On the X-ray reflectivity of elastically bent perfect crystals

Kálmán, Z. H.; Weissmann, S.

doi:10.1107/s002188988301047x

Cited by 23 publications

(22 citation statements)

References 8 publications

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“…Diffracting planes may be expected to have a mean angle 0~plan e with respect to the surface for flat or curved crystals. Use of a constant %ta,e assumes either that materials are isotropic or that %~ane is small, but is generally the first-order contribution (el Kalman & Weissman, 1983). Then,…”

Section: Allowance For Depth Penetration and Johann Geometrymentioning

confidence: 99%

X-ray diffraction of bent crystals in Bragg geometry. I. Perfect-crystal modelling

Chantler¹

1992

J Appl Cryst

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X-ray reflectivity, widths, centroid shifts and profiles for curved perfect crystals are calculated from a model. The crystal is approximated by a stack of perfectcrystal lamellae or blocks with a gradually changing (mean) orientation. A computer program has been developed to calculate the above quantities in the Johann geometry for the composite crystal from the dynamic theory of diffraction. Focusing and defocusing aberrations and the use of photographic detection methods are included. Correction of omissions from earlier theory and modelling is noted, together with observed effects. Incoherent scattering can give dramatic changes in diffracted intensities and significant shifts of final parameters. Effects of depth penetration on shifts, cosine ratios and other parameters are included. Assumptions of the model and implementation are detailed. It is shown that interference effects between waves of roughly equal amplitudes require use of lamellar thicknesses greater than those corresponding to the Darwin range. Internal tests demonstrate agreement with the literature at extremes. The theory is applied to first-and fourth-order spectra in differential Lyman ~ wavelength measurements. Resuits for pentaerythritol 002 crystals are presented. Paper II of this series extends this model to nonideally imperfect crystals and other crystals of interest and discusses experimental agreement.

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Section: Allowance For Depth Penetration and Johann Geometrymentioning

confidence: 99%

X-ray diffraction of bent crystals in Bragg geometry. I. Perfect-crystal modelling

Chantler¹

1992

J Appl Cryst

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“…5 Many methods to model the diffraction profiles of bent crystals have been developed and implemented in the XCRYSTAL BENT module 6 of the XOP package. 7 The multi-lamellar (ML) approximation [8][9][10][11] and the Penning-Polder (PP) method [12][13][14][15] both provide reasonable results in the case of crystals bent cylindrically in the diffraction plane. However, the applications of these theories to simulate the rocking curves of sagittally bent crystals are not yet studied in detail.…”

Section: Introductionmentioning

confidence: 98%

Simulation of diffraction profiles for sagittally bent Laue crystals

Shi

2011

SPIE Proceedings

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The diffraction profiles (or rocking curves) of sagittally bent Laue crystals are known to be significantly wider than those of perfect crystals as a result of the lattice distortion introduced by the sagittal bending. The existing analytical model explains the rocking curve broadening as well as the reflectivity observed. Many theoretical methods were developed for calculating diffraction profiles of meridionally (in the diffraction plane) bent crystals. In this work, we extend these methods to accommodate sagittally bent crystals. The total lattice distortion angle for anisotropic crystals under sagittal bending is adapted into the multi-lamellar approximation using the rotating crystal method, in which the incident angle changes through each lamella. The Penning-Polder theory is examined for bent crystals with the uniform strain gradient. In addition, the Takagi-Taupin equations are solved numerically for sagittally bent Laue crystals. Finally, examples of these simulation results are presented and the merit of each method is discussed.

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“…A lamellar model has been used successfully in both the Bragg and Laue geometries to explain X-ray and neutron diffraction by such crystals (Egert & Dachs, 1970;Albertini et al, 1976;Boeuf et al, 1978;Mikula et al, 1984). Similar treatments were worked out for meridionally bent anisotropic crystals (Kalman & Weissmann, 1983;. Theories have been derived for the angular acceptance of meridionally bent isotropic crystals (Caciuffo et al, 1987;Popovici et al, 1988;Erola et al, 1990;Hiraoka et al, 2001).…”

Section: Introductionmentioning

confidence: 99%

“…From results obtained for plates of constant cross section bent by equal but opposite moments, the displacement ®eld is (Voigt, 1928;Kalman & Weissmann, 1983;Chukhovskii et al, 1994) u x Ex 2 S 13 À y 2 S 23 À zyS 43 À z 2 S 33 u y Ex 2 S 63 2xyS 23 xzS 43 1 u z Ex 2 S 53 xyS 43 2xzS 33 Y where S ij are the elastic compliances of the crystal for a speci®c orientation, u xYyYz are displacements in the x, y and z directions, respectively, and E is a constant related to the bending moment and moment of inertia of the crystal. Thus, it is necessary to take into consideration the anisotropy of the crystal.…”

Section: Introductionmentioning

confidence: 99%

Rocking-curve width of sagittally bent Laue crystals

et al. 2002

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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.

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On the X-ray reflectivity of elastically bent perfect crystals

Cited by 23 publications

References 8 publications

X-ray diffraction of bent crystals in Bragg geometry. I. Perfect-crystal modelling

X-ray diffraction of bent crystals in Bragg geometry. I. Perfect-crystal modelling

Simulation of diffraction profiles for sagittally bent Laue crystals

Rocking-curve width of sagittally bent Laue crystals

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