Coulomb Wave Functions in Repulsive Fields

Quantitative discussion of nuclear reactions due to bombardment with charged particles requires the knowledge of wave functions in repulsive inverse square fields of force. It is customary to approximate these functions by formulas of the type due to Wentzel, Kramers and Brillouin (referred to as WKB). Errors of unknown amount are frequently introduced by such approximations. Formulas necessary for the exact calculation of the needed functions are derived and discussed in the present paper. Results of numerical calculations with estimated total errors are tabulated in a companion article in the Journal of Terrestrial Magnetism and Atmospheric Electricity. The proton energy range covered is from 0 to 2 MEV for Li and from 0 to 8 MEV for $C$. For the partial wave with 0 angular momentum the range of radii in proton collisions is covered for Li from 0 to ${10}^{\ensuremath{-}12}$ cm and for $C$ from 0 to 0.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}12}$ cm. For the partial wave with angular momentum $\ensuremath{\hbar}$, the range of radii extends to 4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}12}$ cm in proton collisions with Li and to 2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}12}$ cm in proton collisions with $C$. The tables are applicable to other reactions as well. Phase shifts necessary for the theory of anomalous nuclear scattering are readily calculated by means of them. The calculation of the regular function and its derivative by means of the tables is easier than that of the irregular solution. The tables are therefore supplemented by graphs in the present paper. By means of these it is possible to calculate the irregular function quickly even though less accurately than by means of the tables. The regular and irregular functions are given, respectively, by ${F}_{L}={C}_{L}{\ensuremath{\rho}}^{L+1}{\ensuremath{\Phi}}_{L}$, ${G}_{L}={D}_{L}{\ensuremath{\rho}}^{\ensuremath{-}L}{\ensuremath{\Theta}}_{L}$. Here the angular momentum of the partial wave is $L\ensuremath{\hbar}$, and ${C}_{L}$ and ${D}_{L}$ depend only on the energy, $\ensuremath{\rho}$ is $\frac{2\ensuremath{\pi}r}{\ensuremath{\Lambda}}$ where $r$ is the radius and $\ensuremath{\Lambda}$ is the de Broglie wave-length. It is found empirically that for low energies the quantities ${\ensuremath{\Phi}}_{L}$, ${\ensuremath{\Theta}}_{L}$ depend only on the radius and not on the energy. This fact is useful in applications and it is explained analytically by means of an apparently new expansion of the confluent hypergeometric function into series of Bessel functions. The successive terms of the series are arranged in ascending powers of the energy and these expansions furnish an independent way of calculating the functions. They are useful for low energies. The exact solutions are compared numerically with the WKB approximations to the functions and to their logarithmic derivatives. The ordinary WKB method is found to give only a crude approximation to the exact solutions in the needed range of energies and radii. The WKB formulas modified by changing $L(L+1)$ into ${(L+\frac{1}{2})}^{2}$ are very much better for small energies and small radii but they are not reliable as the region of positive kinetic energies is approached. In some cases the ${(L+\frac{1}{2})}^{2}$ method is worse than that using $L(L+1)$. The superiority of the ${(L+\frac{1}{2})}^{2}$ method for low energies and constant radii is traced to the fact that in such cases it is identical with the Carlini and Laplace approximations to Bessel functions. From this relationship and graphical comparisons given below errors in the $\mathrm{WKB}{(L+\frac{1}{2})}^{2}$ approximations can be estimated.