Laue Geometry Research Articles

Focusing characteristics of diamond crystal x-ray monochromators. An experimental and theoretical comparison (abstract)a)

Perfect crystals in transmission (Laue) geometry can effectively be used for x-ray monochromators. Perfect Laue crystals show an interesting focusing effect when the incident beam is white and divergent. This focusing is highly dependent on the beam divergence and on the energy bandpass of the crystal. The aim of this work is to study whether this property can be used for focusing a synchrotron x-ray beam, and to obtain quantitatively the beam dimensions of the resulting monochromatic beam. We have experimentally measured the size of an undulator beam after diffraction from a diamond crystal in Laue geometry, and we analyzed and explained the results by comparison with ray tracing.

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.

X-ray polarimetry with phase plates

Diamond crystals were used as quarter-wave plates (QWPs) and half-wave plates (HWPs) to produce and completely analyze circular and vertical linear polarization of monochromatic x rays from an undulator source (ID10) at the European Synchrotron Radiation Facility. The horizontal linear polarization state of the forward diffracted beam in the Bragg or Laue geometry was efficiently converted into circular (99%) and vertical polarization (97%) by tuning the crystal conveniently away from the Bragg peak. Also, the combination of two QWPs was used to obtain linear polarization at any desired azimuth (e.g., 95% at 45°). Further, the second QWP and the linear polarimeter were used to completely characterize the polarization state of the circularly polarized beam after the first QWP. High efficiencies at QWPs and HWPs and high transmitted intensity (40%) were achieved with the diamond crystal in the asymmetric Laue geometry.

P e p o: A program for the calculation of the reflectivity of cylindrically bent Laue crystal monochromators

p e p o is a program for the calculation of the reflection profiles of cylindrically bent Laue crystal monochromators based on the Penning and Polder theory of x-ray diffraction in slightly strained crystals. The creation of new wave fields as well as an anisotropic model for the crystal deformation has been added to the original theory. The program runs under the computing environment IDL and is widget based. The calculation of perfect flat crystal reflectivity curves in Bragg and Laue geometry using the formulas presented by Zachariasen is included. The agreement with experimental results is excellent, allowing the derivation of focal properties from the calculated reflectivity curves.

The resolution function of a triple-crystal diffractometer for high-energy synchrotron radiation in nondispersive Laue geometry

The k-space resolution function of a triple-crystal diffractometer is calculated for an arrangement of three perfect silicon single crystals Bragg diffracting in nondispersive Laue geometry. A comparison is made with the results of measurements using synchrotron radiation in the energy range from 80 to 150 keV. In this case, absorption is very weak and according to dynamical theory the width of the diffraction pattern of thick perfect single crystals is proportional to the wavelength λ, whereas its Lorentzian tails are proportional to λ2. Together with the fact that the Bragg angles are only of the order of 2°, this leads to a concentration of the starlike k-space resolution function into a narrow band parallel to the reciprocal-lattice vector G. For diffraction of 80 keV synchrotron radiation at the silicon 111 reflection, the full width at half-maximum (FWHM) of the intensity distribution in the scattering plane is 1.1 × 10−5 Å−1 perpendicular to G and 2.2 × 10−4 Å−1 parallel to G. The observed differences in the contributions from monochromator and analyzer crystal to the resolution function are explained by the finite width of the electronic window of the detector counting chain and the non-Bragg scattering contribution from the crystals. If annealed Czochralski-grown silicon single crystals with a mosaicity of ~3′′ are used as monochromator and analyzer, the resolution is reduced by one order of magnitude, but for studies of imperfect samples or of diffuse scattering large gains in intensity can be accomplished this way.

Multiple station beamline at an undulator X-ray source

The undulator X-ray source is an ideal source for many applications: the beam is brilliant, highly collimated in all directions, quasi-monochromatic, pulsed and linearly polarized. Such a precious source can feed several independently operated instruments by utilizing a downstream series of X-ray transparent monochromator crystals. Diamond in particular is an attractive monochromator as it is rather X-ray transparent and can be fabricated to a high degree of crystal perfection. Moreover, it has a very high heat conductivity and a rather small thermal expansion so the beam X-ray heat load problem is easily overcome. We describe a possible setup with three successive monochromators. We present data and analysis for focusing the beam in both the horizontal and vertical direction and discuss the focusing properties in both the Laue and Bragg geometry.

Polarized Neutron Glancing-Angle Diffraction Study of Magnetic Structure at the Y/Gd(0001) Interface

ABSTRACTBy exciting diffraction in the Laue geometry by specularly reflected beams, penetrating x-rays and neutrons can be used to probe surface and interfacial atomic structure. We present the first such measurement utilizing polarized neutrons with both incident- and exit-angle resolution. We have observed the evolution of magnetic order by monitoring the in-plane (1010) Bragg peak from an epitaxially grown film consisting of 150 Å Y(0001) on 5000 Å Gd(0001). Magnetic scattering increases as the film is cooled from the Gd bulk Curie temperature (293K) to 100 K. The gross features of the data can be captured by a simple two-layer model, but sufficient disagreement remains that one may yet hope to extract non-trivial structure at the Y/Gd interface.

Gamma-ray focusing concentrators for astrophysical observations by crystal diffraction in Laue geometry

The diffraction and optical properties of several crystal concentrator configurations which could be adopted in hard x-ray astrophysical observations are discussed and compared in terms of effective area, focusing characteristics, and overall dimensions. First, a concentrator consisting of a set of circular rings covered with asymmetric mosaic crystals in Laue geometry is considered; focusing is achieved by an appropriate choice of the geometrical parameters and the asymmetry angles. Specific effective areas larger than 600 cm2 can be obtained in the energy range 120–190 keV by using Si (422) Bragg reflections in a device with a diameter of 2.5 m and a focal length of 13.5 m. Afterwards, a spherical-zone crystal concentrator is analyzed where asymmetric Laue geometry leads to focusing. With a focal length of 8 m and Si (422) reflections, the mean effective area in the energy range 100–300 keV is about 6500 times larger than the focal spot while the separation power of the concentrator is of the order of 10 arcsec. Finally, a concentrator consisting of several coaxial focusing paraboloids is analyzed. The surfaces of the paraboloids are covered by mosaic crystal tiles with lattice planes disposed normally to the meridians. By using Ge crystals with a mosaic width of ≊350 μrad and assuming a focal length f=10 m with a total aperture 2α=16 deg, it is possible to obtain effective areas larger than 1000 cm2 for energies smaller than about 150 keV. On the other hand, the ratio between the mean effective area and the size of the focal spot in the energy range 200–500 keV is higher than 10 000.

Two-dimensional diffraction focusing of X-Ray radiation in Laue geometry

Two-dimensional diffraction focusing of X-Ray radiation in Laue geometry

Perfect crystal X-ray phase retarders

Abstract X-ray phase retarders of perfect crystals are discussed on the basis of a dynamic theory of diffraction. Although all conceivable diffraction geometries can be used for a phase retarder, transmission of less absorptive crystals in both Laue and Bragg geometries as well as reflection and transmission of absorptive crystals in Laue geometry give practical solutions for circularly polarized X-ray generation with synchrotron radiation (SR). Some SR beamline optics using these phase retarders are presented.

The emergence of the synchrotron Laue method for rapid data collection from protein crystals

Synchrotron X-radiation (SR) is intense, polychromatic and collimated. It is widely exploited, in macromolecular crystallography, particularly using a monochromatized short wavelength beam. The spectral curve of SR, however, ideally lends itself to use of Laue geometry, i. e. the original diffraction experimental arrangement based on a stationary crystal and a polychromatic X-ray beam. Rapid exposure times and time-resolved crystallography studies, e. g. of enzymes, are now possible. Historical objections to the use of Laue diffraction data, particularly the multiplicity distribution, have been found not to be as limiting as once thought. The credentials of the Laue method have been established through a variety of Laue crystal structure analyses, involving photographic film as detector. Recently a three-dimensional arrangement of films, known as a toast-rack, has been used to alleviate problems with spatially overlapping spots. This paper provides a review of these results and then reports several developments. In particular, one of the first Laue analyses using an image plate as detector, namely of a cobalt substituted concanavalin A crystal, is discussed. Recent experimental developments, also at the Daresbury synchrotron, are then described. First, a large toast-rack has been used to record Laue data from a protein crystal. Secondly, a transmission X-ray mirror has been constructed from thin mylar (1.5 μm) and used to provide a λ max filter instead of using aluminium foils. Thirdly, since the Laue method suffers from poor sampling of the low resolution data, a new method (known as LOT) has been introduced.

Nonimaging characterization of imperfect single crystals by means of a three-crystal diffractometer for high energy synchrotron radiation

A three-crystal diffractometer for synchrotron radiation with energies of 100 keV and higher has been used in a nondispersive setting to map out the distribution of Bragg scattered intensity in the scattering plane with very high resolution. By using perfect silicon crystals in symmetrical Laue geometry for reflection 220 as monochromator and analyzer, respectively, a resolution of ≊2×10−4 Å−1 in the direction parallel to the reciprocal lattice vector G220 and of ≊10−5 Å−1 in the direction perpendicular to G220 has been achieved at high counting rates. As a first example the mosaic structure of a plastically deformed silicon crystal has been characterized with respect to its mosaic distribution and lattice parameter fluctuations. Second, the study of a silicon-germanium gradient crystal, produced by means of the chemical vapor deposition technique, demonstrated that the substrate peak could be well separated from the intensity distribution of the gradient crystal. A two-dimensional contour plot of the intensity distribution in the vicinity of 220 shows the variation of the lattice parameters in the gradient crystal, as well as the lattice plane tilts of the substrate and the silicon-germanium layers.

X-ray focusing by the zone plate in Laue geometry

The experimental results on diffraction of a spatially inhomogeneous X-ray beam made in the Laue geometry are presented. A diffraction lens with a full aperture A=168 mu m, number of zones N=23 and gold coating thickness tAu=1.5 mu m was fabricated to modulate the incident beam at the crystal entrance surface and focus the diffracted beam at the exit surface. A numerical simulation of the obtained image was produced and comparative analysis was carried out.

Towards a rigorous treatment of the wave-field propagation according to the statistical theory of dynamical diffraction

Boundary conditions in the statistical theory of dynamical diffraction are shown to be closely related to the interference effect between the incident wave and the forward-diffracted wave. The propagation of the incoherent diffracted beams is analysed in the case of Laue geometry. For E=0 (E, static Debye-Waller factor), the authors then obtain generalized energy transfer equations which are valid in the whole influence region of a point radiation source. The most significant physical difference from previous formulations concerns the behaviour of integrated intensities for crystals such that chi 2 tau T>>1 ( chi is the reciprocal of extinction distance, tau the correlation length of the lattice distortion and T the crystal thickness along the incident direction).

Neutron diffraction in flat and bent mosaic crystals for asymmetric geometry

Exact solutions of power transfer equations for plane-parallel and bent crystals in symmetric and asymmetric Bragg and Laue geometries are obtained, for the nonabsorbing case. Conceptual and mathematical errors in the solutions for asymmetric diffraction given by Bacon & Lowde [Acta Cryst. (1948). 1, 303–314] are pointed out. A very significant result is that, for asymmetrical Laue geometry, over 50% reflecting power is proved to be possible when b < 1. The integrated reflecting power and expressions for the optimum value of the curvature radius for bent mosaic crystals in the Bragg and Laue cases are also given.

Light pulse dispersion under Laue diffraction from a spatial holographic grating

A theory of interaction of phase-modulated light pulse with a plane-parallel spatial grating under two-wave dynamical diffraction in transmission (Laue) geometry is presented. The temporal profiles of the transmitted and diffracted pulses outside the phase diffraction plate have been calculated. It is shown that time dependence of the output pulses exhibits a two-peak structure in the general case. The phenomenon of time compression of the phase-modulated incident pulse under passage through the grating is discussed.

Synchrotron X-ray crystallography techniques: time-resolved aspects of data collection

Synchrotron X-radiation (SR) is intense, polychromatic and collimated. These properties are exploited routinely now to measure data at high resolution from proteins or from large unit cells (e.g. viruses), particularly using a monochromatized short wavelength beam. The time needed to measure protein crystal data-sets (in rotation geometry) can be quick (hours or minutes) compared with laboratory sources (weeks or days). Even so, more rapid data collection is of interest for timeresolved macromolecular crystallography. White beam : stationary crystal (Laue) geometry at SR sources offers shorter exposure times (seconds and less with film). Laue data-sets can be sensitive to subtle structural differences. Technical challenges still presented by SR Laue patterns include the energy overlap of low-resolution data and the spatial overlap of spots, both of which affect the completeness of data-sets. Some energy deconvolution is already possible by the use of multiple films. The spatial overlap problem can be alleviated by the use of three-dimensional detectors, such as a ‘toast-rack’ of plates. Monitoring of a crystalline process, via the Laue pattern, requires time slicing detector systems (e.g. based on CCDS) to be developed.

Bent-crystal monochromator for 150 keV synchrotron radiation

The 220 reflectivity curve of a 4 mm thick Si crystal is measured in asymmetric transmission (Laue) geometry. The crystal is bent to a radius varying from 24 to 138 m, and the analyzer is a nearly perfect Si(220) crystal in non-dispersive (+, −) arrangement. The full width at half-maximum of the reflectivity curves varies from 61 to 10 μrad, while that of a flat perfect crystal is 1.2 μrad, and the observed maximum reflectivity is about 60%. The calculated value is even higher, exceeding 80%. The bent crystal is used as an inverse-Cauchois monochromator, where the source of radiation is on the focusing circle. The effect of the beam divergence on the energy bandpass is eliminated, leaving only the contribution of the reflectivity curve. This can be matched to the desired resolution of the experiment by the choice of the thickness and asymmetry angle of the crystal. Compton-profile measurements from Ag and Au foils are used for a comparison of different monochromators. The bent-crystal monochromator provides 50 times more flux than the perfect Si(220) crystal and a better energy resolution than heat-treated Si(220) or mosaic Ge(220) crystals.

X-ray beam sheet deflection for liquid surface spectrometry

An x-ray beam incident on a liquid surface at glancing angles may be specularly reflected (XR), diffracted (GID), or yield fluorescent radiation (GIF). The three mechanisms, encompassing liquid surface x-ray spectrometry, give complementary information about the liquid surface structure. In common is the incident monochromatic beam: It has the shape of a downward sloping thin sheet with the sheet normal and the propagation direction ideally spanning a vertical plane. Methods to produce such a beam from a horizontal, polychromatic synchrotron radiation beam are discussed. For a high brilliance undulator source the simplest solution turns out to be Bragg reflection from a single tilted monochromator crystal in Laue geometry.

Monochromator on a synchrotron undulator source for liquid surface studies

For liquid surface studies a monochromatic beam of relative bandwidth between 0.1% and 1% at a variable angle in the vertical plane between 0° and 10° is needed. The beam should be like a sheet some tens of μm thick and some mm wide, and as intense as possible. We discuss a monochromator made of a beryllium mosaic crystal using the (002) reflection in Laue geometry placed in undulator beams of DORIS III at the Hamburger Synchrotronstrahlungslabor and of the European Synchrotron Radiation Facility. An analysis of the diffraction properties in terms of mosaic spread, heat load, optical quality of the monochromatic beam, and, in particular, the expected flux, gives an optimum value of about 0.2 mrad for the mosaic spread. We show by experiments that presently available crystals a few hundred μm thick have the required quality and are capable of reflecting up to 45% of the useful x rays, in full agreement with diffraction theory including secondary extinction. Heat load experiments reported elsewhere in this conference prove that Be crystals can withstand high beam power density and that the thermal deformation is small compared to the mosaic spread. The results of the flux calculations are most encouraging: a gain of order 103 to 104 can be expected as compared to existing liquid surface spectrometers.