Peculiarities of the diffraction contrast in plane-wave X-ray topographs of weakly deformed crystals in the bragg geomet
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RACTION AND SCATTERING OF IONIZING RADIATIONS
Peculiarities of the Diffraction Contrast in PlaneWave XRay Topographs of Weakly Deformed Crystals in the Bragg Geometry A. E. Voloshin Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskii pr. 59, Moscow, 119333 Russia email: [email protected] Received March 28, 2011
Abstract—The regularities of the topographic contrast formation in weakly deformed nonabsorbing crystals in the Bragg geometry for the diffraction of a plane Xray wave have been investigated by the method of Rie mann functions. It is shown that in this case the extinction contrast has an interference character, alternates, and is proportional to the strain gradient rather than to the squared strain (like in the case of strong lattice distortions). The data of the analysis are compared with the results of a model numerical experiment. The possibility of implementing an Xray–acoustic resonance in the Bragg geometry is shown. DOI: 10.1134/S1063774511050233
INTRODUCTION As was shown in [1–12], planewave Xray topo graphs can be used for a quantitative analysis of weak strains in crystals. In particular, based on the solution of the inverse 1D problem of the elasticity theory [13– 21], one can analyze the variations in the lattice parameter that are caused by nonuniform impurity distribution in the crystal. In this case, to perform a diffraction experiment, it is essential to correctly choose the conditions under which the orientational contrast dominates significantly over the extinction contrast, because in all aforementioned studies the analysis of the dependence that the Xray topographic contrast has on the effective lattice misorientation αg was based on the Bonse approximation [22], which is valid specifically for the orientational contrast: ΔI I = k α g .
(1)
Here, k = (Δ I I ) Δθ is the proportionality factor, which determines the rocking curve slope, and Δθ is the angular displacement of the working point along the rocking curve. The effective lattice misorientation α g can be written as a convolution of the components ∂u wij = i of the distortion tensor W: ∂x j αg =
λ ∂(g ⋅ u) = λ s w g , i ij j sin 2θ ∂s1 sin 2θ
(i, j = 1, 2, 3).(2)
Here, g is the diffraction vector, s1 is the unit vector in the incidentwave direction, and si and gj are the com ponents of the vectors s1 and g. If (1) is valid, we have the relation Δθ = α g .
(3)
With allowance for the fact that the inverse diffraction problem has not yet been solved, this approach allows one to quantitatively analyze (relatively easily and with fairly high precision) weak strains in crystals. Thus, to make the subsequent calculations correct, the measurement conditions should provide the dom inance of the orientational contrast above the extinc tion contrast. A numerical model experiment on the inverse reconstruction of the impurity striation profile in a Si crystal was performed in [20]. This experiment was based on processing three series of topographs at different dynamic deviations of the sample
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