Search

We have studied with the sphere-in-contact model the structure of rhombohedral graphite and found that there are molecular channels 22.7 degrees with respect to the material symmetry axis. We show via theoretical simulations with the Laue equation, Kramers equation and Duane-Hunt law that X-ray radiation (λ < 1.34 Å) that is traveling along these molecular channels would have a larger mean-free-path. This suggests that stacks of rhombohedral graphite can become effective filters for X-ray sources, such as the ones used in Laue photography and other X-ray techniques that use Bremsstrahlung.

A semi-analytical approach based on the influence functions of a point source located on the crystal surface has been adopted to show that the focusing ability of cylindrically bent Laue crystals may be strongly enhanced by replacing symmetrically cut crystals with asymmetrically cut crystals. This approach is generally applicable to any distance between the X-ray source and the focusing bent crystal. A mathematically straightforward method to simplify the derivation of the already known expression of the influence functions in the case of deformed crystals with a constant strain gradient (e.g.cylindrically bent crystals) is also presented.

High-speed (submicrosecond) translation and rotation of crystals are tracked in real time via simultaneous synchrotron X-ray imaging and Laue diffraction., Dynamic compression experiments are performed on single-crystal Si under split Hopkinson pressure bar loading, together with simultaneous high-speed (250–350 ns resolution) synchrotron X-ray Laue diffraction and phase-contrast imaging. A methodology is presented which determines crystal rotation parameters, i.e. instantaneous rotation axes and angles, from two unindexed Laue diffraction spots. Two-dimensional translation is obtained from dynamic imaging by a single camera. High-speed motion of crystals, including translation and rotation, can be tracked in real time via simultaneous imaging and diffraction.

Fil: Piro, Oscar Enrique. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - La Plata. Instituto de Fisica La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Fisica La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Fisica; Argentina

The effect of two-crystal focusing of neutrons at Laue diffraction from large perfect silicon crystals has been studied. It has been shown that the focusing effect makes it possible to reach an angular resolution better than 0.03″, which is about 0.01 of the width of a Bragg reflection. This circumstance allows a new ultraprecision neutron-spectrometry method based on the method of spin-echo small-angle neutron scattering in combination with Laue diffraction.

Spherical wavefields in the Laue case are obtained when the real part of the crystal structure factor is zero or small compared with the imaginary part. The results are the same as in the conventional case where the real part is large. Through this work, it is shown that virtual wavefields on the vacuum side should be taken into account to explain the obtained results. It is also shown that, to obtain the virtual wavefields, a modification of the conventional spherical wave theory is necessary.

(311) curved planes can be exploited for efficiently focus hard X-rays. With this purpose, a self-standing bent crystal was manufactured at the Sensor and Semiconductor Laboratory of Ferrara (Italy). The crystal was designed as an optical component for a X-ray concentrator such as a Laue lens. The curvature of (311) planes was obtained through the quasi-mosaic effect. The diffraction efficiency of the sample was tested at the Institut Laue Langevin of Grenoble (France) by using a collimated monochromatic X-ray beam. This was the first prove of the diffraction properties of (311) quasi-mosaic planes. Diffraction efficiency resulted 35 % with a 182 keV X-ray beam, in agreement with the theoretical expectation. It corresponded to a reflectivity of 33 %. While the chosen orientation is not the most performing lying of planes, it can be used, in addition to smaller-index planes, in order to raise the total effective area of a Laue lens. To quantify it, a Laue lens based on quasi-mosaic silicon and germanium crystals, exploiting (111), (422) and (311) diffracting planes, was achieved and simulated with the LaueGen code.

Symmetrical Laue diffraction in a perfect crystal with a plane entance surface is considered. The two dimensional curvature of the wave front of the incident beam is taken into account. Using the corresponding Green function, a general expression for the amplitude of diffracted wave in the crystal is presented. Based on this expression, the concept of a locally plane wave is analyzed taking into account two-dimensional curvature of the wave front. This concept is used for obtaining the rocking curves depending on the angles of deviation from the chosen exact Bragg direction in both the diffraction plane and in perpendicular direction. The obtained result is compared with the result of the standard dynamical diffraction theory.

An experimental setup for demonstrating the Bragg diffraction of light by a 2D-photon structure (two-dimensional photon crystal) is described. The conditions for formation of the optical Laue X-ray diffraction pattern are considered for the light reflected from the surface of the photon crystal. The procedure for determining the lattice spacing of the photon crystal from its optical Laue X-ray diffraction pattern is described. The lattice spacing data of the photon crystal with a hexagonal close-packed structure are obtained.

We propose a novel experiment to test the weak equivalence principle (WEP) for the Laue diffracting neutron. Our experiment is based on an essential magnification of an external affect on neutron diffracting by Laue for the Bragg angles close to the right one in couple with additional enhancement factor which exists due to the delay of the Laue diffracting neutron at such Bragg angles. This enhancement phenomena is proposed to be utilized for measuring the force which deviates from zero if WEP is violated. The accuracy of measuring inertial to gravitational neutron masses ratio for the introduced setup can reach ∼10−5, which is more than one order superior to the best present-day result.

Phase-matched second-harmonic generation (SHG) under the Bragg diffraction in the Laue geometry in one-dimensional photonic crystal (PhC) is studied theoretically and experimentally. We demonstrate that the phase-matched SHG can be realized in a PhC by compensation of the material dispersion of the PhC constituent layers of adjustable thickness. The second-order nonlinear susceptibility is introduced in the porous quartz-based PhC by its infiltration by sodium nitrite. We observed that two second-harmonic (SH) beams appear after passing through the PhC under the phase-matched process, which correspond to the transmission and diffraction angular directions. The appearance of the phase-matched SHG is confirmed by a pronounced SH spectral dependence and a narrow SH angular distribution, with the FWHM of the SH peak of approximately 3.5 times smaller as compared to the case of non-phase-matched SHG.

Second-harmonic generation (SHG) in the Laue scheme of the dynamical Bragg diffraction in one-dimensional photonic crystal (PhC) is studied. The experiments are performed for partially annealed porous-silicon PhC containing 250 periods of the structure. Our measurements confirm that the phase-matched optical SHG is observed under the Bragg conditions, which is evidenced by a narrow angular and spectral distribution of the diffracted SHG outgoing the PhC. This is confirmed by both the analytical description of the SHG process performed in the two-wave approximation, and by direct calculations of the PhC dispersion curves for the fundamental and SHG wavelengths by the revised plane wave method. Possible types of phase- and quasi-phase-matching realized in the studied PhC under the Laue diffraction scheme are discussed.

Synchrotron X-ray Laue microdiffraction is used to investigate the microstructure of deformed quartz, which has trigonal symmetry. The unambiguous indexing of a Laue diffraction pattern can only be achieved by taking the intensities of the diffraction peaks into account. The intensities are compared with theoretical structure factors after correction for the incident X-ray beam flux, X-ray beam polarization, air absorption, detector response and Lorentz factor. This allows mapping of not only the grain orientation but also the stress tensor. The method is applicable for correct orientation determination of all crystals with trigonal symmetry and is indispensable for structure refinements of such materials from Laue diffraction data.

A new method for determination of the orientation matrix of Laue X-ray data is presented. The method is based on matching of the experimental patterns of central reciprocal lattice rows projected on a unit sphere centered on the origin of the reciprocal lattice with the corresponding pattern of a monochromatic data set on the same material. This technique is applied to the complete data set and thus eliminates problems often encountered when single frames with a limited number of peaks are to be used for orientation matrix determination. Application of the method to a series of Laue data sets on organometallic crystals is described. The corresponding program is available under a Mozilla Public License-like open-source license.

In sample-scanning Laue microdiffraction, the local crystal orientation and local deviatoric strain tensor are obtained by illuminating the polycrystalline sample with a broadband `white' (5–30 keV) X-ray microbeam and analyzing the spot positions in the resulting local Laue pattern. Mapping local hydrostatic strain is usually slower, owing to the need to alternate between white and tunable-energy monochromatic microbeams. A technique has been developed to measure hydrostatic strain while keeping the white beam. The energy of one of the Laue spots of the grain of interest is measured using an energy-dispersive point detector, while simultaneously recording the Laue pattern on the two-dimensional detector. The experimental spot energy,Eexp, is therefore measured at the same time asEtheor, the theoretical spot energy for zero hydrostatic strain, which is derived from the analysis of the Laue pattern. The performance of the technique was compared with that of the monochromatic beam technique in two test cases: a Ge single crystal and a micrometre-sized UO2grain in a polycrystal. Accuracies on the hydrostatic strain Δa/aof ±0.4 × 10−4and ±1.3 × 10−4were obtained for Ge and UO2, respectively. Measurement strategies to limit the remaining uncertainties onEtheorare discussed.

Several formulae are suggested allowing the measurement of angles between the primary X-ray beam and the zone axes. This makes it possible to derive a single crystal lattice and determine its orientation from a single Laue photograph.

Recently using the theory of Afanasev-Aginian (1977-1978) it has been reconsidered theoretically the properties of the contributions of PXR, DTR and their interference into the radiation intensity produced by particles in single crystals in Laue geometry, essentially differing from those of Bragg geometry. After a short review of theoretical and experimental works devoted to DTR in this work we discuss the properties of Laue geometry and make comparison of the predictions of the above theory with the results of the second experiment performed by Avakian et al in 1987 with the help of 4.5 GeV electrons.

Parametric X-ray radiation and parametric X-ray radiation at a small angle with respect to the velocity of a relativistic electron crossing a single crystal plate in Laue scattering geometry are studied based on the dynamic X-ray diffraction theory. Analytical expressions for the spectral-angular density of radiations are derived in the general case of asymmetric reflection. The ratio of the contributions of these radiation mechanisms is considered.

The phenomenon of a neutron phase shift due to Laue transmission in a perfect crystal blade is discussed. Quantitative measurements of this phase shift are presented in the vicinity of the Bragg condition well in agreement with numerical calculations. The phase shift shows a strong angular sensitivity and might constitute an interesting opportunity for precision measurements of fundamental quantities like the neutron-electron scattering length or gravitational short-range interactions.

X-ray rocking curves from side surface of a thin plane-parallel Ge crystal are measured under the condition where the dispersion of the refracted angle is quite large. In the diffracted beams from side surface are observed, which are analysed by using Wagner's approach for Bragg-(Bragg) m -Laue case. The origin of the fringes is explained by the interference between diffraction beams in Bragg-Laue case and that in Bragg-Bragg-Laue case.

Detailed study of a neutron Laue diffraction in a noncentrosymmetric crystal was carried out using a prototype of the experimental setup to measure the neutron electric dipole moment (nEDM) by Laue diffraction method (LDM). A new kind of spin rotation effect was observed for the first time. We attribute this effect to a difference of the two Bloch waves amplitudes in the crystal, which is caused by the presence of the definite small crystal deformation. In a revised scheme of the experiment, this effect could be exploited for a purposeful manipulation of the Bloch waves.

Quantitative analysis of the focusing performance of bent Laue crystals is carried out using the rigorous wave-optical formalism.Vianumerical solution of the Takagi equations and the Fresnel integral, the influence of diffraction on the formation of the image of an X-ray source is analysed. The results are compared with those obtained using a formalism based on the geometrical optics approximation and significant differences are found. A formula for the geometrical aberration of a cylindrically bent crystal is derived. The effect of the aberration on image formation is analysed.

Accurate shapes of the accessible reciprocal-space region for non-coplanar Bragg and Laue geometries are given. Vector formulae for finding these regions and for experimental geometry calculations for obtaining diffraction from any accessible (using a given wavelength) reciprocal node are presented. A method for converting these formulae to scalar equations is proposed. Solutions are given and illustrated as three-dimensional figures in reciprocal space.

Experimental studies of X-ray diffraction from reflecting atomic planes (111) of X-cut quartz single crystal in Laue geometry influenced by the temperature gradient were carried out. It is shown that by using the temperature gradient it is possible to reflect a hard X- ray beam with photon energy near the 100 keV with high efficiency. It has been experimentally proved that the intensity of the reflected beam can be increased by more than order depending on the value of the temperature gradient.

An analytical study of the geometrical properties of the X-ray diffraction applied to the case of Laue lenses is reported in this paper. The results of this investigation are analytical equations that describe the point in which a photon is diffracted as a function of the crystal position and of the X-ray source angular coordinates. In addition, the contributions to the lens Point Spread Function of each crystal is described, helping in understanding the optical properties of a Laue lens. The results obtained can be inserted in simulation and image deconvolution softwares for speeding up the calculations.

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.

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.

Contrast in a systematic arrangement of lower order Laue zone (LOLZ) beams is reported and analysed using a Bloch wave description. Observations are reported for hexagonal barium ruthenium zirconate (Ba4Ru3ZrO12) and barium ruthenium titanate (Ba3Ti2RuO9), both near the c-axis orientation. The specific scattering dynamics invoked by this diffraction geometry may have novel uses in the exploration of crystallographic parameters.

The main limitation of Laue geometry for the achievement of a small focus size is the focus broadening caused by the intrinsic Darwin width and the spread of the beam in the Borrmann triangle, resulting from propagation inside the crystal. A method, based on dynamical focusing, is suggested that allows improvement of the quality of high-energy polychromatic focusing by bent crystals in Laue geometry.

A new formalism is presented concerning the dynamics of X-rays in crystals. It is based on Takagi's equations and Fourier optics; it also offers an alternative to the usual Ewald-von Laue approach. The article does not give new results but shows a new way to formulate the dynamical theory of X-ray diffraction. In addition, it proposes a novel description of X-ray propagation based on the analogy between the dynamics of X-rays in crystals and that of two-level quantum systems.

The investigation of twin structure in La0.95Sr0.05Ga0.9Mg0.1O2.92perovskite-type crystals has been undertaken using the technique of white-beam X-ray microdiffraction. The sample shows definite twinning which is reflected in the Laue patterns as apparent splitting of the peaks. Reflections from the different domains were indexed separately. Positional shifts of twin domain spots with respect to the reference matrix spot, arising from the mutual misorientation of diffraction vectors, were measured on Laue patterns and compared with calculated values in order to solve the twin laws in this system. The dominant twin operations were found to be (12\bar{1}) and (10\bar{1}) mirror reflections.

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.

The use of bent asymmetric Laue crystals to sagittally focus high-energy synchrotron X-rays calls for an understanding of the mechanisms affecting X-ray diffraction by such crystals. The rocking-curve width, a measurable quantity directly related to the distortion of the lattice planes, is the necessary first step towards such an understanding. A model is formulated for assessing the rocking-curve widths of sagittally bent Laue crystals, considering the elastic anisotropy. A method for depth-resolved measurement of the rocking curves was also developed to verify the model. The model successfully explains the wide range of rocking-curve widths of a large number of reflections from silicon crystals with two different orientations.

The theory of the dynamic 2D focusing of an X-ray wave upon its Laue diffraction by a system of two bent crystals is developed. The reflecting planes of the crystals are perpendicular to their surfaces. 2D focusing is shown to be highly sensitive to the bending radius of the crystals and to the difference in their thicknesses. The effect of astigmatism on 2D Laue focusing is described, and conditions for stigmatic focusing are found. One-dimensional focusing inside the crystals is discussed. The spectral characteristics of a double-crystal 2D-focusing spectrometer are considered.

Based on an analytical expression of electron dynamical diffraction derived recently, the exit wavefunctions of nonperiodic samples have been calculated. The results have shown that the amplitude and phase obtained by the analytical expression were in good agreement with those obtained using the multi-slice method. Besides, the influence of the higher-order Laue zone on the wavefunction for different sample thickness was studied in detail and the results showed that the exit wavefunction is dominated by the zero-order Laue zone. When the sample is very thin, the influence of the higher-order Laue zone should be considered.

The theory for the Bragg dynamical X-ray diffraction and the yield of the secondary radiation scattered via incoherent channels under conditions of X-ray diffraction (X-ray standing wave technique) is developed for the crystal systems composed from many layers. The formulae are derived in a form suitable for a computer simulation of the experimental angular or energy spectra as well as for a determination of unknown parameters via fitting. The Bragg case (reflection) and the Laue case (transmission) are considered within the same approach. The universal computer program is created and its operation is demonstrated on two examples. It is shown that the model of the multilayered crystal system may be a useful tool for analyzing a lot of problems where the one-dimensional crystal-lattice distortions of different kinds influence the rocking curve and the curve of the secondary radiation yield. In the first example the In 0.5 Ga 0.5 P epitaxial film on the GaAs substrate is considered. The experimental angular spectra of the X-ray reflectivity and the yield of the In L α and P K α fluorescence in the Bragg case are described by fitting the parameters of the structure. In the second example the Laue case of X-ray diffraction in a sample having a monotonously increasing (decreasing) shift of the Bragg angle with increasing depth is analyzed. The phenomenon of the complete switching of the X-ray beam from the incident direction to the direction of reflection is calculated and discussed.

This study investigates the phase shift induced by Laue transmission in a perfect Si crystal blade in unprecedented detail. This `Laue phase' was measured at two wavelengths in the vicinity of the Bragg condition within a neutron interferometer. In particular, the sensitivity of the Laue phase to the alignment of the monochromator and interferometer (rocking angle) and beam divergence has been verified. However, the influence of fundamental quantities, such as the neutron–electron scattering length, on the Laue phase is rather small. The fascinating steep phase slope of 5.5° [(220) Bragg peak] and 11.5° [(440) Bragg peak] per 0.001 arcsec deviation from the Bragg angle has been achieved. The results are analysed using an upgraded simulation tool.

The second German edition of Werner Massa’s crystallography textbook appears in English for the first time, in a highly readable and lucid translation by Robert Gould of Edinburgh University. This book presents a high level and quite specific treatment of crystallography, aimed at advanced students or researchers who wish to learn more about what lies beyond automated ‘black box’ laboratory techniques. Essentially, the book is in two parts: a general introduction to crystallography and single-crystal X-ray diffraction in particular (i.e. data acquisition), followed by more detailed chapters on processing diffraction data in order to ‘solve’ a crystal structure. We begin with some basic crystallographic concepts: unit cells and lattices. Diffraction of X-rays from a virtually infinite lattice ‘grating’ is analysed using the Laue equations and the Bragg equation. …

A preliminary analysis is presented of neutron Laue diffraction data from a single crystal of La 2 Co 1.7 at temperatures between 295 and 15 K, recorded on neutron-sensitive image plates using a thermal neutron beam. The data can be interpreted in terms of a charge (nuclear) density wave with a propagation vector q n =(q x , 0, q z ) with q x =0.113(1) a * and q z =0.203(2) c *. It has also been confirmed by using a band-pass filter with a cold neutron beam that antiferromagnetic Bragg reflections which have a magnetic propagation vector q m =( 1 3 , 1 3 , 0) exist below 144(1) K and correspond to a cobalt atom moment of 0.80(15) μ B . Magnetic satellite reflections with wave vectors corresponding to q n ± q m have also been observed on the Laue diagrams below this temperature. The values of the propagation vector and magnetic moment have been confirmed from neutron powder diffraction diagrams taken at 150 and 1.5 K.

The Ga K fluorescent X-rays emitted from the incident surface of a GaAs perfect crystal were observed around the (200) diffraction peak in the Laue case, tuning the energy of incident X-rays from a synchrotron radiation source between the K absorption edges of Ga and As atoms. With the change of the energy of incident X-rays, the asymmetry of the fluorescent X-ray curves will change accordingly, the tendency is similar to that in the Bragg case. However, the variation in the Laue case cannot be explained by moving the nodal planes of X-ray standing wave (XSW) with respect to (200) diffraction plane of GaAs. In the Laue case, the variation of the amplitudes of XSW with the incident angle is greatly related to the phase of the structure factor whose change will result in the change of the asymmetry of the fluorescence curves. Under the assumption that the fluorescence yield is proportional to the electric-field intensity at the position where the atoms are located, the formulae for calculating the fluorescence emission in the Laue case for perfect crystals are derived. The agreement of the experimental result with the theoretical calculation is quite good.

By analysing in detail the existing plane wave dynamical theory of X-ray diffraction for thin (μt sec θ ∼ 0.30) and perfect crystal we demonstrate the presence of forward diffracted beam when the crystal is oriented for Laue diffraction. Recently, these features have been demonstrated experimentally by Lal et al. (Solid State Comm. 81 (1992) 461–465 & 96 (1995) 33–36) in the case of diamond crystals. We also show that a residual direct beam is implicitly built in the expression for forward diffracted beam intensity. It has been shown that there is a substantial decrease in absorption coefficient, μ, of thin perfect crystals for exploring X-ray beam around Laue diffraction maximum. The computed fractional change in μ, Δμ/μ, at the [unk]20 diffraction peak of a thin (111) diamond crystal for MoK α1 exploring radiation is found to be in reasonably good agreement with the experimentally observed value (Lal et al. Acta Crystallogr. A52 (1996) C-493). The results reported here are of general nature and applicable to all crystals.

In protein crystallography, the use of low-resolution reflections is important in defining the molecular mask and polypeptide backbone. However, in Laue data collection, the loss of low-resolution reflection data (>2dmin) can be as high as 40–50%, even after the deconvolution of multiples. To estimate the reflection intensities that are not recorded in data collection, a new method is presented based on maximizing the entropy of the Patterson function subject to the constraints imposed by the intensities of the observed reflections. The method has been tested with Laue diffraction data from hen egg-white lysozyme. All unobserved reflections within 5 A resolution were estimated, and their inclusion in the electron-density-map calculation significantly improved the connectivity. This method could also be applied to improve the completeness of monochromatic data.

The forbidden reflections in Laue geometry are shown to have the same properties as nonforbidden ones. The only difference is their width which is nearly four orders of magnitude less than the width of nonforbidden reflections.

In this work we present an improved dynamical theory within the Laue formalism for the two-beam case, which is valid for X-ray Bragg and Laue diffraction on ideal crystals of finite thickness. In our calculations no approximations, which are generally used in the conventional dynamical theory, are made. We limit ourselves to the two-beam case and neglect only the quadratic terms of the dielectric susceptibility. We take into account: i) the asymptotic sphericity of the dispersion surface and all the four solutions of the secular equation; ii) the difference between electric and displacement fields; iii) the boundary conditions of continuity of the tangential components of the electric and magnetic fields at the two crystal-vacuum interfaces. With the equations derived it is possible to describe the interaction of the X-ray beam with the crystal in a dynamical way in the whole angular range from 0 to π/2. Our improved theory can be applied for describing: i) symmetric and asymmetric reflections both near the Bragg condition and at the far tails of the Bragg peaks; ii) Bragg diffraction of X-rays at very small glancing angles of the order of the critical angle; iii) Bragg and Laue diffractions in which the diffracted beam travels almost parallel to the crystal surface.

The Darwin equations, which describe the multiple Bragg reflection of X-rays or neutrons in a mosaic crystal slab, have previously been solved only for special cases. Here, the complete and exact analytical solution of these equations is obtained for both the Bragg case (reflection geometry) and the Laue case (transmission geometry) with the help of a computer algebra program and it is shown that the resulting general expressions for both the reflectivity R and the transmissivity T can each be expressed in a compact form. It is found, for example, that for a mosaic crystal anomalous absorption occurs only in the Bragg case and not in the Laue case. This is in contrast to the dynamical theory of diffraction, which applies to an ideally perfect crystal, where anomalous absorption (due to the Borrmann effect) is found in both Laue and Bragg cases. With this new general expression for R, the Fankuchen gain is calculated for a crystal of finite thickness, taking correctly into account the effects of both absorption and secondary extinction. General expressions for the optimum crystal thickness are also obtained for both Bragg and Laue cases. In a companion paper, these general results are applied to a detailed numerical calculation of the reflecting properties of various neutron monochromator crystals.

Order in γ-phase TiAl alloys is investigated by convergent beam diffraction of 200 keV electrons. The separation of different branches of the dispersion surface, as observed in quantum state rings in beams from higher order Laue zones, is shown to provide a direct measure of order. Theory and experiment are shown to be in good agreement.

A series of quasi-mosaic curved crystals has been fabricated for the Laue project to build up a prototype of a Laue lens. Curvature has been imparted to the samples thanks to the grooving method, which makes it possible to realize self-standing curved plates. Samples are 10 × 30 × 2 mm 3 Ge crystals characterized by two curvatures of different crystallographic planes. The primary curvature allows the focalization of the diffracted X-rays on a focal spot smaller than the crystal dimension. The secondary curvature, i.e. quasi-mosaic curvature, amplifies the diffraction efficiency by more than one order of magnitude with respect to an equivalent crystal without quasi-mosaic curvature.

Starting from the discovery of Roentgen’s X-ray (Rontgen 1895), I here reconstruct the various attempts carried out to find X-ray diffraction using very thin wedge-shaped slits. These experiments allowed A. Sommerfeld to achieve, at the beginning of 1912, a qualitative estimate of the wavelength of X-rays around 4 × 10−9 cm. This value was at the base, together with the adoption of the space-lattice model for the crystal, of Laue idea of using a crystal as a diffraction grating for X-ray. Then we will discuss the feasibility of the Laue project, as well as the realization of the experiment, and the first results of W. Friedrich and P. Knipping, with the theoretical interpretation given to them by Laue. Laue’s discovery of X-ray diffraction was immediately taken up by William Henry Bragg and re-interpreted by his son William Lawrence in terms of X-ray reflection by the planes of the crystal, thus becoming, in a few months, a formidable tool for the study of crystal structures, as well as for the precise determination of the wavelengths of X-rays (to Bragg father we owe the construction of the first X-ray ionization spectrometer). Particular attention will be given to the works of C. G. Barkla on X-ray absorption, which resulted in the discovery in 1906 of the Roentgen characteristic radiation of the elements. These works strengthened the hypothesis of the wave nature of X-rays, at the expense of the “corpuscular hypothesis” (supported notably by J. Stark and by W.H. Bragg), and represented a solid reference point for the project of Laue.