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An X-ray diffractometer using mirage diffraction

aSaitama Institute of Technology, Fukaya, Saitama 369-0293, Japan, bInstitute of Material Structure Science, KEK-PF, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan, and cUniversity of Yamanashi, Kofu, 400-8510, Japan
*Correspondence e-mail: tomoe-f@wonder.ocn.ne.jp

(Received 11 September 2013; accepted 26 May 2014; online 19 July 2014)

Some characteristics are reported of a triple-crystal diffractometer with a (+, −, +) setting of Si(220) using mirage diffraction. The first crystal is flat, while the second and third crystals are bent. Basically, the first crystal is used as a collimator, the second as a monochromator and the third as the sample. The third crystal also works as an analyzer. The advantages of this diffractometer are that its setup is easy, its structure is simple, the divergence angle from the second crystal is small and the energy resolution of the third crystal is high, of the order of sub-meV.

1. Introduction

The refracted beam of an X-ray in a bent perfect crystal propagates along a hyperbolic trajectory and comes back to the incident surface in the Bragg geometry. The refracted beam is referred to a beam representing the Poynting vector of the X-ray in this paper. Authier (2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.], p. 355) pointed out that the behavior of the refracted beam in a bent crystal resembled a mirage in optics. We call the diffracted beam coming out of the crystal a mirage diffraction beam. Under anomalous transmission conditions, the divergence angle of the refracted beam is quite large compared with that of the incident beam. The refracted beam can be regarded as a quasi-spherical wave, even when the divergence angle of the incident beam is smaller than 1′′ (Authier, 2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.], p. 313). When mirage diffraction beams interfere with each other, this results in interference fringes, which are called mirage interference fringes (Fukamachi et al., 2010[Fukamachi, T., Tohyama, M., Hirano, K., Yoshizawa, M., Negishi, R., Ju, D., Hirano, K. & Kawamura, T. (2010). Acta Cryst. A66, 421-426.]). Such mirage interference fringes were first observed by Zaumseil (1978[Zaumseil, P. (1978). Krist. Tech. 13, 983-990.]). Mirage interference fringes have been used for evaluation of the strain gradient in a bent crystal (Jongsukswat et al., 2012[Jongsukswat, S., Fukamachi, T., Hirano, K., Ju, D., Negishi, R., Shimojo, M., Hirano, K. & Kawamura, T. (2012). Jpn. J. Appl. Phys. 51, 076702.]).

In this paper, we will describe a triple-crystal diffractometer using mirage interference fringes of Si(220) and some characteristics of the diffractometer.

2. Theoretical basis

According to Gronkowski & Malgrange (1984[Gronkowski, J. & Malgrange, C. (1984). Acta Cryst. A40, 507-514.]), the trajectory of the refracted beam in a bent crystal is given for the symmetric Bragg geometry as

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

for [| {{W_{\rm s}}} | \ge 1], where s(Ws) = −1 for Ws < −1 and s(Ws) = 1 for Ws > 1. [{\theta _{\rm{B}}}] is the Bragg angle, and x and z are the coordinates parallel and normal to the crystal surface in the incident plane, respectively, with the origin at the incident point of the X-ray on the surface. The parameter Ws is the initial value of the deviation from the Bragg condition W, which is defined by

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

with [\alpha] being the incident glancing angle and C the polarization factor. [{\chi _h} = |{\chi _h}|\exp (i{\varphi _r})] is the hth Fourier coefficient of the X-ray polarizability and [\beta] is a parameter corresponding to the strain gradient in the crystal, defined by

[\beta = {\lambda \over {C{{({\chi _h}{\chi _{ - h}})}^{1/2}}\cos {\theta _{\rm B}}}}{{{\partial ^2}({\bf{h}} \cdot {\bf{u}})} \over {\partial {x_0}\partial {x_{\rm h}}}}. \eqno(3)]

Here, [\lambda] is the X-ray wavelength, h the reciprocal vector corresponding to the hth reflection and [{\bf{u}}] the displacement vector; x0 and xh are the coordinates in the directions of the transmitted and diffracted beams, respectively. Under anomalous transmission conditions ([{W_{\rm s}} \le - 1]), the trajectory of the refracted beam has a hyperbolic form, as shown in Fig. 1[link] for [{W_{\rm s}}\beta] < 0. The eccentricity of the hyperbola is related to [\beta]. By taking derivatives of equation (2)[link], the divergence angle ([\Delta \alpha]) of the incident beam is related to the change of W as

[|\Delta \alpha | = \left| {\Delta W{{C{{({\chi _h}{\chi _{ - h}})}^{1/2}}} \over {\sin 2{\theta _{\rm{B}}}}}} \right| \eqno(4)]

when [{\theta _{\rm{B}}}] is fixed. In Fig. 1[link](a), the angle [\gamma] between the refracted beam and the surface is given by

[\tan \gamma = {{1 - |{r^2}|} \over {1 + |{r^2}|}}\tan \theta {}_{\rm{B}}. \eqno(5)]

Here the reflectivity (r) is defined by [r = | {{\bf{D}}_{{\rm h}}^{(1)}} |/| {{\bf{D}}_0^{(1)}} |] with [{\bf{D}}_0^{(1)}] and [{\bf{D}}_{{\rm h}}^{(1)}] being the electric displacement vectors of the incident and diffracted beams, respectively. The superscript on [{\bf{D}}] represents the branch index. For a non-absorbing crystal, r is expressed by

[r = \exp (i{\varphi _{r}}) [- W - {({W^2} - 1)^{1/2}}]. \eqno(6)]

For W = −1, [|r{|^2}] = 1 and [\gamma] = 0, and for W = −2, [|r{|^2}] = 0.07 and [\gamma] is approximately equal to [{\theta _{\rm{B}}}]. The angle amplification rate A between the changes in [\gamma] and [\alpha] in reflection geometry is given by Authier (2001[Authier, A. (2001). Dynamical Theory of X-ray Diffraction. Oxford University Press.], p. 313) as

[A = \left| {{{\Delta \gamma } \over {\Delta \alpha }}} \right| = {{2\cos^2\theta {}_{\rm{B}}\sin ^2\theta {}_{\rm{B}}} \over {C{{({\chi _h}{\chi _{ - h}})}^{1/2}}({W^2} - \sin ^2\theta {}_{\rm{B}}){{({W^2} - 1)}^{1/2}}}}.\eqno(7)]

Since A is between 104 and 106, even when [\Delta \alpha] is less than 1′′ and the incident beam can be regarded as a quasi-plane wave, [\Delta \gamma] is nearly equal to the Bragg angle, and the refracted beam can be regarded as a quasi-spherical wave if [| W |] is close to 1. This means that this single crystal works as a lens. This angle amplification can also be applied to a monochromator reflecting X-rays with very small divergence angle as well as to an analyzer with high energy resolution. Authier (1960[Authier, A. (1960). C. R. Acad. Sci. Paris, 251, 2003-2005.]) obtained a highly collimated incident beam by using this angle amplification in the transmission geometry so as to verify the double refraction.

[Figure 1]
Figure 1
Schematic illustrations of beam geometries in a bent single crystal. (a) The trajectory of the refracted beam in a bent crystal. (b) Trajectories of mirage diffraction beams emitted from A2, showing interference fringes. (c) Trajectories of the refracted beams for Ws from −1.1 to −1.5, when the X-ray energy is 11 100 eV and [\beta] is 1.0 mm−1 for the Si(220) reflection.

In Fig. 1[link](b) is shown a schematic illustration of interference fringes between two mirage diffraction beams. The refracted beams [{{\bf{S}}_{\rm m1}}] and [{{\bf{S}}_{\rm m2}}] correspond to the incident beams for the values of W1 and W2, respectively, satisfying the relation W2 = - (W12 + 3)1/2/2. In Fig. 1[link](c), trajectories are shown of the refracted beams when Ws changes from −1.1 to −1.5 in the case of the Si(220) reflection. The X-ray energy is 11 100 eV and [\beta] is 1 mm−1.

The intensities of the mirage diffraction beams are measured as a function of the distance x from the incident point of the beam to the emission point of the mirage diffraction beam. By using equation (1)[link], the deviation ([\delta W]) of the parameter of Ws from −1 is given by

[\delta W = {W_{\rm s}} + 1 = 1 - {\left({1 + {{{\beta ^2}{x^2}} / 4}} \right)^{1/2}}.\eqno(8)]

It is possible to get the value of [\delta W] by measuring the position of mirage interference fringes x. Fig. 2[link] shows the diffraction geometry for an X-ray with energy E and glancing angle [{\alpha _0}]. The thick solid line shows the dispersion surface in the crystal for the Bragg angle [{\theta _{\rm B}}] in the Bragg arrangement. Lo is the Lorentz point, La is the Laue point and c1 on the X axis is the point corresponding to Ws = −1. The nomenclature of the points Lo and La is adopted according to the books by Pinsker (1977[Pinsker, Z. G. (1977). Dynamical Scattering of X-rays in Crystals, pp. 31-32. Berlin: Springer.]) and Authier (1960[Authier, A. (1960). C. R. Acad. Sci. Paris, 251, 2003-2005.], pp. 68–71). The line T0′ represents the dispersion surface in a vacuum and is parallel to the asymptote T0 of the hyperbola of the dispersion surface in the crystal. [{{\bf{K}}_0}(\alpha {}_0)] is the wavevector of the incident beam whose glancing angle is [\alpha {}_0] and energy is E. We assume that the perpendicular line v1 passing the tie point corresponding to [{{\bf{K}}_0}(\alpha {}_0)] crosses the dispersion surface at c1. The refracted beam [{{\bf{S}}_1}] excited at the point c1 runs parallel to the crystal surface. The distance X0 from Lo to c1 is given by

[{X_0} = {{{K_0}\left| {{\chi _h}} \right|} \over {2\cos {\theta _{\rm{B}}}}}, \eqno(9)]

where [{K_0} = |{{\bf{K}}_0}({\alpha _0})|], C = 1 for [\sigma] polarization and [{({\chi _h}{\chi _{ - h}})^{1/2}} = |{\chi _h}|]. When the glancing angle changes from [\alpha {}_0] to [{\alpha _0} - \delta \alpha], the corresponding perpendicular line changes from v1 to v2 and the parameter Ws changes from −1 to [-1+ \delta W]. The tie point in a vacuum moves from a1 to a2 and that in the crystal moves from c1 to c2. From the tie point c2, the refracted beam [{{\bf{S}}_2}] is excited. In Fig. 2[link], the relation

[|\delta W{X_0}| = \left| {\delta {{\bf{K}}_\alpha }} \right|\sin {\theta _{\rm{B}}} \eqno(10)]

is obtained.

[Figure 2]
Figure 2
Dispersion surface in a Bragg geometry. La is the Laue point, Lo the Lorentz point and c1 the tie point for Ws = −1. The solid line [T_0'] represents the dispersion surface in a vacuum for an incident beam of energy E and the solid curve the corresponding dispersion surface of branches (1) and (2) in the crystal. The dot–dashed curve T0 represents the asymptote of the dispersion surface in the crystal. The dashed line represents the dispersion surface in a vacuum for an X-ray of energy [E - \delta E]. The angle change [\delta \alpha] gives the same change in [\delta W{X_0}] as the change in the wavevector [\delta {{\bf{K}}_E}] gives when the energy changes from E to [E - \delta E]. The relations [|{{\bf{K}}_0}({\alpha _0})| = |{{\bf{K}}_0}({\alpha _0} - \delta \alpha)|] and [| {{{\bf{K}}_0}({\theta _{\rm{B}}})} | =] [| {{\bf{K}}_0'({\theta _{\rm{B}}} + \delta {\theta _{\rm{B}}})} | +] [| {\delta {{\bf{K}}_E}} |] hold.

3. Experimental

The experiment was carried out by using X-rays from synchrotron radiation at BL-15C, KEK-PF, Tsukuba, Japan. The optical system is shown in Fig. 3[link](d). The X-rays were [\sigma] polarized and the energy was tuned to 11 100 ± 0.5 eV by using a double-crystal Si(111) monochromator. After Slit 1, the first crystal was basically used as a collimator, the second as a monochromator and the third as the sample, as shown in Fig. 3[link](d). The first crystal was flat. The second and third crystals were bent by applying force in the backward and forward directions of gravity, as shown in Figs. 3[link](b) and 3[link](c), respectively. The three plane parallel crystals were prepared by non-disturbance polishing at Sharan Instrument Corporation. The crystals were 50 mm long, 15 mm wide and 0.28 mm thick. The usual Bragg diffracted beam (1Ph) passed through the second slit (Slit 2), and the first peak of the mirage interference fringes ([^2 P_{\rm m}^{\langle 1 \rangle}]) passed through the third slit (Slit 3). Here the left superscript on P represents either the first (1), second (2) or third (3) crystal. The right superscript on Pm represents a serial peak number of the mirage interference fringes.

[Figure 3]
Figure 3
(a) Photograph of the cantilever jig and the crystal. (b) Geometry of the second crystal. The force is applied in the opposite direction to gravity. (c) Geometry of the third crystal. The force is applied in the direction of gravity. (d) Schematic diagram of the triple-crystal diffractometer with (+, −, +) setting. The vertical width of Slit 2 is 0.02 mm, and those of Slit 3 and Slit 4 are 0.04 mm.

The mirage diffraction intensities of [^2 P_{\rm m}^{\langle n \rangle}] from the second crystal are shown as a function of the distance x in Fig. 4[link](a), where the glancing angle was fixed and β was 0.73 mm−1. The intensities were measured by moving Slit 3 in front of the scintillation counter SC2 in Fig. 3[link](d) after removing the third crystal. The value of β was determined by measuring the position of the third peak as reported by Jongsukswat et al. (2012[Jongsukswat, S., Fukamachi, T., Hirano, K., Ju, D., Negishi, R., Shimojo, M., Hirano, K. & Kawamura, T. (2012). Jpn. J. Appl. Phys. 51, 076702.]). The mirage diffraction intensities of [^3P_{\rm m}^{\langle n \rangle}] from the third crystal are shown as a function of x in Fig. 4[link](b), where the glancing angle was fixed and β was 0.63 mm−1. The intensities were measured by moving Slit 4 in front of the scintillation counter SC1. The vertical width of Slit 2 was equal to 0.02 mm and those of Slit 3 and Slit 4 were 0.04 mm.

[Figure 4]
Figure 4
Intensities of mirage interference fringes as a function of the distance x. (a) Intensities of mirage interference fringes from the second crystal and (b) those from the third crystal.

Rocking curves of 1Ph from the first crystal and 2Ph from the second crystal are shown in Fig. 5[link](a). The FWHM of the curve 1Ph is 9′′, which is twice as large as that of the curve 2Ph of 4.5′′. The rocking curve of the first peak ([^2P_{\rm m}^{\langle 1 \rangle}]) of the mirage interference fringes, with an FWHM of 3.8′′, is shown in Fig. 5[link](b) (dots). It has an asymmetric form characteristic of the rocking curve from a weakly absorbing crystal. The slopes of both shoulders of the peak are steeper than those of 2Ph. The rocking curve of 3Ph from the third crystal is shown in Fig. 5[link](c) (dots). It also has an asymmetric form and is in good agreement with the curve (solid line) calculated by taking the absorption effect into account. The FWHM of the peak is 4′′. The rocking curves of 3Ph, [^3P_{\rm m}^{\langle 1 \rangle}] and 3Pt +3Plt from the third crystal are shown in Fig. 6[link]. The ordinate is the intensities and the abscissa is the incident angle. The origin of the angle (0) is taken at the center of 3Ph, which corresponds to the Lorentz point. The mirage fringe intensity [^3P_{\rm m}^{\langle 1 \rangle}] is measured by the scintillation counter SC1 after setting Slit 4 at the peak position of the curve of [^3P_{\rm m}^{\langle 1 \rangle}]. 3Pt is the intensity of the transmitted beam and 3Plt is the intensity of the emitted beam in the direction of the transmitted beam from the lateral surface of the crystal. These intensities are measured by the scintillation counter SC2. The sharp peak of [^3P_{\rm m}^{\langle 1 \rangle}], with an FWHM of 0.6′′, appears between the peaks of 3Ph and 3Pt +3Plt. The peaks of [{}^3P_{\rm m}^{ \langle 1 \rangle }] and 3Pt + 3Plt appear on the negative angle side where the anomalous transmission occurs.

[Figure 5]
Figure 5
(a) Rocking curve of 1Ph from the first crystal (open circles) and that of 2Ph from the second crystal (solid circles). (b) Rocking curve of [{}^2P_{\rm m}^{ \langle 1 \rangle }] from the second crystal (solid circles). (c) Rocking curve of 3Ph from the third crystal (solid circles). The solid curves in (b) and (c) are the curves of Ph for the Si(220) reflection calculated by taking the absorption effect into account.
[Figure 6]
Figure 6
Rocking curves of 3Ph (open circles), 3Pt + 3Plt (triangles) and [{}^3P_{\rm m}^{ \langle 1 \rangle }] (solid circles) from the third crystal. The solid line is the calculated curve of Ph.

In Fig. 7[link] are shown the four rocking curves [^3P_{\rm m}^{\langle n \rangle}] for n = 1, 2, 3 and 4 from the third crystal, together with the rocking curves of 3Pt + 3Plt. The origin of the angle is taken at the center of the peak of [{}^3P_{\rm m}^{ \langle 2 \rangle }]. Each curve of [^3P_{\rm m}^{\langle n \rangle} ] was measured by the scintillation counter SC1 after setting Slit 4 at the peak position of the mirage interference fringes, and the curve of 3Pt + 3Plt was measured simultaneously by the scintillation counter SC2, as shown in Fig. 3[link](d). When n increases, the peak position of [^3P_{\rm m}^{\langle n \rangle}] moves to the lower incident angle side, close to the peak of the curve 3Pt + 3Plt. The angle difference between the two peaks [^3P_{\rm m} ^{\langle 1 \rangle}] and [^3P_{\rm m} ^{\langle 4 \rangle}] is 0.3′′. The average angle difference between two adjacent peaks is 0.1′′. The curve of 3Pt + 3Plt shows two peaks: one, corresponding to the peak of 3Plt, appears around the origin, and the other, corresponding to the peak of the transmitted beam 3Pt, appears around the angle of −1.3′′. When the rocking curves of the nth (n = 1, 2, 3 and 4) peak of the interference fringes [{}^3P_{\rm m}^{ \langle n \rangle }] are measured after setting Slit 4 at its peak position, the measured peak of 3Pt + 3Plt stays at the same position as shown in Fig. 7[link]. The peak of 3Pt + 3Plt is a good reference point for the incident angle. It is possible to determine a very small angle difference between two peaks of [{}^3P_{\rm m}^{ \langle n \rangle }] and [{}^3P_{\rm m}^{ \langle n + 1 \rangle }] by measuring the curves of [{}^3P_{\rm m}^{ \langle n \rangle }] and 3Pt + 3Plt simultaneously.

[Figure 7]
Figure 7
Rocking curves of [{}^3P_{\rm m}^{ \langle n \rangle }] and 3Pt + 3Plt from the third crystal. Solid squares shows [{}^3P_{\rm m}^{ \langle 1 \rangle }], open circles [{}^3P_{\rm m}^{ \langle 2 \rangle }], solid circles [{}^3P_{\rm m}^{ \langle 3 \rangle }] and open squares [{}^3P_{\rm m}^{ \langle 4 \rangle }]. The rocking curves of 3Pt + 3Plt (left) were measured simultaneously with the curve of [{}^3P_{\rm m}^{ \langle n \rangle }] with n = 1, 2, 3 and 4. The peak angles of [{}^3P_{\rm m}^{ \langle n \rangle }] for n = 1, 2, 3 and 4 correspond to the angles of the refracted beam [{{\bf{S}}_{\rm m1}}] in Fig. 1[link](b). The shoulder structures of [{}^3P_{\rm m}^{ \langle n \rangle }] on the high-angle side are probably caused by the component of the refracted beam [{{\bf{S}}_{\rm m2}}]

4. Discussion

Equation (1)[link] derived by Gronkowski & Malgrange (1984[Gronkowski, J. & Malgrange, C. (1984). Acta Cryst. A40, 507-514.]) is applicable only to a monochromatic X-ray of a plane wave. In the present experiment, however, since the X-rays from synchrotron radiation are emitted from a source of finite size and are monochromated by a crystal monochromator, they have a small energy bandwidth as well as a small divergence angle. It is noted that [\Delta \alpha] denotes the divergence angle for monochromatic X-rays and [\delta \alpha] denotes the angle shift corresponding to the energy shift [\delta E], which can be estimated from the observed range of mirage interference fringes. [\Delta \alpha] is equal to the maximum value of [\delta \alpha]. It is necessary to have the relation between the diffraction of monochromatic X-rays and that of X-rays with a finite energy bandwidth and a finite divergence angle. If the X-ray energy changes from E to [E - \delta E] and the amplitude of the wavevector changes from [| {{{\bf{K}}_0}} |] to [| {{{\bf{K}}_0}} | - | {\delta {{\bf{K}}_E}} |] while the glancing angle is fixed, the dispersion surface T0′ moves to T0′′ and the Bragg angle changes from [{\theta _{\rm{B}}}] to [{\theta _{\rm{B}}} + \delta {\theta _{\rm{B}}}] as shown in Fig. 2[link]. Here [|\delta {{\bf{K}}_E}|] is the amplitude of the wavevector [\delta {{\bf{K}}_E}], which is related to the energy deviation [\delta E] by

[|\delta {{\bf{K}}_E}|/{K_0} = \delta E/E. \eqno(11)]

In Fig. 2[link], we have the relation between [\delta W] and [|\delta {{\bf{K}}_E}|] as

[\delta W{X_0} = \left| {\delta {{\bf{K}}_E}} \right|\cos {\theta _{\rm{B}}}. \eqno(12)]

Using equations (9)[link], (11)[link] and (12)[link], [\delta E/E] is expressed by

[{{\delta E} \over E} = {{\left| {{\chi _h}} \right|} \over {2\cos ^2\theta _{\rm{B}}}}\delta W. \eqno(13)]

When the dispersion angle changes from [{\alpha _0}] to [{\alpha _0} - \delta \alpha], the relation [| {\delta {{\bf{K}}_\alpha }} | = | {{{\bf{K}}_\alpha }} |\delta \alpha] holds in Fig. 2[link]. [\delta E] is related to [\delta \alpha] as

[\delta E = E\,\delta \alpha \tan {\theta _{\rm{B}}}, \eqno(14)]

by using the relation [| {\delta {{\bf{K}}_E}} |\tan {\theta _{\rm{B}}} = {K_0}\,\delta {\theta _{\rm{B}}}] or [\delta E\tan \theta _{\rm{B}} =] [E\,\delta {\theta _{\rm{B}}}].

By inserting equation (13)[link] into equation (8)[link], the energy shift [\delta E] is related to the distance x by

[\delta E = {{|{\chi _h}|E} \over {2\cos ^2{\theta _{\rm{B}}}}}\left [{1 - {{\left({1 + {{{\beta ^2}{x^2}} \over 4}} \right)}^{1/2}}} \right].\eqno(15)]

It is possible to estimate [\delta E] by measuring x. In the present experiment, since the measured maximum value of x is 3 mm, as shown in Fig. 4[link](b), the maximum value of [|\delta E|] is obtained as 11 meV by using equation (15)[link] and the maximum value of [\delta \alpha] is obtained as 0.65′′ by using equation (14)[link]. By differentiating equation (15)[link], we have the energy resolution dE obtained from the position of the mirage interference fringes with the position resolution dx as

[|{\rm d}E| = {{|{\chi _h}|} \over {2\cos ^2{\theta _{\rm{B}}}}}{\left({1 + {{{\beta ^2}{x^2}} \over 4}} \right)^{ - 1/2}}{{{\beta ^2}} \over 4}Ex\,{\rm d}x. \eqno(16)]

When dx is 0.1 mm and [\beta] = 0.63 mm−1 in the case of Si(220), [|{\rm d}E|] is approximately 0.4 meV at x = 1.5 mm. This means that the mirage interference fringes from the third crystal can be used for energy analysis of the beams with an energy resolution of sub-meV. The value of 0.1 mm for dx is nearly the same as the projected width of Slit 4 on the sample surface ([0.04\,/\sin {\theta _{\rm{B}}} = 0.12] mm).

In the case of angle dispersive diffractometry, an asymmetric reflection is usually used as a monochromator or an analyzer with high energy resolution from meV to sub-meV. For example, two asymmetrically cut crystals with (+n, +m, −m, −n) setting were used as a monochromator with the energy bandwidth of meV (Ishikawa et al., 1992[Ishikawa, T., Yoda, Y., Izumi, K., Suzuki, C. K., Zhang, X. W., Ando, M. & Kikuta, S. (1992). Rev. Sci. Instrum. 63, 1015-1018.]; Yabashi & Ishikawa, 2000[Yabashi, M. & Ishikawa, T. (2000). SPring-8 Annual Report 1999, pp. 151-152. SPring-8, Hyogo-ken, Japan.]). According to the relation [|{\rm d}E/E| = {\rm d}{\theta _{\rm{B}}}\,/] [\tan {\theta _{\rm{B}}}], it is also possible to have X-rays with a small energy bandwidth by using back reflection. X-rays with an energy resolution of 0.45 meV were obtained by using Si(13 13 13) reflection with [{\theta _{\rm{B}}}] = 89.98° for X-rays of 25.70 keV (Verbeni et al., 1996[Verbeni, R., Sette, F., Krisch, M. H., Bergmann, U., Gorges, B., Halcoussis, C., Martel, K., Masciovecchio, C., Ribois, J. F., Ruocco, G. & Sinn, H. (1996). J. Synchrotron Rad. 3, 62-64.]). Another monochromator with an energy resolution of sub-meV was designed by combining asymmetric reflection with back reflection (Baron et al., 2001[Baron, A. Q. R., Tanaka, Y., Ishikawa, D., Miwa, D., Yabashi, M. & Ishikawa, T. (2001). J. Synchrotron Rad. 8, 1127-1130.]; Stoupin et al., 2013[Stoupin, S., Shvyd'ko, Y. V., Shu, D., Blank, V. D., Terentyev, S. A., Polyakov, S. N., Kuznetsov, M. S., Lemesh, I., Mundboth, K., Collins, S. P., Sutter, J. P. & Tolkiehn, M. (2013). Opt. Exp. 21, 30932-30946.]). Stoupin et al. designed a monochromator with high spectral efficiency by combining asymmetric reflection from an asymmetrically cut diamond with back reflection from a silicon crystal. In these angle dispersive monochromators, it is necessary to choose an appropriate crystal and its Bragg reflection after setting the X-ray energy. In contrast, it is possible to adjust the energy resolution for any energy of X-rays just by choosing a Bragg reflection and changing the strain gradient parameter [\beta] for the current monochromator using mirage interference fringes.

As the FWHM of the rocking curve of [{}^3P_{\rm m}^{ \langle 1 \rangle }] in Fig. 6[link] is 0.6′′, which is much smaller than that of 1Ph (9′′) in Fig. 5[link](a), the rocking curve of [{}^2P_{\rm m}^{ \langle 1 \rangle }] is regarded to be the intrinsic curve of the primary diffraction beam of 1Ph. The value of 0.6′′ actually corresponds to the FWHM of [{}^2P_{\rm m}^{ \langle 1 \rangle }]. However, according to the estimation by using the beam width passing through Slit 2 with a width of 0.02 mm, the FWHM of the rocking curve of [{}^2P_{\rm m}^{ \langle 1 \rangle }] should be 0.2′′. The measured FWHM is three times larger than this estimated value. The cause of this difference is probably the source size (0.06 mm) of the synchrotron radiation X-rays from the bending magnet. If the source size is assumed to be 0.06 mm, the FWHM of the rocking curves of [{}^2P_{\rm m}^{ \langle 1 \rangle }] becomes 0.6′′, which agrees with the observed maximum value of [\delta \alpha] ≃ 0.65′′. If we use an undulator instead of the bending magnet, we can obtain a beam with a much smaller divergence angle by using this mirage diffractometer, because the source size of the undulator is approximately ten times smaller than that of the bending magnet.

Fig. 8[link] shows a DuMond diagram to demonstrate the angle and energy resolutions of the current triple-crystal difftactometer with (+, −, +) setting. The reflection indices of 220 are the same for the three Si crystals. The glancing angle of the incident beam on each crystal is fixed when the mirage interference fringes are observed by moving Slit 4 in Fig. 3[link](d). The divergence of the Bragg angle [\delta {\theta _{\rm{B}}}] (= 0.07′′) and the bandwidth of the wavelength [\delta {\lambda _{\rm b}}] (= 1.1 × 10 - 7 nm) of the second crystal used as a monochromator are determined according to the source size of the synchrotron radiation X-rays as described above. The divergence angle d[{\theta _1}] of the beam from the first peak of the mirage interference fringes in Fig. 4[link](a) is found to be 0.003′′ by using equations (4)[link], (14)[link] and (16)[link], after passing Slit 3 of 40 µm width. The value of d[{\theta _1}] corresponds to the angular width of the [{\theta _{\rm{B}}}][\lambda] curve. As the mirage interference fringes from the third crystal are observed by moving Slit 4 with a vertical width of 40 µm, the angular width d[{\theta _2}] of the beam from Slit 4 is 0.003′′ and the bandwidth of the wavelength d[\lambda] is 4 × 10 - 9 nm, which is much smaller than [\delta {\lambda _{\rm b}}], as shown in Fig. 8[link]. Slit 4 is located at [{\theta _{\rm{B}}}] when x = 0 mm and at [{\theta _{\rm{B}}} + \delta {\theta _{\rm{B}}}] when x = xmax. The energy width dE is approximately 0.4 meV by using the relation [|{\rm d}\lambda /\lambda | = |{\rm d}E/E|].

[Figure 8]
Figure 8
DuMond diagram. [\delta {\theta _{\rm{B}}}] (= 0.07′′) is the angular width of the incident beam on the second crystal used as a monochromator through Slit 2, d[{\theta _1}] (= 0.003′′) that on the third crystal used as the sample through Slit 3 and d[{\theta _2}] (= 0.003′′) the angular resolution of the analyzer obtained by moving Slit 4. [\delta {\lambda _{\rm b}}] (= 1.1 × 10−7 nm) is the bandwidth of the X-ray wavelength through Slit 2 and d[\lambda] (= 4 × 10−9 nm) the wavelength resolution of the analyzer obtained by moving Slit 4.

5. Summary

In the present experiment, the divergence angle of the beam from the first Si crystal is 4′′. When this beam is incident on the second Si crystal, the divergence angle of the mirage interference fringes is reduced to 0.6′′. The second crystal works as a monochromator to obtain a small divergence angle and thus high energy resolution. Table 1[link] shows the peak positions x and the corresponding values of Ws1 (Ws2), [\delta E] and [\delta \alpha (n)] of mirage interference fringes from the third crystal. Ws1 and Ws2 are the values of Ws obtained by using equation (8)[link] for the refracted beams [{{\bf{S}}_{\rm m1}}] and [{{\bf{S}}_{\rm m2}}] in Fig. 1[link](b). The intensity of the diffracted beam corresponding to [{{\bf{S}}_{\rm m1}}] is approximately five times larger than that corresponding to [{{\bf{S}}_{\rm m2}}], as shown in Fig. 7[link]. This is because beam [{{\bf{S}}_{\rm m1}}] directly reaches A2, while beam [{{\bf{S}}_{\rm m2}}] reaches point A2 after being once reflected from the top surface. Then we use the value of Ws1 for estimating [\delta E] and [\delta \alpha (n)]. The average value of the difference between two adjacent values of [\delta \alpha (n)] given by [\textstyle\sum_{n = 1}^3 [\delta \alpha (n + 1) - \delta \alpha (n)]/3] is 0.1′′, which agrees with the angular difference between two adjacent peaks of [{}^3P_{\rm m}^{ \langle n \rangle }] in Fig. 7[link]. The value of energy width [\delta E] obtained in Table 1[link] shows that the third crystal works as an analyzer with high energy resolution.

Table 1
The peak position x of mirage interference fringes of [{}^3P_{\rm m}^{ \langle n \rangle }] for n = 1, 2, 3 and 4, and the corresponding values of Ws, [\delta E] and [\delta \alpha (n)]

n x (mm) Ws1 (Ws2) [\delta E] (meV) [\delta \alpha (n)] (′′)
1 1.41 −1.10 (−1.03) −3.0 −0.18
2 1.96 −1.17 (−1.05) −5.0 −0.30
3 2.27 −1.23 (−1.06) −6.7 −0.40
4 2.51 −1.27 (−1.07) −7.9 −0.48

In order to have X-rays with a very small divergence angle, an asymmetric crystal monochromator is widely used. When we use an Si(220) asymmetric monochromator for an X-ray energy of 11 100 eV, for example, the Bragg angle is 16.9° and the asymmetric factor b [= [\sin ({\theta _{\rm{B}}} + a)/\sin ({\theta _{\rm{B}}} - a)] with a = 15.9°] is 31. The divergence angle from the asymmetric crystal monochromator is 1/5.6 (= 1/b1/2) of the divergence angle from a symmetric crystal, and the X-ray intensity is reduced by 1/31. If we use the mirage interference fringes from Si(220) as a monochromator, on the other hand, the divergence angle is 1/8 of the incident X-ray and the intensity is reduced by 1/50, because the number of photons of the incident beams through the second crystal is approximately 25 000 s−1 and that of the mirage interference fringes from the third crystal is 500 s−1 according to the present experiment, as shown in Fig. 4[link]. The intensity is the same as or slightly weaker than that from an asymmetric crystal. It is an advantage of this monochromator using the mirage interference fringes that the setup is quite easy and its structure is simple, although the divergence angle and the intensity are the same orders of magnitude as those of an angle dispersive monochromator using an asymmetric crystal.

Fukamachi et al. (2011[Fukamachi, T., Jongsukswat, S., Kanematsu, Y., Hirano, K., Negishi, R., Shimojo, M., Ju, D., Hirano, K. & Kawamura, T. (2011). J. Phys. Soc. Jpn, 80, 083002.]) reported a monochromator with a very small divergence angle using the multiple-Bragg Laue mode diffraction from a lateral surface. The characteristics and usability of this instrument are nearly the same as those of the present monochromator using mirage interference fringes, but the divergence angle from the monochromator using the multiple-Bragg Laue mode is about twice that using mirage diffraction.

In this experiment, we have measured the intensities of mirage diffraction with a scintillation counter by moving a slit of very small width in front of it. The intensity distribution of the mirage interference fringes as a function of x is regarded as a spectrum of the incident X-rays projected onto the crystal according to equation (15)[link]. If we use an X-ray CCD camera or a position sensitive detector instead of a scintillation counter, the energy resolution of the spectrum should be improved to a large extent and the measuring time should be reduced. As we have very large angle amplification in the mirage diffraction, in the near future we can expect to achieve high energy resolution (less than meV) by using this diffractometer if it is combined with X-rays from an undulator beamline and a CCD camera.

Acknowledgements

This work was carried out under the approval of the Program Advisory Committee of PF (proposal No. 2012G606). This work was partially supported by the Nano Technology Project for Private Universities (from 2011), with matching fund subsidy from the Ministry of Education, Culture, Sports, Science and Technology.

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