Influence of Spin-Dependent Level Densities and Gamma-Ray Competition in the Decay of Highly Excited Nuclei

Excitation functions for the ${\mathrm{Fe}}^{56}(\alpha , pxn)$ reactions ($x=1\mathrm{} \mathrm{to} 4$) have been calculated with the statistical theory of nuclear reactions, using optical-model transmission coefficients for neutrons, protons, and $\alpha$ particles and a level density of the form $\rho (E, J)\propto \mathrm{}(2J+1)\mathrm{}\rho (E-E_{\mathrm{rot}})$, where $E_{\mathrm{rot}}=\frac{J(J+1\mathrm{})\mathrm{}{\hslash }^2}{2{\mathcal{I}}_{\mathrm{rig}}R}$. Here ${\mathcal{I}}_{\mathrm{rig}}$ is the rigid-body moment of inertia, and the dimensionless parameter $R$ was taken in different calculations as 1 or $\infty$. Where a rigid body moment of inertia was used, two assumptions were made concerning $\gamma$-ray de-excitation: It was assumed that (a) there was no $\gamma$-ray competition if the excitation energy exceeded the minimum nucleon binding energy, or (b) there was no $\gamma$-ray competition if the excitation exceeded the binding energy plus rotational energy for each spin. Of the three sets of calculations, the latter set gave the best over-all agreement with experimental ($\alpha , \mathrm{pn}$) and ($\alpha , p2n$) excitation functions. Calculations were also performed for the ${\mathrm{Ti}}^{48}({\mathrm{C}}^{12}, pxn)$ excitation functions (where ${\mathrm{Ti}}^{48}$+${\mathrm{C}}^{12}$ forms the same compound nucleus as ${\mathrm{Fe}}^{56}$+$\alpha$), where $x=1\mathrm{} \mathrm{to} 4$, with $R=1.0\mathrm{}$, and assumption (b) concerning $\gamma$-ray-nucleon emission competition. It is concluded that excitation-function measurements to test the influence of angular momentum on the independence hypothesis should show observable differences in shape and energy dependence, but that a good knowledge of $E$ and $\frac{\mathrm{dE}}{\mathrm{dX}}$ for the heavy ion is required if one is to be confident of the interpretation of the results. The influence of $\gamma$-ray competition based on assumption (b) is considered as a function of mass number of the compound nucleus; the qualitative differences expected for actual excitation functions with respect to the predictions of the Weisskopf-Ewing evaporation model with no $\gamma$-ray competition should decrease with an increase in mass number. Specifically, excitation functions from lower-mass compound nuclei should be broader, should be displaced to higher energies, and should have more pronounced high-energy tails than those from heavier-mass systems.