Moments in nuclear resonant inelastic x-ray scattering and their applications

Sum rules of the moments of the nuclear resonant inelastic x-ray scattering (NRIXS) spectrum provide means of data analysis and information on atomic dynamics. We extend existing work by calculating the third and fourth moment beyond the harmonic approximation and show that NRIXS can provide a direct measurement of the anharmonic terms in lattice potentials. Projected partial phonon density of states (ppDOS) extracted from measured spectra provide vibrational mode-specific information on lattice dynamics. Furthermore, unique contributions to thermodynamic properties can be defined and calculated as the moments and other weighted integrals of ppDOS. A summary of some of these thermodynamic quantities is given. The directional dependence of NRIXS and its effects are emphasized. We derive explicit relationships between the moments of phonon excitation probability function and those of ppDOS. The comparison between the two sets of moments provides a consistency check and insights to the lattice dynamics of the system under study.

Figure 1

The measured NRIXS spectra of goethite (the solid line), hydronium-jarosite (the dashed line), and potassium-jarosite (the dotted line). Here, it only shows data in an energy range of $\pm$75 meV, while the actual data collection range was $-$120 to +130 meV.

Figure 2

The ppDOS of goethite (the solid line), hydronium-jarosite (the dashed line), and potassium-jarosite (the dotted line). Here, it only shows data in an energy range up to 100 meV.

Figure 3

The relative error estimates of the moments due to counting statistics (circles), energy variations (squares), and energy scaling error (triangles). The estimates are based on the data taken for a sample of goethite.