Determination of small level splittings in highly charged ions via angle-resolved measurements of characteristic x rays

The angular distribution and the photon-photon angular correlation have been investigated for the x-ray emission from two-step radiative cascades that proceed via overlapping intermediate resonances. In particular, density matrix theory is applied in order to explore how the splitting of these intermediate levels affects the subsequent x-ray emission and whether measurements of photon angular distributions may help reveal level crossings in highly charged ions, if analyzed along isoelectronic sequences. Detailed computations within the multiconfiguration Dirac-Fock method were performed especially for the two-step $1s2p^2{J}_i=1/2,3/2\to 1s2s2pJ=1/2,3/2+{\gamma }_1\to 1s^{2}2s{J}_f=1/2+{\gamma }_1+{\gamma }_2$ cascade of lithiumlike ions, for which a level crossing of the two $1s2s2pJ=1/2,3/2$ intermediate resonances occurs in the range $74\le Z\le 79$. For this cascade, a remarkably strong depolarization effect, associated with the finite lifetime of these intermediate levels, is found for the angular distribution and the photon-photon correlation function for all level splittings $\Delta \omega \lesssim 0.2\text{a.u.}\approx 5.4$ eV. We therefore suggest that accurate angle-resolved measurements of the x-ray emission may serve also as a tool for determining small level splittings in excited highly charged ions.

Figure 1

Level scheme of the two-step radiative decay cascade (1). The ion is initially assumed to be in the doubly excited level ${\alpha }_i{J}_i$ and decays radiatively via the two overlapping intermediate resonances $\alpha J$ and ${\alpha }^{\prime }{J}^{\prime }$ into the ground level ${\alpha }_f{J}_f$. While the two photons ${\gamma }_1$ and ${\gamma }_2$ can be distinguished energetically, no further line structures could be resolved individually because of the (partial) overlap of the levels $\alpha J$ and ${\alpha }^{\prime }{J}^{\prime }$.

Figure 2

Level scheme and dominant decay channels of the doubly excited $1s2p^2{J}_i=1/2,3/2$ resonances of lithiumlike ${\mathrm{W}}^{71+}$ ions along with their estimated branching fractions. About $0.01\%$ proceeds via the (blue) radiative cascade (6) of interest.

Figure 3

Energy splitting $\Delta \omega$ between the two intermediate $1s2s2p_{1/2}J=1/2,3/2$ levels as function of the atomic number Z along the lithium isoelectronic sequence. The crossing of these two levels occurs approximately in the range $74\le Z\le 79$.

Figure 4

Anisotropy parameters (12) of the angular distribution of the second-step photons as functions of the level splitting $\Delta \omega$. Results are shown for an initially aligned $1s2p^2{J}_i=3/2$ resonance of lithiumlike ${\mathrm{W}}^{71+}$ ions and by assuming four different alignment parameters for this resonance owing to its prior excitation: ${\mathcal{A}}_2=$ 1.0 (black solid line), 0.5 (red dashed lined), $-0.5$ (blue dash-dotted line), and $-1.0$ (gray dotted line).

Figure 5

Angular distribution of the second-step photon emission for the cascade (6). Results are shown for an initially aligned $1s2p^2{J}_i=3/2$ resonance with alignment parameters ${\mathcal{A}}_2=-1.0$ (left panel) and 1.0 (right panel) of lithiumlike ${\mathrm{W}}^{71+}$ ions and for four assumed level splittings of the intermediate $1s2s2p_{1/2}J=1/2,3/2$ levels: $\Delta \omega =0.01$ a.u. (black solid line), 0.1 a.u. (red dashed line), 0.5 a.u. (blue dash-dotted line), and 1.0 a.u. (pink dotted line).

Figure 6

Anisotropy parameters (8) and (10) for the photon-photon angular correlation function of the initial $1s2p^2{J}_i=1/2$ (left panel) and $1s2p^2{J}_i=3/2$ (right panel) resonances of lithiumlike ${\mathrm{W}}^{71+}$ ions. These parameters are shown as functions of energy splitting $\Delta \omega$ and for four different alignment parameters of the $J_i=3/2$ resonance: ${\mathcal{A}}_2=-1.0$ (black solid line), $-0.5$ (red dotted line), 0.0 (blue dash-dotted line), and 0.5 (pink dotted line).

Figure 7

Photon-photon angular correlation functions for the photon cascade (6) are shown here for the initial $1s2p^2{J}_i=1/2$ level (left panel) calculated using Eq. (7) as well as the $1s2p^2{J}_i=3/2$ level with initial alignment ${\mathcal{A}}_2=-1.0$ (middle panel) and 0.0 (right panel) of lithiumlike ${\mathrm{W}}^{71+}$ ions calculated using Eq. (9). Results are shown for the same level splittings of the intermediate resonances and using the same style as in Fig. 4.