Laue Geometry Research Articles

A role of second-order derivatives of amplitudes in X-ray beam dynamical diffraction equations

Dynamical diffraction equations for X-ray beams are considered with retaining the second-order derivatives of amplitudes in the direction perpendicular to the diffraction plane. For the dynamical diffraction problem, the retarded Green function is determined in case of a perfect crystal. Amplitudes of transmitted and diffracted waves in the crystal are represented as convolution over the crystal surface with use of determined Green function. Such representation can be used for solving diffraction problems in Laue and Bragg geometries in perfect crystals with both plane and not plane entrance and exit surfaces.

Curved focusing crystals for hard X-ray astronomy

A lens made by a properly arranged array of crystals can be used to focus x-rays of energy ranging from 30 to 500 keV for x-ray astronomy. Mosaic or curved crystals can be employed as x-ray optical elements. In this work self standing curved focusing Si and GaAs crystals in which the lattice bending is induced by a controlled damaging process on one side of planar crystals are characterized. Diffraction profiles in Laue geometry have been measured in crystals at x-ray energies E = 17, 59 and 120 keV. An enhancement of diffraction efficiency is found in asymmetric geometries.

Borman effect in resonant diffraction of X-rays

A dynamic theory of resonant diffraction (occurring when the energy of incident radiation is close to the energy of the absorption edge of an element in the composition of a given substance) of synchronous X-rays is developed in the two-wave approximation in the coplanar Laue geometry for large grazing angles in perfect crystals. A sharp decrease in the absorption coefficient in the substance with simultaneously satisfied diffraction conditions (Borman effect) is demonstrated, and the theoretical and first experimental results are compared. The calculations reveal the possibility of applying this approach in analyzing the quadrupole-quadrupole contribution to the absorption coefficient.

Polarization effects in diffraction-induced laser pulse splitting in one-dimensional photonic crystals

The polarization effects in the diffraction-induced pulse splitting (DIPS) observed under the dynamical Bragg diffraction in the Laue geometry in linear one-dimensional photonic crystals (PCs) are studied theoretically and experimentally. It is demonstrated that the characteristic length of the laser pulse path in a PC, or splitting length, used to describe the temporal pulse splitting, as well as the number of the outgoing femtosecond pulses, are influenced significantly by the polarization of the incident laser pulse. We have observed that the characteristic splitting time in porous quartz PCs for the s-polarized probe pulse is approximately 1.5 times smaller as compared with that measured for the p-polarized radiation. These results are supported by the theoretical description and ensure that the polarization sensitivity of the DIPS effect is due to a large lattice-induced dispersion of the PC. It is also shown that the number of output pulses can be varied from two up to four in both transmission and diffraction directions depending on the polarization of incident femtosecond pulses.

Phase-specific high temperature creep behaviour of a pre-rafted Ni-based superalloy studied by X-ray synchrotron diffraction

The phase-specific high temperature creep behaviours of the γ and γ′ phases of a rafted Ni-based single crystal superalloy were investigated by a combination of in situ creep experiments and diffraction of high-energy X-ray synchrotron radiation. In situ experiments were performed at constant temperatures in a 930–1125 °C temperature range and under variable applied stress in order to study the material’s response (plastic strain, load transfer) to stress jumps. Using three crystal diffractometry in transmission (Laue) geometry, it was possible to measure the average lattice parameters of both the matrix and the rafts in the [1 0 0] direction at intervals shorter than 300 s. The absolute precision on the measurement of the constrained transverse mismatch (in the rafts’ plane) is better than 10−5. Plastic strain occurs within the γ corridors as soon as the Von Mises stress exceeds the Orowan stress. The plasticity of the γ′ rafts apparently depends on the transverse stress (i.e. perpendicular to the tensile axis) exceeding a threshold value of 60 MPa.

Stigmatic X-ray imaging using a single spherical Laue crystal

We propose a crystal configuration using a single Laue spherical crystal for imaging applications. A crystal in Laue geometry set to focus a divergent beam in the meridional (diffraction) plane, but does not focus in the sagittal plane. A transmission object placed in the beam is imaged with different horizontal and vertical aspect ratio, but it is possible to find a configuration with similar aspect ratio. This system is studied using ray tracing, which permit to reproduce preliminary experimental data [1]. The concept of a stigmatic focusing by a single optical element may have applications in imaging, like in transmission microscopy or for hard X-ray backlighting and self-imaging in high energy density plasma experiments. Inertial confinement fusion experiments and large tokamak projects such as ITER may benefit from this optical configuration.

Calculation of diffraction efficiency for curved crystals with arbitrary curvature radius

A model is proposed to calculate the diffraction efficiency of X-rays in Laue geometry for curved crystals with an arbitrary value of the curvature radius. The model generalizes the results based on the dynamical theory of diffraction, which are valid only for crystals with a radius of curvature lower than the critical curvature. The model is proposed for any kind of crystal, and its efficiency tends to one-half in the limit of a thick flat crystal. On the basis of this model, it was possible to reconsider the results of recently observed diffraction efficiency for curved crystals. Finally, the model sets an upper limit for diffraction efficiency of low-curvature curved crystals, this latter being useful in applications such as the construction of a hard X-ray Laue lens.

Bent perfect crystals as X-ray focusing polychromators in symmetric Laue geometry

The focusing properties of cylindrically bent crystals in symmetric Laue geometry are discussed using the formalism of Fresnel diffraction and the analytical solution of the Takagi-Taupin equations for a point source on the entrance surface. The existence of a focal shift in the dynamical focusing effect is pointed out and discussed. The present theoretical framework is applied to experiments performed at the energy-dispersive X-ray absorption spectroscopy beamline of the European Synchrotron Radiation Facility concerning the position and the size of the focal spot obtained from a polychromatic source at a large distance from the bent crystal.

Dynamical Bragg diffraction of optical pulses in photonic crystals in the Laue geometry: Diffraction-induced splitting, selective compression, and focusing of pulses

A theory for the dynamical Bragg diffraction of a spatially confined laser pulse in a linear photonic crystal with a significant modulation of the refractive index in the Laue geometry has been developed. The diffraction-induced splitting of a spatially confined pulse into the Borrmann and anti-Borrmann pulses localized in different regions of the photonic crystal and characterized by different dispersion laws is predicted. The selective compression or focusing of one of these pulses with the simultaneous broadening or defocusing of the other pulse is shown to be possible.

Preparation of bent crystals as high-efficiency optical elements for hard x-ray astronomy

Curved crystals, instead of flat mosaic crystals, can be used as optical elements of a Laue lens for hard x- and gamma-ray astronomy to increase the diffraction efficiency. We propose to achieve the bending of the crystals by a controlled surface damaging, which introduces defects in a layer of few tens nanometers in thickness undergoing a highly compressive strain. Several oriented silicon and gallium arsenide wafer crystals have been treated. The local and mean curvature radii of each sample have been determined by means of high resolution x-ray diffraction measurements in Bragg condition at low energy (8 keV). Silicon samples showed spherical curvatures, whereas GaAs-treated samples evidenced elliptical curvatures with major axes corresponding to the 〈110〉 crystallographic directions. Curvature radii between 3 and 70 m were easily obtained in wafers of different thicknesses. The characterization of GaAs samples performed in Laue geometry at gamma-ray energy of 120 keV confirmed the increase of the diffraction efficiency in the bent crystals.

High-resolution x-ray characterization of mosaic crystals for hard x-ray astronomy

GaAs, Cu, CdTe, and CdZnTe crystals have been studied as optical elements for lenses for hard x-ray astronomy. High-resolution x-ray diffraction at 8 keV in Bragg geometry and at synchrotron at energies up to 500 keV in Laue geometry has been used. A good agreement was found between the mosaicity evaluated in Bragg geometry at 8 keV with x-ray penetration of the order of few tens of micrometers and that derived at synchrotron in transmission Laue geometry at higher x-ray energies. Mosaicity values in a range between a few to 150 arcsec were found in all the samples but, due to the presence of crystal grains in the cm range, CdTe and CdZnTe crystals were found not suitable. Cu crystals exhibit a mosaicity of the order of several arcmin; they indeed were found to be severely affected by cutting damage which could only be removed with a very deep etching. The full width at half maximum of the diffraction peaks decreased at higher x-ray energies showing that the peak broadening is affected by crystallite size. GaAs crystals grown by Czochralski method showed a mosaic spread up to 30 arcsec and good diffraction efficiency up to energies of 500 keV. The use of thermal treatments as a possible method to increase the mosaic spread was also evaluated.

Observation of parametric X-ray radiation by an imaging plate

We have demonstrated experimentally the application of an imaging plate for registering the angular distribution of parametric X-ray radiation. The imaging plate was used as a two-dimensional position-sensitive X-ray detector. High-quality images of the fine structure in the angular distributions of the yield around the reflection of the parametric X-ray radiation produced in a silicon crystal by a 255-MeV electron beam from a linear accelerator have been observed in the Laue geometry. A fairly good agreement between results of measurements and calculations by the kinematic theory of parametric X-ray radiation is shown. Applications of the imaging plates for the observation of the angular distribution of X-rays produced by accelerated particles in a crystal are also discussed.

High-efficiency x-ray gratings with asymmetric-cut multilayers

We present the fabrication and analysis of efficient and highly dispersive gratings for the x-ray and extreme ultraviolet (EUV) regime. We show that an asymmetric-cut multilayer structure can act as a near-perfect blazed grating. The precision and high line density are achieved by layer deposition of materials, which can be controlled to the angstrom level. We demonstrate this in the EUV regime with two structures made by cutting and polishing magnetron-sputtered multilayer mirrors of over 2000 bilayers thick, each with a period of 6.88 nm. These were cut at angles of 2.9° and 7.8° to the surface. Within the 3% bandwidth rocking curve of the multilayer, the angular dispersion of the diffracted wave was in agreement with the grating equation for elements with 7250 and 19,700 line pairs/mm, respectively. The dependence of the measured efficiency was in excellent agreement with a formulation of dynamical diffraction theory for multilayered structures. At a wavelength of 13.2 nm, the efficiency of the first-order diffraction was over 95% of the reflectivity of the uncut multilayer. We predict that such structures should also be effective at shorter x-ray wavelengths. Both the Laue (transmitting) and Bragg (reflecting) geometries are incorporated in our formalism, which is applied to the analysis of multilayer Laue lenses and focusing and dispersing Bragg optics.

Computer simulation on spatial resolution of X-ray bright-field imaging by dynamical diffraction theory for a Laue-case crystal analyzer

Recently, dark-field imaging (DFI) and bright-field imaging (BFI) have been proposed and applied to visualize X-ray refraction effects yielded in biomedical objects. In order to clarify the spatial resolution due to a crystal analyzer in Laue geometry, a program based on the Takagi-Taupin equation was modified to be used for carrying out simulations to evaluate the spatial resolution of images coming into a Laue angular analyzer (LAA). The calculation was done with a perfect plane wave for diffraction wave-fields, which corresponded to BFI, under the conditions of 35 keV and a diffraction index 440 for a 2100 μm thick LAA. As a result, the spatial resolution along the g-vector direction showed approximately 37.5 μm. 126 μm-thick LAA showed a spatial resolution better than 3.1 μm under the conditions of 13.7 keV and a diffraction index 220.

Study of oxide precipitates in silicon using X‐ray diffraction techniques

AbstractThe results of a study of oxide precipitates in Czochralski (CZ) grown silicon using two X‐ray diffraction methods are reported. The diffuse scattering around the Bragg diffraction maxima was measured on a series of samples after various two‐stage annealing treatment. Combining the analysis of diffuse scattering with other experimental techniques we were able to determine mean precipitate size and deformation field around the precipitates. The obtained data show that the deformation field is proportional to the precipitate volume and independent on the annealing temperature or annealing time. The dynamical diffraction in Laue geometry was used to measure precipitate concentration. The results are compared to the selective etching concentration measurement.

Influence of diffraction in crystals on the coherence properties of X-ray free-electron laser pulses

The spatial and temporal evolution of the field of random X-ray femtosecond pulses and their coherent properties upon pulse propagation in free space and under dynamical diffraction in perfect crystals in the Bragg and Laue geometries has been analyzed on the basis of the formalism developed in statistical optics. Particular attention is paid to the influence of large pulse propagation distances, which are characteristic of lengthy channels of X-ray free-electron lasers.

Precipitation in silicon wafers after high temperature preanneal studied by X-ray diffraction methods

We have investigated oxygen precipitation in Czochralski silicon wafers focusing on influence of nucleation temperature and high temperature pre-anneal during common three step treatment. Thick Si wafers were studied mainly by x-ray diffraction in Laue transmission geometry using Mo x-ray tube, but were also compared to reciprocal space maps obtained in Bragg reflection geometry. The analysis of measured diffraction scans in Laue geometry was performed by means of Takagi equations and statistical dynamical theory of diffraction. From the simulated Laue diffraction curves we find the size of the individual defect area and the fraction of strain area volume in the wafer. The results obtained from x-ray diffraction were compared to loss of interstitial oxygen according to infrared absorption spectroscopy and the size of SiO2 precipitate core was estimated. These techniques are in agreement with transmission electron microscopy images.

Oxygen Precipitation Studied by X-Ray Diffraction Techniques

We report on study of oxygen precipitates grown in Czochralski silicon wafers investigated by x-ray diffraction in Bragg reflection geometry and Laue transmission geometry. The analysis of diffraction curves in Laue geometry was done using Takagi equations and statistical dynamical theory of diffraction. These techniques allow us to determine as the radius of defect area as the defect concentrations from measurement in Laue geometry. These results obtained on silicon wafers exposed to two-step and three-step treatments were compared with other experimental techniques including transmission electron microscopy and infrared absorption spectroscopy, while only the largest precipitates are detected by other techniques. The results of all methods are in good agreement.

Effect of epilayer microstructure on shape of X-Ray diffraction peaks

The X-Ray diffraction (XRD) reflections from epitaxial layers with various microstructures have been measured in the Bragg and Laue geometries. The shapes of XRD peaks measured in the θ and θ–2θ scan modes have been analyzed using the approximation by the Voigt function. It is shown that, for the structures with more regular dislocation systems, the Gaussian component predominates, whereas a more chaotic distribution of dislocations leads to an increase in the Lorentzian component. Far on the wings of XRD peaks, the rate of intensity decrease exceeds that predicted by the Voigt function.

Strain evaluation in AlInN/GaN Bragg mirrors by in situ curvature measurements and ex situ x-ray grazing incidence and transmission scattering

Strain in lattice matched and mismatched AlInN/GaN Bragg mirror structures were studied by in situ curvature and various ex situ x-ray measurements. In the case of lattice mismatched structures considerable deviations of the in-plane lattice parameters were evidenced near the surface region as well as in depth using x-ray grazing incidence and x-ray transmission scattering in Laue geometry. The experimental findings are explained in terms of partial stress relaxation of the AlInN/GaN Bragg layer stack with respect to the underlying GaN buffer and a mutual tensioning of the GaN and AlInN layers with respect to each other.