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

Yoshimura, J.

doi:10.1107/S2053273319006557

addenda and errata

FOUNDATIONS
ADVANCES

ISSN: 2053-2733

Volume 75| Part 4| July 2019| Pages 652-654

https://doi.org/10.1107/S2053273319006557

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

Jun-ichi Yoshimura ^a ^*

^aSakai 5-13-2-A322, Musashino-shi, Tokyo 180-0022, Japan
^*Correspondence e-mail: j-yoshimura@voice.ocn.ne.jp

Edited by K. Tsuda, Tohoku University, Japan (Received 17 April 2019; accepted 8 May 2019; online 26 June 2019)

Seven corrections are made and several supplementary equations are added to the article by Yoshimura [Acta Cryst. (2015), A71, 368–381].

Keywords: diffraction moiré fringes; Pendellösung oscillation; phase jump; gap phase.

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On p. 371, left column, near the top, just after `the index (i) = (1, 2) … dispersion surface', the following comment should be added: `the upper sine in equations (5a), (5b) refers to the case of i = 1, and the lower sine to the case of i = 2'. On p. 371, right column, near the bottom, just after `the indices (i, j) = (1, 2) …, respectively', the following comment should be added: `the upper sign in equations (15a) to (15d) refers to the case of j = 1, and the lower sine to the case of j = 2'. Equations (14b), (14c) are incorrect. They must be corrected to

$[\eqalignno {E_{og^\prime}^{i,j}({\bf r})& = C_o^iC^{i,j}_{og^\prime} \exp 2\pi i ({\bf g}^\prime \cdot {\bf r}_b) \exp i \bigl \{ ({\bf K}_e \cdot {\bf r}_a) + [{\bf k}^i_o \cdot({\bf r}_{a^\prime} - {\bf r}_a)]&\cr &\quad+ [{\bf K}_{o}^i \cdot ({\bf r}_b - {\bf r}_{a^\prime})] + [{\bf k}_{og^\prime}^{i,j} \cdot ({\bf r}_{b^\prime} - {\bf r}_b)]&\cr &\quad + [{\bf K}^{i,j}_{og^\prime} \cdot ({\bf r} - {\bf r}_{b^\prime})]\bigr \}&\cr &= C_o^iC^{i,j}_{og^\prime} \exp i\bigl [ -K\delta ^i_at_1 - K\delta ^{i,j}_{b,oo^\prime}t_2 + u_nT_{b^\prime}/ \gamma_g&\cr &\quad- 2\pi (\Delta{\bf g} \cdot {\hat{\bf K}}_g)T_{b^\prime}/\gamma_g\bigr ] \exp i [({\bf K}_e + 2\pi {\bf g}^\prime) \cdot {\bf r}], & (14b)}]$

$[\eqalignno { E_{go^\prime}^{i,j}({\bf r}) &= C_g^iC^{i,j}_{go^\prime} \exp 2\pi i [({\bf g} \cdot {\bf r}_a) - ({\bf g}^\prime \cdot {\bf r}_b)] \exp i \bigl \{ ({\bf K}_e \cdot {\bf r}_a)&\cr &\quad+ [{\bf k}^i_g \cdot ({\bf r}_{a^\prime} - {\bf r}_a)] + [{\bf K}_{g}^i \cdot ({\bf r}_b - {\bf r}_{a^\prime})] + [{\bf k}^{i,j}_{go^\prime} \cdot ({\bf r}_{b^\prime} - {\bf r}_{b})]&\cr &\quad +[{\bf K}^{i,j}_{go'}\cdot({\bf r}-{\bf r}_{b'})]\bigr \}& \cr & = C_g^iC^{i,j}_{go^\prime} \exp i\bigl [ -K\delta ^i_at_1 - K\delta ^{i,j}_{b,gg^\prime}t_2 + u_n (T_{a^\prime} - T_{b^\prime})/ \gamma_g&\cr &\quad+ 2\pi (\Delta {\bf g} \cdot {\hat{\bf K}}_o) T_{b^\prime}/\gamma_o\bigr ] \exp i [({\bf K}_e - 2\pi \Delta{\bf g}) \cdot {\bf r}]. & (14c)}]$

In these corrections, the last terms in the first exponential functions on the right-hand side in the second equations were corrected.

On p. 372, right column, an error is involved in equation (22a). It must be corrected to

$[\eqalignno{{I_{og'}}({\bf{r}}) &= {1 \over 4}{{\gamma _o} \over {\gamma _g}}\exp \left [ - {1 \over 2}{\mu _o}\left({1 \over {\gamma _o}} + {1 \over {\gamma _g}} \right)({t_1} + {t_2}) \right]{{U_r^2} \over {u_{or}^2 + U_r^2}}&\cr &\times \Bigg[\left(1 + {{u_r^2} \over {u_r^2 + U_r^2}} \right) \cosh (2K\alpha _{21,i}{t_1})+ {{2u_r} \over {(u_r^2 + U_r^2)^{1/2} }}&\cr &\quad\times \sinh (2K\alpha _{21,i}{t_1}) + {{U_r^2} \over {u_r^2 + U_r^2}}\cos (2K\alpha _{21,r}{t_1})\Bigg]&\cr &\quad \times \left[\cosh (2K\beta _{o,i}{t_2}) - \cos (2K\beta _{o,r}{t_2}) \right].&(22a)}]$

In this correction, the last term in the second bracket on the right-hand side was corrected. On p. 373, left column, equation (24b) is incorrect. It must be corrected to

$[K{\alpha _{21,i}} = {1 \over 2}{{(u_ru_i + U_rU_i)} \over {(u_r^2 + U_r^2)^{1/2} }}{1 \over {\gamma _g}}.\eqno(24b)]$

On p. 376, left column, the description of `t_gap = 0.024 mm' is incorrect. It must be corrected to `t_gap = 0.24 mm'. The mentioned errors in equations (14b), (14c) and (24b) do not influence the calculation of equations (20) to (23b), since the correct expressions as above were used in deriving them. The mentioned error in equation (22a) does not influence the computations of the presented images and graphs in the paper, since the computations were all made correctly using the correct expression as mentioned above; the errors are only in the text.

As a supplement to the previous presentation of the equation for the diffracted – or G – wave image intensity $[{I_g}({\bf{r}} )]$ in equation (20), the equation for the corresponding transmitted – or O – wave image intensity $[{I_o}({\bf{r}} )]$ is added as in the following:

$[\eqalignno{{I_o}({\bf{r}} )& = \left| \textstyle\sum\limits_{i,j} \left[E_{oo'}^{i,j}({\bf{r}} ) + E_{go'}^{i,j}({\bf{r}} )\right] \right|^2&\cr &=I_{oo'}({\bf{r}} ) + {I_{go'}}({\bf{r}} ) + {A_o}({\bf{r}} )\cos{\Psi _o}({\bf{r}} ) + {B_o}({\bf{r}} )\sin {\Psi _o}({\bf{r}} )&\cr &&(49)}]$

with

$[\eqalignno{\Psi_o({\bf r})&=2\pi[\Delta{\bf g}\cdot({\bf r}-{\bf r}_o)]+K\alpha_{go}t_2 -u_nt_{\rm gap}/\gamma_g&\cr &\quad -2\pi(\Delta{\bf g}\cdot{\hat{\bf K}}_o)\cdot (T_{b^\prime}/\gamma_o)&(50)}]$

$[\eqalignno{{I_{oo'}}({\bf{r}} )& = {1 \over 4}\exp\left [ - {1 \over 2}{\mu _o}\left({1 \over {\gamma _o}} + {1 \over {\gamma _g}} \right)({t_1} + {t_2}) \right]&\cr &\quad\times \Bigg[ \left(1 + {{u_r^2} \over {u_r^2 + U_r^2}} \right)\cosh(2K\alpha _{21,i}{t_1}) + {{2u_r} \over {(u_r^2 + U_r^2)^{1/2} }}&\cr &\quad\times \sinh(2K\alpha _{21,i}{t_1}) + {{U_r^2} \over {u_r^2 + U_r^2}}\cos(2K\alpha _{21,r}{t_1})\Bigg]&\cr &\quad\times\Bigg[\left(1 + {{u_{or}^2} \over {u_{or}^2 + U_r^2}} \right) \cosh(2K\beta _{o,i}{t_2} ) + {{2u_{or}} \over {(u_{or}^2 + U_r^2)^{1/2} }}&\cr &\quad\times \sinh (2K\beta _{o,i}{t_2}) + {{U_r^2} \over {u_{or}^2 + U_r^2}}\cos(2K\beta _{o,r}{t_2} )\Bigg]&(51a)}]$

$[\eqalignno{I_{go'}({\bf{r}} )& = {1 \over 4}\exp\left [ - {1 \over 2}{\mu _o}\left({1 \over {\gamma _o}} + {1 \over {\gamma _g}} \right)({t_1} + {t_2}) \right]&\cr &\quad \times {{U_r^2} \over {(u_r^2 + U_r^2 )}}{{U_r^2} \over {(u_{gr}^2 + U_r^2)}}\big[ \cosh(2K\alpha _{21,i}{t_1} )&\cr &\quad - \cos(2K\alpha _{21,r}{t_1})\big] \big[ \cosh (2K\beta _{g,i}{t_2}) - \cos(2K\beta _{g,r}{t_2})\big]&\cr &&(51b)}]$

$[\eqalignno{{A_o}({\bf{r}} )& = {1 \over 2}\exp\left [- {1 \over 2}{\mu _o}\left({1 \over {\gamma _o}} + {1 \over {\gamma _g}} \right)({t_1} + {t_2} ) \right]&\cr &\quad\times{{{U_r}} \over {(u_r^2 + U_r^2)^{1/2} }}{{{U_r}} \over {(u_{gr}^2 + U_r^2)^{1/2} }} \Big\{ \{ \sinh(2K{\alpha _{21,i}}{t_1})&\cr &\quad\times[\cos(K{\beta _{ -, r}}{t_2})\sinh(K{\beta _{ +, i}}{t_2} ) + \cos(K{\beta _{ +, r}}{t_2})&\cr &\quad\times\sinh(K{\beta _{ -, i}}{t_2})] - \sin(2K{\alpha _{21,r}}{t_1})[\sin(K{\beta _{ -, r}}{t_2})&\cr &\quad\times\cosh(K{\beta _{ +, i}}{t_2}) + \sin(K{\beta _{ +, r}}{t_2})\cosh(K{\beta _{ -, i}}{t_2} ) ]\} &\cr &\quad + {{u_{or}} \over {(u_{or}^2 + U_r^2)^{1/2} }}\{ \sinh(2K{\alpha _{21,i}}{t_1}) [\cos(K{\beta _{ -, r}}{t_2})&\cr &\quad\times\cosh(K{\beta _{ +, i}}{t_2}) - \cos(K{\beta _{ +, r}}{t_2})\cosh (K{\beta _{ -, i}}{t_2}) ]&\cr &\quad -\sin(2K\alpha_{21,r}t_1)\cdot[\sin(K\beta_{-,r}t_2)\sinh(K\beta_{+,i}t_2)&\cr &\quad -\sin(K\beta_{+,r}t_2)\sinh(K\beta_{-,i}t_2)]\}&\cr &\quad + {{u_r} \over {(u_r^2 + U_r^2)^{1/2} }} [\cosh(2K{\alpha _{21,i}}{t_1}) - \cos(2K{\alpha _{21,r}}{t_1})]&\cr &\quad \times [\cos(K{\beta _{ -, r}}{t_2})\sinh(K{\beta _{ +, i}}{t_2})+ \cos(K{\beta _{ +, r}}{t_2})&\cr &\quad\times\sinh(K{\beta _{ -, i}}{t_2})] + {{u_r} \over {(u_r^2 + U_r^2)^{1/2} }}{{{u_{or}}} \over {(u_{or}^2 + U_r^2)^{1/2} }} &\cr &\quad\times[\cosh(2K{\alpha _{21,i}}{t_1}) - \cos (2K{\alpha _{21,r}}{t_1}) ]&\cr &\quad\times [\cos(K{\beta _{ -, r}}{t_2})\cosh(K{\beta _{ +, i}}{t_2}) - \cos(K{\beta _{ +, r}}{t_2})&\cr &\quad\times\cosh(K{\beta _{ -, i}}{t_2})]\Big\}&(52a)}]$

$[\eqalignno{{B_o}({\bf{r}} )& = {1 \over 2}\exp \left[ - {1 \over 2}{\mu _o}\left({1 \over {\gamma _o}} + {1 \over {\gamma _g}} \right)({t_1} + {t_2})\right]&\cr &\quad\times{{{U_r}} \over {(u_r^2 + U_r^2)^{1/2} }}{{{U_r}} \over {(u_{gr}^2 + U_r^2)^{1/2} }}\Big\{\{-\sinh (2K{\alpha _{21,i}}{t_1})&\cr &\quad\times [\sin (K{\beta _{ -, r}}{t_2})\cosh(K{\beta _{ +, i}}{t_2}) + \sin(K{\beta _{ +, r}}{t_2})&\cr &\quad\times\cosh (K{\beta _{ -, i}}{t_2}) ]- \sin(2K{\alpha _{21,r}}{t_1}) [\cos(K{\beta _{ -, r}}{t_2})&\cr &\quad\times\sinh (K{\beta _{ +, i}}{t_2}) + \cos(K{\beta _{ +, r}}{t_2})\sinh (K{\beta _{ -, i}}{t_2})] \}&\cr &\quad + {{{u_{or}}} \over {(u_{or}^2 + U_r^2)^{1/2} }}\{ - \sinh(2K{\alpha _{21,i}}{t_1}) [\sin(K{\beta _{ -, r}}{t_2})&\cr &\quad\times\sinh(K{\beta _{ +, i}}{t_2}) - \sin(K{\beta _{ +, r}}{t_2})\sinh (K{\beta _{ -, i}}{t_2}) ]&\cr &\quad-\sin(2K\alpha_{21,r}t_1)\cdot[\cos(K\beta_{-,r}t_2)\cosh(K\beta_{+,i}t_2)&\cr &\quad -\cos(K\beta_{+,r}t_2)\cosh(K\beta_{-,i}t_2)]\}&\cr &\quad- {{{u_r}} \over {(u_r^2 + U_r^2)^{1/2} }} [\cosh(2K{\alpha _{21,i}}{t_1}) - \cos(2K{\alpha _{21,r}}{t_1})]&\cr &\quad\times [\sin (K{\beta _{ -, r}}{t_2})\cosh (K{\beta _{ +, i}}{t_2}) + \sin (K{\beta _{ +, r}}{t_2})&\cr &\quad\times\cosh (K{\beta _{ -, i}}{t_2})] - {{{u_r}} \over {(u_r^2 + U_r^2)^{1/2} }} {{{u_{or}}} \over {(u_{or}^2 + U_r^2)^{1/2} }}&\cr &\quad\times[\cosh (2K{\alpha _{21,i}}{t_1}) - \cos (2K{\alpha _{21,r}}{t_1})]&\cr &\quad\times [\sin (K{\beta _{ -, r}}{t_2})\sinh (K{\beta _{ +, i}}{t_2})- \sin (K{\beta _{ +, r}}{t_2})&\cr &\quad\times\sinh (K{\beta _{ -, i}}{t_2})]\Big\}.&(52b)}]$

The numbering of the equations here is continued from the last equation (48) in the original paper (Yoshimura, 2015 ). $[E_{oo'}^{i,j}({\bf{r}} )]$ and $[E_{go'}^{i,j}({\bf{r}} )]$ in equation (49) are as given in equations (14a), (14c), respectively.

Through similar calculations to those written in the right-hand side column on p. 373, the term of interference phase $[{\Psi _o}({\bf{r}} )]$ in equation (50) can be reduced to

$[\Psi_o({\bf r})=\Psi_o({\bf r}_{b^\prime})=\{2\pi\Delta{\bf g}_{\parallel}\cdot[({\bf r}_{b^\prime}-{\bf r}_o)_{\parallel}-{\bf I}_{\parallel}] -u_nt_{\rm gap}/\gamma_g\}\eqno(53)]$

which is the same as $[{\Psi _g}({\bf{r}} )]$ in equation (34) for the G-wave image intensity (here, the symbol || denotes the component parallel to the specimen surfaces). In the present case that $[({{\bf{r}} - {{\bf{r}}_{b'}}} )\parallel {{\hat{\bf K}}_o}]$ , part of the first term and the fourth term in equation (50) cancel each other as follows:

$[\eqalign{&2\pi[\Delta{\bf g}\cdot({\bf r}-{\bf r}_{b^\prime})]-2\pi (\Delta{\bf g}\cdot{\hat{\bf K}}_o)(T_{b^\prime}/\gamma_o)\cr &=2\pi[\Delta {\bf g}\cdot({\bf r}-{\bf r}_{b^\prime})]-2\pi(\Delta{\bf g}\cdot{\hat {\bf K}}_o)\cdot[({\bf r}-{\bf r}_{b^\prime}) \cdot {\bf n}]/({\hat{\bf K}}_o\cdot{\bf n})\cr &=0.}]$

The moiré images of the O-wave in Figs. 14(a) and 14(b) and the curves concerned in Figs. 15(a) and 15(b) were computed using these equations (49), (51a)–(52b) and (53).

References

Yoshimura, J. (2015). Acta Cryst. A71, 368–381. CrossRef IUCr Journals Google Scholar

This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-BY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.

FOUNDATIONS
ADVANCES

ISSN: 2053-2733

Volume 75| Part 4| July 2019| Pages 652-654

https://doi.org/10.1107/S2053273319006557

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addenda and errata\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

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

References

addenda and errata