Theory of Relativistic Magnetic Dipole Transitions: Lifetime of the Metastable<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>2</mml:mn><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow></mml:math>State of the Heliumlike Ions

It has recently been established that the radiative lifetime of the metastable $2^{3}S$ state of helium and the heliumlike ions is determined by single-photon magnetic dipole ($M1$) transitions to the ground state, rather than the two-photon process proposed by Breit and Teller. The theory of $nl\ensuremath{-}{n}^{\ensuremath{'}}l$ $M1$ transitions with $n\ensuremath{\ne}{n}^{\ensuremath{'}}$ is developed in the Pauli approximation and extended to two-electron systems. Terms arising from relativistic energy corrections and finite-wavelength effects are included. The results for hydrogenic systems are shown to be identical to those obtained in the relativistic four-component Dirac formulation. The coefficients in the ${Z}^{\ensuremath{-}1}$ perturbation expansion of the $1s2s^{3}S\ensuremath{-}1{s}^{2}^{1}S$ $M1$ transition integral are evaluated through ninth order and used to calculate the $M1$ emission probabilities from the $2^{3}S$ state of the two-electron ions up to Fe XXV. The emission probability for neutral helium is 1.27 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ ${\mathrm{sec}}^{\ensuremath{-}1}$. The results are compared with recent solar coronal observations by Gabriel and Jordan, and with a measurement of the $2^{3}S$ state lifetime in Ar XVII by Schmieder and Marrus.