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We propose formulas of the nuclear beta-decay rate that are useful in apractical calculation. The decay rate is determined by the product of thelepton and hadron current densities. A widely used formula relies upon the factthat the low-energy lepton wave functions in a nucleus can be well approximatedby a constant and linear to the radius for the $s$-wave and $p$-wave wavefunctions, respectively. We find, however, the deviation from such a simpleapproximation is evident for heavy nuclei with large $Z$ by numerically solvingthe Dirac equation. In our proposed formulas, the neutrino wave function istreated exactly as a plane wave, while the electron wave function is obtainedby iteratively solving the integral equation, thus we can control theuncertainty of the approximate wave function. The leading-order approximationgives a formula equivalent to the conventional one and overestimates the decayrate. We demonstrate that the next-to-leading-order formula reproduces well theexact result for a schematic transition density as well as a microscopic oneobtained by a nuclear energy-density functional method.

Beta-decay formulas revisited (I): Gamow--Teller and spin-dipole contributions to allowed and first-forbidden transitions