Inelastic X-Ray Scattering

**Chapter 1**

Fluids

**Abstract**

molecular systems.

**1. Introduction**

**3**

*Alessandro Cunsolo*

High-Resolution Inelastic X-Ray

Density Fluctuations in Simple

Scattering: A Probe of Microscopic

The explicit form of the inelastic X-ray scattering, IXS, cross-section is derived within a time-dependent perturbative treatment of the scattering process. In this derivation, the double differential cross-section is obtained from the Fermi Golden Rule within a plane wave expansion of the vector potential. Furthermore, it is assumed throughout that the Thompson term of the perturbative Hamiltonian yields the overwhelming contribution to the scattering. The achievement of an explicit form for the double differential scattering cross-section rests on the validity of the adiabatic or Born-Oppenheimer approximation. As a result, it is here shown that that the IXS double differential cross-section is proportional to the spectrum of density fluctuations of the sample, which is thus the sample variable directly accessed by IXS measurements. Although the whole treatment is valid for monatomic systems only, under suitable approximations, it can be extended to

Inelastic scattering measurements are among the most powerful tools to investigate the collective terahertz dynamics of disordered systems [1, 2]. Although this subject has been the focus of intense scrutiny in the past few decades, it still presents many challenging aspects. In a spectroscopic measurement, the dynamic response of the target system is stimulated via the exchange of an energy ℏω and momentum ℏ*Q* where ℏ is the reduced Planck's constant. A suitable choice of the exchanged wavevector amplitude *Q* = |*Q*| and ω enables to tune the probe to dynamic events occurring over different scales. For infinitesimal *Q* and *ω* values, the measurement probes slowly decaying, hydrodynamic, density fluctuation modes either propagating or diffusing throughout the system, which resembles a continuous and homogeneous medium [3]. Upon increasing *Q's* and *ω's*, probed dynamic events become gradually faster and involve fewer atoms until the extreme, single-particle limit is reached. In this limit, the probe couples with the free recoil of

**Keywords:** inelastic X-ray scattering, theory of the scattering, theory of the line-shape, double differential scattering cross-section
