determination of nuclear cross sections by the collective model and generalized nuclear shell theory

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Published: | September 24, 2009 |
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An analysis of total nuclear cross sections of various nuclei is presented, which provides detailed knowledge on the physical processes, e.g. potential/resonance scatter and nuclear reactions. The physical basis for potential/resonance scatter and the threshold energy E_{Th} resulting from Coulomb repulsion of nuclei are collective/oscillator models. These models only use as parameters the nuclear charge number Z and the number of all nucleons A_{N}. The outermost interaction distance R_{strong} of strong interactions amounts to R_{strong} = A_{N}
^{1/3}∙1.2∙10^{-13 }cm. For nuclei with Z ≤ 30 the threshold energy E_{Th} does not exceed 8.5 MeV; E_{Th} (oxygen): 6.9998 MeV.

The part pertaining to the nuclear reactions can only be determined by the microscopic theory. This theory is based on the nonlinear/nonlocal Schrödinger equation containing a combination of different types of Gaussian convolution kernels. They are used to describe strong interactions (π – and K – mesons), spin-orbit couplings between nucleons, and electrostatic repulsion of protons. The principle calculation procedure is the Hartree-Fock theory extended and generalized by configuration interactions (HF-CI). A low order approximation leads to the nuclear shell model based on the harmonic oscillator approach (models of Jensen/Mayer-Göppert and Elliot). Since harmonic oscillators are only reasonable for the ground state and lower excited states, we have also accounted for higher order solutions of excited/virtually excited states, which describe tensor forces and symmetry violations. Some essential calculation parameters of the collective model have been determined by that way. The determination of transition probabilities (*contributions to the* *S matrix*) yields possible reaction channels via excited and virtually excited states of all possible nuclear configurations represented by Slater determinants.

A physical impact is the fluence decrease of proton beams in different media, the contributions of secondary particles to Bragg curves, and their scatter behavior. A particular feature is that the resonance domain of the total nuclear cross section results from primary protons with energies E < 100 MeV inducing different kind of excitations, whereas for E > 100 resonance and potential scatter at the nucleus becomes a decreasing importance. With regard to secondary protons we have to differ between reaction protons resulting from a change of the isospin and non-reaction protons, which may either change the spin multiplicity or induce collective vibrations/rotations of the nucleus. The total cross sections Q^{tot} of nuclear interactions are characterized by following features: 1. Q^{tot} = 0 (if E < E_{Th}); 2. Gaussian behavior of Q^{tot} in the resonance domain; 3. Sequence of exponential functions up to the asymptotic behavior of Q^{tot}. The asymptotic behavior of Q^{tot} approximately obeys A_{N}
^{2/3} (geometric cross-section).