Symmetrical Laue Case Research Articles

Phase of Pendellösung oscillations in X-ray dynamical diffraction for perfect crystals

Formulations are given for the intensities of transmitted and diffracted waves in the Laue case of a perfect crystal. This is applicable irrespective of the magnitudes of both the real and imaginary parts of the Fourier components of the crystal polarizability. The phase shift of the Pendellösung oscillations of the transmitted and diffracted waves is analyzed in detail for a symmetrical Laue case. The phase is determined to shift continuously from out of phase to in phase for absorbing crystals.

X-ray dynamical diffraction analogues of the integerand fractional Talbot effects.

The X-ray integer and fractional Talbot effect is studied under two-wave dynamical diffraction conditions in a perfect crystal, for the symmetrical Laue case of diffraction. The fractional dynamical diffraction Talbot effect is studied for the first time. A theory of the dynamical diffraction integer and fractional Talbot effect is given, introducing the dynamical diffraction comb function. An expression for the dynamical diffraction polarization-sensitive Talbot distance is established. At the rational multiple depths of the Talbot depth the wavefield amplitude for each dispersion branch is a coherent sum of the initial distributions, shifted by rational multiples of the object period and having its own phases. The simulated dynamical diffraction Talbot carpet for the Ronchi grating is presented.

Eikonal approximation of dynamical diffraction equations for X-ray beams with a two-dimensional curvature of the wave front. II. Focusing of an X-ray beam by a crystal with non-plane entrance and exit surfaces. Symmetrical Laue case

On the basis of eikonal approximation developed in the first part of this work, diffraction of X-ray beam with a two-dimensional curvature of wave front and its focusing by a crystal with nonplane entrance and exit surfaces is studied in symmetrical Laue case. The expressions for eikonal and focusing distances inside and outside the crystal in the vacuum, are obtained. The size of the focal spot and the intensity gain in the spot, are estimated. The condition of point focusing is obtained. The conditions for obtaining practically achievable focusing distances are investigated.

A sensitive method for detecting long‐range strain in nearly perfect crystals using X‐ray dynamical diffraction

AbstractA very sensitive method is proposed for the detection of long‐range strain in otherwise perfect crystalline materials. In this method, the specimen is rotated around the scattering vector in the symmetrical Laue case. Long‐range strains due to a uniform bending or torsion, that are not detectable with the symmetrical Laue condition, are sensitively detected by the proposed method of rotating the specimen. Several demonstrative topographic observations are presented. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Energy-dispersive diffraction with synchrotron radiation and a germanium detector

The response of an intrinsic Ge detector in energy-dispersive diffraction measurements with synchrotron radiation is studied with model calculations and diffraction from perfect Si single-crystal samples. The high intensity and time-structure of the synchrotron radiation beam leads to pile-up of the output pulses, and the energy distribution of the pile-up pulses is characteristic of the fill pattern of the storage ring. The pile-up distribution has a single peak and long tail when the interval of the radiation bunches is small, as in the uniform fill pattern, but there are many pile-up peaks when the bunch distance is a sizable fraction of the length of the shaping amplifier output pulse. A model for the detecting chain response is used to resolve the diffraction spectrum from a perfect Si crystal wafer in the symmetrical Laue case. In the 16-bunch fill pattern of the ESRF storage ring the spectrum includes a large number of ;extra reflections' owing to pile-up, and the model parameters are refined by a fit to the observed energy spectrum. The model is used to correct for the effects of pile-up in a measurement with the 1/3 fill pattern of the storage ring. Si reflections (2h,2h,0) are resolved up to h = 7. The pile-up corrections are very large, but a perfect agreement with the integrated intensities calculated from dynamical diffraction theory is achieved after the corrections. The result also demonstrates the convergence of kinematical and dynamical theories at the limit where the extinction length is much larger than the effective thickness of the perfect crystal. The model is applied to powder diffraction using different fill patterns in simulations of the diffraction pattern, and it is demonstrated that the regularly spaced pile-up peaks might be misinterpreted to arise from superlattices or phase transitions. The use of energy-dispersive diffraction in strain mapping in polycrystalline materials is discussed, and it is shown that low count rates but still good statistical accuracy are needed for reliable results.

High-pass-filtered diffraction microtomography by coherent hard x rays for cell imaging: theoretical and numerical studies of the imaging and reconstruction principles

This paper presents theoretical and numerical studies of diffraction tomography using hard x rays, from the viewpoint of imaging and reconstruction methods for cell imaging. The proposed system employs a single-perfect-crystal analyzer in symmetric Laue-case transmission geometry to efficiently detect the higher spatial frequency components of an object's refractive-index distribution, and to effectively suppress interference between the unperturbated wave field and the wave field diffracted by the object. This system features acquisition of a single projection by a single exposure using a simple geometry and aggressive use of diffracted x rays. We present the physical description of the imaging method using the Fourier diffraction theorem derived from the Born approximation. First, we demonstrate that the reconstruction leads to the phase-retrieval problem. We then describe a reconstruction algorithm based on the classical Gerchberg-Saxton-Fienup algorithm. Finally, we show the efficacy of this system by computer simulation. Our simulation demonstrates that the imaging system delineates microstructure 3.5 microm in diameter in a phase object 400 microm in diameter.

Plane-wave synchrotron x-ray topography observation of grown-in microdefects in a slowly pulled CZ-silicon crystal

Plane-wave x-ray topography was used to characterize grown-in defects in silicon crystals grown by the Czochralski method at a slow speed. The 220 reflection rocking curves of the sample crystal were measured in a symmetric Laue case. Oscillatory profiles were clearly observed in the rocking curves and were in good agreement with those calculated from the dynamical theory of x-ray diffraction. Images of the grown-in defects were observed in the plane-wave x-ray topographs of the 220 reflection. The density of the defects was determined as 300 cm−3. The lattice strain around the defects was evaluated as 10−6 from the dependence of the defect image contrast on the x-ray incident angles. The defect images are discussed using a grown-in defect model of entangled dislocation loop clusters.

Diffraction and imaging study of imperfections of crystallized lysozyme with coherent X-rays.

Phase-contrast X-ray diffraction imaging and high-angular-resolution diffraction combined with phase-contrast radiographic imaging were employed to characterize defects and perfection of a uniformly grown tetragonal lysozyme crystal in the symmetric Laue case. The full-width at half-maximum (FWHM) of a 4 4 0 rocking curve measured from the original crystal was approximately 16.7 arcsec and imperfections including line defects, inclusions and other microdefects were observed in the diffraction images of the crystal. The observed line defects carry distinct dislocation features running approximately along the <1 1 0> growth front and have been found to originate mostly in a central growth area and occasionally in outer growth regions. Inclusions of impurities or formations of foreign particles in the central growth region are resolved in the images with high sensitivity to defects. Slow dehydration led to the broadening of a fairly symmetric 4 4 0 rocking curve by a factor of approximately 2.6, which was primarily attributed to the dehydration-induced microscopic effects that are clearly shown in X-ray diffraction images. The details of the observed defects and the significant change in the revealed microstructures with drying provide insight into the nature of imperfections, nucleation and growth, and the properties of protein crystals.

A dynamical Bragg equation for high-order Laue zone reflections

A dynamically corrected Bragg equation for high-order Laue zone (HOLZ) reflections is derived directly from the Bloch wave formalism instead of the geometric argument used to deduce the kinematical Bragg condition. It differs from the kinematical Bragg equation by replacing the plane wave vector in the kinematical equation with the Bloch wave vectors. This dynamical equation reduces to the kinematical equation when the crystal potential is zero. It also demonstrates the occurrence of dynamical shifts for the HOLZ reflections but their absence for the zero-order Laue zone (ZOLZ) reflections in the symmetrical Laue case.

X-ray standing wave as a result of only the imaginary part of the atomic scattering factor.

The X-ray standing wave has been studied when the real part of the scattering factor is zero. In the symmetric Laue case, the phase of the standing wave advances by pi when the deviation parameter W changes from -1 to 1, which is the same variation as in the usual symmetric Bragg case when only the real part of the scattering factor exists. However, the phase in the former case is different from that in the latter by pi/2. By using the standing waves, the origins of the anomalous transmission and anomalous absorption effects reported by Fukamachi & Kawamura [Acta Cryst. (1993), A49, 384-388] have been analysed. The standing wave in the Laue case can give rise to a more accurate method of site determination of a specified impurity atom as well as a wider range of applications than a conventional standing-wave approach.

Effect of Strong Borrmann Absorption on X-Ray Fluorescence Yield Curves

The Ga K X-ray fluorescence yield from GaAs was measured around the 200 reflection peak in the symmetric Laue case, by tuning the incident X-ray energy from a synchrotron radiation source between the K-absorption edges of Ga and As. It was found that the fluorescence yields from the crystal surface become minimum at the exact Bragg angle, when the diffraction is induced only by χhi, i.e., the imaginary part of the Fourier coefficient of the X-ray polarizability (χhr being the real part). In the case of |χhr|=|χhi|, which occurs at two energy points, both yield curves are asymmetrical with respect to the exact Bragg angle and their asymmetries are opposite to each other.

On the divergence of rays Bragg diffracted by perfect stationary crystals illuminated with collimated polychromatic radiation

The properties of three diffraction geometries, viz symmetric Bragg-case reflection from a crystal surface, symmetric Laue-case transmission through a parallel-sided crystal and asymmetric Bragg-case reflection from a crystal surface, have been examined under the conditions that the crystal (which is assumed to be perfect) is held in fixed orientation relative to an ideally collimated incident beam of white X-rays. Symmetric Bragg-case reflection has the valuable property of invariant direction of reflected rays for all orders of reflection excited, and for strong and weak reflections alike. For the symmetric Laue case, and for the asymmetric Bragg case approximately, for small asymmetry, expressions are given for the angular range of diffracted rays, scaled in terms of the extinction distance of the reflection concerned. The findings have practical applications in the design and interpretation of continuous-radiation topographic experiments.

Determination of Anomalous Scattering Factor by Using Transmitted Rocking Curves near the Absorption Edge

The rocking curves of transmitted and diffracted beams for GaAs 200 reflection in the symmetric Laue case were measured near the Ga K-absorption edge, using X-rays from synchrotron radiation. The changes in the transmitted rocking curves become conspicuous as compared with those of the diffracted curves near the edge, which is used for determining anomalous scattering factors of Ga by a profile fitting method.

DIRECTION OF THE ENERGY FLOW IN PERFECT CRYSTALS-NEAR THE ATOMIC ABSORPTION EDGE

The direction of the energy flow in perfect crystals near the atomic absorption edge is studied in detail in the symmetrical Laue and Bragg cases. It is found that in the symmetrical Laue case, the direction of the energy flow corresponding to a tie point on the dispersion surface deviates from the normal of the real part of the dispersion surface, only when the contribution of the imaginary part of the atomic scattering factor to the diffraction is comparable to that of the real part. However, in the symmetrical Bragg case, the deviation takes place in the so-called “total reflection” region even when the contribution of the imaginary part of the atomic scattering factor to the diffraction is negligible. When diffraction is induced only by the imaginary part of the atomic scattering factor, the deviation becomes the largest in the Laue case, but the least in the Bragg case.

The quantitative study of X-ray diffraction dynamics near atomic absorption edge

The diffracted and transmitted rocking curves for GaAs 200 reflection in the symmetric Laue case have been measured by the use of synchrotron radiation. The quantitative comparison between the measured curves and the corresponding calculated ones has been carried out near the K-absorption edge of Ga. The measured results agree well with the calculated ones to some extent and the X-ray absorption near edge structure (XANES) can be clearly seen in the rocking curves.

Observation of Phase Change of X-ray Polarizability in Rocking Curves

By tuning the X-ray energy from synchrotron radiation very close to the K-absorption edges of Ga and As, rocking curves were measured of the GaAs 200 diffraction and the transmitted beams in the symmetric Laue case. The rocking curve of the transmitted beam is asymmetric with respect to the angle of the exact Bragg condition. The asymmetry reverses as the sign of the Fourier transform of either the real or the imaginary part of the X-ray polarizability changes. From the turning point of asymmetry, the accuracy of the existing calculated values of anomalous scattering factors was examined.

DETERMINATION OF ANOMALOUS SCATTERING FACTOR FOR Ga BY USING ROCKING CURVES OF TRANSMITTED BEAM IN GaAs

The diffracted and the transmitted rocking curves for GaAs(200) in the symmetric Laue case are observed near the Ga K-absorption edge by using the synchrotron radiation. The quantitative comparison between the measured transmitted rocking curves and the corresponding calculated ones based on the dynamical theory of X-ray diffraction has been cattied out, and the anomalous scattering factors for Ga K-absorption edge in GaAs have been determined.

Angular amplification in neutron or X-ray dynamical diffraction by perfect crystals

A theoretical analysis of the properties of the dispersion surface related to the dynamical theory of diffraction of neutron or X-rays by perfect crystals leads to the prediction of an angular amplification effect when an asymmetric cut of the crystal is performed. Contrary to the well-known angular amplification effect occurring inside the crystal and disappearing at the outer surface in the symmetrical Laue case, the angular amplification discussed in the present paper concerns the beam emerging from the crystal, so that several angular amplifiers in cascade can be used. A possible application of this effect is discussed.

On the applicability of bloch’s theorem to the total wave function in the dynamical theory of electron diffraction

The dynamical theory of electron diffraction is widely used in materials science problems such as determining the contrast in electron microscope images of crystal defects, calculations of structure images and calculations of diffracted beam intensities in electron diffraction patterns. In carrying out these calculations, the high energy approximation is normally made and it is usually assumed that the crystal is in a symmetrical Laue orientation. In practice, however, a specimen in the electron microscope will generally be oriented so that the non-symmetrical Laue case is obtained. Even in those special cases where the symmetrical Laue case is obtained for the zero order Laue zone reflections, non-symmetrical Laue effects may occur if reflections in higher order zones are important. It has been shown in the literature that the Bloch functions are not orthogonal in more general forms of the dynamical theory in which the high energy approximation is not made and the nonsymmetrical Laue case is considered,.

On long-range lattice perfection in natural type 1a diamond

Abstract Several methods have been applied to examine the lattice perfection in a natural diamond that was uncommonly low in density of dislocations and other sources of long-range strains, but which belonged to the commonest optical type 1a. Infrared absorption microscopy and cathodoluminescence topography demonstrated an exceptionally uniform distribution of nitrogen impurity content, with a concentration of about 0.1%. Defect-imaging examinations by birefringence, X-ray single-crystal and double-crystal topography (which were facilitated by optically polished surfaces on the specimen) were applied to select the most strain-free volume, which was then probed by synchrotron radiation double-crystal diffractometry. A 220-reflection, symmetrical Laue case rocking curve only about 0.25 arc second wider than that calculated for perfect diamond was recorded. Analysis was expressed in terms of apparent value of ε = F″g/F″0, the ratio of imaginary parts of the structure factor for orders g and 0. The observed profile for g = 220 and wavelength 0.154 nm was well fitted by apparent ε = 0.69 (theoretical ε = 0.96). Minor asymmetry in the flanks of the experimental profile was explained by the presence of a small component of incoherent scattering in the diffracted beam.