Statistical-Model Analysis of Isomeric Ratios in (α, xn) Compound Nuclear Reactions

Abstract

Cross-section ratios of isomeric pairs produced in the compound nuclear reactions $^{41}\mathrm{K}(\ensuremath{\alpha}, n)^{44}\mathrm{Sc}$, $^{55}\mathrm{Mn}(\ensuremath{\alpha}, n)^{58}\mathrm{Co}$, $^{93}\mathrm{Nb}(\ensuremath{\alpha}, n)^{96}\mathrm{Tc}$, $^{93}\mathrm{Nb}(\ensuremath{\alpha}, 2n)^{95}\mathrm{Tc}$, $^{93}\mathrm{Nb}(\ensuremath{\alpha}, 3n)^{94}\mathrm{Tc}$, and $^{136}\mathrm{Ba}(\ensuremath{\alpha}, 3n)^{137}\mathrm{Ce}$ are analyzed in terms of the statistical model. In this context Vandenbosch and Huizenga have proposed a formalism for calculating isomer formation from compound nuclei of relatively low excitation energy and spin. It is assumed that the density of levels of spin $J$ in residual nuclei is proportional to ($2J+1$) $\mathrm{exp}[\ensuremath{-}\frac{J(J+1)}{2{\ensuremath{\sigma}}^{2}}]$ and that in the case of a product formed by neutron evaporation the emission of charged particles is not of importance. Here $\ensuremath{\sigma}$ is the spin cutoff parameter and is related to the effective moment of inertia $\mathcal{I}$. In this paper the extension of such a model to compound nuclei of much higher excitation energy and spin is considered. Typically by this method our experimental isomeric ratios imply a value of $\mathcal{I}$ much smaller than that of a rigid sphere, ${\mathcal{I}}_{R}$, even for nuclei excited to well above nucleon binding energies. The introduction of two additional factors in the formalism leads to more reasonable values of $\frac{\mathcal{I}}{{\mathcal{I}}_{R}}$. First, we invoke a principle of limiting spin. That is, the level density of a residual nucleus is not described by the expression above for all $J$; rather, above some critical $J$ value, determined by a Fermi-gas model, the level density is taken to be zero. Secondly, we find charged-particle emission, particularly of $\ensuremath{\alpha}$ particles, to be of importance in many cases. With the inclusion of these factors experimental isomeric ratios are consistent with an $\frac{\mathcal{I}}{{\mathcal{I}}_{R}}$ value of unity when the excitation energy ${E}_{f}$ is greater than about 10 MeV. Below 10 MeV, $\frac{\mathcal{I}}{{\mathcal{I}}_{R}}$ has essentially the same dependence on ${E}_{f}$ for all of the reactions analyzed.