Effects of Radioactive Disintegrations on Inner Electrons of the Atom

TLDR: In this article, the probability that nuclear emission of an alpha- or beta-particle causes ionization of a $K$ or $L$ electron of the atom is calculated by time-dependent perturbation theory using nonrelativistic Coulomb wave functions.

Abstract: The probability that nuclear emission of an alpha- or beta-particle causes ionization of a $K$ or $L$ electron of the atom is calculated by time-dependent perturbation theory using nonrelativistic Coulomb wave functions. Beta-emission (electron or positron) causes an ionization probability of $\frac{0.64}{{Z}^{2}}$ and $\frac{2.1}{{Z}^{2}}$ per beta in the $K$ and $L$ shells, respectively. (The $K$ shell result agrees with Migdal and Feinberg; the $L$ shell result disagrees with Migdal.) The use of nonrelativistic wave functions causes an appreciable underestimate in the ionization probability for $K$ electrons of heavy atoms. Screening corrections for the use of Coulomb wave functions would increase the ionization probabilities by a factor of 1.4 for $K$ electrons and by a factor of 3 or 4 for $L$ electrons. Migdal's result for dipole electronic transitions caused by nuclear alpha-decay are reduced by a factor 25 (for the case of ${\mathrm{Po}}^{210}$) because of nuclear recoil. Quadrupole matrix elements such as ${({r}^{\ensuremath{-}3})}_{1s,{n}^{\ensuremath{'}}d}$ are evaluated by a new method developed by H. A. Bethe. This method uses the Sommerfeld integral representation for the continuum ${n}^{\ensuremath{'}}d$ wave function. Quadrupole transitions are negligible for $K$ electrons, but are the predominant effect for $L$ electrons. The calculated ionization probabilities for ${\mathrm{Po}}^{210}$ are ${10}^{\ensuremath{-}7}$ and 1.1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ per alpha in the $K$ and $L$ shells, respectively. For alpha-decay, screening corrections and higher multipole transitions would both increase the ionization probability for $L$ electrons. Madansky and Rasetti's measurements of photons from RaE are consistent with our calculations, but Bruner's measurements on ${\mathrm{Sc}}^{44}$ are not. Grace's interpretation of $K$ x-rays from ${\mathrm{Po}}^{210}$ is consistent with the calculation of this paper, while Barber and Helm's interpretation is not. Rubinson and Bernstein find 8 times the $L$ x-ray yield from ${\mathrm{Po}}^{210}$ we have calculated for Coulomb wave functions.