# Inelastic X-ray scattering (IXS)

### From Online Dictionary of Crystallography

Inelastische Röntgenstreuung (*Ge*). X線非弾性散乱 (*Ja*). Dispersión inelástica de rayos X (*Sp*).

A scattered photon may transfer some of its energy to the sample. The transferred energy can excite the atomic lattice (phonons), single electrons or an ensemble of electrons (collective excitations). IXS considers the energy and the momentum of an incoming and a scattered (emitted) photon. The difference is the energy and momentum transfer, respectively. An **IXS** process can occur after non-resonant or resonant X-ray excitation.

**X-ray Raman scattering** (**XRS**) is non-resonant inelastic scattering of X-rays from core electrons. It is analogous to Raman scattering, which is a widely-used tool in optical spectroscopy, with the difference being that the wavelengths of the exciting photons fall in the X-ray regime and the corresponding excitations are from deep core electrons. The process is in principle analogous to **X-ray absorption**, but the energy transfer plays the
role of the X-ray photon energy in X-ray absorption. XRS can be used to measure absorption edges of low *Z* elements using hard X-rays.

The cross section of non-resonant inelastic X-ray scattering is

[math]{d^2\sigma\over d\Omega dE} = \Big({d\sigma\over d\Omega}\Big)_{Th} \times S(q,E)[/math]

with [math](d\sigma/d\Omega)_{Th}[/math] being the Thomson cross section, which signifies that the scattering is that of electromagnetic waves from electrons. The physics of the system under study is buried in the dynamic structure factor [math]S(q,E)[/math], which is a function of momentum transfer [math]q[/math] and energy transfer [math]E[/math]. The dynamic structure factor contains all non-resonant electronic excitations, including not only the core-electron excitations observed in XRS but also *e.g.* plasmons (the collective fluctuations of valence electrons) and Compton scattering. At small momentum transfer [math]\mathbf{q} = \mathbf{k} - \mathbf{k}^\prime[/math] the XRS signal is proportional to the X-ray absorption cross-section. A large momentum transfer allows access to transition matrix elements of higher order, *e.g.* quadrupole transitions.

**Resonant inelastic X-ray scattering** (**RIXS**) or **resonant X-ray emission spectroscopy** (**RXES**) is frequently used to study the electronic structure. It requires the incident energy to be close to an absorption edge. It is possible to distinguish between two cases of resonant scattering:

(1) A fluorescence line can be measured after resonant excitation. This is referred to as resonant X-ray emission spectroscopy (RXES) or direct RIXS. In a one-electron picture one could refer to this process as a spectator decay: the photo-excited electron remains in a bound, previously unfilled orbital and an electron from a filled orbital decays to fill the core hole.

(2) The photo-excited electron decays to fill the core hole (participator decay). The modification of the Coulomb potential due to photo-excitation and decay may give rise to an excited final state, *i.e.* the system does not return to its ground state. The final state excitation can be a local (*e.g.* *dd*-excitation), including the ligand (charge transfer excitation) or higher coordination spheres and/or the long-range order of the system (plasmons, magnons, orbitons). The technique is often referred to as (indirect) resonant inelastic X-ray scattering (RIXS).

The general usage in the literature does not strictly distinguish between the two cases.

RIXS/RXES is a second-order process that is theoretically described (Kotani and Shin, 2001) by the Kramers–Heisenberg equation:

[math]F(\Omega, \omega) = \sum\limits_f \Bigg|\sum\limits_n{{\langle f|T_2^\dagger|n\rangle \langle n|T_1|g\rangle}\over{E_g - E_n + \Omega -i{\Gamma_n/2}}}\Bigg|^2 * {{\Gamma_f/2\pi}\over{(E_g-E_f+\Omega-\omega)^2 + {\Gamma_f^2/ 4}}}[/math]

with the ground ([math]g[/math]), intermediate ([math]n[/math]) and final ([math]f[/math]) state electron wavefunctions, their energies [math]E_g[/math], [math]E_n[/math] and [math]E_f[/math], the lifetime broadenings [math]\Gamma_n[/math] and [math]\Gamma_f[/math] (full width at half maximum) as well as the transition operators [math]T_1[/math] and [math]T_2[/math] for absorption and emission of an X-ray photon, respectively. The difference between incident and emitted X-ray energy is the energy transfer [math]\Omega - \omega[/math]. The Kramers–Heisenberg equation is often simplified by neglecting interference effects, multi-electron excitations and/or the angular dependence.

RIXS allows spectra to be recorded broadened only by [math]\Gamma_f[/math] and not [math]\Gamma_n[/math]. In most cases it holds that [math]\Gamma_n \gt \Gamma_f[/math] and RIXS thus enables the observation of spectral features with sharper line width than for an absorption spectrum. The method requires that the instrumental energy band width is sufficiently small.

## Reference

- Kotani, A. and Shin, S. (2001).
*Rev. Mod. Phys.***73**, 203–246