Xray scattering by an arbitrary structure Coherence Kinematic

X-ray scattering by an arbitrary structure • • Coherence Kinematic and static approximation Interferences Calculation of scattered intensity

Mutual correlation function P 1 P 2 For Gaussian source, plane quasi-monochromatic wave, small divergence : Distance over which the wave is monochromatic Distance over which the wave is planar

Coherence: effect on diffraction Transverse coherence Longitudinal coherence Transverse coherence is the more important: All sources are coherent enough to make atoms interfere but not enough or the whole crystal (Synch. XFEL)

Classical approximations • Kinematic approximation • Scattered beam intensity is negligible compared to incident beam intensity • No multiple scattering, no intensity decay (does not conserve energy! § Born) • Good approximation for small crystals • No approximation Dynamical theory • Static approximation • X-ray frequencies: 1018 Hz • Atomic vibrations frequencies: 1012 Hz (THz) z t t • Atomic displacements negligible compared to X-ray wavelengths.

Interferences (Th. Young) Plane wave Fresnel Fraunhofer

Two regimes of diffraction Fresnel regime Near-field Augustin Fresnel (1788 -1827) 150 µm Fraunhofer regime Far-field Joseph von Fraunhofer (1787 -1826) 10 µm

Fraunhofer regime: the phase difference

Examples: Colloïd hard spheres (117 nm) of PMMA in decalin Water E=11 ke. V E=8 ke. V P. Wochner et al. , PNAS 106, 11511 (2009). T. Head-Gordon et al. , Chem. Rev. 102, 2651 (2002). 2 °C 77 °C

Kinematical aprroximation: Calculation of scattered intensity volume of the crystal T: time of experiment

Amplitude scattering Electron density • Set of identical atoms

Intensity expression Limited volume of scattering : -r

Density-density correlation function Statistical average

Role of coherence Nano-particles 390 nm-Si in water-lutidin solution Phase transitions ü 33°C ‘‘Repulsive’’ glass ü 33. 39°C Liquid: smoothed speckles ü 33. 6°C ‘‘Attractive’’ glass X. Lu et al. Soft Matter, 6, 2010, 6160

Role of the volume Separation of the effects of the volume and of the microscopic structure

Total intensity Intensity scattered by a arbitrary homogenous body is:

Interpretation Small angle scattering Size, shape and long distance density fluctuations I(q) Large/wide angle scattering Atomic structure of sample SAXS WAXS q

Scattering function For a isotropic system:

Example 1 Liquid argon 85 K Neutrons Solid argon f. c. c 6. 5 Å 3. 75 Å 5. 31 Å

Example 2: water Ice Ih (P 63/mmc) (Bernal-Fowler Ice) Oxygen tetrahedra E. D. Isaacs et al. , Phys. Rev. Lett. 82 (1999) 600 Low density ( r = 0. 0295 mol. A 3) O-O Correlation function Extrapolation 26 MPa 209 MPa 400 MPa Breaking of H bonding Extrapolation High density ( r = 0. 0402 mol. A 3) A. Soper et al. , Phys. Rev. Lett. 84 (2000) 2881

• Ferroelectric KNb. O 3/Ba. Ti. O 3 • Perovskite ABO 3, ferroelectric • T > 120 °C, Cubic Pm 3 m, paraelectric • 0°C < T < 120 °C, Tetragonal P 4 mm, ferroelectric P 4 mm Pm 3 m, 1 st order transition (domains). • -90°C < T < 0 °C, Orthorhombic Cmm 2 P 4 mm, 1 st order transition. • T < -90 °C, Trigonal R 3 m Cmm 2, 1 st order transition. Ba 2+/K+, Ti 4+/Nb 3+, O 24 Å O 1 st Ti Trigonal Ba Orthorhombic Tetragonal 1 st

Ferroelectric perovkite: KNb. O 3 FT Coll. Claire Laulhé, Françoise Hippert, Gabriel Cuello

Pair distribution function in solids K-O O-O Nb-O a K-Nb a

Determination of Nb-O distances O R Longue [111] T Courte C

Lead liquid structure H. Reichert, Nature 408, (2000) 839 Local order: 5 -fold symmetry Icosahedral?

Diffraction by thermotropic liquid crystals Isotropic liquid Smectic A Nematic Smectic C

Small Angle Scattering SAXS-SANS is used to determine • the shape • the size • the organisation… …of small objects (clusters, macromolecules, precipitates, bubbles) of nano(micro)metric size (20– 1000 Å) Applications : • Polymer science, colloids, soft matter • Metallurgy, earth science • Biology

SAXS Scattered intensity/object: Effective density Average on all orientations

Guinier’s law L 6 Guinier ’s law:

Guinier’s law Example of a sphere RG/a ~ 0. 77

Set of particles of total surface S Porod’s law Deviation to the Porod regime: interface roughness. . .

Fractals SANS on sedimentary rock