On the theory of beta-radioactivity | the use of linear combinations of invariants in the interaction Hamiltonian

The theory of β-radioactivity, based on the use of a linear combination of all five relativistic invariants, is developed for allowed transitions. The interaction Hamiltonian is taken as H_β = GΣ_k=1 ⁵C_kJ_k, where the C_k are coefficients that determine the combination of the invariants J_k with 1 = scalar, 2 = vector, 3 = tensor, 4 = axial vector and 5 = pseudoscalar. The formula for the shape of β⁺ and β^--spectra is calculated, taking into account the nuclear charge Z ( E energy of the electron; P and q are the momenta of the electron and the neutrino respectively): ? The probability for K-capture becomes for this interaction: ? The angular correlation between electron and neutrino is determined by (neglecting the influence of the nuclear charge): ? A general principle, viz. complete symmetry for the processes of β⁺ and β^--emission, is proposed, which has as a consequence that only two types of combinations can exist: a) combinations of the invariants 1, 4 and 5, b) combinations of the invariants 2 and 3. We then obtain the result that in both cases the 1/ E-term in the β-spectrum drops out. Some considerations are given on the comparison of theory and experiment. Thus it is discussed that recoil-experiments with ⁶He are not yet sufficient to decide if the interaction is given by a “pure” invariant or by a linear combination. They should be completed by recoil-experiments with other nuclei e.g. ¹⁹Ne.