The Computation of Quadrupole and Magnetic-Dipole Transition Probabilities

The theory of quadrupole radiation is formulated in such a fashion as to make available all the methods of computation and sum rules which have been developed for electric-dipole radiation. Formulas for the strengths are given for the $\mathrm{SL}{M}_S{M}_L$ scheme as well as the $\mathrm{SLJM}$ scheme. The problem of the relative strengths of the multiplets in a transition array is treated, and formulae are given that explicitly solve this problem for all possible transitions between two-electron configurations. Values for $\mathrm{np}{n}^{\prime }{p}^{\prime }\leftrightharpoons \mathrm{np}{n}^{\prime \prime }p$, $\mathrm{np}{n}^{\prime }p\leftrightharpoons n{p}^2$, $\mathrm{np}{n}^{\prime }p\to \mathrm{np}{n}^{\prime }P$ and $n{p}^2\to n{p}^2$ are tabulated. The spectroscopic stability method and the eigenfunction method of obtaining relative multiplet strengths are given a convenient formulation; and values are computed for $n{p}^3\to n{p}^3$. A root $S^{\frac{1}{2}}$ of the line strength is defined in such a way that it transforms like a Hermitian matrix component. This enables convenient computation of line strengths in intermediate coupling. The $J$-file sum rule is shown to hold in intermediate coupling with the same degree of generality as for the electric-dipole case. Finally, closed formulas for the magnetic-dipole strengths in $\mathrm{LS}$ coupling are given and the method of transformation to intermediate coupling is indicated.