research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Journal logoJOURNAL OF
SYNCHROTRON
RADIATION
ISSN: 1600-5775

Refraction in general asymmetric X-ray Bragg diffraction

CROSSMARK_Color_square_no_text.svg

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221 Praha 8, Czech Republic
*Correspondence e-mail: hrdy@fzu.cz

(Received 21 February 2001; accepted 6 September 2001)

When the surface of a single-crystal monochromator is not parallel to the diffracting crystallographic planes, the diffracted beam is generally deviated from the plane of diffraction and the angle between the diffracted beam and the diffracting planes is different from the angle between the incident beam and the diffracting planes. The angular diffraction regions for the incident and diffracted beams are also different. This is the manifestation of the refraction occurring during Bragg diffraction. Very simple formulae are presented which describe this situation in a general case (e.g. for a rotated-inclined X-ray monochromator). These formulae allow sagittally focusing monochromators for synchrotron radiation to be easily designed, based on X-ray diffraction–refraction phenomena. Some important properties of such types of monochromators are deduced.

1. Introduction

From a geometrical point of view, Bragg-case X-ray diffraction on a symmetrically cut crystal behaves like reflections on a mirror. The angle between the incident beam and the surface of the crystal equals the angle between the diffracted beam and the surface of the crystal. This is no longer valid if the surface of the crystal is not parallel to the diffracting crystallographic planes. Here we can have two extreme cases. First, the impinging X-ray beam, the normal to the diffracting planes and the normal to the surface are in one plane. This is the well known asymmetric Bragg-case diffraction. Second, starting from the asymmetric case, the surface is rotated around the normal to the diffracting planes by 90°. Then the plane containing the impinging beam and the normal to the diffracting planes is perpendicular to the plane determined by the normals to the diffracting planes and to the surface. This situation is often called inclined diffraction. If the angle of rotation is different from 90°, we have a mixture of the asymmetric and the inclined cases, and here this situation will be called general asymmetric or rotated-inclined diffraction. (This is an analogy of a general asymmetric or rotated-inclined monochromator which is based on this kind of diffraction.)

In asymmetric Bragg-case diffraction, not only the angle between the incident beam and the surface differs from the angle between the diffracted beam and the surface, but also the angle between the incident beam (more precisely the centre of the diffraction region) and the diffracting crystallographic planes (θB + Δθ0) differs from the angle between the diffracted beam and the diffracting planes (θB + Δθh). The angular width ω0 of the diffraction region of the incident beam for a certain wavelength λ and the width ωh of the diffraction region of the reflected (diffracted) beam are also different. This is a consequence of the dynamical theory of diffraction on perfect crystals and the situation is described well by, for example, Matsushita & Hashizume (1983[Matsushita, T. & Hashizume, H. (1983). Handbook of Synchrotron Radiation, Vol. 1a, pp. 261-314. Amsterdam: North-Holland.]). The following simple relations are stated for the asymmetric diffraction between the widths and positions of the centres of the diffraction regions (crystal functions):

[\eqalign{\omega_0&=\omega_{\rm s}b^{-1/2},\cr\omega_{\rm s}&=\left[2r_e\lambda^2P\left|F_{hr}\right|\exp(-M)\right]/{\pi}V\sin2\theta_{\rm{B}},\cr\Delta\theta_0&=(1/2)(1+1/b)\Delta\theta_{\rm s},\cr\Delta\theta_{\rm s}&=r_e\lambda^2F_{0r}/{\pi}V\sin2\theta_{\rm{B}},\cr\omega_h&=\omega_{\rm s}b^{1/2},\cr\Delta\theta_h&=(1/2)(1+b)\Delta\theta_{\rm s},\cr\theta_0&=\theta_{\rm{B}}+\Delta\theta_0,\cr\theta_h&=\theta_{\rm{B}}+\Delta\theta_h,\cr b&=\sin(\theta_{\rm{B}}-\alpha)/\sin(\theta_{\rm{B}}+\alpha).}\eqno(1)]

Here, V is the unit-cell volume, re = e2/mc2, Fhr is the real part of the structure factor Fh (h stands for Müller indices hkl), P is the polarization factor, and exp(−M) is the temperature factor. The index s stands for symmetrical diffraction. The angle α is the angle between the diffracting planes and the surface and is taken as positive for grazing incidence. The typical values of Δθ and ω are from fractions to tens of angular seconds. For the cross sections CS0 and CSh of the incident and reflected beams, it holds that

[{\rm{CS}}_h={\rm{CS}}_0/b,\eqno(2)]

and, together with (1)[link],

[\omega_h{\rm{CS}}_h=\omega_0{\rm{CS}}_0.\eqno(3)]

From the above it is seen that in the case of the asymmetric Bragg-case diffraction the angle between the incident beam (the centre of the diffraction region) and the diffracting planes is different from the angle between the diffracted beam (again the centre of the diffraction region) and the diffraction planes, and their (meridional) difference δm,asym is given by (Hrdý & Hrdá, 2000[Hrdý, J. & Hrdá, J. (2000). J. Synchrotron Rad. 7, 78-80.])

[\delta_{\rm{m,asym}}=\Delta\theta_0-\Delta\theta_h=2\Delta\theta_{\rm{s}}\tan\theta_{\rm{B}}\tan\alpha/\left(\tan^2\theta_{\rm{B}}-\tan^2\alpha\right),\eqno(4)]

or simply

[\delta_{\rm{m,asym}}=(1/2)(1/b-b)\Delta\theta_{\rm{s}}.\eqno(5)]

The deviation from the mirror-like behaviour, δasym, was used for the proposal of the meridional focusing of diffracted synchrotron radiation on the crystal with a transversal groove on its surface (Hrdý & Hrdá, 2000[Hrdý, J. & Hrdá, J. (2000). J. Synchrotron Rad. 7, 78-80.]).

In the case of inclined diffraction the situation is different and was probably first studied by Hrdý & Pacherová (1993[Hrdý, J. & Pacherová, O. (1993). Nucl. Instrum. Methods, A327, 605-611.]), and later, with respect to possible application for focusing, in some of our preceding papers (Hrdý, 1998[Hrdý, J. (1998). J. Synchrotron Rad. 5, 1206-1210.]; Artemiev et al., 2000[Artemiev, N., Busetto, E., Hrdý, J., Pacherová, O., Snigirev, A. & Suvorov, A. (2000). J. Synchrotron Rad. 7, 355-419.]). It was shown that the diffracted beam is sagittally deviated from the plane of diffraction (i.e. the plane determined by the impinging beam and the normal to the diffracting planes) by an angle δs,incl given by

[\delta_{\rm{s,incl}}=K\tan\beta,\eqno(6)]

where β is the angle between the surface and the diffracting planes (angle of inclination) and K for silicon crystals is given by

[K=1.256\times10^{-3}\,d_{hkl}\,\,{\rm{[nm]}}\,\,\lambda\,\,{\rm{[nm]}}.\eqno(7)]

It has been shown both theoretically (Hrdý, 1998[Hrdý, J. (1998). J. Synchrotron Rad. 5, 1206-1210.]) and experimentally (Hrdý & Siddons, 1999[Hrdý, J. & Siddons, D. P. (1999). J. Synchrotron Rad. 6, 973-978.]) that, owing to the tangential dependence (6)[link], an X-ray synchrotron radiation beam diffracted on a crystal with a longitudinal parabolic groove may be sagittally focused.

Korytár, Boháček & Ferrari (2000[Korytár, D., Boháček, P. & Ferrari, C. (2000). Czech. J. Phys. 50, 841-850.], 2001[Korytár, D., Boháček, P. & Ferrari, C. (2001). Czech. J. Phys. In the press.]) suggested that a substantial increase of this sagittal deviation may be achieved if the asymmetric diffraction component is present, i.e. in the case of general asymmetric (or rotated-inclined) diffraction. They also performed a numerical calculation of this effect taking into account the exact shapes of surfaces in the reciprocal space. By diffraction on an asymmetrically cut crystal with a longitudinal W-shaped groove we have unambiguously proved that this effect exists (Korytár, Hrdý et al., 2001[Korytár, D., Hrdý, J., Artemiev, N., Ferrari, C. & Freund, A. (2001). J. Synchrotron Rad. 8, 1136-1139.]).

In fact, the sagittal deviation of a beam in the case of rotated-inclined geometry was studied even earlier (Kashihara et al., 1998[Kashihara, Y., Yamazaki, H., Tamasaku, K. & Ishikawa, T. (1998). J. Synchrotron Rad. 5, 679-681.]; Yabashi et al., 1999[Yabashi, M., Yamazaki, H., Tamasaku, K., Goto, S., Takeshita, K., Mochizuki, T., Yoneda, Y., Furukawa, Y. & Ishikawa, T. (1999). Proc. SPIE, 3773, 2-13.]; Blasdell & Macrander, 1994[Blasdell, R. C. & Macrander, A. T. (1944). Nucl. Instrum. Methods Phys. Res. A, 347, 320-323.]). The results of these works (including the papers of Korytar), however, are in rather complicated form and do not allow the focusing properties to be studied directly.

In this paper a very simple formula for the sagittal deviation of the beam in the case of general asymmetric (or rotated-inclined) diffraction is derived which allows sagittally focusing monochromators based on this kind of diffraction to be studied and designed easily. Some important properties of these monochromators are also deduced.

2. Derivation of the formula for sagittal deviation of an X-ray beam for general asymmetric diffraction

Let us consider as a starting position an asymmetrical Bragg-case diffraction. The situation in the reciprocal space for this kind of diffraction (Batterman & Cole, 1964[Batterman, B. W. & Cole, H. (1964). Rev. Mod. Phys. 36, 681-717.]) is shown in Fig. 1[link]. P and P′ are the starting points of the wave vectors for the incident and the diffracted wave vectors for the symmetrical diffraction. SpO and SpH are spheres of radius 1/λ centred on the reciprocal space points O and H. LP = LP′ = Δθs(1/λ). For asymmetric diffraction the starting points for the incident and diffracted wave vectors are P1 and P2, respectively. Obviously, LP1 = Δθ0(1/λ) and LP2 = Δθh(1/λ). The distance P1P2 = AP2/cosα and AP2 = LP1cosθB + LP2cosθB. From this and from (1)[link] it follows that

[P_1P_2=2\Delta\theta_{\rm s}\cos\theta_{\rm{B}}(1/\lambda)(2+b+1/b)/4\cos\alpha.\eqno(8)]

So far we have been working in a plane. To take into account the rotated-inclined diffraction it is necessary to work in a three-dimensional space. With a reasonable approximation we can consider the spheres passing through the points L and Q as planes perpendicular to the plane of the drawing. Similarly, the dispersion surfaces are taken as hyperbolical cylinders perpendicular to the plane of the drawing. We have used the same approximation in deriving (6)[link] and it proved to be good for β up to about 85°. Creating a groove in the surface of such an asymmetrically cut crystal means that the plane of the surface is rotated by some angle β about the axis o which is parallel to the surface (see Fig. 1[link]). It also means that the normal P1P2 to the surface is rotated by the angle β and now it intersects the plane LP2 (or SpH) at some point Pβ which is above the point P2 (above the plane of diagram); thus in Fig. 1[link] both points coincide. The distance P2Pβ = P1P2tanβ. This implies that the diffracted wavevector starts now at Pβ and for the sagittal deviation of the diffracted beam it holds that

[\delta_{\rm{s,r{\hbox{-}}i}}=P_2P_{\!\beta}/(1/\lambda).\eqno(9)]

In the inclined case (α = 0) the sagittal deviation would obviously be

[\delta_{\rm{s,incl}}=PP^{\,\prime}\tan\beta/(1/\lambda),\eqno(10)]

where PP′ = 2ΔθscosθB(1/λ). Now it is possible to write the final formula for the sagittal deviation of the diffracted beam in the rotated-inclined case,

[\delta_{\rm{s,r{\hbox{-}}i}}=\delta_{\rm{s,incl}}(2+b+1/b)/4\cos\alpha.\eqno(11)]

[We can create the rotated-inclined case differently. Starting from the asymmetric case, we can rotate the surface about the axis which is parallel not to the surface as considered above but to the diffracting planes by a certain angle β. Then we will obtain practically the same formula as (11)[link]; only cosα will be missing.]

[Figure 1]
Figure 1
Schematic representation of the situation in reciprocal space for asymmetric Bragg-case diffraction. After the rotation of P1P2 about the axis o parallel with a surface by an angle β, this line will intersect the sphere SpH at a point Pβ which will be the starting point of the diffracted wave vectors for general asymmetric Bragg-case diffraction. Pβ is above the plane of the drawing and its projection onto this plane coincides with P2.

The experimental value of the sagittal deviation found by Korytár, Hrdý et al. (2001[Korytár, D., Hrdý, J., Artemiev, N., Ferrari, C. & Freund, A. (2001). J. Synchrotron Rad. 8, 1136-1139.]) was within the precision of the method, in good agreement with (11)[link].

From the presented geometrical derivation and by using Fig. 1[link] it is obvious that the meridional component δm,r-i of the beam deviation in the rotated-inclined case equals δm,asym given by (4)[link] and (5)[link]. Then from the above it follows that if the rotated-inclined diffraction is created in the above-described way, then, if the impinging beam spans its diffraction region ω0, the diffracted beam is deviated in the meridional direction (from a mirror-like reflection) by an angle δm,r-i given by (5)[link], and the meridional component of the diffracted beam span is the angular interval ωh. At the same time the beam is sagittally deviated from the plane of diffraction by the angle δs,r-i given by (11)[link], and the sagittal component ωs,r-i of the diffracted beam span is given by (12)[link].

3. Conclusions

From (6)[link] and (11)[link] it is seen that δr-i is proportional to tanβ and that also for the rotated-inclined (or general asymmetric) case the sagittal focusing of diffracted radiation can be achieved by a longitudinal parabolic groove. δr-i has its minimal value for b = 1 (symmetrical case). The increase of the sagittal deviation occurs for both b < 1 and b > 1, i.e. for the grazing-incidence case and grazing-emergence case, and is independent of the sign of α. To design the shape of the focusing parabolic groove the procedure is the same as that described by Hrdý (1998[Hrdý, J. (1998). J. Synchrotron Rad. 5, 1206-1210.]) and Hrdý & Siddons (1999[Hrdý, J. & Siddons, D. P. (1999). J. Synchrotron Rad. 6, 973-978.]), though it is necessary to replace K with K(2 + b +1/b)/4cosα. We have designed a sagittally focusing monochromator based on the above theory and have performed an experiment at the ESRF (Artemiev et al., 2001[Artemiev, N., Hoszowska, J., Hrdý, J. & Freund, A. (2001). To be published.]). The focusing properties observed were in agreement with our expectation.

So far the treatment has only been concentrated on the central beams. As in the pure inclined case (see, for example, Artemiev et al., 2000[Artemiev, N., Busetto, E., Hrdý, J., Pacherová, O., Snigirev, A. & Suvorov, A. (2000). J. Synchrotron Rad. 7, 355-419.]), when the incident beam spans the diffracting region ω0 then the diffracted beam spans a certain angle, here ωr-i. From the geometry shown in Fig. 1[link] and from (1)[link], it follows that

[\omega_{\rm{s,r{\hbox{-}}i}}/\delta_{\rm{s,r{\hbox{-}}i}}=\omega_0/\Delta\theta_0=\omega_h/\Delta\theta_h.\eqno(12)]

From (1)[link] it follows that (12)[link] has its maximum value for b = 1. For |α| approaching θB, the left-hand side of (12)[link] approaches zero. This means that the sagittal focusing for the rotated-inclined case is sharper than for the pure inclined case.

Acknowledgements

The author wishes to express his thanks to N. Artemiev for valuable discussions. This research was financially supported by the Grant Agency of the Academy of Sciences of the Czech Republic (A1010104/01) and by the Ministry of Industry and Trade of the Czech Republic (PZ-CH/22).

References

First citationArtemiev, N., Busetto, E., Hrdý, J., Pacherová, O., Snigirev, A. & Suvorov, A. (2000). J. Synchrotron Rad. 7, 355–419. Web of Science CrossRef IUCr Journals
First citationArtemiev, N., Hoszowska, J., Hrdý, J. & Freund, A. (2001). To be published.
First citationBatterman, B. W. & Cole, H. (1964). Rev. Mod. Phys. 36, 681–717. CrossRef CAS Web of Science
First citationBlasdell, R. C. & Macrander, A. T. (1944). Nucl. Instrum. Methods Phys. Res. A, 347, 320–323.  CrossRef Web of Science
First citationHrdý, J. (1998). J. Synchrotron Rad. 5, 1206–1210. Web of Science CrossRef IUCr Journals
First citationHrdý, J. & Hrdá, J. (2000). J. Synchrotron Rad. 7, 78–80. Web of Science CrossRef IUCr Journals
First citationHrdý, J. & Pacherová, O. (1993). Nucl. Instrum. Methods, A327, 605–611.
First citationHrdý, J. & Siddons, D. P. (1999). J. Synchrotron Rad. 6, 973–978. Web of Science CrossRef IUCr Journals
First citationKashihara, Y., Yamazaki, H., Tamasaku, K. & Ishikawa, T. (1998). J. Synchrotron Rad. 5, 679–681. Web of Science CrossRef CAS IUCr Journals
First citationKorytár, D., Boháček, P. & Ferrari, C. (2000). Czech. J. Phys. 50, 841–850. CAS
First citationKorytár, D., Boháček, P. & Ferrari, C. (2001). Czech. J. Phys. In the press.
First citationKorytár, D., Hrdý, J., Artemiev, N., Ferrari, C. & Freund, A. (2001). J. Synchrotron Rad. 8, 1136–1139. Web of Science CrossRef CAS IUCr Journals
First citationMatsushita, T. & Hashizume, H. (1983). Handbook of Synchrotron Radiation, Vol. 1a, pp. 261–314. Amsterdam: North-Holland.
First citationYabashi, M., Yamazaki, H., Tamasaku, K., Goto, S., Takeshita, K., Mochizuki, T., Yoneda, Y., Furukawa, Y. & Ishikawa, T. (1999). Proc. SPIE, 3773, 2–13. CrossRef

© International Union of Crystallography. Prior permission is not required to reproduce short quotations, tables and figures from this article, provided the original authors and source are cited. For more information, click here.

Journal logoJOURNAL OF
SYNCHROTRON
RADIATION
ISSN: 1600-5775
Follow J. Synchrotron Rad.
Sign up for e-alerts
Follow J. Synchrotron Rad. on Twitter
Follow us on facebook
Sign up for RSS feeds