Spherical X-ray Wave Research Articles

Features of Spherical X-Ray Wave Diffraction by a Low Angle Twist Boundary Perpendicular to the Crystal Surface

Features of Spherical X-Ray Wave Diffraction by a Low Angle Twist Boundary Perpendicular to the Crystal Surface

Study of phase objects by dynamical diffraction of a spherical X-ray wave in a crystal using a temperature gradient

The possibility of reconstructing the internal structure of a phase object by dynamical diffraction of a spherical X-ray wave in a crystal using a temperature gradient was investigated. It is shown that the use of a temperature gradient directed perpendicular to the crystal surface makes it possible to obtain a focal distribution of all point inhomogeneities in the phase object on the exit surface of the crystal. By moving the phase object along the direction of the diffraction vector and registering the entire map of obtained data, one can restore the additional phase of the wave and the internal structure of the phase object section.

Determination of the superlattice structure factor by X-ray diffraction using a temperature gradient.

The influence of a temperature gradient directed perpendicular to the crystal surface on the diffraction focusing of a spherical X-ray wave in a superlattice is studied for the Laue geometry. It is shown that different satellites can be focused on the exit surface of the crystal by a smooth change in the gradient value, which can become the basis for the experimental determination of the structure factor of the superlattice.

X-ray dynamical diffraction Talbot effect behind a crystal in free space.

The dynamical diffraction Talbot effect takes place inside a crystal, when a periodic object is illuminated by a plane or spherical X-ray wave which then falls on the crystal at an angle close to the Bragg angle for some reflection. Both theoretical consideration and numerical calculations show that the dynamical diffraction Talbot effect also takes place behind the crystal. The effect is accompanied by the dynamical diffraction pendulum effect and wave focusing. Expressions are found for the dynamical diffraction Talbot distance for areas before and after focusing. The spatial Fourier spectrum of the periodic object is obtained on the focusing plane. Detailed analysis when the periodic object is illuminated by a plane wave has shown new features of this effect. The dynamical diffraction Talbot effect in free space can be used to determine the structure of a periodic object, to determine the structure of an arbitrary object placed before or after the periodic object, and to determine structural defects and deformations of the crystal.

Theory of the Laue Diffraction of X Rays in a Thick Single Crystal with an Inclined Step on the Exit Surface. I: Numerical Solution

The analytical solution of the problem of the Laue diffraction of an X-ray spherical wave in a single crystal with an inclined step on the exit surface has been obtained. The general equations are used for the specific case of plane wave diffraction in a thick crystal under the Borrmann conditions. It is shown that, provided that the crystal thickness increases from the side of the reflected beam, the reflected-wave relative amplitude is determined by three complex terms. This may formally lead to interference and an increase in the intensity in maxima by a factor of 9 as compared with the crystal without a step. The equation for the transmitted beam contains only two terms, and the corresponding increase in intensity cannot be by more than a factor of 4. The results of analytical calculations coincide with the results obtained by numerical methods and presented in the first part of the work.

Spherical-wave X-ray dynamical diffraction Talbot effect inside a crystal.

Two-wave dynamical diffraction of an X-ray spherical wave in a crystal, when the wave passes through an object with a periodic amplitude transmission function, is considered. The behavior of the diffracted wave (spherical-wave Talbot effect) in the crystal is investigated. The Talbot effect inside the crystal is accompanied by the focusing effect and the pendulum effect. Peculiarities of the effect before the focus point, in the focusing plane and in the region after the focus point inside the crystal are revealed. An expression is found for the Talbot depth and the spherical-wave Talbot effect in these three regions is investigated. The spherical-wave dynamical diffraction Talbot effect in a crystal is compared with the classical spherical-wave Talbot effect and also with spherical-wave effects inside the crystal without a periodic object.

New kind of interference in the case of X-ray Laue diffraction in a single crystal with uneven exit surface under the conditions of the Borrmann effect. Analytical solution.

The analytical solution of the problem of X-ray spherical-wave Laue diffraction in a single crystal with a linear change of thickness on the exit surface is derived. General equations are applied to a specific case of plane-wave Laue diffraction in a thick crystal under the conditions of the Borrmann effect. It is shown that if a thickness increase takes place at the side of the reflected beam, the related reflected wave amplitude is calculated as a sum of three terms, two of which are complex. If all three terms have a comparable modulus, it can lead to an increase in the reflected beam intensity by up to nine times due to interference compared with the value for a plane parallel shape of the crystal. The equation for the related transmitted wave amplitude contains only two terms. Therefore, the possibility to increase intensity is smaller compared with the reflected beam. The analytical solution is obtained after a solution of the integral equations by means of the Laplace transformation. A general integral form of the Takagi equations derived earlier is used. The results of relative intensity calculations by means of analytical equations coincide with the results of direct computer simulations.

Influence of Interdiffusion of Heteromaterials on the Diffraction Focusing of Spherical X-Ray Wave in a Superlattice

The influence of interdiffusion of heteromaterials on the diffraction focusing of spherical X-ray wave in a superlattice is investigated. It is shown that in case of dynamical diffraction of spherical X-ray wave in a short period superlattice, the focal lengths of satellites vary depending on the degree of interdiffusion of heteromaterials of the bilayer. The measurement of focal lengths of satellites allows the determination of structural factors of the superlattice and eventually the calculation of the thickness of transition diffuse layer.

Computer simulations of X-ray spherical wave dynamical diffraction in one and two crystals in the Laue case.

This article reports computer simulations of X-ray spherical wave dynamical diffraction in one and two single crystals in the Laue case. An X-ray compound refractive lens (CRL) as a secondary radiation source of spherical waves was considered for the first time and in contrast to previous simulations with the assumption of the use of a slit. The main properties of the CRL as a secondary source are discussed and two focusing phenomena are analysed. The first one is the diffraction focusing effect for one single crystal in the reflected beam and in the case of a large source-to-detector distance. The second one is the same but for two single crystals and for the twice-reflected beam in the case of a short distance between the source and detector. The first effect is well pronounced in the case of strong absorption. However, it may also be used as an element of an energy spectrometer in the medium and even weak absorption case. The second effect will appear in the case of weak absorption. It is shown that it is not effective to use it in an energy spectrometer. In the case of weak absorption the transverse size of the diffraction focused beam will oscillate together with the reflected beam integral intensity. The oscillation period is close to the extinction length.

X-ray Phase Contrast at Diffraction Focusing of a Spherical Wave in a Short-Period Superlattice

The possibility of reconstructing the internal structure of one-dimensional phase objects at diffraction focusing of X-ray spherical wave was investigated. As a crystal analyzer a strongly absorbing wedge-shaped short period superlattice with a rib parallel to the diffraction vector was used in symmetric Laue geometry case. It is shown that by moving a phase object along the direction of diffraction vector one can bring different satellites in the focusing condition. It is possible to restore an additional wave phase acquired in the phase object by recording the entire map of obtained data.

FEATURES OF DIFFRACTION FOCUSING OF A SPHERICAL X-RAY WAVE ON A SHORT PERIOD SUPERLATTICE

The problem of a dynamic diffraction of spherical X-ray wave on a short period superlattice is considered theoretically. It is shown that a focusing of satellites occurred both at different depths of the superlattice and at different distances from the crystal in vacuum depending on the structural factors of the superlattice. The larger the structural factor of the satellite, the more deeply the wave in the crystal is focused and, respectively, the less is the focal length out of the crystal in vacuum.

Dynamical diffraction of spherical X-ray wave on a superlattice

It was shown that at the dynamical diffraction of spherical X-ray wave on a short period superlattice, a focusing of satellites occurred both at different depths of the superlattice and at different distances from the crystal in vacuum depending on the structural factors of the superlattice. Based on the obtained results a method for investigation of a superlattice by means of dynamical diffraction focusing of a spherical X-ray wave was proposed. The method is based on the measurement of focal distances of different satellites.

Bragg diffraction of spherical X-ray wave with allowance for two-dimensional curvature of wavefront

Symmetrical Bragg diffraction of a spherical X-ray wave in a perfect semi-infinite crystal with a plane entrance surface is considered. As distinct from the standard theory, the two-dimensional curvature of the incident wavefront is taken into account. Using the corresponding Green function, a general expression for the amplitude of reflected wave on the entrance surface of the crystal is presented. Based on this expression, different cases of source-crystal distance are analyzed: the source is placed near the crystal and the crystal is in the Fresnel and Fraunhofer zones (large distance). The intensity distributions of the reflected wave are presented with allowance for two-dimensional curvature of the incident wavefront. Numerical calculations confirm the correctness of the used approximations.

10.1007/s11445-008-2002-6

The effect of total reflection (switching) of a spherical X-ray wave in the case of Laue diffraction in a crystal with bending deformation is analyzed by the trajectory method. Qualitative analytical description and computation of the spatial structure of the reflected beam for large and small distances between the spherical-wave source and the crystal are performed. The mechanism of much more efficient reflection of an X-ray beam by a deformed crystal in comparison with a perfect crystal is clearly demonstrated. It is also shown that the trajectory method is very convenient for description of the total reflection phenomenon.

Trajectory method in the theory of Laue diffraction of X rays in crystals: II. Effect of total reflection at bending deformation

The effect of total reflection (switching) of a spherical X-ray wave in the case of Laue diffraction in a crystal with bending deformation is analyzed by the trajectory method. Qualitative analytical description and computation of the spatial structure of the reflected beam for large and small distances between the spherical-wave source and the crystal are performed. The mechanism of much more efficient reflection of an X-ray beam by a deformed crystal in comparison with a perfect crystal is clearly demonstrated. It is also shown that the trajectory method is very convenient for description of the total reflection phenomenon.

Birefringence in multiple X-ray scatterings deduced from the alternative theory of Pendelloesung beat

The study of the inner structure of the weak-absorbing noncrystalline materials as well as the biological objects is carried out by the method of the X-ray phase contrast which at present is substantially in progress. In the present work the possibility of reconstructing the inner structure of the one-dimensional phase objects under the diffraction focusing of the X-ray spherical wave is reported.The fact that with the incidence of the monochromatic X-ray spherical wave on the crystal in Laue geometry the diffracted radiation is focused inside the crystal is used. The focal line is perpendicular to the diffraction plane. Before entering the crystal the radiation is supposed to pass through the phase object where the inhomogeneities have one-dimensional distribution along the perpendicular diffraction plane.With the wave transmission trough the phase object the conditions of the focusing are broken: because of the refraction on the inhomogeneities the X-rays change their directions and the wave front is deformed. The intensity distribution analysis in the image diffracted from a strong-absorbing wedge-like crystal with rib parallel to the diffraction vector shows that the focusing is possible for every point of the phase object if one compensates the deviation from Bragg condition along the diffraction vector and the wave front deformation along the perpendicular diffraction plane. The focusing is possible if the source-crystal distance is smaller than the ratio of the inhomogeneity size and the difference of the index of refraction from the unit. m35.p03

The formation of the X-ray phase contrast under the diffraction focusing of the spherical wave

The study of the inner structure of the weak-absorbing noncrystalline materials as well as the biological objects is carried out by the method of the X-ray phase contrast which at present is substantially in progress. In the present work the possibility of reconstructing the inner structure of the one-dimensional phase objects under the diffraction focusing of the X-ray spherical wave is reported.The fact that with the incidence of the monochromatic X-ray spherical wave on the crystal in Laue geometry the diffracted radiation is focused inside the crystal is used. The focal line is perpendicular to the diffraction plane. Before entering the crystal the radiation is supposed to pass through the phase object where the inhomogeneities have one-dimensional distribution along the perpendicular diffraction plane. With the wave transmission trough the phase object the conditions of the focusing are broken: because of the refraction on the inhomogeneities the X-rays change their directions and the wave front is deformed. The intensity distribution analysis in the image diffracted from a strong-absorbing wedge-like crystal with rib parallel to the diffraction vector shows that the focusing is possible for every point of the phase object if one compensates the deviation from Bragg condition along the diffraction vector and the wave front deformation along the perpendicular diffraction plane. The focusing is possible if the source-crystal distance is smaller than the ratio of the inhomogeneity size and the difference of the index of refraction from the unit.

Approximate analytical solution of the Takagi equations for X-ray diffraction in the backscattering mode from a cylindrically bent crystal

The dynamic diffraction of a spherical X-ray wave from a cylindrically bent crystal in the backscattering mode has been theoretically studied. An approximate analytical solution of the system of Takagi differential equations is obtained under certain conditions imposed on the coordinates of atoms inside the crystal.

On the Theory of X-Ray Diffractional Laue Focusing on Two Bent Crystals

The Laue focusing of a spherical x-ray wave onto dynamical diffraction by a double-bent crystal system is studied. The geometrical equations of two-dimensional Laue focusing and the intensity distribution formula are derived. A high sensitivity of the two-dimensional Laue focusing to the crystal thickness and bending radii difference is revealed. It is shown that this sensitivity can be used to control the crystal thickness with accuracy up to 100 A and bending radii up to ∼ 10-2 m. The focusing process of the spherical wave inside crystals as well as in vacuum is investigated. The possibility of using the two-crystal Laue system as a lens for x-ray microscopy is studied. The spectral characteristics of a twice-diffracted wave are discussed. The spectral resolution is demonstrated to be 106.

Theory of two-dimensional focusing of a spherical X-ray wave by Bragg diffraction by two bent crystals

The dynamical theory of the diffractional Bragg focusing in a system of cylindrically bent crystals is developed. Two-dimensional focusing `point-to-point' is shown to be possible under certain conditions. The influence of geometrical aberrations (astigmatism, spherical aberration) on the focusing is discussed. The influence of X-ray source sizes and temperature on the focusing is treated theoretically. A double-crystal focusing spectrometer is analysed. Focusing of X radiation by crystals with thin epitaxic films is investigated theoretically. Sensitivity of this focusing to the film thickness (∼1 µm) is demonstrated.