![]() ![]() To overcome these problems, it has been proposed to use multiple-beam diffraction to calibrate photon energies ( e.g. In other words, Bragg’s law of 2BD is mainly a one-dimensional (1D) equation that is unable to generate two-dimensional (2D) diffraction patterns required for precise determination of the 2D position and direction of an X-ray beam in space. The other disadvantage of Bragg diffraction is that it is only very sensitive to the incidence angle θ but insensitive to the lateral φ angle along the direction perpendicular to the plane of incidence (the plane containing both the principal incident wavevector and the diffraction vector). This may make it difficult to achieve adequate precision so that one has to seek other alternatives. In some unfavarouble cases, however, the absorption edges can be very wide (up to a few tens of electronvolts) (Keski-Rahkonen & Krause, 1974 ▸) without clear sharp features. For elements with sharp absorption features, the absorption method may give high energy precision (at the eV level or better) (Kraft et al., 1996 ▸). As an alternative, X-ray absorption edges of elements have been widely used for wavelength calibration, particularly for double-crystal monochromators (DCMs) of synchrotron beamlines. Without references, therefore, it is challenging to use Bragg diffraction to accurately calibrate/measure X-ray wavelengths (photon energies) or beam directions since it is usually difficult to measure θ with very high precision. If λ can continuously vary, Bragg diffraction may occur at any angle = λ, where θ is the Bragg angle, λ is the X-ray wavelength, and d is the spacing of the diffracting lattice planes. It should be noted that the gradient of the phase-shift against specimen thickness depends on the sign of the excitation error as discussed in the simulation part, i.e., when s 220 0, the gradient of the phase-shift is low.X-ray diffraction from crystals is generally based on Bragg’s law of two-beam diffraction (2BD), 3(h)–(n) s 220 is positive (the reciprocal lattice point of 220 is located inside the Ewald sphere). 3(a)–(g), the excitation error of the 220 reflection s 220 is negative (which means that the reciprocal lattice point of 220 is located outside the Ewald sphere) and for Fig. ![]() The tilting was changed in constant steps (0.39 mrad), such that for the tilting condition of Fig. 3(a) to 3(n) is estimated to be 5.04 mrad. 3(o), (p), the total tilting angle from Fig. ![]() From a movement of a cross point of Kikuchi lines indicated by black arrows in Figs. Figures 3(o) and 3(p) are diffraction patterns obtained in the diffraction condition of Figs. The bright field images, the reconstructed phase images and the phase shift profiles for each tilt conditions are shown in Fig. Next, we systematically tilted the direction of the incident electron beam around the 220 Bragg diffraction condition using a beam tilt function of the illumination system. The thicknesses of the positions are about 50 nm and 150 nm which were estimated by our calculations. The red arrows indicate the positions of dark thickness fringes. (b) Bragg diffraction condition with 220 reflection is exactly satisfied. (a) Strong Bragg reflections are not excited. 1(c).īright-field images, reconstructed phase images, and phase shift profiles of a wedge shaped Si crystal. But they can be explained by dynamical theory of electron diffraction as shown in Fig. Under the exact 220 Bragg diffraction condition, thickness fringes appear in the bright-field image, and the phase shift jumps by π (indicated by red arrows in the reconstructed phase image and the phase-shift profile) where the dark thickness fringes appear in the corresponding bright-field image. Under the non-Bragg condition, there are no thickness fringes in the bright field image, and the phase shift increases in proportion to the specimen thickness. 2(b) is under an exact 220 Bragg diffraction condition. 2(a) is under a non-Bragg condition and that shown in Fig. (1).įigure 2 shows bright-field images and reconstructed phase images of a wedge shape Si specimen, and the phase shift profiles in the region of the specimen indicated by the red rectangles in the bright-field and reconstructed phase images. (4) explains the characteristic phase shift around the Bragg condition that is not explained by eq. 1(d-i)), the small loop does not include the origin of the complex plane, which means the increase of the phase shift is less than 2π. This means the increase of the phase shift is larger than 2π. 1(b-i)), the small loop includes the origin of the complex plane. When the sign of s 220 is negative ( Fig. \sigma = \frac)$, shown as red bold lines for thicknesses from 20 nm to 80 nm, forms a shape of a small loop. ![]()
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