research papers
Xray Laue diffraction by sectioned multilayers. I. Pendellösung effect and rocking curves
^{a}Institute of Physics and Mathematics, Federal Research Center "Komi Scientific Center", The Ural Branch of the Russian Academy of Sciences, Syktyvkar 167982, Russian Federation
^{*}Correspondence email: vpunegov@dm.komisc.ru
Using the Takagi–Taupin equations, Xray Laue Pendellösung effect and rocking curves is studied. Numerical simulation of Laue diffraction in multilayer structures W/Si and Mo/Si is carried out. It is shown that the determination of sectioned depths based on the period of the interference fringes of the experimental rocking curves of synchrotron radiation is not always correct.
in flat and wedge multilayers is theoretically considered. Recurrence relations are obtained that describe Laue diffraction in structures that are inhomogeneous in depth. The influence of sectioned depth, imperfections and nonuniform distribution of the multilayer period on theKeywords: sectioned multilayers; dynamical Xray Laue diffraction; recurrence relations; Pendellösung effect; rocking curves.
1. Introduction
Elements of multilayer Xray optics are widely used in synchrotron radiation installations for transporting Xray beams (Rack et al., 2010) and focusing radiation (Lyatun et al., 2020), in extremeultraviolet lithography (Chkhalo et al., 2017) and in astronomy (Tamura et al., 2018). Such multilayers mainly refer to grazingincidence reflectors or, in terms of Xray diffraction in crystals, to Bragg geometry (Authier, 2001).
To focus hard Xrays, it was proposed to create multilayer Laue lenses (Maser et al., 2004), which, like Fresnel zone plates (Kagoshima & Takayama, 2019), transport radiation in transmission geometry. The fabrication of Laue lenses, which are depthsectioned and thicknessgraded multilayers, is a complex problem (Kang et al., 2007). Therefore, the first step in the study of Laue diffraction was presented by a synchrotron Xray study of laterally bounded (sectioned) multilayers with a constant period (Kang et al., 2004, 2005).
The ). Theoretically, asymmetric diffraction is also possible for sectioned multilayers; however, technologically this is a difficult problem associated with oblique sectioning of a multilayer structure.
of Xray scattering in the Laue case is well developed for crystals, both for symmetric and asymmetric geometry (Authier, 2001Dynamical Xray Laue diffraction in a multilayer differs compared with diffraction in grazingincidence geometry. One of the main features is the pendulum (Pendellösung) effect (Authier, 2001), when the intensity of the Xray beam of the transmission wave is transmitted into the diffraction beam and then, with increasing depth, to the contrary, the intensity of the diffraction wave is transmitted into the transmission beam.
In the absence of an algorithm for simulating dynamical Laue diffraction, the analysis of experimental results using the interferencefringe spacing of the rocking curve was not entirely correct (Kang et al., 2005, 2007). Therefore, the present work is devoted to a consistent consideration of the Laue of diffraction in multilayers.
2. Dynamical Laue diffraction in a sectioned multilayer with a constant period
Multilayer Laue lenses are graded multilayers. Unfortunately, on the basis of the Takagi–Taupin equations (Takagi, 1962, 1969; Taupin, 1964; Afanas'ev & Kohn, 1971; Kato, 1973), in the general case, it is impossible to obtain analytical solutions describing Laue diffraction in structures with period variation. Only some laws of changes in the period of the structure allow analytical solutions; in particular, such solutions were previously obtained for crystals with quadratic (Kolpakov & Punegov, 1985) and exponential (Andreev, 2001) displacement fields, and crystals with a (Chukhovskii & Khapachev, 1985; Kato, 1990). In most cases, one has to limit oneself to the numerical solution of the Takagi–Taupin equations in an oblique coordinate system (Punegov, 2020; Lomov et al., 2021), for example, using the `halfstep derivative' method (Epelboin, 1985).
Therefore, let us consider Xray dynamical Laue diffraction in a sectioned multilayer with a constant period d = d_{t} + d_{b}, where d_{t} is the thickness of the upper layer and d_{b} is the thickness of the lower layer (Fig. 1).
A plane Xray wave is incident on the left side of the sectioned multilayer at an angle θ = θ_{B} + ω, where ω is the deviation of the Xray beam from the θ_{B} (Fig. 1). In contrast to the Ewald–Laue approach (Authier, 2001), we will proceed from the onedimensional Takagi–Taupin equations for the periodic structure, which in the Cartesian coordinate system have the form
where E_{0}(η, x) and E_{1}(η, x) are the amplitudes of the transmission and diffraction Xray waves, respectively, a_{0} = , a_{1}= , a_{−1} = a_{1}, is the angular parameter, λ is the wavelength of the Xray radiation in vacuum, and C is the polarization factor. The Fourier coefficients of the Xray polarizability for a structure with a twolayer period in the directions of transmission χ_{0} and diffraction χ_{1} are written as
In relation (2), χ_{t} and χ_{b} are the Fourier coefficients of polarizabilities and thicknesses of the upper (t) and lower (b) layers of the period of the multilayer. Xray polarizabilities of chemical elements are calculated using the tabular values of optical constants: χ_{j} = 2(δ_{j} + iβ_{j}), where ; ; j = t or b, which indicates the corresponding layer in the period of the multilayer; r_{0} = e^{2}/mc^{2} is the classical radius of an electron, where e and m are the charge and mass of an electron, respectively; N_{j} is the atomic density; Z is the number of electrons in an atom; and and are dispersion corrections to the atomic amplitude. Equation (1) contains an attenuation factor f, which describes the attenuation of Xray reflection in a multilayer. This coefficient characterizes the disturbances in the periodic structure of multilayers during their creation. In particular, contribution to the attenuation factor is given by the roughness of the boundaries layers in the structure (Névot & Croce, 1980; de Boer, 1994), errors in layer thickness of the structure (Spiller & Rosenbluth, 1985, 1986), diffusion blurring of boundaries between layers (presence of transition layers) (Stearns, 1989), random loss in a shortperiod multilayer of a heavy or light layer (Kopylets et al., 2019), local bending of layers during polishing (Kang et al., 2007), etc.
For example, accounting for interlayer roughness in Parratt's method of recurrence relations (Parratt, 1954) was considered using the Névot–Croce and Debye–Waller attenuation factors (Bushuev & Sutyrin, 2001; Kohn, 2003). Unfortunately, without an analysis of diffuse scattering, it is difficult to determine the contribution of each type of defect to the attenuation factor. Therefore, in our theoretical consideration of dynamical Laue diffraction, the attenuation factor will take on the value 0 ≤ f ≤ 1 without determining the specific types of defects.
We consider the model generally accepted in the dynamical Laue diffraction theory (Authier, 2001), in which the front of a plane incident Xray wave and the size of the multilayer are spatially unrestricted in the vertical direction (Fig. 1), hence the amplitudes of Xray waves in equation (1) depend only on one horizontal coordinate x. Such an Xray diffraction model can be considered if only the rocking curves are analyzed. In the case where the Xray beam incident on the multilayer is spatially restricted, it is necessary to use the twodimensional Takagi–Taupin equations (Punegov et al., 2017; Punegov & Karpov, 2021). This approach allows one to analyze Xray reciprocalspace maps (RSMs).
The solution of coupled equation (1) for Laue diffraction can be obtained using the boundary conditions = = 1 and E_{1}(η, 0) = 0. Applying these boundary conditions, we can obtain expressions for the amplitudes of the transmission E_{0}(η, x) and diffraction E_{1}(η, x) waves,
and
where ξ = −(η^{2} + 4f^{2}a_{1}a_{−1})^{1/2} and ψ = a_{0} + η/2.
2.1. Pendellösung effect in homogeneous sectioned multilayer
Let us first consider the pendulum (Pendellösung) effect under Laue diffraction conditions. If the exact Bragg condition is satisfied (η = 0) and taking into account a_{1} = a_{−1}, the intensity distributions of the transmission and diffraction Xray waves can be written as
and
where is the linear coefficient of absorption of Xrays in the multilayer. The dynamical coefficient in equation (1) is represented as a_{1} = a′_{1} + ia′′_{1}, where is a real part and is an imaginary part. Considering and , solutions (5) and (6) can be rewritten in a more visual form for attenuation Pendellösung oscillations as
and
Expressions (7) and (8) clearly describe the Pendellösung effect of Laue diffraction when the exact Bragg condition is satisfied. The pendulum beat period for a perfect multilayer depends on the Fourier polarizability coefficient χ_{1}, which characterizes the interaction of Xray waves in a periodic medium.
Fig. 2 shows the intensity distributions over the depth of a perfect (f = 1) Mo/Si multilayer with the same thicknesses of alternating layers, d_{Mo} = d_{Si} = 3.5 nm, calculated using exact solutions (5) and (6), as well as using approximate solutions (7) and (8). In the numerical calculations, the synchrotron radiation wavelength λ = 0.1305 nm was used. The optical constants are presented in Tables 1 and 2. The total depth of the multilayer is L_{x} = 2l_{Pen} = 76.4 µm, which corresponds to two Pendellösung distances. At an exact (η = 0), the transmission intensity at a depth of x = 19.1 µm, which corresponds to half the Pendellösung distance, almost completely (corrected for photoelectric absorption) transforms into a diffraction wave. Furthermore, the reverse process is observed. The visible difference between the curves of the pendulum effect in the case of exact and approximate solutions is observed at large sectioned depths of the multilayer (Fig. 2).
Fig. 3 demonstrates Pendellösung oscillations for a perfect and imperfect Mo/Si multilayer. The attenuation factor for the defect structure is f = 0.8. It can be seen from the figure that the period of Pendellösung oscillations increases in the case of an imperfect multilayer. This is due to the fact that the presence of defects in the multilayer reduces the reflectivity of the periodic structure. A similar behavior of Pendellösung oscillations was observed in the case of dynamical Laue diffraction in a crystal with defects (Punegov & Pavlov, 1992).
Solutions (3) and (4) make it possible to obtain the intensity distributions of the transmission I_{0}(x)= and diffraction I_{1}(x)= waves inside the multilayer, depending on the value of the angular parameter η (Fig. 4). So, for example, with an angular deviation of the incident Xray wave from the by ω = 0.18 mrad, a decrease in the period of the Pendellösung oscillations is observed (Fig. 4). The transmission Xray wave is not completely transmitted into the diffraction beam (Fig. 4, curve 3). Therefore, the diffraction intensity inside the multilayer is lower than at the exact (compare curves 3 and 4).
If in equation (1) a_{−1} = 0, this means that there is no transfer of intensity from the diffraction beam back to the transmitted beam, solutions (3) and (4) are transformed into expressions corresponding to the kinematical approximation,
and
In the kinematical approximation, for the intensities of the transmission and diffraction waves, we obtain
and
where is the Laue interference function (James, 1950). In the case of kinematical diffraction, the intensity of the transmitted wave, according to equation (11), exponentially attenuates due to photoelectric absorption in the multilayer. The intensity of the diffraction wave has a typical angular distribution of kinematical diffraction. Solutions (9)–(12) are valid for multilayers with a very small sectioned depth.
2.2. Rocking curves of a sectioned multilayer with constant period
In experimental works (Kang et al., 2004, 2005, 2007), rocking curves of Laue diffraction from sectioned multilayers are presented as angular dependences of q_{x} scans in In our consideration, the deviation from the in solutions (3) and (4) is determined by the parameter η, which corresponds to the q_{x} scan in (Kang et al., 2005; Punegov et al., 2014, 2016).
In Fig. 5, rocking curves of Mo/Si multilayers of different depths L_{x} are shown. In the case when the depth of the multilayer is equal to half the Pendellösung distance (L_{x} = l_{Pen}/2 = 19.1 µm), the intensity of the diffraction wave, according to Fig. 2, is maximum and is equal to , where is the intensity of the incident radiation. The intensity of the transmitted wave in the exact Bragg orientation is I_{0}(η = 0) ≃ 0 [Fig. 5(a)].
The rocking curves of the transmitted and diffracted intensities from an Mo/Si multilayer with a depth of L_{x} = l_{Pen} × 3/4 = 28.7 µm are shown in Fig. 5(b). The maximum of the diffracted intensity is . In the case where L_{x} = l_{Pen} = 38.2 µm, the gap is observed in the profile of the rocking curve of the diffraction wave at the exact Bragg orientation [Fig. 5(c)] and the minimum of the diffracted intensity is . This behavior of the rocking curve is typical for all periodic media in the case of Laue diffraction, when the depth of the sectioned structure is equal to the Pendellösung distance.
3. Dynamical Laue diffraction in an inhomogeneous sectioned multilayer: recurrence relations
In addition to flat multilayer Laue lenses, to improve focusing, it has been proposed to manufacture wedged multilayer Laue lenses (Yan et al., 2007; Conley et al., 2008), in which the period of the structure changes not only in thickness z but also along the depth x of the sectioned multilayer system. Equation (1) is written for a multilayer with a constant period and does not allow taking into account the change in the structure period along the x coordinate. A multilayer with wedge layers is a structure whose period varies along the x axis (Fig. 6). Since equation (1) describes diffraction by a structure with a constant period, we divide the wedge multilayer into vertical elementary sections, within which the period of the structure is constant (Fig. 6). We denote the depth of the vertical sections in the direction of the x axis as x^{p} − x^{p−1} = l^{p}. Here x^{p} and x^{p−1} are the coordinates of the left and right boundaries of the vertical elementary section with a number p, respectively, and l^{p} is the depth of this section, while the sections are numbered from left to right (p = 0, 1, 2,…, P). The superscript of all parameters corresponds to the elementary section number. Let the diffraction in the first vertical section be described by equation (1), then for the section with the number p instead of the parameter η in (1) one should write η^{p} = η + Δη^{p}, where Δη^{p} = 2π Δd^{p}/d^{2} and Δd^{p} = d − d^{p} is the mismatch of the period of the pth vertical elementary section relative to the first vertical section with a multilayer period d^{1} = d.
Xray diffraction in the pth elementary vertical section of the multilayer is described by a system of equations of the form
Let and represent the amplitudes of the Xray fields at the boundary (p − 1) of the pth vertical elementary section. Then, taking into account these boundary conditions, we can obtain solutions for the transmission and diffraction waves inside the elementary section with number p,
and
where
and
At the boundary between the pth and the (p + 1)th elementary vertical sections, the solutions for the amplitudes of Xray waves have the form
and
The amplitudes and serve as the boundary conditions for diffraction in the (p + 1) elementary vertical section. Performing sequentially the procedure of recurrent calculations with the initial boundary conditions on the input face, and , we obtain the amplitudes of the transmitted and diffracted waves on the output face of the multilayer inhomogeneous in depth L_{x}, where .
3.1. Pendellösung effect in the case of a wedge sectioned multilayer
To calculate the intensity distribution of the transmission and diffraction waves in the wedge multilayer (Fig. 6), we used the recurrence relations (16) and (17). We will consider cases of weak and strong linear period variation. In the first case, the period varies with the depth of the Mo/Si multilayer from 7.000 to 6.997 nm. In the case of a strong gradient, the period changes from 7.000 to 6.986 nm. Fig. 7 shows the depth distributions of intensities in Mo/Si multilayers with variations in the periods. In the case of a weak gradient of the period variation at the initial depth interval from 0 to 10 µm, the behavior of the transmission and diffraction intensities coincides with the Pendellösung effect of a multilayer with a constant period. Furthermore, due to the gradient of the period variation of the Mo/Si system, the classical Pendellösung effect is violated. As a result, the distance of the Pendellösung fringe decreases and the intensity of the transmission wave is not completely transferred to the diffraction beam [Fig. 7(a)]. For a strong gradient of the period variation of the Mo/Si multilayer, the Pendellösung profiles of the transmission and diffraction Xray waves do not intersect. The period of the Pendellösung fringe decreases monotonically with the depth of the sectioned multilayer [Fig. 7(b)].
The intensity distributions of the transmission and diffraction waves also depend on the lateral disturbance gradient of the multilayer structure. The Pendellösung effect depending on the weak and strong gradient of the attenuation factor is shown in Fig. 8. The attenuation factor varies linearly from 1 to 0.8 in the case of a weak gradient and from 1 to 0.6 in the case of a strong gradient. Since the perfection of the multilayer structure deteriorates along the depth of the Mo/Si multilayer, the period of Pendellösung oscillations in the direction of the x axis increases. We see that the stronger the attenuationfactorvariation gradient, the more visible the increase in the Pendellösung distance along the depth of the multilayer (Fig. 8).
3.2. Rocking curves of wedge sectioned multilayers
We calculated Xray rocking curves of Mo/Si multilayers that are nonuniform over depth x (L_{x} = l_{Pen} = 38.2 µm) using the recurrent solutions (16) and (17). In Fig. 9, the rocking curves of transmission and diffraction waves with the strong variation gradient of the multilayer period for different numbers of fragmentations P of the structure depth into elementary vertical sections are shown. Starting from P = 30, an increase in the number of fragmentations into elementary sections does not change the rockingcurve profile.
Fig. 10 demonstrates the rocking curves of wedge Mo/Si multilayers with weak and strong gradients of period variation along the depth of the sectioned structure. On the profile of the rocking curve of the diffracted wave in the case of the weak gradient, a gap is observed [Fig. 10(a)], which is less deep compared with the splitting of the rocking curve of a flat multilayer [Fig. 5(c)]. The rocking curve of the multilayer with the strong periodvariation gradient has two symmetrical gaps [Fig. 10(b)].
The influence of linearly varying attenuation factors along the depth of the sectional Mo/Si multilayer with a constant period d on the profiles of the rocking curves is shown in Fig. 11. The weak gradient of the attenuationfactor variation insignificantly affects the rocking curves [compare Fig. 11(a) with Fig. 5(c)]. On the other hand, a strong attenuationfactor gradient strongly changes the profile of the diffraction curve [Fig. 11(b)].
4. Determination of the depth of sectioned multilayers
An experimental study of Laue diffraction in sectioned W/Si and Mo/Si multilayers with hard synchrotron radiation energies (9.5 and 19.5 keV) was performed for sectioned depths from 2 to 17 µm (Kang et al., 2005). The period of the W/Si multilayer was 29 nm (sample A). The of tungsten in the W/Si period was 60%. The second Mo/Si structure had a period of 7 nm with an equal proportion of Mo and Si (sample B). Since in the article (Kang et al., 2005) the results of diffraction were mainly discussed using synchrotron radiation with a wavelength of λ = 0.1305 nm (energy E = 9.5 keV), in this work numerical calculations were performed using this wavelength for structures A and B.
The optical constants of tungsten, molybdenum and silicon with respect to the wavelength λ = 0.1305 nm of synchrotron radiation were obtained using the Xray Server computer program (Stepanov, 1997, 2013; Stepanov & Forrest, 2008), and are shown in Table 1. Based on these data, the Fourier coefficients of Xray polarizability, χ_{0} and χ_{1}, for the case of Laue diffraction in W/Si and Mo/Si multilayers, were calculated (Table 2).


The Pendellösung distance for the W/Si multilayer is more than four times less than the corresponding value for the Mo/Si structure (). There are two options for determining the sectioned depth of multilayers L_{x}. The first approach is based on the fact that the depths of the sectioned multilayers L_{x} are determined by measuring the period of interference fringes on the rockingcurve profile. There are two drawbacks to this approach. First, this method is precisely realized within the kinematical approximation. Second, good angular resolution is needed in the measurement of the rocking curves. The second approach is based on the numerical simulation of the rocking curves for Laue diffraction from multilayers using solutions. Even with the absence of information on the structural perfection of multilayers, the second approach gives an insignificant error in numerical simulation and subsequent comparison with experimental measurements.
For a synchrotron radiation energy of 9.5 keV in the Laue diffraction experiment involving W/Si and Mo/Si multilayers, the angular distribution of the scattered intensity was investigated by scanning the scattering vector in the direction of the projection Q_{x} (Kang et al., 2005), which corresponds to the angular parameter η = Q_{x} in our consideration. The W/Si and Mo/Si samples were wedge shaped in the y direction perpendicular to the diffraction plane (x, z), which gave a change in the sectioned depth L_{x}. The rocking curves were calculated using solution (4) for the diffracted wave.
As in the experimental work (Kang et al., 2005), in the numerical simulation, we used the W/Si structure with a period d = d_{W} + d_{Si} = 29 nm, where d_{W} = 17.4 nm and d_{Si} = 11.6 nm. In the case of the Mo/Si multilayer, the period was d = d_{Mo} + d_{Si} = 7 nm with d_{Mo} = d_{Si} = 3.5 nm. The optical constants of these multilayers are presented in Table 2.
It is known that the rockingcurve profiles always have a gap when the sectional depth of the sample is close to the Pendellösung distance (Authier, 2001). The calculated Xray rocking curves of the sectioned W/Si and Mo/Si multilayers for different depths are shown in Fig. 12.
For W/Si multilayers with depths L_{x} equal to 2.6, 4.4, 6.6 and 8.8 µm, the calculated rocking curves [Fig. 12(a)] are close to the experimental data (Kang et al., 2005), with the characteristic gap in the profile of the diffraction curve appearing for L_{x} = 8.8 µm (). For a relatively large depth L_{x} = 10.7 µm, the calculated rocking curve differs from the experimental diffraction curve (Kang et al., 2005).
The diffraction curves of the Mo/Si multilayers with small sectioned depths of 3.9, 6.3 and 8.6 µm [Fig. 12(b)] are consistent with experimental data (Kang et al., 2005). However, the calculated rocking curves for sectioned depths of 11.2 and 15.0 µm [Fig. 12(b)] differ significantly from the experimental results (Kang et al., 2005). For example, the experimental diffraction curve from an Mo/Si multilayer with a depth of 15.0 µm has a gap in the middle of the rockingcurve profile. Such a gap is only possible when the sectional depth L_{x} is close in value to the Pendellösung distance (). Thus, analysis of experimental data to determine the sectioned depth (Kang et al., 2005) using the value of the interferencefringe spacing η_{0} is not always justified. For example, for the depth L_{x} = 15.0 µm and η_{0} = 0.42 µm^{−1}, and for L_{x} = 38.2 µm and η_{0} = 0.165 µm^{−1}.
The question then arises: why for small sectioned depths of multilayers are the calculated and experimental rocking curves close to each other, but for large L_{x} they are significantly different? Fig. 13 shows the calculated rocking curves of the Mo/Si multilayer structure for the sectioned depth L_{x} = 38.2 µm with different angular resolutions. We also show the influence of the instrumental function on Xray diffraction from multilayers in the Laue geometry (curves 2, Fig. 13). In our calculations, we used a diffraction scheme containing a fourbounces Ge(220) monochromator and a threebounces Ge(220) analyzer.
In the case of a large depth L_{x}, interference oscillations are located close to each other; therefore, all oscillations are observed only at a high angular resolution Δη = 0.02 µm^{−1} [Fig. 13(a)] and the period of oscillations is η_{0} = 0.1646 µm^{−1}. In the case of a low resolution, for example, η_{0} = 0.335 µm^{−1}, not all interference oscillations are registered; therefore, in this case, the distance between the oscillations is η_{0} = 0.335 µm^{−1} [Fig. 13(b)]. From this value, we can obtain the sectioned depth L_{x} = 18.8 µm, which is approximately two times less than in the case of high angular resolution. Consequently, the analysis of the experimental data of Xray Laue diffraction to determine the sectional depth of multilayer structures is possible only under the condition of a high angular resolution of the rocking curves.
5. Concluding remarks
We have theoretically investigated in detail Xray Laue diffraction in multilayers in the case of an incident plane wave. The influence of sectional depth, imperfections (defects) and nonuniform distribution of the period of multilayers on the Pendellösung effect and rocking curves has been shown. However, more complete information on the structure of multilayers is provided by the use of tripleaxis diffractometry (Iida & Kohra, 1979; Punegov, 2015). Therefore, the next step will be to consider the Laue diffraction in the case of restricted Xray beams (Punegov et al., 2017; Punegov & Karpov, 2021). This will make it possible to calculate RSMs in Laue geometry, as well as compare the calculated q_{x} and q_{z} sections of RSMs with experimental data (Kang et al., 2005).
References
Afanas'ev, A. M. & Kohn, V. G. (1971). Acta Cryst. A27, 421–430. CrossRef CAS IUCr Journals Web of Science Google Scholar
Andreev, A. V. (2001). JETP Lett. 74, 6–9. Web of Science CrossRef CAS Google Scholar
Authier, A. (2001). Dynamical Theory of Xray Diffraction. Oxford University Press. Google Scholar
Boer, D. K. G. de (1994). Phys. Rev. B, 49, 5817–5820. CrossRef Web of Science Google Scholar
Bushuev, V. A. & Sutyrin, A. G. (2001). Surf. Investig. 16, 121–126. Google Scholar
Chkhalo, N. I., Gusev, S. A., Nechay, A. N., Pariev, D. E., Polkovnikov, V. N., Salashchenko, N. N., Schäfers, F., Sertsu, M. G., Sokolov, A., Svechnikov, M. V. & Tatarsky, D. A. (2017). Opt. Lett. 42, 5070–5073. Web of Science CrossRef CAS PubMed Google Scholar
Chukhovskii, F. N. & Khapachev, Yu. P. (1985). Phys. Status Solidi A, 88, 69–76. CrossRef CAS Web of Science Google Scholar
Conley, R., Liu, C., Qian, J., Kewish, C. M., Macrander, A. T., Yan, H., Kang, H. C., Maser, J. & Stephenson, G. B. (2008). Rev. Sci. Instrum. 79, 053104. Web of Science CrossRef PubMed Google Scholar
Epelboin, Y. (1985). Mater. Sci. Eng. 73, 1–43. CrossRef CAS Web of Science Google Scholar
Iida, A. & Kohra, K. (1979). Phys. Status Solidi A, 51, 533–542. CrossRef CAS Web of Science Google Scholar
James, R. W. (1950). The Optical Principles of the Diffraction ofXrays. London: G. Bell and Sons. Google Scholar
Kagoshima, Y. & Takayama, Y. (2019). J. Synchrotron Rad. 26, 52–58. Web of Science CrossRef CAS IUCr Journals Google Scholar
Kang, H. C., Stephenson, G. B., Liu, C., Conley, R., Khachatryan, R., Wieczorek, M., Macrander, A. T., Yan, H., Maser, J., Hiller, J. & Koritala, R. (2007). Rev. Sci. Instrum. 78, 046103. Web of Science CrossRef PubMed Google Scholar
Kang, H. C., Stephenson, G. B., Liu, C., Conley, R., Macrander, A. T., Maser, J., Bajt, S. & Chapman, H. N. (2004). Proc. SPIE, 5537, 127–132. CrossRef CAS Google Scholar
Kang, H. C., Stephenson, G. B., Liu, C., Conley, R., Macrander, A. T., Maser, J., Bajt, S. & Chapman, H. N. (2005). Appl. Phys. Lett. 86, 151109. Web of Science CrossRef Google Scholar
Kato, N. (1973). Z. Naturforsch. 28, 604–609. CrossRef CAS Web of Science Google Scholar
Kato, N. (1990). Acta Cryst. A46, 672–681. CrossRef Web of Science IUCr Journals Google Scholar
Kohn, V. G. (2003). Poverkhnost, 1, 23–27. Google Scholar
Kolpakov, A. V. & Punegov, V. I. (1985). Solid State Commun. 54, 573–578. CrossRef CAS Web of Science Google Scholar
Kopylets, I., Devizenko, O., Zubarev, E., Kondratenko, V., Artyukov, I., Vinogradov, A. & Penkov, O. (2019). J. Nanosci. Nanotechnol. 19, 518–531. Web of Science CrossRef CAS PubMed Google Scholar
Lomov, A. A., Punegov, V. I. & Seredin, B. M. (2021). J. Appl. Cryst. 54, 588–596. Web of Science CrossRef CAS IUCr Journals Google Scholar
Lyatun, I., Ershov, P., Snigireva, I. & Snigirev, A. (2020). J. Synchrotron Rad. 27, 44–50. Web of Science CrossRef CAS IUCr Journals Google Scholar
Maser, J., Stephenson, G. B., Vogt, S., Yun, W., Macrander, A., Kang, H. C., Liu, C. & Conley, R. (2004). Proc. SPIE, 5539, 185–194. CrossRef Google Scholar
Névot, L. & Croce, P. (1980). Rev. Phys. Appl. (Paris), 15, 761–779. Google Scholar
Parratt, L. G. (1954). Phys. Rev. 95, 359–369. CrossRef Web of Science Google Scholar
Punegov, V. I. (2015). Phys. Usp. 58, 419–445. Web of Science CrossRef CAS Google Scholar
Punegov, V. I. (2020). JETP Lett. 111, 376–382. Web of Science CrossRef CAS Google Scholar
Punegov, V. I. & Karpov, A. V. (2021). Acta Cryst. A77, 117–125. Web of Science CrossRef IUCr Journals Google Scholar
Punegov, V. I., Kolosov, S. I. & Pavlov, K. M. (2014). Acta Cryst. A70, 64–71. Web of Science CrossRef IUCr Journals Google Scholar
Punegov, V. I., Kolosov, S. I. & Pavlov, K. M. (2016). J. Appl. Cryst. 49, 1190–1202. Web of Science CrossRef CAS IUCr Journals Google Scholar
Punegov, V. I. & Pavlov, K. M. (1992). Sov. Tech. Phys. Lett. 18, 390–391. Google Scholar
Punegov, V. I., Pavlov, K. M., Karpov, A. V. & Faleev, N. N. (2017). J. Appl. Cryst. 50, 1256–1266. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rack, A., Weitkamp, T., Riotte, M., Grigoriev, D., Rack, T., Helfen, L., Baumbach, T., Dietsch, R., Holz, T., Krämer, M., Siewert, F., Meduňa, M., Cloetens, P. & Ziegler, E. (2010). J. Synchrotron Rad. 17, 496–510. Web of Science CrossRef CAS IUCr Journals Google Scholar
Spiller, E. & Rosenbluth, A. E. (1985). Proc. SPIE, 0563, 221–236. CrossRef CAS Google Scholar
Spiller, E. & Rosenbluth, A. E. (1986). Opt. Eng. 25, 954–963. CrossRef CAS Web of Science Google Scholar
Stearns, D. G. (1989). J. Appl. Phys. 65, 491–506. CrossRef CAS Web of Science Google Scholar
Stepanov, S. (2013). J. Phys. Conf. Ser. 425, 162006. CrossRef Google Scholar
Stepanov, S. A. (1997). Xray Server, https://xserver.gmca.aps.anl.gov/. Google Scholar
Stepanov, S. & Forrest, R. (2008). J. Appl. Cryst. 41, 958–962. Web of Science CrossRef CAS IUCr Journals Google Scholar
Takagi, S. (1962). Acta Cryst. 15, 1311–1312. CrossRef CAS IUCr Journals Web of Science Google Scholar
Takagi, S. (1969). J. Phys. Soc. Jpn, 26, 1239–1253. CrossRef CAS Web of Science Google Scholar
Tamura, K., Kunieda, H., Miyata, Y., Okajima, T., Miyazawa, T., Furuzawa, A., Awaki, H., Haba, Y., Ishibashi, K., Ishida, M., Maeda, Y., Mori, H., Tawara, Y., Yamauchi, S., Uesugi, K. & Suzuki, Y. (2018). J. Astron. Telesc. Instrum. Syst. 4, 011209. Google Scholar
Taupin, D. (1964). Bull. Soc. Fr. Miner. Crist. 87, 469–511. CAS Google Scholar
Yan, H. F., Maser, J., Macrander, A., Shen, Q., Vogt, S., Stephenson, G. B. & Kang, H. C. (2007). Phys. Rev. B, 76, 115438. Web of Science CrossRef Google Scholar
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