Relativistic evaluation of the two-photon decay of the metastable $1s^{2}2s2p{}^3{P}_0$ state in berylliumlike ions with an effective-potential model

The two-photon $1s^{2}2s2p{}^3{P}_0\to 1s^2{s}^2{}^1{S}_0$ transition in berylliumlike ions is investigated theoretically within a fully relativistic framework and a second-order perturbation theory. We focus our analysis on how electron correlation, as well as the negative-energy spectrum, can affect the forbidden $E1M1$ decay rate. For this purpose, we include the electronic correlation via an effective local potential and within a single-configuration-state model. Due to its experimental interest, evaluations of decay rates are performed for berylliumlike xenon and uranium. We find that the negative-energy contribution can be neglected at the present level of accuracy in the evaluation of the decay rate. On the other hand, if contributions of electronic correlation are not carefully taken into account, it may change the lifetime of the metastable state by up to 20%. By performing a fully relativistic $jj$-coupling calculation, we find a decrease of the decay rate by two orders of magnitude compared to nonrelativistic $LS$-coupling calculations, for the selected heavy ions.

Figure 1

Energy atomic structure of berylliumlike uranium relative to the final state $2s^2{}^1{S}_0$. The four paths composed of the blue $\left(M1\right)$ and thicker green $\left(E1\right)$ arrows connect the initial state to the final state, through the first intermediate states of $S^{1/2}(2,1),S^{3/2}(2,1),S^{1/2}(1,2)$, and $S^{3/2}(1,2)$ of Eq. (2), which are allowed by selection rules. Solid and dashed arrows show the $S^{j_n}(1,2)$ and $S^{j_n}(2,1)$ amplitudes, respectively.

Figure 2

Diagram of paths, or allowed matrix elements, connecting the initial state to the final state through ${\mathrm{\Psi }}^{\text{nonexc}}\left(1M_n\right)$ and ${\mathrm{\Psi }}^{\text{exc}}\left(1M_n\right)$ possible intermediate states. Blue and green arrows represents $M1$ and $E1$ multipoles, respectively. Path (a) (solid arrows) corresponds to an intermediate state ${\mathrm{\Psi }}^{\text{nonexc}}\left(1M_n\right)$ with a dominant contribution of ${\mathrm{\Phi }}_{2s2p}\left(1M_n\right)$. Path (b) (dashed arrows) connects an intermediate state ${\mathrm{\Psi }}^{\text{exc}}\left(1M_n\right)$ to the final and initial states via the nonorthogonality of the spectator electrons. Path (c) (dot arrows) connects correlation configurations of the initial and final states to ${\mathrm{\Psi }}^{\text{exc}}\left(1M_n\right)$. The dashed line means a link between the configurations with higher $c_0$ and minor $c_1$ contributions in the $jj$ expansion.