Level sequence and splitting identification of closely spaced energy levels by angle-resolved analysis of fluorescence light

The angular distribution and linear polarization of the fluorescence light following the resonant photoexcitation is investigated within the framework of density matrix and second-order perturbation theory. Emphasis has been placed on “signatures” for determining the level sequence and splitting of intermediate (partially) overlapping resonances, if analyzed as a function of photon energy of incident light. Detailed computations within the multiconfiguration Dirac–Fock method have been performed, especially for the $1s^{2}2s^{2}2p^{6}3s,J_i=1/2+{\gamma }_1\to {\left(1s^{2}2s2p^{6}3s\right)}_{1}3p_{3/2},J=1/2,3/2\to 1s^{2}2s^{2}2p^{6}3s,J_f=1/2+{\gamma }_2$ photoexcitation and subsequent fluorescence emission of atomic sodium. A remarkably strong dependence of the angular distribution and linear polarization of the ${\gamma }_2$ fluorescence emission is found upon the level sequence and splitting of the intermediate ${\left(1s^{2}2s2p^{6}3s\right)}_{1}3p_{3/2},J=1/2,3/2$ overlapping resonances owing to their finite lifetime (linewidth). We therefore suggest that accurate measurements of the angular distribution and linear polarization might help identify the sequence and small splittings of closely spaced energy levels, even if they cannot be spectroscopically resolved.

Figure 1

Level scheme for (combined) process of photoexcitation and subsequent fluorescence emission via two overlapping resonances. An atom (or ion) in the initial ground level ${\alpha }_i{J}_i$ absorbs the photon ${\gamma }_1$, is excited to the overlapping $\alpha J$ and ${\alpha }^{\prime }{J}^{\prime }$ resonances, and subsequently decays to some low-lying levels ${\alpha }_f{J}_f$ via fluorescence emission of photon ${\gamma }_2$.

Figure 2

Geometry of the photoexcitation and subsequent radiative decay. The incident light ${\gamma }_1$ propagates along the $z$ axis (chosen as the quantization axis) with its polarization vector ${\epsilon }_{{\lambda }_1}$ in the $x$ axis, while the fluorescence photon ${\gamma }_2$ is described by the two angles $(\theta ,\phi )$.

Figure 3

Level scheme of the $2s\to 3p$ inner-shell photoexcitation (red) and the subsequent radiative decay (blue) of atomic sodium. In the computations, we here use the experimentally known data for the total linewidth ${\mathrm{\Gamma }}_{\mathrm{tot}}$ and the central excitation energy $E_{\mathrm{exc}}$ [40].

Figure 4

Anisotropy parameter $\beta$ for the angular distribution of the ${\left(1s^{2}2s2p^{6}3s\right)}_{1}3p_{3/2},J=1/2,3/2\to 1s^{2}2s^{2}2p^{6}3s,J_f=1/2$ fluorescence emission of sodium as functions of the photon energy $\omega$ of the incident ${\gamma }_1$ light. Results are shown for several assumed level splittings of the two $J=1/2,3/2$ overlapping resonances: $▵E=0.01$ eV (black solid line), 0.06 eV (red dotted line), 0.10 eV (blue dashed line), and 0.12 eV (magenta dash-dotted line).

Figure 5

Anisotropy parameter $\beta$ for the angular distribution of the ${\left(1s^{2}2s2p^{6}3s\right)}_{1}3p_{3/2},J=1/2,3/2\to 1s^{2}2s^{2}2p^{6}3s,J_f=1/2$ fluorescence emission of sodium as functions of the photon energy $\omega$ of the incident ${\gamma }_1$ light. Results are shown for two assumed level splittings $▵E=0.10$ eV (blue dashed line) and $-0.10$ eV (black dash-dot-dotted line), which indicate that the (absolute) splitting of the two overlapping resonances remains the same but the sequence becomes opposite in both cases.

Figure 6

The same as Fig. 4 but for the linear polarization of the fluorescence ${\gamma }_2$ light emitted perpendicular ($\theta ={90}^{\circ }$) to the propagation direction of the incident ${\gamma }_1$ light.

Figure 7

The same as Fig. 5 but for the linear polarization of the fluorescence ${\gamma }_2$ light emitted perpendicular ($\theta ={90}^{\circ }$) to the propagation direction of the incident ${\gamma }_1$ light.