A case study of the transition from isolated resonances to the continuum: the ${}^{34}{\rm Ar}(\alpha,p){}^{37}{\rm K}$ reaction

Many nuclear reactions of astrophysical importance are modeled by Hauser-Feshbach (HF) calculations, the well-established approach for computing average cross sections when many resonant levels are involved. This approach assumes that the density of levels is sufficient that only average properties, such as optical potentials and level densities, are sufficient to model the reaction. However, for intermediate masses or near the drip lines, these assumptions may break down. One example where these considerations are likely important is ${}^{34}{\rm Ar}(\alpha,p){}^{37}{\rm K}$, a reaction which may provide a pathway beyond the waiting point nucleus ${}^{34}{\rm Ar}$ in $x$-ray bursts. This reaction is also the focus of a number of recent and ongoing experiments. We have modeled this reaction using both a HF calculation and an $R$-matrix calculation using resonance parameters sampled from distributions which are consistent with the level density and optical potentials involved. The $R$-matrix calculation includes interference effects between levels with the same spin and parity. These results provide a measure of the statistical error in the cross section or reaction rate prediction arising from low density of states.