DIFRACTIONAL STRUCTURE ANALYSIS
(Specialty course, 4 year, 8 semester)
The goal of studying the discipline
Diffraction of radiation (x-ray or neutron) on the crystal is the best developed experimental method of analysis of the atomic structure. With its help one can restore the structure with the precision, sufficient for the localization of separate atoms, and study the structural reshaping under the action of external conditions. The general name of the set of experimental and mathematical methods letting one to determine the structure of the crystal, and also of the means of representation of the received information is “diffraction structure analysis”. Application of the diffraction of heat neutrons for these purposes usually is called in Russia the structural neutronography. In the course of lectures the questions are considered necessary for understanding the modern state of the structural neutronography, and, first of all, of the neutronography on impulse sources of neutrons. Correspondingly, the course of lectures includes three main themes: symmetry of crystals, diffraction of radiation on the crystal lattice, basics of the structural analysis of crystals. First of them is traditional for the courses of crystallography, two others are described with taking into account the specific character of interaction of neutrons with the matter and its diffraction on the periodic structures.
The course content
(15 lectures for 2 hours, 30 hours)
Introduction to the program course. General notions on symmetries of periodic structures and diffraction of radiation on periodic structures. The principles of structural analysis of crystals. Specifics of the neutronography on impulse sources.
Crystals: symmetry, inner periodicity, anisotropy. Special role of the translation for the structure analysis. Levels of symmetries of the crystals: categories, systems, bravais lattices, point groups, space groups. Crystal lattice and the crystallographic systems of coordinates. Bravais cells: selection rules, simple and centered bravais cells.
Dual vector basis and the reciprocal lattice. Relation between the translations and angles in direct and reciprocal lattices. Vector of the reciprocal lattice. Intersurface distance. Miller indices. Calculation of the intersurface distances in crystal lattice. General properties of the reciprocal lattice vector.
Simple finite symmetry elements. International notation for symmetry elements. Asymmetric point. Multiplication of generic position by the finite symmetry elements. Matrix representation of the symmetry elements. Sequential action of the symmetry elements. Commutative and noncommutative symmetry elements.
Group properties of the symmetry elements. Point groups of symmetries of the crystals. Nomenclature of the point groups: tables, notations of Schonflies, educational, of German-Morgan, generators of the groups, polar groups, central-symmetric, Laue groups.
Spatial groups of symmetries. Symmetry elements of the crystal structures. Mathematical representation of the spatial symmetry elements. Combination of the spatial symmetry elements. Action of the symmetry elements not passing through the origin of the coordinates. International crystallographic tables. Limit groups of symmetries (Curie groups).
Kinematical theory of diffraction. General conditions for emergence of the diffraction picture. Influence of the coherency of the scattering process and of the spatial periodicity of scattering centers on the contrast. Scattering on the ideal periodic structure. Structure factor of the elementary crystal cell. The specifics of the x-rays, electrons and neutrons for diffraction on crystals.
Geometry of the diffraction picture. Vector triangle. Interpretation of the diffraction on the crystal as a reflection from crystallographic planes. The Wolf-Bragg formula. Evald structure. Spheres of reflection and restrictions.
Peculiarities of the interaction of heat neutrons with the crystal. Cross-section for scattering of the neutrons by the crystal: coherent, non-coherent, full. Influence of the thermal motion on the diffraction picture. Specifics of the coherent scattering of neutrons. Main tasks of the structure neutronography.
Methodical peculiarities of the neutron diffraction experiment. Peculiarities of the neutronography on impulse sources of neutrons. Geometry of the diffraction picture. Influence of the finite size and mosaic nature of the crystals on the shape of knots of the reciprocal lattice. Methods of scanning of the reciprocal space of the crystal. Main experimental methods of the neutronography. multidimensional neutron diffractometry.
The problems of the structure analysis of crystals. Structure analysis of the monocrystals. Structure analysis of the polycrystals. Common consideration of x-ray and neutron structure analysis.
Analysis of the intensivity of the diffraction spectra. Intensivity of the diffraction peak as a convolution of the scattering cross-section and resolution function. Integral peak intensivity.
addition of the corrections to the formulae for integral intensivity. Taking into account of the effective spectrum of neutrons. Absorption factor. Taking into account of the secondary extinction. thermal factor. Isotropic and anisotropic approximations. The dependence B(T) in the Debae approximation. Relation of the thermal factor with the phonon spectrum and the heat capacity of the crystal.
Analysis of the structure of powders using the Rietveld method. Determination of the geometrical parameters. The parametric formula for the intensivity. A Criterion of the quality of processing of the diffraction spectrum. The problem of correlation of parameters.
Magnetic properties of the electron, neutron, atom, matter. Description of the magnetic structure of the crystals. Magnetic scattering of the neutrons. Magnetic neutronography (analysis of the magnetic structure of the crystal).
Magnet properties of electron, neutron, atom, substance. Description of the magnet structure of crystals. Magnet scattering of neutrons. Magnet neutronography(analisys of magnet structure of crystal).
Seminars (3 seminars, 6 hours).
Diffraction as a Fourier transformation. The relation between direct and reciprocal spaces. The scheme of the diffractional experiment in the direct and reciprocal spaces. Qualitative analysis of the shape of knots of the reciprocal lattice. Relation between the convultion and product of functions and their Fourier transformations. Modern neutron diffractometers. Their structure, main parameters, classification by the types of tasks.