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Theoretical study of the properties of X-ray diffraction moiré fringes. I.

Yoshimura J - Acta Crystallogr A Found Adv (2015)

Bottom Line: Acta Cryst.Then, the properties of moiré fringes derived from the above theory are explained for the case of a plane-wave diffraction image, where the significant effect of Pendellösung intensity oscillation on the moiré pattern when the crystal is strained is described in detail with theoretically simulated moiré images.Although such plane-wave moiré images are not widely observed in a nearly pure form, knowledge of their properties is essential for the understanding of diffraction moiré fringes in general.

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Affiliation: Photon Factory, Institute of Materials Structure Science, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan.

ABSTRACT
A detailed and comprehensive theoretical description of X-ray diffraction moiré fringes for a bicrystal specimen is given on the basis of a calculation by plane-wave dynamical diffraction theory. Firstly, prior to discussing the main subject of the paper, a previous article [Yoshimura (1997). Acta Cryst. A53, 810-812] on the two-dimensionality of diffraction moiré patterns is restated on a thorough calculation of the moiré interference phase. Then, the properties of moiré fringes derived from the above theory are explained for the case of a plane-wave diffraction image, where the significant effect of Pendellösung intensity oscillation on the moiré pattern when the crystal is strained is described in detail with theoretically simulated moiré images. Although such plane-wave moiré images are not widely observed in a nearly pure form, knowledge of their properties is essential for the understanding of diffraction moiré fringes in general.

No MeSH data available.


Calculated curves of ,  and , and of the as-calculated and the corrected PL phases, analogous to Fig. 7 ▸. The graphs in (a) are for the moiré image in Fig. 8 ▸(b) (inset), and those in (b) are for the moiré image in Fig. 8 ▸(a) (main figure).
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fig9: Calculated curves of , and , and of the as-calculated and the corrected PL phases, analogous to Fig. 7 ▸. The graphs in (a) are for the moiré image in Fig. 8 ▸(b) (inset), and those in (b) are for the moiré image in Fig. 8 ▸(a) (main figure).

Mentions: First, Fig. 6 ▸ shows a computed moiré image when crystal absorption was imaginarily assumed zero (, ). The crystal thicknesses and the deviation angle were assumed to be  mm and . Although opposite to the main convention, the images in this paper are presented so that white contrast indicates higher intensity. The aforementioned abrupt fringe jumps can be observed at 0.6, 2.7, 5.1 and 8.1 mm. The fringe jumps in this case are exactly the half-a-period positional change, and fringe lines are discontinuous between facing image regions. The magnified image in the inset shows details of the fringe jumps and discontinuity. Fig. 7 ▸ shows the curves of (fringe contrast), and calculated by equations (20), (23a), (23b) and (41), for the moiré image in Fig. 6 ▸. The curve of in the middle figure by equation (41) is corrected to the curve as in the bottom figure, in the way described earlier. Fringes in Fig. 6 ▸ change their position in accordance with this corrected phase curve, on the basis of the fringe-line equation (42). Discontinuous phase jumps are clearly recognized in this phase curve. Fig. 8 ▸ shows a moiré image computed with the real value of absorption, with other conditions being the same as those for Fig. 6 ▸. However, since the image is much changed from that in Fig. 6 ▸, an image imaginarily computed with half the real value of absorption is appended in the inset in Fig. 8 ▸(b). (The change in the value for absorption by changing the wavelength also causes an unwanted change in , making an easy comparison difficult.) Fig. 9 ▸(a) shows the calculated curves of , , and for the inset image, Fig. 8 ▸(b). Although the condition = 0 no longer holds, abrupt changes of the phase approximately occur where and comes close to zero. Fig. 9 ▸(b) shows calculated curves of , , and for the image in the main figure, Fig. 8 ▸(a). For this image, the condition nowhere holds, and phase jumps do not occur. Nevertheless, oscillations of , and occur though not an abrupt change, and the fringes undulate correspondingly. Figs. 10 ▸(a), 10 ▸(b) show moiré images at and for comparison with the image in Fig. 8 ▸(a) at ; they were computed with all conditions other than taken to be the same as for Fig. 8 ▸(a). As can be seen in the three images, when the angular position (i.e., deviation angle) varies from the positive to negative side, the fringes become nearly flat in the vicinity of ; as further goes on in the negative region, the fringes begin to undulate again with a short interval.


Theoretical study of the properties of X-ray diffraction moiré fringes. I.

Yoshimura J - Acta Crystallogr A Found Adv (2015)

Calculated curves of ,  and , and of the as-calculated and the corrected PL phases, analogous to Fig. 7 ▸. The graphs in (a) are for the moiré image in Fig. 8 ▸(b) (inset), and those in (b) are for the moiré image in Fig. 8 ▸(a) (main figure).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4487425&req=5

fig9: Calculated curves of , and , and of the as-calculated and the corrected PL phases, analogous to Fig. 7 ▸. The graphs in (a) are for the moiré image in Fig. 8 ▸(b) (inset), and those in (b) are for the moiré image in Fig. 8 ▸(a) (main figure).
Mentions: First, Fig. 6 ▸ shows a computed moiré image when crystal absorption was imaginarily assumed zero (, ). The crystal thicknesses and the deviation angle were assumed to be  mm and . Although opposite to the main convention, the images in this paper are presented so that white contrast indicates higher intensity. The aforementioned abrupt fringe jumps can be observed at 0.6, 2.7, 5.1 and 8.1 mm. The fringe jumps in this case are exactly the half-a-period positional change, and fringe lines are discontinuous between facing image regions. The magnified image in the inset shows details of the fringe jumps and discontinuity. Fig. 7 ▸ shows the curves of (fringe contrast), and calculated by equations (20), (23a), (23b) and (41), for the moiré image in Fig. 6 ▸. The curve of in the middle figure by equation (41) is corrected to the curve as in the bottom figure, in the way described earlier. Fringes in Fig. 6 ▸ change their position in accordance with this corrected phase curve, on the basis of the fringe-line equation (42). Discontinuous phase jumps are clearly recognized in this phase curve. Fig. 8 ▸ shows a moiré image computed with the real value of absorption, with other conditions being the same as those for Fig. 6 ▸. However, since the image is much changed from that in Fig. 6 ▸, an image imaginarily computed with half the real value of absorption is appended in the inset in Fig. 8 ▸(b). (The change in the value for absorption by changing the wavelength also causes an unwanted change in , making an easy comparison difficult.) Fig. 9 ▸(a) shows the calculated curves of , , and for the inset image, Fig. 8 ▸(b). Although the condition = 0 no longer holds, abrupt changes of the phase approximately occur where and comes close to zero. Fig. 9 ▸(b) shows calculated curves of , , and for the image in the main figure, Fig. 8 ▸(a). For this image, the condition nowhere holds, and phase jumps do not occur. Nevertheless, oscillations of , and occur though not an abrupt change, and the fringes undulate correspondingly. Figs. 10 ▸(a), 10 ▸(b) show moiré images at and for comparison with the image in Fig. 8 ▸(a) at ; they were computed with all conditions other than taken to be the same as for Fig. 8 ▸(a). As can be seen in the three images, when the angular position (i.e., deviation angle) varies from the positive to negative side, the fringes become nearly flat in the vicinity of ; as further goes on in the negative region, the fringes begin to undulate again with a short interval.

Bottom Line: Acta Cryst.Then, the properties of moiré fringes derived from the above theory are explained for the case of a plane-wave diffraction image, where the significant effect of Pendellösung intensity oscillation on the moiré pattern when the crystal is strained is described in detail with theoretically simulated moiré images.Although such plane-wave moiré images are not widely observed in a nearly pure form, knowledge of their properties is essential for the understanding of diffraction moiré fringes in general.

View Article: PubMed Central - HTML - PubMed

Affiliation: Photon Factory, Institute of Materials Structure Science, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan.

ABSTRACT
A detailed and comprehensive theoretical description of X-ray diffraction moiré fringes for a bicrystal specimen is given on the basis of a calculation by plane-wave dynamical diffraction theory. Firstly, prior to discussing the main subject of the paper, a previous article [Yoshimura (1997). Acta Cryst. A53, 810-812] on the two-dimensionality of diffraction moiré patterns is restated on a thorough calculation of the moiré interference phase. Then, the properties of moiré fringes derived from the above theory are explained for the case of a plane-wave diffraction image, where the significant effect of Pendellösung intensity oscillation on the moiré pattern when the crystal is strained is described in detail with theoretically simulated moiré images. Although such plane-wave moiré images are not widely observed in a nearly pure form, knowledge of their properties is essential for the understanding of diffraction moiré fringes in general.

No MeSH data available.