Abstract
We have previously developed for nuclear cross-sections of therapeutic protons a calculation model, which is founded on the collective model as well as a quantum mechanical many particle problem to derive the S matrix and transition probabilities. In this communication, we show that the resonances can be derived by shifted Gaussian functions, whereas the unspecific nuclear interaction compounds can be represented by an error function, which also provides the asymptotic behavior. The energy shifts can be interpreted in terms of necessary domains of energy to excite typical nuclear processes. Thus the necessary formulas referring to previous calculations of nuclear cross-sections will be represented. The mass number A N determines the strong interaction range, i.e. R Strong = 1.2×10 -13 ·A N 1/3 cm. The threshold energy E Th of the energy barrier is determined by the condition E strong = E Coulomb . A linear combination of Gaussians, which contain additional energy shifts, and an error function incorporate a possible representation of Fermi-Dirac statistics, which is applied here to nuclear excitations and reaction with release of secondary particles. The new calculation formula provides a better understanding of different types of resonances occurring in nuclear interactions with protons. The present study is mainly a continuation of published papers.
Highlights
The knowledge of the total nuclear cross-section Qtot of protons is an important impact with regard to sophisticated features of therapy planning, since Qtot provides essential information of the following aspects: Decrease of the fluence of primary protons Φpp and release of secondary particles and their transport
With regard to secondary protons we have to differ between two kinds, namely protons resulting from nuclear reactions with production of heavy recoils and those protons, Corresponding author: Waldemar Ulmer; Strahlentherapie Nordwürttemberg and Research Group Radiation Physics and Medical Physics, University of Zürich, Zürich Switzerland
According to the results presented in 2, 3 we need for the calculation of Qtot(E), at first, the threshold energy ETh as a function of Z and AN
Summary
The knowledge of the total nuclear cross-section Qtot of protons is an important impact with regard to sophisticated features of therapy planning, since Qtot provides essential information of the following aspects: Decrease of the fluence of primary protons Φpp and release of secondary particles and their transport (secondary protons, neutrons, clusters like H21, H31, He32, He43, heavy recoil nuclei, which usually undergo either a β+ or β - decay with additional emission of a γ-quant). Nuclear reaction types cannot be regarded as simple resonances, they mainly occur for proton energies E > Eres
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