The Theory of Alpha-Radioactivity

A number of authors have developed the theory of $\alpha$-radioactivity in the case of zero angular momentum of the escaping particle. For $l\ne 0$, formulae have also been given but it is shown that these are inaccurate. In order to discuss the case of non-zero values of $l$, the one-body theory with a rectangular potential well is treated rigorously. A suitable integral representation of the confluent hypergeometric function is developed by the method of steepest descent into an asymptotic series which represents the solution of the wave equation outside the nucleus for $l=0\mathrm{}$. Solutions for $l\ne 0$ are obtained from this by recursion operators. Boundary conditions at the nuclear radius give two equations linking the radius of the nucleus and the depth of its potential hole with the decay constant and the energy of the emitted $\alpha$-particles. The usefulness and reliability for this problem of the one-body model are examined (Section 6). A re-evaluation of nuclear radii has been performed and the results are tabulated. The possibility that the decay constant, for fixed energy, is not a monotonic function of $l$ but has an initial rise is discussed. A quantitative study is made of the $\lambda -E_{\alpha }$ relationships in a few $\alpha$-ray spectra with "fine structure."