Effect of Competition between Gamma-Ray and Particle Emission on Excitation Functions

A procedure has been devised for calculating cross sections for nuclear reactions within about 2 Mev of threshold, where the effect of competition between gamma-ray and particle emission is often important. The requisite formulas depend upon assumptions embodied in the spin-dependent statistical theory of nuclear reactions, so the treatment is most valid for medium to heavy nuclei at moderate bombarding energies. Input data required by the formulas are (1) the level density parameter $a$, (2) an effective nuclear moment of inertia $\mathcal{I}$, (3) the ratio of radiation width to level spacing ($\frac{{\Gamma }_{\gamma }}{D}$) evaluated at some convenient energy, spin, and parity in the excited nucleus immediately preceding the product, (4) transmission coefficients $T_l\left(\epsilon \right)$ for the range of energies and type of particle in the final evaporation step, and (5) the energies, spins, and parities of the first few excited states in the product nucleus. Using reasonable estimates for $\mathcal{I}$, $\frac{{\Gamma }_{\gamma }}{D}$, and $T_l\left(\epsilon \right)$, experimental excitation functions near threshold for the reactions ${\mathrm{Bi}}^{209}(p, 2n){\mathrm{Po}}^{208}$ and ${\mathrm{Sm}}^{144}(\alpha , 3n){\mathrm{Gd}}^{145}$ were analyzed to find the corresponding values of $a$. The results are consistent with $a\sim 0.1$ $A$ ${\mathrm{Mev}}^{-1\mathrm{}}$ ($A$ is the mass number), but inconsistent with $a\sim 2 \mathrm{to} 3$ ${\mathrm{Mev}}^{-1\mathrm{}}$ (independent of $A$), in contrast to the result often obtained when competition from gamma-ray emission is ignored. Also, a semiqualitative argument is given to suggest that competitive gamma-ray emission often seriously influences excitation functions even several Mev above threshold.