Exact Bragg Condition Research Articles

Observation of dislocations in thick β-Ga2O3 single-crystal substrates using Borrmann effect synchrotron x-ray topography

We performed Borrmann effect x-ray topography (XRT) to observe dislocations and other structural defects in a thick β-Ga2O3 (001) substrate. The Borrmann effect was realized by working in a symmetrical Laue geometry (g = 020). Anomalous transmission occurred under the exact Bragg condition, producing a strong diffraction beam that allowed us to image defects across the entire thickness of the substrate. The analysis clearly revealed straight b-axis screw-type and curved dislocations and allowed assessing the corresponding behaviors. Other structural defects, including pipe-shaped voids and dislocation loops produced by mechanical damage, were also observed. Finally, we compared Borrmann effect transmission topography and conventional reflection topography and explained the appearance of some characteristic defects in the two modes. Our results show that Borrmann effect XRT is a powerful and effective technique to study the spatial distribution and structural properties of defects in highly absorbing β-Ga2O3.

Transmitting volume Bragg grating with longitudinal moiré apodization

The apodized transmitting volume Bragg grating (TVBG) with the amplitude of the refractive index modulation (RIM) dropped to zero at both the surfaces is discussed. The proposed method for apodization is based on the recording of two identical uniform TVBGs with slightly different directions of their Bragg wave vectors in a single wafer. As a result, slow sinusoidal moire modulation occurs in the direction perpendicular to the averaged direction of the two wave vectors of Bragg modulation. The cutting of a specimen at the planes of the two closest zeros of the moire envelope produces the apodized TVBG. The longitudinal sinusoidal semi-period profile of RIM requires its specific maximum value to achieve 100% diffraction efficiency at the exact Bragg condition. Such an apodized TVBG demonstrates a significant suppression of the sidelobes in the diffraction efficiency depending on the angular or the wavelength detuning, in comparison to the diffraction operation of a uniform TVBG.

The real part of the dispersion surface in X-ray dynamical diffraction in the Laue case for perfect crystals.

The real part of the dispersion surface in X-ray dynamical diffraction in the Laue case for perfect crystals is analysed using a Riemann surface. In the conventional two-beam approximation, each branch or wing of the dispersion surface is specified by one sheet of the Riemann surface. The characteristic features of the dispersion surface are analytically revealed using four parameters, which are the real and imaginary parts of two quantities that specify the degree of departure from the exact Bragg condition and the reflection strength. The present analytical method is generally applicable, irrespective of the magnitudes of the parameters with no approximation. Characteristic features of the dispersion surface are also revealed by geometrical considerations with respect to the Riemann surface.

Crystal acceleration effect for cold neutrons in vicinity of Bragg resonance

A new mechanism of neutron acceleration in the accelerated perfect crystal is proposed and studied experimentally. The effect arises due to the resonance energy dependence of the neutron refraction index in vicinity of the Bragg's energy. During the neutron time-of-flight through the accelerated crystal the relative neutron-crystal velocity will be changing (with the deviation from the exact Bragg condition) and so the refraction index and the velocity of outgoing neutron will change as well.

The crystal acceleration effect for cold neutrons

A new mechanism of neutron acceleration is discussed and studied experimentally in detail for cold neutrons passing through the accelerated perfect crystal with the energies close to the Bragg one. The effect arises due to the following reason. The crystal refraction index (neutron-crystal interaction potential) for neutron in the vicinity of the Bragg resonance sharply depends on the parameter of deviation from the exact Bragg condition, i.e. on the crystal-neutron relative velocity. Therefore the neutrons enter into accelerated crystal with one neutron-crystal interaction potential and exit with the other. Neutron kinetic energy cannot vary inside the crystal due to its homogeneity. So after passage through such a crystal neutrons will be accelerated or decelerated because of the different energy change at the entrance and exit crystal boundaries.

Rocking curves in the Laue geometry depending on the departure of incident plane X-ray wave from the Bragg condition in the diffraction plane and in the direction perpendicular to the diffraction plane

Using the method of Green functions in the symmetric Laue geometry the rocking curves of plane X-ray wave were theoretically investigated depending on the departure of a beam from exact Bragg condition in the diffraction plane and in the direction perpendicular to the diffraction plane. The Si(220) reflection of MoKα radiation was considered by way of illustration.

Bragg resonance behavior of the neutron refractive index and crystal acceleration effect

The energy dependence of neutron refraction index in a perfect crystal for neutron energy, close to the Bragg ones, was studied. The resonance shape of this dependence with approximately the Darwin width was found. As a result, the value of deviation from the exact Bragg condition can change during the neutron time of flight through the accelerated crystal and so the refraction index and the velocity of outgoing neutron can change as well. Such new mechanism of neutron acceleration in the accelerating perfect crystal was proposed and found experimentally. This mechanism is march more effective then known one concerning with the neutron acceleration in the accelerating usual media.

Formulation of dynamical theory of X-ray diffraction for perfect crystals in the Laue case using the Riemann surface.

The dynamical theory for perfect crystals in the Laue case was reformulated using the Riemann surface, as used in complex analysis. In the two-beam approximation, each branch of the dispersion surface is specified by one sheet of the Riemann surface. The characteristic features of the dispersion surface are analytically revealed using four parameters, which are the real and imaginary parts of two quantities specifying the degree of departure from the exact Bragg condition and the reflection strength. By representing these parameters on complex planes, these characteristics can be graphically depicted on the Riemann surface. In the conventional case, the absorption is small and the real part of the reflection strength is large, so the formulation is the same as the traditional analysis. However, when the real part of the reflection strength is small or zero, the two branches of the dispersion surface cross, and the dispersion relationship becomes similar to that of the Bragg case. This is because the geometrical relationships among the parameters are similar in both cases. The present analytical method is generally applicable, irrespective of the magnitudes of the parameters. Furthermore, the present method analytically revealed many characteristic features of the dispersion surface and will be quite instructive for further numerical calculations of rocking curves.

Crystal acceleration effect for cold neutrons in the vicinity of the Bragg resonance

A new mechanism of neutron acceleration is studied experimentally in detail for cold neutrons passing through the accelerated perfect crystal with the energies close to the Bragg one. The effect arises due to the following reason. The crystal refraction index (neutron–crystal interaction potential) for neutron in the vicinity of the Bragg resonance sharply depends on the parameter of deviation from the exact Bragg condition, i.e. on the crystal–neutron relative velocity. Therefore the neutrons enter into the accelerated crystal with one neutron–crystal interaction potential and exit with the other. Neutron kinetic energy cannot vary inside the crystal due to its homogeneity. So, after passage through such a crystal, neutrons will be accelerated or decelerated because of the different energy change at the entrance and exit crystal boundaries.

Polarization control of an X-ray free-electron laser with a diamond phase retarder.

A diamond phase retarder was applied to control the polarization states of a hard X-ray free-electron laser (XFEL) in the photon energy range 5-20 keV. The horizontal polarization of the XFEL beam generated from the planar undulators of the SPring-8 Angstrom Compact Free-Electron Laser (SACLA) was converted into vertical or circular polarization of either helicity by adjusting the angular offset of the diamond crystal from the exact Bragg condition. Using a 1.5 mm-thick crystal, a high degree of circular polarization, 97%, was obtained for 11.56 keV monochromatic X-rays, whereas the degree of vertical polarization was 67%, both of which agreed with the estimations including the energy bandwidth of the Si 111 beamline monochromator.

Fine structure of the diffraction image of an edge dislocation in high-resolution section topography

The propagation of an x-ray wave field in an elastic field of an edge dislocation crossing a scattering triangle exactly along the bisector of the scattering angle has been considered. The scattering of the x-ray wave field by a complex elastic field of the edge dislocation has been analyzed using the methods of geometrical optics. It has been established that the fine structure of a diffraction image of defects in thick crystals is determined by the differences in scattering of the normal and anomalous modes of the x-ray wave field in the vicinity of the Bragg reflection. In the case of thick crystals, the x-ray diffraction image of defects can have a symmetry different from the symmetry of the function of local misorientations of the crystal lattice. X-ray wave scattering by local distortions of the crystal lattice can occur according to two different mechanisms depending on the gradient of space changes in the deformation field. In the crystal regions where the elastic field varies slowly with a change in the distance, the x-ray wave field has had time to adjust itself to follow the course of deviations of the crystal lattice from the exact Bragg condition. In the crystal region where the elastic field changes significantly at distances of the order of the extinction length, this region leaves the reflecting position and interference scattering occurs at the interface of the region. It is important that the form of the deformation field in this case is of no significance.

Contribution of electron precession to the study of perovskites displaying small symmetry departures from the ideal cubic ABO3 perovskite: applications to the LaGaO3 and LSGM perovskites

Electron microscopy and electron diffraction are well adapted to the study of the fine-grained, faulted pure and doped LaGaO(3) and LSGM perovskites in which the latter is useful for fuel cell components. Because these perovskites display small symmetry departures from an ideal cubic ABO(3) perovskite, many conventional electron diffraction patterns look similar and cannot be indexed without ambiguity. Electron precession can easily overcome this difficulty mainly because the intensity of the diffracted beams on the precession patterns is integrated over a large deviation domain around the exact Bragg condition. This integrated intensity can be trusted and taken into account to identify the 'ideal' symmetry of the precession patterns (the symmetry which takes into account both the position and the intensity of the diffracted beams). In the present case of the LaGaO(3) and LSGM perovskites, the determination of the 'ideal' symmetry of the precession patterns is based on the observation of weak 'superlattice' reflections typical of the symmetry departures. It allows an easy and sure identification of any zone axes as well as the correct attribution of hkl indices to each of the diffracted beams. Examples of applications of this analysis to the characterizations of twins and to the identification of the space groups are given. This contribution of electron precession can be easily extended to any other perovskites or to any crystals displaying small symmetry departures.

Excess coincidences of reflected and refracted x rays from a synchrotron-radiation beamline

The coincidence rate between x rays reflected and refracted from a silicon crystal after a high-resolution monochromator was measured at the long undulator beam line of SPring-8. The data show excess coincidences relative to the random background, which vanish when the detectors of the reflected and refracted beams probe widely separated parts of the incoming x-ray wave front. The measured coincidence rate with different detector sizes and distances between the probed wave front positions, perpendicular to the scattering plane, agree well with fitting functions derived for the present experimental conditions. Furthermore, it is shown that the coincidence rate measured between probing positions in the scattering plane depends strongly on the angular spread of reflected and refracted x rays. Small angular spread is obtained by detuning from the exact Bragg condition, which results in good quantitative agreement of the coincidence data and theoretical predictions.

Approximation of local reflection in meridional schemes of Bragg focusing

The wave amplitudes in the vicinity of the focus in monochromatic focusing are calculated in the approximation where each point of the surface of a weakly bent single crystal reflects an X-ray wave as a planar crystal, while the reflection coefficient is a coordinate function of the deviation from the exact Bragg condition. These amplitudes are calculated for all the meridional diffraction geometries (those for plane and spherical waves and the Johann and aberration-free schemes). For all these schemes, the quantitative characteristic of the process of polychromatic focusing—the intensity integrated over the pass band—is introduced and calculated. The analytical expressions of polyprofiles necessary for the consideration of these schemes are also derived.

Probability of Seeing (001) Cross-Fringes in a Random Cubic Nanocrystal Image

Abstract As a crystal gets smaller, diffraction spots stay visible for larger deviations from the Bragg condition. The upper limit of such deviations is connected to the threshold for getting lattice fringes in TEM images. This in turn allows one to quantify the probability of seeing cross lattice fringes along a certain zone axis. In this abstract we examine a simple semi-empirical model for the probability of detecting (001) zone cross-fringes of a spherical crystal of cubic lattice. The upper limit for the deviation of crystal orientation from the exact Bragg condition, without losing cross fringes down a given zone, is expressed as the maximum half-angle θ1 between the zone and the electron beam. The solid angle σ subtended by a cone with this half-angle is proportional to the probability px that a randomly-oriented crystal will show the cross-fringes associated with that zone. A schematic, illustrating the principle used to calculate the probability of seeing cross-fringes, is given in Figure 1.

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.

Wave-front conversion between a Gaussian beam with a cylindrical phase and a plane wave for on-axis off-Bragg incidence

The theoretical model for an off-Bragg on-axis conversion between a Gaussian beam with cylindrical phase function and a plane wave by a volume aperiodic nonplanar inhomogeneous holographic grating is presented. A two-wave first-order coupled-wave theoretical framework is adopted. Analytical solutions for the amplitudes of two space harmonics of the field inside the grating zone are derived. Both the chromatic and the geometric deviations from the exact Bragg condition are studied. Numerical evaluations show that some anomalous phenomena (Pendellosung fringes, angle amplification effect, achromatism) can arise. High diffraction efficiency (≈1) is predicted even for relatively large off-Bragg deviations. The deterioration of the reconstruction fidelity due to the Pendellosung effect is discussed.

Transformation of X-Ray polarization under dynamical grazing incidence diffraction

The transformation of the X-ray polarization under grazing incidence diffraction is studied. The dependencies of the parameters of the polarization ellipses for specularly reflected and diffracted waves in vacuum on the corresponding parameters of the incident wave with arbitrary polarization are found. Strong spatial crystal anisotropy close to the exact Bragg condition and the existence of two critical reflection angles manifest themselves in a steep change of the polarization state of the secondary waves near these points. It is shown that diffraction conditions exist such that an incident linearly polarized beam of X-ray quanta can acquire a high degree of circular polarization during the reflection process. [Russian Text Ignored].

Collinear interaction of Gaussian acoustic and light beams

A general expression for the amplitude of diffracted light under collinear interaction of Gaussian acoustic and light beams is obtained. It is shown that, if the radius of the acoustic beam has a finite value, the central frequency of the transmission curve shifts with respect to a point corresponding to the exact Bragg condition. The value of the light beam radius can increase or decrease this shift for upshifted or downshifted diffraction respectively. An optimum relationship between the interaction length and the acoustic beam radius is found. This optimum condition is related to a minimum of input acoustic power.

Focusing Properties of a Linear-Phase Bragg-Fresnel Lens

Hard X-ray focusing by a linear phase Bragg-Fresnel crystal lens was performed with an efficiency close to 30%. The phase shift of an incident beam was controlled by deviating the crystal lens from the exact Bragg condition. Analytical expressions for the intensity and the image contrast at the focal point was presented.