= Solution
<Beta decay> converts a <neutron> to a <proton> or vice versa through the <weak interaction>, leaving the <mass> number fixed:
$$
n\to p+e^-+\bar\nu_e,\qquad
p\to n+e^++\nu_e.
$$
Inside a nucleus the full initial and final nuclear masses determine whether either process is possible. <Electron capture>, $p+e^-\to n+\nu_e$, is the related bound-electron process. The emitted charged-lepton spectrum is continuous because <energy> is shared with the <neutrino> and nuclear recoil.
For <allowed beta decay>, the lepton <wavefunctions>' leading spatial terms are constant over the nucleus. A <Fermi transition> has $\Delta J=0$ and no <parity> change. A <Gamow-Teller transition> has $\Delta J=0,1$ and no <parity> change, excluding $0\to0$. Thus a same-parity transition with nonzero initial and final spins and $\Delta J=0$ can contain both contributions. First-forbidden transitions change <parity> and permit $|\Delta J|=0,1,2$; the $|\Delta J|=2$ case is <unique forbidden beta decay>. Higher forbiddenness requires higher powers of the small lepton <momentum> times nuclear size. “Forbidden” denotes suppression of the leading allowed operator, not an impossible decay.<Nuclear Data Sheets General Policies by National Nuclear Data Center>[Physical PDF page 3, General Policies — Theory, Beta Transitions]{ref}
<Fermi's golden rule> expresses a rate as a squared nuclear <matrix element> times leptonic <phase space>. Neglecting charged-lepton <mass> and Coulomb distortion, a constant allowed <matrix element> gives
$$
\lambda\propto |M|^2\int_0^Q E^2(Q-E)^2dE
=\frac{|M|^2Q^5}{30}.
$$
This is the approximate <Sargent rule>, $\lambda\propto Q^5$ for comparable allowed strengths; it is not an exact law across arbitrary transitions. Finite <electron> <mass>, the <Fermi function>, forbidden shape factors and nuclear structure change the comparison. A microscopic first-forbidden unique operator introduces additional <momentum> factors.
Using bare nuclear masses $M_i,M_f$, the kinetic-energy release is
$$
Q_{\beta^-}=(M_i-M_f-m_e)c^2,
\qquad Q_{\beta^+}=(M_i-M_f-m_e)c^2.
$$
The daughter is different in the two cases: its charge is $Z+1$ for beta minus and $Z-1$ for beta plus. The relevant <mass> difference must exceed one <electron> <mass>. For neutral atomic masses, to leading order in small electronic binding corrections, the corresponding expressions are $Q_{\beta^-}=(M_i^{\rm at}-M_f^{\rm at})c^2$ and $Q_{\beta^+}=(M_i^{\rm at}-M_f^{\rm at}-2m_e)c^2$. <Electron capture> is energetically possible approximately when $M_i^{\rm at}>M_f^{\rm at}$, with available shell binding <energy> and daughter excitation included. Nuclear masses and atomic masses must not be mixed.
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