Origin of irregular X-ray mirage fringes from a bent, thin crystal

Fukamachi, T.; Kawamura, T.

doi:10.1107/S2053273322006143

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ISSN: 2053-2733

Volume 78| Part 5| September 2022| Pages 422-429

https://doi.org/10.1107/S2053273322006143

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Origin of irregular X-ray mirage fringes from a bent, thin crystal

Tomoe Fukamachi ^a ^* and Takaaki Kawamura ^b,^c

^aSaitama Institute of Technology, Fukaya, Saitama 369-0293, Japan, ^bCMC Center, Tokyo University of Technology, Hachioji, Tokyo 192-0982, Japan, and ^cUniversity of Yamanashi, Kofu, Yamanashi 400-8510, Japan
^*Correspondence e-mail: tomoe-f@wonder.ocn.ne.jp

Edited by I. A. Vartaniants, Deutsches Electronen-Synchrotron, Germany (Received 8 May 2022; accepted 9 June 2022; online 28 July 2022)

The dynamical theory of diffraction is used to analyse irregular X-ray mirage interference fringes observed in Si220 X-ray reflection topography from a weakly bent, thin crystal due to gravity. The origin of the irregular fringes is attributed to the interference between mirage diffracted beams and a reflected beam from the back surface, which is a new type of interference fringe. The irregular fringes are reproduced by calculating the reflected intensities numerically. The effects of absorption and thermal vibration are quite important for the reproduction. The result shows that the interference fringes depend on the strain as well as the thickness of the crystal, which indicates that the fringes should be useful for analysing weak strain in a crystal as an application.

Keywords: interference fringes; mirage fringes; bent crystal; dynamical theory of X-ray diffraction.

1. Introduction

In a weakly bent crystal with a constant strain gradient, the index of refraction is variable with respect to the depth. When the X-ray beam enters the crystal, it propagates along a hyperbolic trajectory and is reflected back to the entrance surface as shown by S₁ in Fig. 1, which is called mirage diffraction (Authier, 2001 $[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.]$ ). Jongsukswat et al. (2012 ) have observed interference fringes of mirage diffracted beams (IFMD) from a weakly bent plane-parallel crystal and measured the strain of the crystal. The IFMD are caused by the interference between two mirage diffracted beams such as S₁ and S₂ shown in Fig. 1(c). Fukamachi et al. (2011b ) have observed another type of mirage interference fringe caused by interference between diffracted beam S₁ and reflected beam $[S_{\rm B}]$ from the back surface (IFMRB) when the strain gradient of a thin crystal is very small. The fringe spacing of IFMD decreases as a function of the distance between the incident point and the exit point (x), whereas that of IFMRB increases. It is easy to distinguish IFMD and IFMRB. However, IFMD and IFMRB can coexist depending on the strain gradient and the thickness of the crystal. We report on irregular mirage interference fringes when IFMD and IFMRB coexist and their origin using the dynamical theory of diffraction.

Figure 1
(a) Observed Si220 topography from a weakly bent plane-parallel crystal. The intensity profile in Fig. 6

(a) is obtained along the horizontal dashed line. (b) Optical geometry around the crystal. P₀ and P_h represent the incident and the reflected beams, respectively. (c) Schematic illustration of the beam trajectories. $[{\varphi _1}]$ is the angle between S₁ and the lattice plane at A₀. (d) The real part of the dispersion surface (thick solid line). The abscissa represents the X axis and the ordinate the $[Z_{\rm r}]$ axis. L₀ is the Lorentz point at ( $[{\kappa _{{\rm r}x}},h/2]$ ). The part of the dispersion surface below d₂ belongs to branch (1) and the part above d₂ belongs to branch (2). The width of the shaded area corresponds to the divergent angle of the incident beam.

2. Experimental results

Fig. 1(a) shows the observed Si220 topography, which is a part of Fig. 3 of Jongsukswat et al. (2013 ). The sample was a plane-parallel Si single crystal 50 mm long, 15 mm wide and 0.28 mm thick. One end of the sample was pasted to an aluminium base with pine resin as shown in Fig. 1(b). The sample was bent due to the gravity force. The diffraction experiments were carried out at the bending-magnet beamline BL-15C, Photon Factory, KEK, Tsukuba, Japan. The incident X-rays were σ polarized and monochromated by a Si(111) double-crystal monochromator. The X-ray energy was 11100 ± 0.5 eV. The beam size was 0.02 mm long and 4 mm wide. The details of the experiment are described by Jongsukswat et al. (2013).

Fig. 1(a) shows the topography in the range of distance (l) between 16 and 20 mm from the free edge of the sample. The dark contrast around 1 mm from the left is the primary diffraction. The fringe contrasts observed on the right side of it are denoted as MIFMD (modified IFMD) to be studied in this paper. Fig. 1(c) shows schematically the trajectories of the refracted beams in the crystal for formation of IFMD and IFMRB. Here the X-ray beam propagating along the energy flow, that is the Poynting vector of the beam, is referred to as the refracted beam. P₀ and P_h represent the incident beam at the point A₀ and the primary diffracted beam, respectively. The refracted beam S₁ propagates along a hyperbolic trajectory and is emitted from the entrance surface at A₂ without reaching the back surface – it is called the first-order mirage diffracted beam. Similarly, the beam S₂ is another mirage diffracted beam reflected once from the entrance surface at A₁, called the second-order mirage diffracted beam. The beam $[S_{\rm B}]$ is reflected from the back surface and emitted from the entrance surface at A₂, called the back-surface reflected beam. The beams S₃ and S₄ should be called the second- and third-order mirage diffracted beams emitted at A₃, respectively. The longest distance x_c from the incident point to the emitted point for observing the first-order mirage diffracted beam is given by the mirage beam S_c whose vertex of the trajectory is at the back surface. Interference of the first-order mirage diffracted beam with higher-order beams and the back-surface reflected beam P_m¹ can be observed for $[x \,\lt\, {x_c}]$ , which is called the first zone hereafter. Interference between mirage diffracted beams higher than the first order P_m² can be observed for $[{x_c} \,\lt\, x \,\lt\, 2{x_c}]$ , which is called the second zone. In Fig. 1(a), the distance from the incident point to the lateral surface $[x_{\rm L}]$ is smaller than 2x_c. The second zone is given by $[{x_c} \,\lt\, x \,\lt\, x_{\rm L}]$ . The fringes on the rightmost side are the diffracted beams emitted from the lateral surface P_lh and are called IFLSD (interference fringes emitted from the lateral surface in the diffracted beam direction).

In the first zone, seven groups of fringes from M₁ to M₇ may be attributed to IFMD as will be shown in Section 4.2. As the crystal is bent due to gravity, the load is constant along the gravity. The strain gradient (β) is proportional to the square of the distance (l) from the free end according to rod theory. By using the procedure described by Jongsukswat et al. (2012), the strain gradients determined from the positions of the third peak (M₃) of IFMD are β = 0.15 mm⁻¹ at the upper end (l = 16 mm) and β = 0.23 mm⁻¹ at the lower end (l = 20 mm) of Fig. 1(a).

In the first IFMD (M₁), there are six fine interference fringes attributable to IFMRB. In the second IFMD (M₂), there are three fine interference fringes attributable to IFMRB. The third fringe M₃ shows dark contrast in the upper side and the contrast becomes darker in the lower side. The fringe M₄ in the upper side disappears around l = 18 mm. The dark contrast M₄ from the lower side disappears around the site (l ≃ 17 mm) in the middle of M₄ and M₅ coming from the upper side. The fringe M₆ in the upper side is continuously connected to the fringe M₅ in the lower side. In the second zone, three wide bands with low contrast can be seen on the lower side, which are labelled as M′₁, M′₂ and M′₃. The behaviours of these interference fringes M₁ to M₆ as well as M′₁ to M′₃ are quite irregular.

3. Theoretical analysis

3.1. Basis

In the symmetric Bragg mode, the deviation parameter (W) from the Bragg condition of the angle $[\theta _{\rm B}]$ is given by

$[W = {{\sin 2\theta _{\rm B}} \over {C{{\left({{\chi _h}{\chi _{ - h}}} \right)}^{1/2}}}}\left [{\alpha - \left({{\theta _{\rm{B}}} + {{|{\chi _0}|} \over {\sin 2\theta _{\rm B}}}} \right)} \right].\eqno(1)]$

Here α is the incident glancing angle, C the polarization factor and $[{\chi _h}]$ the hth Fourier coefficient of dielectric susceptibility of the crystal. According to Gronkowski & Malgrange (1984 ), the beam trajectory of the refracted beam in a weakly bent non-absorbing crystal is given by

$[{\left({{{\beta z} \over {\tan\theta _{\rm B}}} + W_{\rm s}} \right)^2} - {\left[{\beta x + s\left({W_{\rm s}} \right){{\left(W_{\rm s}^2 - 1 \right)}^{{1 / 2}}}} \right]^2} = 1,\eqno(2)]$

when the beam is incident outside the total reflection region ( $[|W_{\rm s}| \ge 1]$ ). $[W_{\rm s}]$ is the value of W at the incident point on the surface. The parameter $[s(W_{\rm s} )]$ is 1 for $[W_{\rm s} \,\gt\, 1]$ and −1 for $[W_{\rm s} \,\lt\, 1]$ . The trajectory shows a hyperbolic form as seen in Fig. 1(c). The origin of the coordinate is taken at the incident point (A₀), the x axis is along the direction from A₀ to A₁ and the z axis along the inward normal to the surface. The strain gradient parameter β is defined as

$[\beta = {\lambda \over {C{{\left({{\chi _h}{\chi _{ - h}}} \right)}^{1/2}}}}{{{\partial ^2}\left({{\bf{h}} \cdot {\bf{u}}} \right)} \over {\partial {x_0}\partial {x_h}}},\eqno(3)]$

where $[{\bf{h}}]$ is the reciprocal vector, $[{\bf{u}}]$ is the atomic displacement vector, λ the X-ray wavelength, x₀ and x_h are the coordinates of the transmitted and diffracted beam directions, respectively.

When $[\beta W \,\lt\, 0\,(\beta \,\gt\, 0)]$ , a beam incident on the crystal at A₀ propagates along the hyperbolic trajectory S₁ in Fig. 1(c) and changes the direction from +z to −z at the vertex ( x_a,z_a) to reach the point A₂. At A₂, a part of the beam is emitted from the surface as the mirage diffracted beam and the rest is reflected back to the crystal. The reflected beam propagates along a similar hyperbolic trajectory starting from A₂. This propagation process repeats. The electric field of the X-ray after n times of reflection is given by

$[{\bf E}_m^{\langle n \rangle }{\rm{exp}}\left({ - in{\gamma _0}} \right) = {\bf E}_0R_m^{\langle n \rangle }\exp \left({ - in{\gamma _m}} \right)\eqno(4)]$

according to Fukamachi et al. (2010 ). Here $[{\bf E}_0]$ is the electric field of the incident X-ray, $[{\gamma _0}]$ is the phase shift in vacuum and real. The phase shift of the refracted beam in the crystal $[{\gamma _m}]$ is complex by taking the absorption effect into account and given by

$[{\gamma _m} = \textstyle\int {\bf k}\cdot \,{\rm d}{\bf r} = \gamma _{m{\rm r}} + i\gamma _{m{\rm i}}.\eqno(5)]$

Here the integration is carried out along the trajectory, k is the wavevector in the crystal and r is the position vector. $[R_m^{\langle n \rangle }]$ is given as

$[R_m^{\langle n \rangle } = r_1^{n - 1}(r_1^2 - 1),\quad n \,\gt\, 0,\eqno(6)]$

where r₁ is expressed as

$[{r_1} = {{D_h^{1(1 )}} \over {D_0^{1(1 )}}}.\eqno(7)]$

D₀^{1(1 )} and D_h^{1(1 )} are the amplitudes of the 0 and hth Fourier coefficients of the electric displacement. The first number in the superscript represents the number of reflections and the second number (1) denotes the branch (1) of the dispersion surface.

We now need to calculate the phase shift $[{\gamma _m}]$ by using the dispersion surface. When the absorption is weak ( $[{k_a} \,\lt\, 0.1]$ ), the complex dispersion surface for the Bragg mode is given as

$[Z = Z_{\rm r} + iZ_{\rm i} \simeq {{G_{\rm B}} \over {2\tan \theta _{\rm B}}}{\left [{{{\left({W + ig} \right)}^2} - \left({1 + i2{k_a}} \right)} \right]^{1/2}},\eqno(8)]$

according to Fukamachi et al. (2002 ). k_a is expressed as

$[{k_a} = {{|{\chi _{h{\rm i}}}|} \over {|{\chi _{h{\rm r}}}|}}.\eqno(9)]$

$[{\chi _{h{\rm r}}}]$ and $[{\chi _{h{\rm i}}}]$ are the real and the imaginary parts of $[{\chi _h}]$ . $[G_{\rm B}]$ and g are given by

$[G_{\rm B} = {{{K_0}C{{\left({{\chi _h}{\chi _{ - h}}} \right)}^{1/2}}} \over {\cos\theta _{\rm B}}}\eqno(10)]$

and

$[g = {{{\chi _{0{\rm i}}}} \over {C{{\left({{{\left| {{\chi _{h{\rm r}}}} \right|}^2} + {{\left| {{\chi _{h{\rm i}}}} \right|}^2}} \right)}^{1/2}}}} \simeq {{{\chi _{0{\rm i}}}} \over {C\left| {{\chi _{h{\rm r}}}} \right|}}.\eqno(11)]$

By squaring both sides of equation (8) and ignoring small terms $[Z_{\rm i}^2]$ , k_a² and g², equation (8) becomes

$[\eqalignno{{\left({{Z_{\rm r}} + i{Z_{\rm i}}} \right)^2} &\simeq {Z_{\rm r}}^2 + i2Z_{\rm r}{Z_{\rm i}}&\cr &= {\left({{{G_{\rm B}} \over {2\tan \theta _{\rm B}}}} \right)^2}\left [{{W^2} - 1 + i2g\left({W - g'} \right)} \right],&(12)}]$

where the parameter [g'] is given by

$[g' = {{{k_a}} \over g}.\eqno(13)]$

The real part of equation (12) becomes

$[Z_{\rm r} = {{ \pm G_{\rm B}} \over {2\tan\theta _{\rm B}}}{\left({{W^2} - 1} \right)^{1/2}}\eqno(14)]$

which agrees with the expression without absorption, indicating the validity of equation (2). The real part of the phase shift $[{\gamma _{m{\rm r}}}]$ can be written as

$[{\gamma _{m{\rm r}}} = {\gamma _{m{\rm r}x}} + {\gamma _{m{\rm r}z}} = \textstyle\int {k_{{\rm r}x}}\,{\rm d}x + \int {k_{{\rm r}z}}\,{\rm d}z. \eqno(15)]$

Here $[{k_{{\rm r}x}}]$ and $[{k_{{\rm r}z}}]$ are x and z components of the real part $[{\bf k}_{\rm r}]$ of the wavevector k

$[{\bf k} = {\bf k}_{\rm r} + i{\bf k}_{\rm i}\eqno(16)]$

with $[{\bf k}_{\rm i}]$ its imaginary part. As the coordinate of the Lorentz point is $[({{\kappa _{{\rm r}x}},h/2} )]$ ( $[h = | {\bf{h}} |]$ ) according to Fig. 1(d), $[{\gamma _{m{\rm r}x}}]$ is rewritten as

$[{\gamma _{m{\rm r}x}}=\textstyle\int k_{{\rm r}x}\,{\rm d}x=\kappa_{{\rm r}x}x+2\textstyle\int\limits_{0}^{x/2}X\,{\rm d}x.\eqno(17)]$

$[{\kappa _{{\rm r}x}}]$ is the real part of the average wavenumber in the crystal. The first term in equation (17) $[{\kappa _{{\rm r}x}}x]$ is the order of π, which is much smaller than the second term and can be neglected. As X in equation (17) is

$[X = - {{G_{\rm B}} \over 2}W\eqno(18)]$

and

$[{\rm d}x = {1 \over \beta }{\rm d}{\left({{W^2} - 1} \right)^{{1 / 2}}}\eqno(19a)]$

$[{\rm d}z = {{\tan\theta _{\rm B}} \over \beta }{\rm d}W,\eqno(19b)]$

by using equation (2), the second term of equation (17) becomes

$[\int\limits_0^{x/2} X\,{\rm d}x = {{ - G_{\rm B}} \over {2\beta }} \int \limits_{W_{\rm s}}^{ - 1} {{{W^2}} \over {{{\left({{W^2} - 1} \right)}^{1/2}}}}\,{\rm d}W.\eqno(20)]$

The phase shift $[{\gamma _{m{\rm r}z}}]$ is obtained as

$[{\gamma _{m{\rm r}z}} = \int {k_{{\rm r}z}}\,{\rm d}z = - 2 \int\limits_0^{{z_a}} \left({{h \over 2} + {Z_{\rm r}}} \right)\,{\rm d}z = - h{z_a} - 2\int\limits_0^{{z_a}} {Z_{\rm r}}\,{\rm d}z\eqno(21)]$

by referring to Fig. 1(d). The first term on the right-hand side hz_a is the order of π and much smaller than the second term. As the second term is given by

$[\int \limits_0^{{z_a}} {Z_{\rm r}}\,{\rm d}z = {{ - G_{\rm B}} \over {2\beta }}\int \limits_{W_{\rm s}}^{ - 1} {\left({W^2 - 1} \right)^{1/2}}\,{\rm d}W,\eqno(22)]$

equation (15) becomes

$[{\gamma _{m{\rm r}}} = {\gamma _{m{\rm r}x}} + {\gamma _{m{\rm r}z}} = {{ - G_{\rm B}} \over \beta } \int \limits_{W_{\rm s}}^{ - 1} {1 \over {{{\left({{W^2} - 1} \right)}^{1/2}}}}\,{\rm d}W = {{ - G_{\rm B}} \over \beta }M\left({W_{\rm s}} \right)\eqno(23)]$

with

$[M({W_{\rm s}} ) = \log \left| {W_{\rm s} + {{\left(W_{\rm s}^2 - 1 \right)}^{1/2}}} \right|.\eqno(24)]$

From the relation of the imaginary part of equation (12),

$[Z_{\rm i} = {{ \pm 1} \over {{Z_{\rm r}}}}{\left({{{G_{\rm B}} \over {2\tan\theta _{\rm B}}}} \right)^2}g\left({W - g'} \right) = {{ \pm gG_{\rm B}} \over {2\tan\theta _{\rm B}}}{{\left({W - g'} \right)} \over {{{\left({{W^2} - 1} \right)}^{1/2}}}}\eqno(25)]$

is obtained.

For the imaginary part of the phase shift $[{\gamma _{m{\rm i}}}]$ , only the z component needs to be considered. By using the relation

$[{{gG_{\rm B}} \over {2\tan\theta _{\rm B}}} = {{{K_0}{\chi _{0{\rm i}}}} \over {2\sin\theta _{\rm B}}} = {{ - {\mu _a}} \over {2\sin\theta _{\rm B}}}\eqno(26)]$

and equation (25), $[{\gamma _{m{\rm i}}}]$ is given by

$[{\gamma _{m{\rm i}}} = \int {k_{{\rm i}z}}\,{\rm d}z = 2 \int \limits_0^{{z_a}} {Z_{\rm i}}\,{\rm d}z = {{\mu _{\rm a}} \over {\beta \cos\theta _{\rm B}}}\int \limits_{W_{\rm s}}^{ - 1} {{\left({W - g'} \right)} \over {{{\left({{W^2} - 1} \right)}^{1/2}}}}\,{\rm d}W\eqno(27)]$

$[= {{ - \mu _{\rm a}x} \over {2\cos\theta _{\rm B}}}\left [{1 + {{2\left| {g'} \right|} \over {x\left| \beta \right|}}M\left({W_{\rm s}} \right)} \right].\eqno(28)]$

Here $[\mu _{\rm a}]$ is the mean absorption coefficient. The exit point (x) of the mirage diffracted beam is given by

$[x = {{2{{\left(W_{\rm s}^2 - 1 \right)}^{1/2}}} \over {\left| \beta \right|}},\eqno(29)]$

using the relation x = 2x_a due to the symmetry of the trajectory. The initial value of the deviation parameter $[W_{\rm s}]$ is obtained from the exit point (x) of the mirage diffracted beam. For a monatomic crystal with its atomic scattering factor being positive, the condition $[W \,\lt\, -\!1]$ corresponds to the anomalous transmission of the Borrmann effect (Fukamachi et al., 2002). In the following, we will study the fringes under this condition. As the X-rays are σ polarized in the experiment, the polarization factor C in equation (1) is 1.

3.2. Mirage interference fringes

In the first zone, the electric field of IFMD at A₂ is given by

$[\eqalignno{{\bf E}_0R_m^{\langle n \rangle }\exp ({ - in\gamma _m^{\langle n \rangle }} )& = {\bf E}_0R_m^{\langle n \rangle }\exp(n\gamma _{m{\rm i}}^{\langle n \rangle })\exp ({ - in\gamma _{m{\rm r}}^{\langle n \rangle }})&\cr &= {\bf A}_m^{\langle n \rangle }\exp ({ - in\gamma _{m{\rm r}}^{\langle n \rangle }} )&(30)}]$

with $[{\bf A}_m^{\langle n \rangle }]$ as

$[{\bf A}_m^{\langle n \rangle } = {\bf E}_0R_m^{\langle n \rangle }\exp({n\gamma _{m{\rm i}}^{\langle n \rangle }} ).\eqno(31)]$

The electric field at A₂ is written by

$[\exp(-i\gamma_{m{\rm r}}^{\langle 1\rangle})[{\bf A}_m^{\langle 1\rangle} +{\bf A}_m^{\langle 2\rangle}\exp(-i\Delta\gamma_{m{\rm r}}^{\langle 2\rangle})+\ldots ]\eqno(32)]$

where the phase shift $[\Delta \gamma_{m{\rm r}}^{\langle n \rangle }]$ is defined by

$[\Delta \gamma_{m{\rm r}}^{\langle n \rangle } =n\gamma_{m{\rm r}}^{\langle n \rangle }-\gamma_{m{\rm r}}^{\langle 1\rangle}.\eqno(33)]$

In the second zone, a similar equation is obtained without the first-order mirage diffracted beam.

IFMRB observed in the first zone has been explained by the interference between the first-order mirage diffracted beam S₁ and a reflected beam from the back surface $[S_{\rm B}]$ in Fig. 1(c) as expressed by

$[\exp(-i\gamma_{m{\rm r}}^{\langle 1\rangle})[{\bf A}_m^{\langle 1\rangle} +{\bf A}_{\rm B}\exp(-i\Delta\gamma_{\rm B r})].\eqno(34)]$

The amplitude ( $[A_{\rm B} = | {\bf A}_{\rm B} |]$ ) is given by

$[A_{\rm B} = {r_1}\left({{W_{\rm Bs}}} \right)\left[{{{{r_1}\left({{W_{\rm Bs}}} \right)} \over {{r_2}\left({{W_{\rm Be}}} \right)}} - 1} \right]\exp \left({{\gamma _{\rm Bi}}} \right)\left| {{\bf E}_0} \right|,\eqno(35)]$

where $[{W_{\rm Bs}}]$ is the value of W of the beam $[S_{\rm B}]$ at the incident point and $[{W_{\rm Be}}]$ is that at the reflection point ( x/2,H) on the back surface. There is a relation between $[{W_{\rm Be}}]$ and $[{W_{\rm Bs}}]$ given by

$[{W_{\rm Be}} = {W_{\rm Bs}} + {{\beta H} \over {\tan\theta _{\rm B}}}.\eqno(36)]$

In equation (34), $[\Delta\gamma_{\rm B r}]$ is given by

$[\Delta\gamma_{{\rm Br}}={G_{\rm B}\over\beta}\{M(W_{1{\rm s}})-[M(W_{\rm Be})-M(W_{\rm Bs})]\},\eqno(37)]$

where $[W_{1{\rm s}}]$ is the value of W at the incident point for the beam S₁. $[\Delta\gamma_{{\rm Bi}}]$ in equation (35) is given by

$[\eqalignno{{\gamma _{{\rm Bi}}}& = {{{k_a}\mu _{\rm a}} \over {\beta g\cos\theta _{\rm{B}}}} \Bigl\{(W_{\rm Be}^2 - 1 )^{1 / 2} - (W_{\rm Bs}^2 - 1)^{1 / 2}&\cr &\quad - | {g'} |\left[ M(W_{\rm Be}) - M(W_{\rm Bs}) \right] \Bigr\}.&(38)}]$

In order to explain irregular mirage interference fringes observed in the experiment, it is necessary to introduce MIFMD in the first zone given by adding a back-surface reflected beam to equation (32) as

$[\eqalignno{&\exp(-i\gamma_{m{\rm r}}^{\langle 1\rangle})[{\bf A}_m^{\langle 1\rangle}+{\bf A}_m^{\langle 2\rangle}\exp(-i\Delta\gamma_{m{\rm r}}^{\langle 2\rangle})+\ldots &\cr &\quad +{\bf A}_{\rm B}\exp(-i\Delta\gamma_{{\rm Br}})].&(39)}]$

MIFMD in the second zone is composed of mirage diffracted beams higher than the first order, as the contribution of the back-surface reflected beam is small enough to be neglected.

3.3. Angular amplification

For observing MIFMD, it is necessary to have a certain width of ΔW to excite the first- as well as the higher-order mirage diffracted beams simultaneously. There is a relation

$[\tan\varphi = {{{{\left({{W^2} - 1} \right)}^{1/2}}} \over {\left| W \right|}}\tan\theta _{\rm B}\eqno(40)]$

between W and φ which is the angle between the refracted beam and the lattice plane; $[\varphi = 0]$ when [W = - 1] and $[\varphi \simeq \theta _{\rm B}]$ when [W = - 3] (ΔW = 2). If $[\Delta W\simeq 2]$ , the divergence angle of the refracted beam is approximately equal to $[\theta _{\rm B}]$ , the refracted beams are excited within the Borrmann triangle. The divergence angle of the incident X-rays corresponding to ΔW = 2 can be derived as Δα = 8.8 µrad by using equation (1). The angle amplification factor (Δφ/Δα) is approximately 3.1 ×10⁴.

As the topography was taken by fixing the crystal as shown in Fig. 1(c), the incident glancing angle α was fixed. In order to observe mirage interference fringes, a finite divergence angle Δα is needed. The divergence angle Δα is related to the divergence angle $[\Delta\theta_{\rm B}]$ of the beam from the monochromator as

$[|\Delta\theta_{\rm B}|=\sin^2\theta_{\rm B}|\Delta\alpha|\eqno(41)]$

(Fukamachi et al., 2014 , 2015 , 2019 ). By using the Bragg condition, the width ΔE of X-ray energy (E) and the angle width $[\Delta\theta_{\rm B}]$ are related as

$[\left|{{\Delta E}\over E}\right| = \left|{{\Delta\theta_{\rm B}}\over{\tan\theta_{\rm B}}}\right|.\eqno(42)]$

The refracted beams involved in the formation of mirage interference fringes have different wavelengths and different path lengths. Since the mirage interference fringes are observed in the experiment, the coherent condition is satisfied for the refracted beams. The details of the coherent condition have already been given in the previous papers by Fukamachi et al. (2014, 2015, 2019).

4. Results of calculation

4.1. Effects of absorption and thermal vibration

Fig. 2 shows the calculated results of MIFMD composed of mirage diffracted beams and a back-surface reflected beam. The used values of the strain gradient β and the parameter [| {g'} |] are 0.2 mm⁻¹ and 1.0, respectively. In the first zone ( $[0 \,\lt\, x \,\lt\, {x_c}]$ ), the interference fringes are composed of mirage diffracted beams from the first to the tenth order and one back-surface reflected beam. In the second zone ( $[x \,\gt\, {x_c}]$ ), the interference fringes are composed of mirage diffracted beams from the second to the tenth order.

Figure 2
The calculated Si220 reflected intensities of MIFMD for k_a = 0.02 (in black) and k_a = 0 (in blue) in the case of β = 0.20 mm⁻¹, x_c = 6.3 mm and $[x_{\rm L}]$ = 8.4 mm.

The effect of the mean absorption $[\mu _{\rm a}]$ is shown in Fig. 2. According to equation (26), we have

$[\mu _{\rm a} = \cos\theta _{\rm B}G_{\rm B}{k_a}.\eqno(43)]$

When the absorption is ignored, k_a is zero. When the imaginary part of the anomalous scattering factor of Si is taken into account, k_a is 0.02. The blue thin and the black thick curves show MIFMD for k_a = 0 and k_a = 0.02, respectively. Peaks indicated as M₁ to M₇ in the first zone and $[{\rm{M}}'_1]$ and $[{\rm{M}}'_2]$ in the second zone are eventually attributed to MIFMD. When k_a = 0, the peak M₅ is about twice higher than M₁, and the height of $[{\rm{M}}'_1]$ is comparable with that of M₁. When k_a = 0.02, all the peak heights are roughly a quarter of those for k_a = 0. The peak heights from M₁ to M₇ are not so much different, showing a similar trend of variations in the experiment. However, $[{\rm{M}}'_1]$ shows a similar peak height to M₁ while $[{\rm{M}}'_1]$ in Fig. 1(a) shows a much lower peak than M₁. It is not possible to reproduce MIFMD observed in the experiment only by taking into account the effect of mean absorption.

The thermal vibration effect can be taken into account through $[{\gamma _{\rm Bi}}]$ in equation (28), as it includes the term [g'] , which is given by

$[\left| {g'} \right| = {{\left| {{\chi _{h{\rm i}}}} \right|} \over {\left| {{\chi _{0{\rm i}}}} \right|}} = \exp \left({ - B{s^2}} \right),\eqno(44)]$

where B expresses the thermal vibration effect and $[s = \sin\theta _{\rm B}/\lambda]$ . The calculated MIFMD are shown in Fig. 3. The orange, blue and black lines show MIFMD for [| {g'} |] = 1.0, 0.98 and 0.96, respectively. When = 1.0, no thermal vibration is taken into account. When [| {g'}|] becomes small, the overall peak heights become small. When = 0.98, the peak height of M′₁ is about 1/3 of M₁. The peak height of M′₁ becomes about 1/10 of M₁ for = 0.96. By comparing with the experimental results, we adopt 0.96 as the value of [| {g'} |] . The value of B corresponding to [| {g'} | =] 0.96 is 0.60 Å², which is certainly larger than reported values such as 0.469 Å² by Flensburg & Stewart (1999 ) and 0.4833 Å² by Sang et al. (2010 ). The large value may come from an anisotropic or anharmonic vibrational effect. A further study is needed to confirm such a vibrational effect quantitatively.

Figure 3
The calculated Si220 reflected intensities of MIFMD by taking the thermal vibration effect into account when β = 0.20 mm⁻¹ and k_a = 0.02. The orange, blue and black lines show the intensities for [| {g'}| = 1]

, 0.98 and 0.96, respectively.

4.2. Comparison of MIFMD with IFMD and IFMRB

In Fig. 4 are shown the calculated intensities of IFMD (orange), IFMRB (black) and MIFMD (blue). The strain gradient β is assumed to be 0.20 mm⁻¹. The intensity of IFMD shows a slow variation as a function of x. The width of a fringe and the interval between neighbouring fringes become small when x increases. The peak height in M_n decreases as n (x) increases. The peak of M₆ is extremely small compared with the neighbouring peaks. IFMRB starts to appear from $[{x_{\min}} = H/\tan\theta _{\rm B}]$ . The width of the fringe and the interval between neighbouring fringes become large as x increases. The intervals between neighbouring fringes of IFMRB are smaller than those of IFMD when $[x - {x_{\min}}]$ is small, but these two intervals are comparable when x becomes close to x_c. For M₁, MIFMD is approximately the sum of IFMD and IFMRB. For M₂ and M₃, two peaks appear in MIFMD by the influence of IFMRB. For M₃, the peak of IFMD and the valley of IFMRB appear at the same x and the peak of MIFMD is lowered. For M₅ to M₇, the intervals of the fringes of IFMD and IFMRB are almost the same and the peak heights of MIFMD are enhanced more than twice those of IFMD.

Figure 4
The calculated Si220 reflected intensities of IFMD (in orange), IFMRB (in black) and MIFMD (in blue) for β = 0.20 mm⁻¹, k_a = 0.02 and [| {g'} | = 0.96]

4.3. Comparison of topographies

Fig. 5 shows the measured topography (a), the calculated topographies of MIFMD (b), IFMD (c) and IFMRB (d). The abscissa is the distance l in (a) and the strain gradient β in (b)–(d). The correspondence between l and β is described in Section 2.

Figure 5
(a) The observed Si220 topography. (b), (c) and (d) show the calculated topographies of MIFMD, IFMD and IFMRB, respectively, for k_a = 0.02 and [| {g'} | = 0.96]

. Fringes in (d) are numbered as B_n ( $[n = 1,2,\ldots]$ ) from x_c to x_min, as the phase $[\Delta\gamma_{\rm Br}]$ is zero at x_c and becomes large as x decreases. The peak appears at a point x corresponding to $[\Delta\gamma_{\rm Br} = \pi(2n-1)]$ in equation (37)

In Fig. 5(a), there are three dark contrasts of M₂ on the left end (l = 20 mm). The uppermost dark contrast becomes the lowest fringe of dark contrast of M₁ on the right end (l = 16 mm). There are two dark contrasts of M₃ on the left end. The upper dark contrast is weaker than the lower one. The upper dark contrast is connected to the lowest dark contrast of M₂ on the right end after showing weak contrast around l = 19 mm. The lower dark contrast of M₃ on the left end becomes the dark contrast of M₃ on the right end after showing weak contrast around l = 17 mm.

In Fig. 5(b), there are three dark contrasts of M₂ on the left end. The dark uppermost contrast becomes weak around β = 0.19 mm⁻¹, then it is connected to the lowest dark contrast of M₁ on the right end. There are two dark contrasts of M₃ on the left end. The upper contrast is weaker than the lower one. The contrast disappears around β = 0.21 mm⁻¹, then becomes dark and the lowest dark contrast of M₂ on the right end. The lowest dark contrast of M₃ on the left end disappears around β = 0.17 mm⁻¹, then becomes the dark contrast of M₃ on the right end. Similar behaviours of fringe contrasts of the measured topography in Fig. 5(a) are obtained in the calculated MIFMD topography in (b).

In the calculated IFMD topography in Fig. 5(c), the fringe M₁ shows a wide band of dark contrast. The higher-order fringes M₂ to M₇ show the narrower bands and the smaller interval between the neighbouring fringes than M₁. In the calculated IFMRB topography in Fig. 5(d), the width of the dark contrasts becomes monotonously large and the distance between the neighbouring fringes becomes large when x increases from x_min. Around x = 3.5 mm, the dark contrast of M₂ appears at the left end in (c) and the dark contrast of B₁₀ appears at the left end in (d). The shift of M₂ is 1.2 mm from the left to the right end, while that of B₁₀ is 0.3 mm. The shift of M₂ is four times larger than that of B₁₀. Similarly, the shift of M₇ is larger than that of B₁. The variations of interference fringes both in IFMD (c) and IFMRB (d) topographies are regular as a function of x. The irregular variations observed in experiment (a) are only reproduced in MIFMD topography (b). The irregular variations are caused by the different variations between IFMD and IFMRB as a function of x as well as β.

5. Discussion and conclusion

The irregular X-ray mirage interference fringes reported by Jongsukswat et al. (2013) were analysed using the dynamical theory of diffraction. It is necessary to take the absorption as well as thermal vibration effects into account. The absorption effect reduces the peak intensities of the high-order fringes in the first zone. The thermal vibration effect reduces mainly the peak intensities of fringes in the second zone. The calculated MIFMD reproduces the observed irregular modulation of the fringes. The origin of the modulation is attributed to the interference of two or more mirage diffracted beams with a beam reflected from the back surface.

There are still two points that are unclear. (i) Fig. 6 shows (a) the line profile of the measured fringes along the dashed line in Fig. 1(a) and (b) that of the calculated reflected intensities of MIFMD. The peak positions of M₁ and M₂ in the calculated profile appear at x closer to the incident point than in the measured one. One of the possible reasons is the dependence of the strain gradient (β) on the distance (x), which is difficult to estimate by using the deflection theory. More precise analysis of the MIFMD is necessary in future work. (ii) The other point is related to IFLSD observed in Fig. 1(a). The intensities of IFLSD are much higher than those of MIFMD in the second zone. Hirano et al. (2008 , 2009a ,b ) have observed IFLSD from a plane-parallel crystal without distortion and pointed out that the fringes are caused by the interference between the beam directly reaching the lateral surface and the beam reflected once from the back surface. In the present geometry of a bent crystal, there is no beam reaching directly the lateral surface as shown in Fig. 1(c). It is necessary to apply the dynamical theory of diffraction for a distorted crystal for analysing the IFLSD and the strain gradient of the crystal. As an application, the X-ray beams of IFLSD can be used as an X-ray waveguide, since they propagate quite a long distance from the incident point to the exit point. Fukamachi et al. (2011a ) carried out an experiment on an X-ray diffractometer by using X-rays of IFLSD from a plane-parallel crystal as a waveguide and beam splitter. Constructive interference between many beams is derived in the present analysis of MIFMD, which should be useful for developing an X-ray waveguide using IFLSD from a bent crystal.

Figure 6
(a) The intensity profile of the observed Si220 topography along the white dashed line in Fig. 1

(a). (b) The corresponding calculated intensity profile of MIFMD for β = 0.20 mm⁻¹, k_a = 0.02 and [| {g'} | = 0.96]

Acknowledgements

The authors thank Professors Donging Ju and Riichirou Negishi, and Doctor Sukswat Jongsukswat for valuable discussions.

References

Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press. Google Scholar
Flensburg, C. & Stewart, R. F. (1999). Phys. Rev. B, 60, 284–291. Web of Science CrossRef CAS Google Scholar
Fukamachi, T., Jongsukswat, S., Ju, D., Negishi, R., Hirano, K. & Kawamura, T. (2014). J. Appl. Cryst. 47, 1267–1272. Web of Science CrossRef CAS IUCr Journals Google Scholar
Fukamachi, T., Jongsukswat, S., Ju, D., Negishi, R., Hirano, K. & Kawamura, T. (2015). J. Appl. Cryst. 48, 312–312. Web of Science CrossRef CAS IUCr Journals Google Scholar
Fukamachi, T., Jongsukswat, S., Ju, D., Negishi, R., Hirano, K. & Kawamura, T. (2019). Acta Cryst. A75, 842–850. Web of Science CrossRef IUCr Journals Google Scholar
Fukamachi, T., Jongsukswat, S., Kanematsu, Y., Hirano, K., Negishi, R., Shimojo, M., Ju, D., Hirano, K. & Kawamura, T. (2011a). J. Phys. Soc. Jpn, 80, 083001. Web of Science CrossRef Google Scholar
Fukamachi, T., Jongsukswat, S., Kanematsu, Y., Hirano, K., Negishi, R., Shimojo, M., Ju, D., Hirano, K. & Kawamura, T. (2011b). J. Phys. Soc. Jpn, 80, 083002. Web of Science CrossRef Google Scholar
Fukamachi, T., Negishi, R., Zhou, S., Yoshizawa, M. & Kawamura, T. (2002). Acta Cryst. A58, 552–558. Web of Science CrossRef CAS IUCr Journals Google Scholar
Fukamachi, T., Tohyama, M., Hirano, K., Yoshizawa, M., Negishi, R., Ju, D., Hirano, K. & Kawamura, T. (2010). Acta Cryst. A66, 421–426. Web of Science CrossRef CAS IUCr Journals Google Scholar
Gronkowski, J. & Malgrange, C. (1984). Acta Cryst. A40, 507–514. CrossRef CAS Web of Science IUCr Journals Google Scholar
Hirano, K., Fukamachi, T., Yoshizawa, M., Negishi, R., Hirano, K. & Kawamura, T. (2009a). Acta Cryst. A65, 253–258. Web of Science CrossRef IUCr Journals Google Scholar
Hirano, K., Fukamachi, T., Yoshizawa, M., Negishi, R., Hirano, K. & Kawamura, T. (2009b). Phys. Status Solidi A, 206, 1855–1859. Web of Science CrossRef CAS Google Scholar
Hirano, K., Fukamachi, T., Yoshizawa, M., Negishi, R., Hirano, K., Xu, Z. & Kawamura, T. (2008). J. Phys. Soc. Jpn, 77, 103707. Web of Science CrossRef Google Scholar
Jongsukswat, S., Fukamachi, T., Hirano, K., Ju, D., Negishi, R., Shimojo, M., Hirano, K. & Kawamura, T. (2012). Jpn. J. Appl. Phys. 51, 076702. CrossRef Google Scholar
Jongsukswat, S., Fukamachi, T., Ju, D., Negishi, R., Hirano, K. & Kawamura, T. (2013). J. Appl. Cryst. 46, 1261–1265. Web of Science CrossRef CAS IUCr Journals Google Scholar
Sang, X. H., Kulovits, A. & Wiezorek, J. M. K. (2010). Acta Cryst. A66, 685–693. Web of Science CrossRef IUCr Journals Google Scholar