The THz Dynamics of Liquids Probed by Inelastic X-Ray Scattering

The following sections are included:

• Acknowledgements
• Contents
• Symbols used throughout the text

The collective dynamics of disordered systems is a subject of prominent scientific interest since the dawn of modern science. However, despite an intensive theoretical, experimental and computational scrutiny, it still presents many puzzling aspects. The main reasons for this limited understanding are the lack of translational invariance in the microscopic structure of amorphous materials and the often exceptionally complex movements of their microscopic constituents…

This chapter discusses some general properties of correlation functions of stochastic variables, which are observables directly accessed by scattering measurements. Correlation functions are mathematical entities, and, to some extent, their properties need to be discussed using a formal language. Nonetheless, these pages will give more emphasis to essential physical concepts, while the reader is often referred to the vast literature in the field for rigorous analytical derivations…

After discussing in the past chapters how the spectral response relates to correlation functions of fluctuating variables of a fluid at equilibrium, the general purpose of this chapter is helping the reader to become familiar with relevant theoretical aspects of Inelastic X-Ray Scattering. Specifically, the main aim of the following pages is to illustrate the analytical derivation of the scattering cross-section and to demonstrate its direct link to the spectrum of density fluctuation of the sample.

The following sections are included:

• Generalities on the INS technique (Lovesey, 1984, Squires, 2012)
• The cross-section of inelastic neutron scattering (see (Lovesey, 1984))
• Kinematic limitations (Squires, 2012)
• The roles of instrumental resolution and spectral contrast
• Three-axis and time of flight techniques (Windsor, 1981, Squires, 2012, Shirane et al., 2002)
• A few critical features of IXS spectrometers
• An example of state-of-the-art spectrometers: ID28 beamline at ESRF
• Towards new generation IXS spectrometers
• A closer comparison between IXS and INS
  • Advantages of IXS
  • Advantages of INS
• References

After discussing theoretical and experimental aspects of the IXS method, we now focus on the choice of an appropriate model for the measured IXS signal and, primarily, for the variable it directly relates to, namely S(Q, ω). The exact analytical form of S(Q, ω) is known in two extreme dynamic regimes only: at the lowest Q, ω’s (hydrodynamic regime) or at the highest ones (single-particle regime). In between these two limits, density fluctuations become strongly coupled with internal degrees of freedom of the fluid; the effects of this coupling on S(Q, ω) are often subtle and currently lack an analytical prediction. The absence of a firm lineshape theory appears especially penalizing as the spectral evolution along this crossover is uniquely informative of the various dynamical events occurring in a system at mesoscopic scales. Currently, the spectrum is mostly described by approximate models, which are often phenomenological and affected by inherent limitations such as their pertinence to the specific dynamic windows or thermodynamic conditions of the system. Also, the comparison between different analytical models is not always straightforward, and an effort toward a unified formalism is highly desirable (Bafile et al., 2006)…

After discussing some general aspects of the memory function formalism, and ultimately setting a formal link between the dynamic structure factor S(Q, ω) and the Laplace transform of the memory function (Eq. 5.28), we now focus on the actual modelling of the latter variable. All models to be discussed in this Chapter strictly refer to a classical description of the spectrum of density fluctuations in which all relevant operators and variables are treated as commuting quantities. Sizable quantum deviations are assumed to only influence the statistical population of states with energy ħω, via the detailed balance principle; specifically, they cause an asymmetry of the lineshape which can be consistently accounted for by the simple insertion of a frequency-dependent prefactor in the model, as discussed at the end of this chapter…

The following sections are included:

• Using THz spectroscopy to detect mesoscopic collective modes: Early results
• Evidence of extended Brillouin peaks at mesoscopic scales
• Further considerations on the different behavior of noble gases and liquid metals
• The kinetic theory approach: A few introductory topics
• The onset of kinetic regime probed by IXS measurements on deeply supercritical neon
• The crossover from the collective to the single-particle regime: Some qualitative aspects
• Using IXS as a probe of the single-particle regime
• Final states effects
• The case of molecular systems
• Gaining insight from spectral moments: The onset of quantum effects
• IXS studies of quantum effects in simple liquids
• Appendix 7A: Brief hints on the Enskog theory formalism (Kamgar-Parsi et al., 1987)
• Appendix 7B: Handling quantum effects analytically (Fredrikze, 1983)
• References

The following sections are included:

• Introductory topics
• Investigating viscoelastic phenomena by mesoscopic spectroscopy: The Q- and T-dependence of transport parameters
• Structural relaxations
• Brief remarks on the temperature dependence of relaxation time
• Quantitative insight on the structural relaxations: The case of water
  • Gaining insight on the relaxation process from the spectral shape
  • An IXS measurement of the structural relaxation time of water
  • The longitudinal viscosity of water
  • The microscopic contribution to the viscosity
• Collisional relaxations
• Other types of relaxation phenomena
• The adiabatic-to-isothermal transition
• Approximating the Rayleigh contribution to the memory function
• Appendix 8A
• References

The following sections are included:

• How do relaxation processes depend on thermodynamic conditions?
• Liquid-like and compressed gas behaviour
• Evidence of (thermo)dynamic boundaries
• The Frenkel line
• To what extent does the dynamics of a disordered system resemble the one of a solid?
  • Sound damping, structural disorder and elastic anisotropy
• Generalities on the propagation of a shear wave in a liquid
  • A transverse mode in the spectrum of water
  • The onset of a transverse dynamics in monatomic systems
• Polyamorphism phenomena in simple systems investigated by IXS
• References

The following section is included:

• Conclusive remarks