ENHANCED NUCLEAR RESONANCE SCATTERING IN DYNAMICAL DIFFRACTION OF GAMMA-RAYS

A massively parallel deterministic method is described for reconstructing shift-invariant complex Green's functions. As a first experimental implementation, we use a single phase contrast x-ray image to reconstruct the complex Green's function associated with Bragg reflection from a thick perfect crystal. The reconstruction is in excellent agreement with a classic prediction of dynamical diffraction theory.

The diffraction of high‐energy characteristic x‐ray lines, which have relatively large intrinsic widths, by perfect single crystals, which have acceptance angles much less than these line widths, is discussed. Experiments using Bi Kα 1 , which has an equivalent FWHM of 0.0013°, and an asymmetrically cut Si (111) crystal, having acceptance angles of 0.0002° and 0.0004°, showed that only a small fraction of the total characteristic line was, in fact, diffracted by the crystal.

The first section of this chapter concerns scattering factors for the diffraction of electrons by crystalline solids. An explanation of the theory of scattering by a perfect crystal is followed by a discussion of the kinematical, two-beam and phase-grating approximations. Relativistic and absorption effects are considered. Extensive tables of atomic scattering amplitudes for electrons for neutral and ionized atoms are presented. The second section of the chapter briefly discusses the parameterization of electron atomic scattering factors. Tables of useful parameters as a function of accelerating voltage and elastic atomic scattering factors for neutral atoms are given. Complex scattering factors for the diffraction of electrons by gases are discussed in the third section of the chapter. This section includes tables of scattering factors of interest for gas-phase electron diffraction from atoms and molecules in the keV energy region. In addition to the tables and a description of their uses, a discussion of the theoretical uncertainties related to the material in the tables is also provided. The tables give scattering factors for elastic and inelastic scattering from free atoms. The theory of molecular scattering based on these atomic quantities is also discussed. Electron energy-loss spectroscopy on solids is discussed in the fourth section of the chapter. Topics covered include: the use of electron beams; single and multiple scattering; the classification of the different excitations in a spectrum; instrumentation; and the excitation spectra of valence and core electrons. The fifth section of the chapter describes oriented texture patterns. Lamellar and fibre texture patterns are discussed and applications to metals and organic materials are mentioned. The computation of dynamic wave amplitudes in then described in the sixth section of the chapter. This section deals first with the multislice method. The numerical procedure is outlined and factors that influence the choice of thickness of the slice are discussed. Two checks that can be performed during a multislice calculation are noted. The Bloch-wave method is then described. The use of Bloch waves to describe electron diffraction and electron imaging in thin crystals is outlined together with the concept of the dispersion surface. These emerge as natural solutions of the Schrodinger equation with a periodic optical potential to generate the elastic scattering and also the loss of intensity from the coherent wave field due to thermal diffuse and inelastic scattering. The Bloch-wave approach is a useful complement to the multislice method and provides a clear picture of wave propagation in perfect and imperfect crystals. In the seventh section of the chapter, the measurement of structure factors and the determination of crystal thickness by electron diffraction are described. The use of convergent-beam electron diffraction to obtain integrated intensities is discussed and the relationship between intensity features and the dispersion surface is explained. The last section of the chapter concerns crystal-structure determination by high-resolution microscopy. This technique allows the arrangement of atomic columns in thin crystals to be observed directly. The resolution of the best instruments is now slightly below 0.1 nm. The images usually show a projection through a slice of crystal about 20 nm thick, however tomographic (three-dimensional) reconstruction is now possible at nanometre resolution. The images show the host of microphases, grain boundaries, twins, line and planar defects which broad-beam methods, such as X-ray diffraction, provide the average scattering from. These defects often control the properties of crystals, engineering materials and electronic devices. Individual nanostructures, such as carbon nanotubes and catalyst particles, may be imaged at atomic resolution. Fine twinning, polytypes, intergrowth of oxide phases etc. can be identified, and increasingly the detailed atomic structure of defects (such as oxide, superconductor and semiconductor interfaces) is being determined. Substitutional dopant atoms have recently been imaged for the first time. In biology the method is limited by radiation damage; however by summing many images of identical randomly oriented macromolecules, tomographic density maps can be reconstructed at subnanometre resolution from hydrated proteins which cannot be crystallized (e.g. membrane proteins). This section reviews the theoretical principles of high-resolution electron microscopy, including few-beam and structure image formation, effects of electron-optical lens aberrations, partial coherence, resolution-limiting factors, image-simulation methods, dynamical effects, and a summary of super-resolution schemes. Keywords: absorption; atomic scattering amplitudes; atomic scattering factors; Bethe theory; Bloch-wave method; core-electron spectroscopy; core-loss spectroscopy; crystal thickness; crystalline solids; dielectric description; dynamical diffraction; dynamical wave amplitudes; EELS; elastic scattering; electron diffraction; electron energy-loss spectroscopy; electron microscopy; electrons; free-electron gas; high-resolution electron microscopy; HREM; hyper-resolution; inelastic scattering; lamellar textures; lattice-fringe images; molecular scattering factors; multiple scattering; multislice method; oriented texture patterns; plasmons; real solids; relativistic effects; STEM; scanning transmission electron microscopes; scattering factors; solid-state effects; spectrometers; structure factors; surface plasmons; texture; wave amplitudes; X-ray diffraction

The dynamical theory of x-ray diffraction is adapted to the case of nuclear resonant diffraction of Mossbauer radiation. It is found that the dispersion surfaces for the Laue and Bragg conditions of incidence are markedly different and that anomalous transmission occurs in both cases. The effect of the accompanying electronic scattering is to increase the anomalous absorption coefficient. The integrated reflecting power in both Laue and Bragg cases is found to be roughly proportional to the modulus of the total scattering amplitude.

Fracture of brittle and ductile crystals in the generalized stress state. Strength and strain fracture criteria

Crystal perfection in Czochralski grown nickel single crystals

Synchrotron radiation has opened new possibilities for experimental investigations in condensed matter physics. The high intensities of the emitted X-rays allows scattering experiments with an energy resolution high enough to study ionic excitations and dynamics, directly. 1–3 So far, this was only possible by means of inelastic neutron scattering. In that method the probe is coupling to the nuclei in the sample, whereas in case of X-rays the coupling occurs to the electrons. Therefore, X-ray scattering experiments with high resolution in energy and momentum allow to determine the dynamical structure factor which is the fourier transform in time and space of the time dependent electron density–density correlation function. Within the validity of the adiabatic approximation this function is proportional to the ion density–density correlation function This means, peaks in of a liquid, for example, can be interpreted either as localized density fluctuations around or as propagating sound modes for The latter excitations correspond to the well known phonon excitations observable in solids. Since the coupling of the energy to the momentum transfer is negligible, inelastic X-ray scattering has almost no restrictions in the accessible space. Hence this method is extremely attractive for investigations at small Q-values of the coherent part of the dynamical structure factor for liquids and amorphous solids with high sound velocities. In contrast to this, inelastic neutron scattering is strongly limited by the mass of the neutron itself just in this part of the space. Conventional triple axis spectrometer can be optimized for high energy resolution by Bragg scattering from perfect single crystals in extreme backscattering geometry at the monochromator and the analyzer. This technique was used to build the inelastic X-ray spectrometer INELAX 2,3 at the synchrotron radiation laboratory at DESY, Hamburg and the second generation instrument 4 at the ESRF in Grenoble. Energy resolution values of 10 meV down to 1 meV can be achieved by using high order reflections from perfect silicon crystals. The energy transfer in an inelastic scattering experiment is established by an energy shift between the monochromator and the analyzer crystals. This is done by thermal tuning of the lattice parameter of the analyzer or the monochromator crystal. Using the (7 7 7) Bragg reflection of silicon with a primary energy of 13.8 keV, a temperature difference of 0.01 K between both crystals corresponds to an energy transfer of 0.35 meV. The resolution function

A proportional electron counter for Mössbauer gamma-ray- and X-ray-crystal interaction studies under diffraction conditions

Point defects in crystals cause displacements of neighbouring atoms and hence become the source of an additional anisotropy of the X-ray resonant scattering amplitude. This anisotropy can induce 'forbidden' Bragg reflections near absorption edges, called point-defect-induced (PDI) reflections, which are absent in non-resonant X-ray scattering and in resonant X-ray scattering by perfect crystals. The deformation of external electron shells is the physical reason for this phenomenon. Some examples of crystals are considered in which the PDI reflections could be observed.

Previously unreported intensity enhancement of x rays obtained by two-mode excitation is observed on the direct incident beam of synchrotron radiation. The excitation utilizes the coherent interaction of a six-beam Borrmann diffraction in a perfect silicon single crystal. The experiment provides a means of obtaining nearly divergenceless, monochromatic, spatially coherent, and intense x-ray sources for x-ray diffraction and imaging studies. Single mode excitation can be achieved by using a thick crystal.

A new approach which clarifies the structure of many-beam dynamical interactions of diffracted waves is developed on the basis of an analytical formula for the diffracted intensity averaged over crystal thickness. In the systematic n -beam case the orientation dependence of the averaged intensity is proved to be expressed as a linear combination of terms of resonance type ( q - a k )/[( q - a k ) 2 + b k 2 ] and b k /[( q - a k ) 2 + b k 2 ] for k =1, …, n ( n -1)/2, where - q is the tangential component of the incident wave vector to the row of reciprocal lattice points. Each term can be associated with Bragg reflections between pairs of plane wave components of Bloch waves. The parameters a k and b k indicate Bragg positions and reflection widths. Modifications of the result are briefly discussed about the critical voltage effect and the non-systematic case.

The X-ray standing wave (XSW) technique is a sensitive tool for determining the position of atoms within a crystal, adsorbed onto a surface, or distributed within the crystal or at the interface. Conventional XSWs are generated using dynamical Bragg reflection from perfect single crystals. In Bragg diffraction XSW measurements, distances are extracted modulo-d so that it is inappropriate to employ this method for structural determinations of systems extending over several tens of angstroms such as ionic distributions at charged interfaces. For Bragg diffraction purposes, LSMs are depth-periodic structures consisting of alternating layers of high and low electron density materials and are of high enough quality to produce strong Bragg diffraction.

Influence of real photon diffraction on parametric X-ray radiation angular distribution in thin perfect crystals

Fluorescence reabsorption in anthracene single crystals: Lifetime variations with emission wavelength and temperature

It is shown that the contribution of thermal diffuse scattering to integrated Bragg intensities from perfect crystals can be much greater than previously expected. Measurements on the (555) reflection of silicon are presented showing a thermal scattering contribution of about 70% of the true Bragg intensity. The special aspect of the effect in perfect crystals is discussed and methods of reducing it or correcting for it are included.