Darwin Width Research Articles

The promise of GaAs 200 in small-angle neutron scattering for higher resolution.

The Q resolution in Bonse-Hart double-crystal diffractometers is determined for a given Bragg angle by the value of the crystallographic structure factor. To date, the reflections Si 220 or Si 111 have been used exclusively in neutron scattering, which provide resolutions for triple-bounce crystals of about 2 × 10-5 Å-1 (FWHM). The Darwin width of the GaAs 200 reflection is about a factor of 10 smaller, offering the possibility of a Q resolution of 2 × 10-6 Å-1 provided crystals of sufficient quality are available. This article reports a feasibility study with single-bounce GaAs 200, yielding a Q resolution of 4.6 × 10-6 Å-1, six times superior in comparison with a Si 220 setup.

Preparation and measurement of an x-ray Laue-type monochromator based on a WSi2/Si multilayer

The Laue-type multilayer monochromator (LMM) is a promising optical element with a small size and high efficiency in a synchrotron radiation facility. By the dynamical diffraction theory, using DC magnetron sputtering technology, an LMM with a total thickness of 47 µm and a periodic thickness of 4.7 nm WSi2/Si multilayer at 26 keV is designed and fabricated. During the preparation, the total number of layers is up to 20000, and every 300th layer of Si is replaced by WSi2 as the marker, so the multilayer is divided into 67 areas. The cross section of the multilayer is measured by a scanning electron microscope (SEM), and the marker region thickness error is 0.28% (RMS). The diffraction test experiment of the LMM is carried out at the Shanghai synchrotron radiation facility (SSRF). The 1st-order peak angle is 5.05 mrad, and the efficiency is 75.0%, which is close to the theoretical calculation result of 5.1 mrad and 79.1%. The Darwin width of the LMM is 0.17 mrad which is equal to the theoretical calculation. Based on the Bragg’s diffraction equation, the energy resolution (ΔE/E) is 3.3%.

X-ray diffraction from strongly bent crystals and spectroscopy of X-ray free-electron laser pulses.

The use of strongly bent crystals in spectrometers for pulses of a hard X-ray free-electron laser is explored theoretically. Diffraction is calculated in both dynamical and kinematical theories. It is shown that diffraction can be treated kinematically when the bending radius is small compared with the critical radius given by the ratio of the Bragg-case extinction length for the actual reflection to the Darwin width of this reflection. As a result, the spectral resolution is limited by the crystal thickness, rather than the extinction length, and can become better than the resolution of a planar dynamically diffracting crystal. As an example, it is demonstrated that spectra of the 12 keV pulses can be resolved in the 440 reflection from a 20 µm-thick diamond crystal bent to a radius of 10 cm.

X-ray diffraction in superabsorbing crystals: absorption intrinsic width.

The several mathematical formulations of X-ray diffraction theory facilitate its understanding and use as a materials characterization technique, since one can opt for the simplest formulation that adequately describes the case being studied. As synchrotrons advance, new techniques are developed and there is a need for simple formulations to describe them. One of these techniques is soft resonant X-ray diffraction, in which the X-rays suffer large attenuation due to absorption. In this work, an expression is derived for the X-ray diffraction profiles of reflections where the linear absorption is far greater than primary extinction; in other words, the crystal is superabsorbing. The case is considered of a parallel plate crystal, for which the diffraction profile of the superabsorbing crystal is computed as a function of crystal size normal to the diffraction planes. For thin crystals or those with negligible absorption, the diffraction profile of a superabsorbing crystal coincides with the result of the kinematical theory. For thick crystals, the absorption intrinsic profile is obtained, described by a Lorentzian function and characterized by the absorption intrinsic width. This absorption intrinsic width is proportional to the linear absorption coefficient and its expression is similar to that for the Darwin width, while the absorption intrinsic profile is a special case of the Laue dynamical theory, and it is similar to the Ornstein-Zernike Lorentzian. The formulation of X-ray diffraction of superabsorbing crystals is simple and provides new perspectives for the soft resonant X-ray diffraction technique.

Quartz conditioning crystal for X-ray rocking curve topography

A large α-quartz crystal designed to condition the monochromatic beam at beamline 1-BM of the Advanced Photon Source is presented. The purpose of this crystal was to provide a precise match to the Bragg angle of quartz crystals that are commonly used to make analyzers for plasma diagnostics. In order to characterize these analyzers, area detectors need to be positioned at distances from the analyzer at upwards of 1000 mm. As a result of this precise matching, a Bragg-reflected beam from such an analyzer is precisely parallel to the beam incident on the conditioning crystal. This situation alleviates the need to adjust the position of the area detector as the distance between the analyzer and the area detector is varied. That is, there is no need to account for vertical displacement of the diffracted-beam image as a function of this distance. Additionally, verification that the analyzer is set to the correct Bragg reflection is obtained by scanning this distance, because only for a spurious reflection will there be a vertical displacement. This is a very useful check. To commission the conditioning crystal, diffraction from a high-quality flat quartz crystal was mapped using a CCD. Bragg diffraction from the 40\overline 40 reflection at 8.5 keV was studied over an area of 23 × 31 mm. The theoretical Darwin width of the flat sample in this case was 4.7 µrad. An FWHM value near 6 µrad was measured over almost the entire mapped area. These data demonstrate that the resolution function for this four-crystal arrangement is ∼4 µrad. Data are also presented for a 0.1 mm-thick α-quartz wafer pressed into a concave form, having a nominal radius of 500 mm and intended for use as an analyzer. Because analyzers are bent crystals, diffraction occurs in narrow bands. When a multiple exposure is made as a function of rocking angle a striped pattern is obtained, which is commonly referred to as a zebra-stripe pattern. A series of zebra stripes from the 30\overline 3\overline 3 Bragg reflection of the bent wafer over its 25 mm-diameter area were recorded on a CCD. The zebra-stripe pattern was analyzed to show a uniform bending to a radius of 497.0 ± 0.3 mm, in support of the nominal value. An r.m.s. slope error of 7 µrad was also obtained from this analysis.

Development of a crystal collimation system for high-resolution ultra-small-angle X-ray scattering applications.

Crystal collimation offers a viable alternative to the commonly used pinhole collimation in small-angle X-ray scattering (SAXS) for specific applications requiring highest angular resolution. This scheme is not affected by the parasitic scattering and diffraction-limited beam broadening. The Darwin width of the rocking curve of the crystals mainly defines the ultimate beam divergence. For this purpose, a dispersive Si-111 crystal collimation set-up based on two well conditioned pseudo channel-cut crystals (pairs of well polished, independent parallel crystals) using a higher-order reflection (Si-333) has been developed. The gain in resolution is obtained at the expense of flux. The system has been installed at the TRUSAXS beamline ID02 (ESRF) for reducing the horizontal beam divergence in high-resolution mesurements. The precise mechanics of the system allows reproducible alignment of the Bragg condition. The high resolution achieved at a sample-detector distance of 31 m is demonstrated by ultra-small-angle X-ray scattering measurements on a model system consisting of micrometre-sized polystyrene latex particles with low polydispersity.

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.

High diffraction efficiency in crystals curved by surface damage

Diffraction profiles of curved Si and GaAs crystals obtained by a controlled damage process on one side of planar crystals have been investigated at X-ray energiesE= 17, 59 and 120 keV. AtE= 17 and 59 keV in the condition of slight curvature, that is when the diffracting plane bending over the extinction length is lower than the Darwin width, the Laue diffraction profiles with lattice planes parallel or inclined with respect to the curvature radiusRshow an enhancement of integrated intensity proportional to 1/R, much larger than in the corresponding perfect bent crystals. AtE= 120 keV, in the condition of strong curvature, the crystals behave as bent perfect crystals with integrated intensity corresponding to that of a mosaic crystal. These crystals are proposed as optical elements for focusing hard X-ray beams.

Sagittal focusing of synchrotron radiation X-rays using a winged crystal.

A Si(111) winged crystal has been designed to minimize anticlastic bending and improve sagittal focusing efficiency. The crystal was thin with wide stiffening wings. The length-to-width ratio of the crystal was optimized by finite element analysis, and the optimal value was larger than the `golden value'. The analysis showed that the slope error owing to anticlastic bending is less than the Darwin width. The X-rays were focused two-dimensionally using the crystal and a tangentially bent mirror. The observed profiles of the focal spot agreed well with the results of a ray-tracing calculation in the energy range from 8 to 17.5 keV. X-ray diffraction measurements with a high signal-to-noise ratio using this focusing system were demonstrated for a small protein crystal.

Evaluation of two methods to minimize the anticlastic curvature of cylindrical bent crystals

A scanning dynamically focusing sagittal X-ray monochromator accepting 3 mrad x 0.1 mrad from a 6 T wiggler has been designed for XAFS measurements. In the energy range 4.1-12.4 keV, the slope error of the second cylindrical crystal caused by anticlastic bending must be less than 1/5 of the Darwin width of the crystal or the photon flux will decrease drastically. Two methods to minimize the anticlastic bending are proposed. Thin crystals with stiffening ribs and thin crystals with an aspect ratio equal to the ;golden value' are evaluated by finite-element analysis and by long-trace-profiler characterization. Both approaches are satisfactory, but the ;golden value' approach is preferred in this case for the second crystal of the new monochromator not only because it is easy to manufacture but also because the surface is smoother than the ribbed crystal.

Bent crystals in Laue geometry: dynamical focusing of a polychromatic incident beam

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.

Thermal neutron optical experiments with tuneable many-bounce channel-cut crystals

We realized the first tuneable channel-cut crystals to be used as monochromator and analyzer in a double-crystal diffractometer and demonstrated the superior operation for ultra small angle scattering. With this new system it was possible to measure the ultra small angle scattering from a single 14-μm glass fibre, the small angle scattering from an Al wedge and the diffraction pattern of a Si phase grating. The last measurement was only possible due to the reduction of the Darwin width of both crystals from 5.7 sec of arc (natural Darwin width) to 1.4 sec of arc.

X-ray diffraction in perfect t x l crystals. Rocking curves.

A general formalism, based on the Takagi-Taupin equations, for calculating rocking curves in perfect t x l crystals is presented. It includes nonsymmetrical scattering, refraction, and ordinary and anomalous absorption. t and l may be varied independently. In the limit of a semi-infinite crystal, the standard results from the fundamental theory are retrieved. For crystal dimensions less than the extinction length, the theory converges to the kinematical limit. Simulations for germanium and silicon show significant influence of crystal finiteness. When dynamical effects are prominent, the curves exhibit various degrees of asymmetry and the full width at half-maximum is generally larger than the corresponding Darwin width. This is attributed to combined Laue and Bragg contributions which are shifted with respect to each other owing to refraction.

Correction for the twist and the conical defects of a sagittaly bent crystal

Abstract The symmetrical bending of the focusing crystal of a double X-ray monochromator is a difficult problem. Indeed, the slope due to the curvature is usually three orders of magnitude higher than the accepted slope error (typically, the Darwin width of the crystal). In these conditions, even a low parasitic slope error induced by the bending process may lead to quite a strong intensity decrease. When the bending moment is applied, the main parasitic distortions which may appear are typically the anticlastic curvature, the inhomogeneous sagittal curvature, the conical shape and the twist. On the D2AM beamline at the ESRF, a program called CHKC2 has been developed to correct on-line the latter two distortions: the conical shape and the twist. On this beamline the X-ray beam, which has been collimated by a grazing angle mirror, is monochromatized first by a flat silicon crystal, and then diffracted by the sagittaly curved crystal. A fluorescent screen gives an image of this diffracted beam. The CHKC2 program records a series of images of the beam diffracted by the sagittaly curved crystal while rotating this crystal with respect to the flat one. An automatic analysis of this topography (plane-wave topography) is then performed in order to evaluate the behavior of the tuned areas and compare it to a model. An accurate correction is proposed to the beamline operator so that optimisation using long multidimensional scanning is no more necessary.

A thermal neutron interferometer of the Fabry-Perot type

A new thermal neutron Fabry-Perot interferometer (FPI) similar to that proposed for hard X-rays by the same authors is proposed. Both devices consist of two thin parallel crystals separated by a nondiffracting gap. Bragg angles in both cases are very close to π/2. While for X-rays the thickness of the plates is of the order of a few μm, for neutrons it is one to two orders of magnitude larger. Then, the actual construction of the new interferometer should not impose the difficulties found for X-rays. The transmission profile shows the usual FPI transmission resonances. By changing the thickness of the mirrors and the dimensions of the gap, the energy width of these resonances can be controlled from being rather broad, of the order of the Darwin width, ΔE/E ≈ 10−6, to the extremely narrow theoretical limit of ΔE/E ≈ 10−9. The values may be changed within these limits, either by the choice of the crystal or of the reflection involved and by changing the dimensions of the gap and plate thicknesses. The examples calculated here concentrate on silicon crystal slabs which are cheap and relatively easy to cut down to the required thickness. While the numerical results given are valid only for this material, the analysis made points out to various features of more general validity. In order to design an experiment the diffraction profiles are studied (a) for constant plate thickness and varying gap and (b) for constant gap and varying plate thickness. The results are presented in the text. It is shown that within limits, the effect of roughness of the crystal surfaces does not significantly alter the interferometer performance.

Perfect Crystals in the Asymmetric Bragg Geometry as Optical Elements for Coherent X-ray Beams

Perfect crystals in the asymmetric Bragg geometry are evaluated as optical elements for manipulating coherent X-ray beams. Such optics can be used to modify the transverse coherence length of a synchrotron X-ray beam, with the intention of increasing the usable coherent flux. The wavelength range, angular divergence and flux of X-rays passing through a pinhole aperture are examined in detail, as functions of source and pinhole size, crystal-to-pinhole separation and the asymmetry factor. In developing this analysis, the behavior of asymmetrically cut crystals is explained in reciprocal space, with reference to the crystal truncation rod associated with the reflection. The results show that, for synchrotron beams that are collimated to a small fraction of the incident Darwin width, the wavelength range accepted by the crystal is typically dispersed into an angular spread in the exit beam. This chromatic aberration greatly reduces the transverse coherence length in a manner that does not conserve the coherent flux. The calculations are in agreement with measurements of the divergence and flux through a micrometer-sized pinhole using a synchrotron wiggler X-ray source.

Asymmetrically cut crystals as optical elements for highly collimated x-ray beams

Asymmetrically cut perfect crystals, in both the Laue and Bragg geometries, are examined as single crystal monochromators for x-ray beams that are collimated to a small fraction of the Darwin width, as is typical in experiments with coherent x rays. Both the Laue and asymmetric Bragg geometries are plagued by an inherent chromatic aberration that increases the beam divergence much beyond that of the symmetric Bragg geometry. Measurements from a recent experiment at the ESRF are presented to compare Si(220) (symmetric Bragg), diamond(111) (asymmetric Laue), and diamond(111) (symmetric Bragg inclined) geometries.

Focusing monochromators for high energy synchrotron radiation

Bent crystals are introduced as monochromators for high energy synchrotron radiation. The reflectivity of the crystal can be calculated reliably from a model where the bent crystal is approximated by a stack of lamellas, which have a gradually changing angle of reflection. The reflectivity curves of a 4 mm thick, asymmetrically cut (χ=9.5°) Si(220) crystal are measured using 150 keV radiation and varying the bending radius from 25 to 140 m. The width of the reflectivity curve is up to 50 times the Darwin width of the reflection, and the maximum reflectivity exceeds 80%. The crystal is used as a monochromator in Compton scattering measurements. The source is on the focusing circle, so that the resolution is limited essentially by the detector/analyzer. A wide bandpass, sharply focused beam is attained when the source is outside the focusing circle in the transmission geometry. In a test experiment. 1012 photons on an area of 2 mm2 was observed. The energy band was about 4 keV centered at 40 keV. A powder diffraction pattern of a few reflections of interest was recorded by an intrinsic Ge detector, and this demonstrated that a structural transition can be followed at intervals of a few milliseconds.

Adaptive crystal optics for high power synchrotron sources

Abstract We report successful operation of the first X-ray monochromator with built-in correction for the thermal strain field created by a high power X-ray beam. The 27 pole wiggler magnet at the National Synchrotron Light Source provides a total power load of 330 W in a beam 60 mm wide. A (220) silicon monochromator system has been tested; the compensated deformation field contributes only 2 arc sec full width at half maximum intensity to the monochromator Darwin width of high order Bragg reflections.

Bragg reflection transmission filters for variable resolution monochromators

There are various methods for improving the angular and spectral resolution of monochromator and analyzer systems. The novel system described here, though limited to higher X-ray energies (> 20 keV), is based on a dynamic effect on the transmitted beam with a thin perfect crystal plate set in the Bragg reflection case. In the case of Bragg reflection from a perfect crystal, the incident beam intensity is reduced as it penetrates the crystal in the range of reflection. This extinction length is of the order of microns. The attenuation length, which determines the amount of normal transmission through the plate is generally much longer. Thus, in the case of the Bragg reflection, the attenuation of the transmitted beam can change by several orders of magnitude with a small change in energy or angle. A thin crystal plate cuts a notch in the transmitted beam with a width equal to its Darwin width, thus acting as a transmission filter. When used in a nondispersive mode with other monochromator crystals, the filter set at the Bragg angle will reflect the entire Darwin width of the incident beam and transmit the wings of the incident beam distribution. When the element is offset in angle by some fraction of the Darwin width, the filter becomes useful in adjusting the angular width of the transmitted beam and removing a wing. Used in pairs with symmetrical offset, the filters can be used to continuously adjust the intrinsic angular divergence of the beam with good wing reduction. Such filters may be useful in improving the angular resolution of a small-angle-scattering camera, for instance. These filters may be added to a Bonse-Hart camera with one pair on the incident beam to reduce the intrinsic beam divergence and a second pair on the analyzer arm to improve the analyzer resolution. Also, these filters could be used in dispersive monochromator systems to improve the energy resolution.