# NEUTRON AND X-RAY DIFFRACTION

## Learning outcomes of the course unit

The aim of the course is to achieve a good understanding of X-ray and neutron diffraction and scattering techniques, and of their main applications to the condensed matter physics.

## Prerequisites

Good working knowledge of advanced calculus: functions of two or more variables, differential equations, functions of complex variables, Fourier transforms.

Good working knowledge of classical mechanics, electricity and magnetism, quantum mechanics.

## Course contents summary

Introduction to X-ray diffraction

X rays: waves and photons, scattering from an electron, from an atom, from a molecule, Compton scattering, absorption, reflection refraction, coherence, magnetic interactions, X-ray sources, laboratory instrumentation, synchrotron radiation, collimation, spectral prperties, wigglers and undulators.

Kinematic theory of X-ray diffraction

Scattering from a crystal lattice, Bragg and Laue equations, Ewald sphere, the structure factor of the unit cell, lattice vibrations, Debye-Waller factor, integrated intensity.

Refraction and reflection of X rays

Index of refraction and absorption, Snell's law and Fresnel equations, reflection from an homogeneous layer with finite thickness, from a multilayer, rough surfaces and interfaces..

Dynamic theory of X-ray diffraction

Reflection and transmission from few atomic layers, Darwin theory, Darwin reflectivity curve, width of the Darwin curve, integrated intensity, standing waves.

X-ray absorption

Photoelectric absorption, scattering and absorption cross sections, X-ray absorption from a single atom, EXAFS, theoretical framework and examples, X-ray dicroism, resonant scattering.

Introduction to neutron scattering

Neutron properties and complementarities between neutrons and X-rays, neutron production, reactors, spallation sources, neutron transport, guides, neutron selection: monochromators and choppers, neutron detectors, instrumentation: diffractometers, small angle scattering instruments and reflectometers, spectrometers at continuous sources: TAS & TOF, spectrometers at pulsed sources, high resolution spectrometers..

Neutron scattering theory

Scattering cross-sections, the dynamic structure factor, coherent and incoherent scattering, scattering functions and correlation functions, energy spectrum accessible to neutrons and atomic motions that can be probed.

Neutron diffraction

diffraction from crystal structures, diffraction from disordered systems, small angle scattering, form factor, structure factor, contrast variation..

Inelastic and quasielastic neutron scattering

Inelastic scattering, normal modes, one-phonon cross section, measure of the dispersion curves, incoherent scattering and incoherent approximation, vibrational spectroscopy, correlation functions and intermediate scattering functions, examples of diffusive motions.

Magnetic neutron scattering

Magnetic interactions of neutrons with electrons, magnetic cross section and form factor, diffraction from magnetically ordered systems, polarised neutrons, inelastic neutron scattering.

## Recommended readings

Neutron and Synchrotron Radiation for Condensed Matter Studies, Vol I, (Springer-Verlag & Les Editions de Physique)

X-ray diffraction, B.E. Warren (1990 - Dover Publ.)

The Optical Principles of the Diffraction of X-rays, R.W. James (1982 - Ox Bow Press)

Introduction to the Theory of Thermal Neutron Scattering, G.L. Squires, (1978 - Cambridge Univ. Press,)

Neutrons and Solid State Physics, L. Dobrzynski, K. Blinowski (1994 – Ellis Horwood)

Neutron Scattering Methods of Experimental Physics, Vol. 23A,(1986 - Academic Press, New York)

## Teaching methods

Teaching:

Theoretical lectures and practical exercises

Evaluation:

Oral exam