Overlap and Exchange Effects in Beta Decay

The change in nuclear charge by one unit in beta decay causes initial and final atomic states to overlap imperfectly. The effect of this imperfect overlap on the shape of allowed electron and positron emission spectra is calculated. The calculated change in the spectrum shape can be simulated by including the average excitation energy of the final atom in the energy balance. The inhibition, due to imperfect atomic overlap, of electron-capture rates, as well as total electron and positron-emission rates, is also determined. In all known cases, imperfect atomic overlap increases beta-decay lifetimes by at most a few percent and usually by an amount less than a few tenths of one percent. Antisymmetrization between decay and bound atomic electrons, in conjunction with the change in nuclear charge, gives rise to exchange effects in electron emission and electron capture. Due to exchange terms, the usual allowed electron spectrum is multiplied by a quantity that is of the order of $1-2Z^{-1\mathrm{}}$ for energies less than the binding energy of a $K$ electron in the initial atom. This exchange correction is negligible for higher energies of the emitted continuum electron. A simple approximate formula is derived that predicts the effect of exchange on $L$ to $K$ capture ratios; this formula predicts a 22% increase over the usual theoretical value for the $L$ to $K$ ratio of ${\mathrm{Ar}}^{37}$. The ${\mathrm{Ar}}^{37}$ prediction is in excellent agreement with recent experiments and with a more complicated calculation by Odiot and Daudel. Exchange effects change total electron emission and electron capture rates by at most a few percent.