Resonant inelastic X-ray scattering

RIXS is a two steps process. First an electron is resonantly excited from a core level, defined by the absorption edge, to an empty state, leaving a core hole. The intermediate state with the core hole has a lifetime of few femtoseconds, then the system radiatively decays into the final state with the filling of the core hole and the emission of another photon. Since the probability of a radiative core hole relaxation is low, the RIXS cross section is very small and a high brilliance X-ray source is needed. Being a second order process, the RIXS cross section is described by the Kramers-Heisenberg formula. The scattering geometry (incidence and scattering angles) determines the momentum transfer \vec{q}. In order to explore the \vec{q} space the spectrometer angle with respect to the incoming beam can be changed, as well as the incident angle to the sample. The net result is a final state with an electron-hole excitation, as an electron was created in an empty valence band state and a hole in a filled shell. If the hole is in the filled valence shell, the electron-hole excitation can propagate through the material, carrying away momentum and energy. Momentum and energy conservation require that these are equal to the momentum and energy loss of the scattered photon. Indirect RIXS File:Indirect RIXS process.jpg|thumb|Indirect RIXS process. An electron is excited from a deep-lying core level into the valence shell. Excitations are created through the Coulomb interaction U_c between the core hole (and in some cases the excited electron) and the valence electrons. ==Experimental details==

In general the natural linewidth of a spectral feature is determined by the life-times of initial and final states. Indeed, as for X-ray absorption and non-resonant X-ray emission spectroscopy the energy resolution is often limited by the relatively short life-time of the final state core-hole. As in RIXS a high energy core-hole is absent in the final state, this leads to intrinsically sharp spectra with energy and momentum resolution determined by the instrumentation. A convolution of the incident X-ray bandpass, defined by the beamline monochromator, and the bandpass of the RIXS spectrometer for the analysis of the scattered photons energy gives the total (combined) energy resolution. Since RIXS exploits high energy photons in the X-ray range, a very large combined resolving power (103-105 depending on the goal of the experiment) is needed to detail the different spectral features. Therefore, in the last two decades efforts have been made to improve RIXS spectrometers performances, gaining orders of magnitude in terms of resolving power. State of the art soft X-rays RIXS beamlines in use at the ESRF, at DLS and at NSLS II, have reached approximately 40000 of combined resolving power, leading to a record energy resolution of 25 meV at Cu L3 edge. As for hard X-rays, the optical design is different and requires the use of Bragg reflection crystal analyzers. Thus, the resolving power is mostly determined by the crystal analyzers in use. Soft X-ray spectrometers of optical elements for X-rays reduces the throughput. In addition to that, a non-negligible contribution to the combined resolving power is due to the imperfections on the surface of mirrors and gratings (slope error). Finally, the lower the number of optical elements to be aligned, the better in terms of setup time. The monochromatized X-rays impinge on the sample with a defined geometry and are scattered and collected by the spectrometer. Collection mirrors are often placed after the sample, the distance (1 cm to 1 m) depends on the optical design. This is useful to increase the acceptance angle of the spectrometer and thus the efficiency. The optical layout for hard X-rays RIXS spectrometers is different. The spectrometers are based on spherical crystal analyzers (typically more than one to increase the solid angle of the spectrometer) exploiting Bragg reflections and on a position sensitive detector, typically in the so called Rowland geometry. This means that the source (X-rays spot on the sample), the analyzers and the detector must sit on the Rowland circle. By scanning the positions of the analyzers and of the detector (the source is fixed for convenience) the Bragg condition is changed and thus the energy of the scattered X-rays can be analyzed. By increasing the radius of the Rowland circle, the energy resolution can be increased, loosing in terms of efficiency. Nevertheless, as opposed to soft X-rays spectrometers, the resolving power of the spectrometer is limited by the crystal analyzers. Thus, increasing too much the dimensions of the spectrometer does not pay off. Depending on the chosen absorption edge (and thus incidence energy), different crystal analyzers are used both on the monochromator side and on the spectrometer side. Thanks to the high penetration depth of hard X-rays, there is no need of UHV. Therefore, the exchange of optical elements, such as crystal analyzers, is less disruptive than for soft X-rays. ==RIXS properties==

Compared to other inelastic scattering techniques as INS, IXS, EELS or Raman scattering that present shortcomings, RIXS has a number of unique features: it covers a large scattering phase-space thanks to the high energy photons, it is polarization dependent, element specific, bulk sensitive and requires only small sample volumes enabling studies on thin films as well as diluted solutions. RIXS is a resonant technique because the energy of the incident photon is chosen such that it coincides with, and hence resonates with, one of the atomic X-ray absorption edges of the system. The resonance greatly enhances the valence contribution to the inelastic scattering cross section, sometimes by many orders of magnitude. Comparing the energy of a neutron, electron or photon with a wavelength of the order of the relevant length scale in a solid - as given by the de Broglie equation considering the interatomic lattice spacing is in the order of Ångströms - it derives from the relativistic energy–momentum relation that an X-ray photon has more energy than a neutron or electron. The scattering phase space (the range of energies and momenta that can be transferred in a scattering event) of X-rays is therefore without equal. In particular, high-energy X-rays carry a momentum that is comparable to the inverse lattice spacing of typical condensed matter systems so that, unlike Raman scattering experiments with visible or infrared light, RIXS can probe the full dispersion of low energy excitations in solids. RIXS can utilize the polarization of the photon: the nature of the excitations created in the material can be disentangled by a polarization analysis of the incident and scattered photons, which allow one, through the use of various selection rules, to characterize the symmetry and nature of the excitations. RIXS is element specific: chemical sensitivity arises by tuning to the absorption edges of the different types of elements in a material. RIXS can even differentiate between the same chemical element at sites with different valencies or at inequivalent crystallographic positions as long as the X-ray absorption edges in these cases are distinguishable. In addition, the type of information on the electronic excitations of a system being probed can be varied by tuning to different X-ray edges (e.g., K, L or M) of the same chemical element, where the photon excites core-electrons into different valence orbitals. RIXS is bulk sensitive: the penetration depth of resonant X-ray photons depends on the material and on the scattering geometry, but typically is of the order of a few micrometers in the hard X-rays regime (for example at transition metal K-edges) and on the order of 0.1 micrometers in the soft X-ray regime (e.g. transition metal L-edges). RIXS needs only small sample volumes: the photon-matter interaction is relatively strong, compared to for instance to the neutron-matter interaction strength. This makes RIXS feasible on very small volume samples, thin films, surfaces and nano-objects, in addition to bulk single crystal, powder samples or diluted solutions. == RIXS spectral features ==

and vibrational modes that are present in a RIXS spectrum through the electron-phonon coupling. Only a portion of phonons modes that characterize the sample are visible through RIXS. Electron-hole continuum and excitons in band metals, doped systems and semiconductors are visible through RIXS, thanks to the enhancement of valence charge excitations guaranteed by the resonance character of the technique. In the charge channel, also plasmons and their dispersion can be measured by RIXS, as well as orbital and crystal field excitations and charge transfer excitations. Moreover, it has been theoretically shown that RIXS can probe Bogoliubov quasiparticles in high-temperature superconductors, and shed light on the nature and symmetry of the electron-electron pairing of the superconducting state. == Pump-probe RIXS with X-ray free electron lasers (XFELs) ==

With the advent of XFELs, sources that can provide extremely brilliant (more than five orders of magnitude larger than synchrotron sources) and extremely short X-ray pulses, X-ray spectroscopies performed in a pump and probe fashion are nowadays available. The power of pump-probe spectroscopies lies in the possibility to study how a system evolves after an external stimulus. The most straightforward example is the study of photoactivated biological process, such as the photosynthesis: the sample is illuminated by an optical laser tuned at the proper wavelength and then its evolution is observed taking snapshots as a function of time. == Applications ==

• Intracellular metal speciation, • Mott insulators • high-temperature superconductors (e.g., cuprates), • Iron-based superconductors, • Semiconductors (e.g. Cu2O) • Metalloproteins (e.g. the oxygen-evolving complex in photosystem II) • Catalysis • Water, aqueous solution, aqueous acetic acid, aqueous glycine • High pressure. ==See also==