Parametrization of the angular correlation and degree of linear polarization in two-photon decays of hydrogenlike ions

The spontaneous two-photon emission in hydrogenlike ions is investigated within the framework of second-order perturbation theory and Dirac's equation. Special attention is paid to the angular correlation of the emitted photons as well as to the degree of linear polarization of one of the two photons, if the second is just observed under arbitrary angles. Expressions for the angular correlation and the degree of linear polarization are expanded in powers of cosine functions of the two-photon opening angle, whose coefficients depend on the atomic number and the energy sharing of the emitted photons. The effects of including higher (electric and magnetic) multipoles upon the emitted photon pairs beyond the electric-dipole approximation are also discussed. Calculations of the coefficients are performed for the transitions $2s_{1/2}\to 1s_{1/2}$, $3d_{3/2}\to 1s_{1/2}$, and $3d_{5/2}\to 1s_{1/2}$, along the entire hydrogen isoelectronic sequence ($1\le Z\le 100$).

Figure 1

Geometry of the two-photon decay. The $z$ axis is collinear with the first photon momentum vector ${\mathit{k}}_{\mathbf{1}}$. The plane $zx$ contains the second photon momentum vector ${\mathit{k}}_{\mathbf{2}}$ The angle $\theta$ is the opening angle between the direction of the two photons.

Figure 2

Values of the parameters $a_i$ defined in Eq. (9), for the $2s_{1/2}\to 1s_{1/2}$ transition as a function of $Z$. The values obtained in this work, which correspond to different energy shares $y=0.1$, 0.25, and 0.5, are represented by a black-solid, red-dashed, and green-dot-dashed curve, respectively. With exception of $a_0$ (given in s${}^{-1}$), all parameters are dimensionless. The same energy shares values given by Au [17] are represented by a blue dotted line ($y=0.5$), a cyan dot-dot-dashed line ($y=0.25$), and by the purple short dashed line ($y=0.1$). The parameters $a_2$ of Ref. [17] are always equal to one, as represented by the purple dotted line. On the bottom right plot, the blue dotted line refers to values of the total decay rate obtained from the parametrization of Ref. [17]. The black line represents the total decay values obtained from our current parametrization. The dark green dashed line contains the values provided by Refs. [18, 19]. The level of precision of the plot does not allow one to distinguish differences between these references. Values of parameters $b_i$ can be obtained from this figure using the relations $b_0\approx -a_0$, $b_1\approx a_3$, and $b_4\approx -a_4$.

Figure 3

Values of the parameters $a_i$ defined in Eq. (9), for the $2d_{3/2}\to 1s_{1/2}$ transition as function of $Z$. The different energy shares $y=0.1$, 0.25, 0.5 are represented by a black-solid, red-dashed, and green-dot-dashed curve, respectively. With exception of $a_0$ (given in s${}^{-1}$), all parameters are dimensionless. Values for parameters $b_i$ can be obtained from this figure (within its resolution) using the relations $b_0\approx -a_0/13$ and $b_1\approx 13a_3$.

Figure 4

Values of the parameters $a_i$ defined in Eq. (9), for the $2d_{5/2}\to 1s_{1/2}$ transition as a function of $Z$. The different energy shares $y=0.1$, 0.25, 0.5 are represented by a black-solid, red-dashed, and green-dot-dashed curve, respectively. With exception of $a_0$ (given in s${}^{-1}$), all parameters are dimensionless.