Spatial Quantum Beats in Vibrational Resonant Inelastic Soft X-Ray Scattering at Dissociating States in Oxygen

Resonant inelastic soft x-ray scattering (RIXS) spectra excited at the $1{\sigma }_g\to 3{\sigma }_u$ resonance in gas-phase ${\mathrm{O}}_2$ show excitations due to the nuclear degrees of freedom with up to 35 well-resolved discrete vibronic states and a continuum due to the kinetic energy distribution of the separated atoms. The RIXS profile demonstrates spatial quantum beats caused by two interfering wave packets with different momenta as the atoms separate. Thomson scattering strongly affects both the spectral profile and the scattering anisotropy.

Figure 1

Experimental (a) and theoretical (b) RIXS spectra for $\theta ={90}^{°}$; $\omega =539$, 538.75, 538.5, 538 eV (from top to bottom) as indicated in the absorption spectrum in the top panel. The theoretical “atomic” peak includes also contribution from the dissociative $1^3{\Pi }_g$ final state which converges to the same dissociative limit as the ground state. The polarization dependence of the RIXS spectrum shown in panel (c) demonstrates that scattering to $\nu =0$ has a different dependence on the angle $\theta =\angle (\mathbf{e},{\mathbf{n}}_1)$ (here $\theta =0^{°}$ and ${90}^{°}$), than scattering to $\nu >0$. The gray bar shows the intensity of the elastic peak intensity for $\theta ={90}^{°}$ without Thomson scattering.

Figure 2

Physical picture of the modulations in the vibrational RIXS spectrum. The calculated RIXS spectrum (gray profile in the right panel) with one intermediate state does not display any envelope oscillations and differs qualitatively from the experimental spectrum. The interference between the $Q$ and $D$ scattering channels results in a spatial modulation of the total wave packet ${\Psi }_Q+{\Psi }_D$ which leads to the beat structure in the RIXS spectrum (“$Q+D$” spectrum in the right panel). The reflection principle explains qualitatively how the spatial modulation of $|{\Psi }_Q+{\Psi }_D{|}^2$ gives the intensity modulation in the RIXS spectrum.