Sum rules for the polarization correlations in photoionization and bremsstrahlung

The polarization correlations in doubly differential cross sections are investigated for photoionization and ordinary bremsstrahlung. These correlations describe the polarization transfer between incident light and ejected photoelectrons as well as between an incoming electron beam and bremsstrahlung light, respectively. They are characterized by a set of seven real parameters $C_{ij}$. We show that the squares of these parameters are connected by simple “sum rules.” These sum rules can be applied for both one-electron systems and also for atoms, if the latter are described within the independent particle approximation. In particular, they are exact in their simplest form (i) for the photoionization of $K$-, $L_{I,II}$-, and $M_{I,II}$-atomic shells, as well as (ii) for bremsstrahlung in which the electron is scattered into $s_{1/2}$ or $p_{1/2}$ states, as in the tip (bremsstrahlung) region. Detailed calculations are performed to verify the derived identities and to discuss their possible applications for the analysis of modern photoionization and bremsstrahlung experiments. In particular, we argue that the sum rules may help to determine the entire set of (significant) polarization correlations in the case when not all $C_{ij}$ are available for experimental observation.

Figure 1

The polarization correlation parameters $C_{ij}$ for $K$-shell ionization of hydrogenlike beryllium $\left(Z=4\right)$ by 100 keV (upper panel), 500 keV (middle panel), and 1 MeV (lower panel) photons. Different curves are used to represent the correlations: $C_{10}$ (red solid line), $C_{33}$ (black short-dashed line), $C_{31}$ (blue dotted line), $C_{23}$ (green long-dash-dotted line), $C_{02}$ (orange short-dash-double-dotted line), $C_{21}$ (maroon long-dashed line), and $C_{12}$ (pink short-dash-dotted line). Moreover, we have additionally labeled various among the curves in different panels.

Figure 2

The same as Fig. 1 but for $K$-shell ionization of hydrogenlike argon $\left(Z=18\right)$.

Figure 3

The same as Fig. 1 but for $K$-shell ionization of hydrogenlike xenon $\left(Z=54\right)$.

Figure 4

The polarization correlation parameters $C_{ij}$ for bremsttrahlung produced by 100 keV (upper panel), 500 keV (middle panel), and 1 MeV (lower panel) electrons scattered by a bare beryllium ion $\left(Z=4\right)$. Different curves are used to represent the correlations: $C_{03}$ (red solid line), $C_{32}$ (black short-dashed line), $C_{12}$ (blue dotted line), $C_{11}$ (green long-dash-dotted line), $C_{31}$ (orange short-dash-double-dotted line), $C_{20}$ (maroon long-dashed line), and $C_{23}$ (pink short-dash-dotted line). Moreover, the linear combination of squares of correlations $R_{\mathrm{br}}$ is shown on an absolute scale by the thin gray solid curve.

Figure 5

The same as Fig. 4 but for electron scattering by the bare argon ion $\left(Z=18\right)$.

Figure 6

The same as Fig. 4 but for electron scattering by the bare xenon ion $(Z=54)$.