Time-resolved X-ray scattering by electronic wave packets: analytic solutions to the hydrogen atom

Abstract

Modern pulsed X-ray sources permit time-dependent measurements of dynamical changes in atoms and molecules via non-resonant scattering. The planning, analysis, and interpretation of such experiments, however, require a firm and elaborated theoretical framework. This paper provides a detailed description of time-resolved X-ray scattering by non-stationary electronic wave packets in atomic systems. A consistent application of the Waller-Hartree approximation is discussed and different contributions to the total differential scattering signal are identified and interpreted. Moreover, it is demonstrated how the scattering signal of wave packets in the hydrogen atom can be expressed analytically. This permits simulations without numerical integration and establishes a benchmark for both efficiency and accuracy. Based on that, scattering patterns of an exemplary wave packet in the hydrogen atom are computed for different points in time. In doing so, distinct features of time-resolved X-ray scattering by non-stationary electronic wave packets are illustrated and accentuated in greater detail than it has been done before.

Electronic Coherence in Ultrafast X-Ray Scattering from Molecular Wave Packets

“…The coherent mixed component is qualitatively different from the other two. It relies on interferences between different coherently populated electronic states and displays the same distinctive features as the scattering predicted for electronic wave packets in atoms [21][22][23]. In a molecule, these features are found to provide a direct signature of transient electronic coherences and could be exploited to characterize the evolution of molecular wave packets at avoided crossings, conical intersections, or during partial revivals.…”

Violation of centrosymmetry in time-resolved coherent x-ray diffraction from rovibrational states of diatomic molecules

“…Consequently, F (−q) = F * (q) and the differential cross section, which is the absolute square of the scattering amplitude, is always centrosymmetric, even if ρ e (x) lacks any symmetry. The violation of centrosymmetry (voc) in nonresonant diffraction images has been found in simulations focused on the detection of electron motion in the hydrogen atom [47,51,52]. In this paper we show, using the same level of approximation as in time-independent theories, that voc is a general phenomenon in time-resolved coherent x-ray diffraction.…”

“…Finally, since asymmetric electronic motions in atoms also produce similar asymmetric diffraction patterns [47,51,52,64], the asymmetry in x-ray diffraction from diatomic molecules may be altered if electronic transitions accompany the ro-vibrational nuclear motion during a molecular reaction. Hence, the voc asymmetry may be a probe for studying the interplay between the electronic and nuclear degrees of freedom in molecular reactions.…”

Theory of ultrafast x-ray scattering by molecules in the gas phase

“…Remarkably, their approach led to equations very similar to those obtained by more recent and fundamental derivations. Most notably, Cao and Wilson 25 could distinguish the same three components to the scattering signal identified in more recent work 28,[35][36][37] : elastic scattering, inelastic scattering, and scattering related to electronic coherences.…”