gms | German Medical Science

48th Meeting of the Particle Therapy Co-Operative Group

Particle Therapy Co-Operative Group (PTCOG)

28.09. - 03.10.2009, Heidelberg

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

Meeting Abstract

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  • W. Ulmer - Medical Systems, Varian International AG, Baden, Switzerland

PTCOG 48. Meeting of the Particle Therapy Co-Operative Group. Heidelberg, 28.09.-03.10.2009. Düsseldorf: German Medical Science GMS Publishing House; 2009. Doc09ptcog211

doi: 10.3205/09ptcog211, urn:nbn:de:0183-09ptcog2110

Published: September 24, 2009

© 2009 Ulmer.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( You are free: to Share – to copy, distribute and transmit the work, provided the original author and source are credited.



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 ETh 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 AN. The outermost interaction distance Rstrong of strong interactions amounts to Rstrong = AN 1/3∙1.2∙10-13 cm. For nuclei with Z ≤ 30 the threshold energy ETh does not exceed 8.5 MeV; ETh (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 Qtot of nuclear interactions are characterized by following features: 1. Qtot = 0 (if E < ETh); 2. Gaussian behavior of Qtot in the resonance domain; 3. Sequence of exponential functions up to the asymptotic behavior of Qtot. The asymptotic behavior of Qtot approximately obeys AN 2/3 (geometric cross-section).