# Elements of reaction theory

# Elements of reaction theory

This chapter derives formulas for gross sections of potential scattering, and shows their generalizations to nuclear reactions. A projection technique is introduced, which allows one to separate in the T-matrix the part which varies smoothly in energy from that for resonances. For the latter, general expressions are deduced which involve partial and total widths. Non-hermitean Hamiltonians for the optical model are obtained by applying suitable energy averages. The doorway mechanism is explained by which the intermediate structure in the resonances can be understood: it employs widths for decays to the continuum and to the compound states. The statistical theory is addressed, for which on the basis of general assumptions, analytic formulas for the fluctuating cross section are derived, which are in accord with N. Bohr's hypothesis of the independence of entrance and exit channels. The Weisskopf-Ewing relations and the Hauser-Feshbach theory are described, which involve branching ratios and formation probabilities. A critique of the statistical model is included.

*Keywords:*
reaction theory, T-matrix, resonances, partial and total widths, optical model, compound states, statistical theory, Bohr's independence hypothesis, Weisskopf-Ewing relations, Hauser-Feshbach theory

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