I. INTRODUCTION
Even though the neutron sources provide currents several orders of magnitude lower than X-ray sources, there is a permanent attention paid to find the ways or methods and techniques helping in increasing the luminosity, and/or resolution of neutron scattering instruments. In addition to construction of high-flux neutron sources, a broad range of efficient neutron techniques, namely based on Bragg diffraction optics, have been recently developed (see, e.g., Popovici and Yelon, Reference Popovici and Yelon1994, Reference Popovici and Yelon1995; Vogt et al., Reference Vogt, Passell, Cheung and Axe1994; Mikula et al., Reference Mikula, Vrána, Wagner and Furusaka2008a, Reference Mikula, Vrána, Pilch, Seok Seong, Woo and Em2014c; Mikula and Vrána, Reference Mikula and Vrána2015; Seong et al., Reference Seong, Em, Mikula, Šaroun and Kang2010). This effort is strengthened not only by a necessity of investigations of samples of smaller dimensions but also by a necessity of using significantly better measurement resolution for investigation of finer effects, e.g., distinguishing and identifying powder diffraction lines corresponding to very close values of lattice spacings, for studies of plastic deformation by means of diffraction profile analysis and for studies of mosaic distribution of mosaic single crystals. Namely, the latter one requires a substantial increase of angular and/or energy resolution of conventional diffractometers/spectrometers operating, where the monochromator plays a key role. Bent-perfect-crystal (BPC) slabs as an excellent alternative of conventional mosaic crystals provide a way to increase luminosity and/or angular/energy resolution of some scattering devices installed at steady-state sources (Popovici and Yelon, Reference Popovici and Yelon1995). An increase of the luminosity is carried out by focusing in real space, while a higher resolution can be achieved by focusing in momentum space in combination with a rather small effective mosaicity of the bent-perfect crystals. In this way, excellent results have been achieved in powder diffractometry and, namely, in the case of the residual stress diffractometer where by using the BPC monochromator the figure of merit of the instrument increased by a factor of 10 or more (Popovici and Yelon, Reference Popovici and Yelon1994; Vogt et al., Reference Vogt, Passell, Cheung and Axe1994; Mikula et al., Reference Mikula, Vrána, Lukáš, Šaroun and Wagner1996; Wimpory et al., Reference Wimpory, Mikula, Šaroun, Poeste, Li, Hoffman and Schneider2008; Seong et al., Reference Seong, Em, Mikula, Šaroun and Kang2010; Woo et al., Reference Woo, Em, Seong, Shin, Mikula, Joo and Kang2011). Together with the construction of new powerful neutron sources, designs of new scattering instruments with improved resolution properties are required. In this way, the so-called dispersive monochromators based on a dispersive double-diffraction process appear as candidates of monochromators for (very) high-resolution neutron diffractometers and spectrometers. It can be realized either by means of two independent crystals (Mikula et al., Reference Mikula, Vrána, Pilch, Šaroun, Seong, Woo and Em2014a, Reference Mikula, Vrána, Seong, Woo, Em and Korytár2014b, Reference Mikula, Vrána, Pilch, Seok Seong, Woo and Em2014c; Mikula and Vrána, Reference Mikula and Vrána2015) or by means of strong multiple Bragg reflections (MBRs) realized inside one crystal (Mikula et al., Reference Mikula, Vrána, Lott and Wagner2006a, Reference Mikula, Vrána, Wagner and Furusaka2008a, Reference Mikula, Vrána, Šaroun, Seong and Em2011a, Reference Mikula, Vrána, Šaroun, Krejčí, Seong, Woo and Furusaka2013). First, high-resolution MBR monochromators have been tested using neutron radiography for investigation of edge refraction effects (Mikula et al., Reference Mikula, Vrána, Wagner, Erko, Idir, Krist and Michette2008b, Reference Mikula, Vrána, Šaroun, Krejčí, Seong, Woo and Furusaka2013).
MBR effects are well known in X-ray diffraction when the positive MBR-diffraction peaks were first observed by Renninger (Renninger, Reference Renninger1937, Reference Renninger1960). Many other related references can be found, e.g., in the books of Pinsker and Chang (Pinsker, Reference Pinsker1982; Chang, Reference Chang1984). Concerning neutrons, relatively small MBR effects were observed in mosaic crystals (Moon and Shull, Reference Moon and Shull1964; Kuich and Rauch, Reference Kuich and Rauch1965, Reference Kuich and Rauch1967). However, only in the case of elastically deformed BPC slabs have very strong MBR effects consisting in one or more double-diffraction processes been observed. Elastic deformation can strongly amplify the MBR effects and their related intensities can be more than one order of magnitude higher than the intensity related to the allowed reflection of a perfect nondeformed crystal (Mikula et al., Reference Mikula, Vrána, Wagner, Erko, Idir, Krist and Michette2008b, Reference Mikula, Vrána, Šaroun, Seong and Em2011a, Reference Mikula, Vrána, Šaroun, Seong, Em and Moon2011b, Reference Mikula, Vrána, Šaroun, Davydov, Em and Seong2012). In the last decade, we have paid systematic attention to studying these effects, namely when it has been found that some strong MBR effects can be considered as alternative to very high-resolution dispersive double-crystal neutron monochromators (Popovici and Yelon, Reference Popovici and Yelon1995; Mikula et al., 2014; Mikula and Vrána, Reference Mikula and Vrána2015). As the MBR is, in fact, the dispersive double-diffraction process realized in one elastically deformed crystal, the aim of the present studies is a further search of strong MBR effects (in this case in relation to the forbidden Si(002) reflection at the constant neutron wavelength of 0.162 nm) which could potentially be an alternative to the standard dispersive double-crystal monochromator (Popovici and Yelon, Reference Popovici and Yelon1995; Mikula and Vrána, Reference Mikula and Vrána2015).
II. SEARCH FOR MBR EFFECTS
It is known that the MBR effects in a single crystal can be observed when more than one set of planes are simultaneously operative for a given wavelength, i.e., when two or more reciprocal lattice points are on the Ewald sphere (see Figure 1). In the search of them, two methods are usually used:
• The method of θ−2θ scan (related usually to weak or forbidden reflection) in the white beam for a fixed azimuthal angle, which is in fact based on changing the radius of the Ewald sphere (see, e.g., Mikula et al., Reference Mikula, Vrána and Wagner2006b, Reference Mikula, Vrána, Wagner, Erko, Idir, Krist and Michette2008b).
• The method of azimuthal rotation of the crystal lattice around the scattering vector of the primary reflection for a fixed wavelength [see Figure 1(b)].
Figure 1. Schematic diagram of a two-step MBR simulating a primary reflection in real (a) and reciprocal space (b) for two reciprocal lattice points on the Ewald sphere. The numbers 1, 2, and 3 represent the primary, secondary, and tertiary reflection planes, respectively.
For the experimental studies of MBR effects, two Si slabs of different crystal cuts were chosen (see Figure 2). During the experiment, in both cases the bent Si slabs were set for diffraction on (002) planes in symmetric transmission geometry at the mean Bragg angle θ = 17.4° (for λ = 0.162 nm). The dimensions of the slabs were 200 × 40 × 3.5 mm3 (length × height × thickness). The thickness of 3.5 mm permitted us to bend the slabs in a rather large range of curvatures. For the search of possible MBR effects, the calculation procedure introduced in our earlier paper (Mikula et al., Reference Mikula, Vrána, Wagner, Erko, Idir, Krist and Michette2008b) was used. Figure 3 shows the related azimuth–Bragg angle relationships for the occurrence of the secondary lattice planes (represented by the reciprocal lattice points determined by the vector g 2,i on the Ewald sphere) accompanied the primary Si(002) one. The index i means that there could be more than one additional reciprocal lattice points on the Ewald sphere simultaneously. It can be seen from Figure 1(b) that due to the crystal symmetry, it follows that when a secondary reflection fulfils the Bragg condition simultaneously with the primary one, there exists automatically the tertiary reflection defined by g 3,i = g 1–g 2i.
Figure 2. Top view on two crystal cuts of the Si slabs that were used in the experimental search of MBR effects in symmetric transmission geometry.
Figure 3. Parts of the azimuth vs. Bragg angle relationship for the 002 primary reflection of the diamond structure as related to the experimental conditions of the slab: (a) the main face is parallel to (100) lattice planes and (b) the main face is parallel to (110) lattice planes.
The experiment was carried out on the neutron optics diffractometer NOP (installed at the Řež nuclear reactor LVR-15 – http://www.ujf.cas.cz/en/departments/department-of-neutron-physics/instruments/lvr15/hk8b/) operating at the fixed neutron wavelength of λ = 0.162 nm provided by the fixed bent-perfect Si(111) pre-monochromator. The Si(002) slabs were situated on the second axis of the diffractometer. The beam incident on the Si(002) slabs was limited to the cross section of 20 × 10 mm2 (width × height). For the rotation of the crystal around the scattering vector, a goniometer was used. Due to the construction of the four-point bending device (Mikula et al., Reference Mikula, Kulda, Lukáš, Vrána, Ono and Sawano2000) and the goniometer head for the azimuthal rotation, only the range of azimuthal angles ϕ from −7° to 12° could be used.
III. EXPERIMENTAL RESULTS
A. The main face of the crystal slab parallel to the (100) lattice planes
First, the rocking curves of the bent Si(002) crystal with respect to the bent Si(111) pre-monochromator for different individual azimuthal angles ϕ were carried out. Figure 4 shows a 3D map of the MBR effects constructed on the base of the individual rocking curves carried out in the vicinity of the mean Bragg angle. In relation to Figure 3, the rocking curves mean short Δθ scans with respect to the horizontal line (representing the mean Bragg angle θ = 17.4° which corresponds to the primary reflection) for each azimuthal angle ϕ. Figure 5 demonstrates several rocking curves at different azimuthal angles which show a contribution of some of MBRs. The effect of the MBR phenomenon can be also shown by the ϕ-rotation of the crystal slab for a constant value of the Bragg angle θ. Figure 6 shows the occurrence of the MBR effect as a function of the azimuthal angle ϕ for two values of Δθ from the mean Bragg angle θ. Finally, Figure 7 shows a 2D map of the MBR effects as constructed from the individual rocking curves in the vicinity of the azimuthal angle ϕ = 6° in more detail. It should be pointed out that the form of the 2D map follows the lines of MBR occurrences from Figure 3(a). Then, after setting the Si(002) slab at the peak position for ϕ = 5.4° (see Figure 5), the divergence of the neutrons obtained by the MBR phenomenon was tested by means of small holes in the Cd sheet inserted into the MBR beam just close to the Si(002) crystal and imaged by imaging plate (IP) at the distance of 60 cm. It can be seen from Figure 8 that the individual images are slightly inclined and perpendicular to the line (331)/(333) + (333)/(331) from Figure 3(a). The images of the holes confirm that the MBR beam is horizontally (in the scattering plane) highly collimated. An estimation provides the value of about 1.5 × 10−3 rad. However, the beam is vertically poorly collimated as coming from the Si(111) crystal.
Figure 4. 3D representation of the MBR effects for different azimuthal angles ϕ and the bending radius of the Si(002) slab of R = 12 m.
Figure 5. Si(002) rocking curves measured in the vicinity of the mean Bragg angle for several different azimuthal angles and R = 12 m.
Figure 6. Intensity of the MBR effect vs. the azimuthal angle ϕ for two values of Δθ from the mean Bragg angle.
Figure 7. 2D map of the MBR effects constructed at the vicinity of the azimuthal angle ϕ = 6°.
Figure 8. Photo of the holes in a Cd sheet (a) as imaged by IP at a distance of 60 cm from the holes (b).
B. The main face of the crystal slab parallel to the (110) lattice planes
A similar experiment was also carried out with another cut of the crystal slab, when the main face was parallel to the planes (110). The related 3D representation of the MBR effects based on the rocking curves of the Si(002) planes for different azimuthal angles ϕ and the bending radius of the Si(002) slab of R = 9 m is shown in Figure 9. Figure 10(a) shows several rocking curves for different azimuthal angles. On the other hand, Figure 10(b) demonstrates the influence of the degree of crystal deformation on the visibility of the MBR effect. As the rocking curves are not of the Gaussian form, in the next step the diffracted beam was imaged by IP at the distance of 30 cm from the MBR crystal for different ϕ positions (see Figure 11). In relation to possible MBR expectations displayed in Figure 3, it can be decided that the images shown in Figure 11 consist principally of two MBR contributions. The vertical part corresponds to the MBR effect related to the cooperative action of the secondary/tertiary planes (111)/(111), while the inclined part corresponds to the MBR effect related to the two pairs of planes (513)/(511) + (511)/(513). In this way, it is necessary to point out that the incident beam from the pre-monochromator has a rather poor collimation (no Soller collimators were used and the width of the incident beam was 20 mm), and therefore, the phase-space element of the incident beam Δθ × Δλ is rather large. It brings about that the related MBR effects shown in the rocking curves or their azimuthal dependence are rather broad. It should be pointed out that in the case of images shown in Figure 11, they are not constructed from the rocking curves (see Figures 4, 7, and 9) but taken by IP at chosen individual points given by the coordinates θ and ϕ. The dimensions of the diffracted images depend on the cross-section of the incident beam on the Si(002) slab as well as the vertical divergence of the beam and of course, on the distance of IP from the slab. However, each local place of the image provides a high spatial resolution as has already been demonstrated in the radiography experiment with other MBR monochromatic beam (see, e.g., Mikula et al., Reference Mikula, Vrána, Wagner, Erko, Idir, Krist and Michette2008b).
Figure 9. 3D representation of the MBR effects in the vicinity of the mean Bragg angle for different angles ϕ and R = 9 m.
Figure 10. Several rocking curves for (a) different azimuthal angles ϕ at a constant R = 9 m and (b) rocking curves for several radii of curvature of the Si slab at the azimuthal angle of ϕ = −6°.
Figure 11. Images of the MBR beam as taken by IP in the peak position of the rocking curve for (a) ϕ = −5°, (b) ϕ = −5.5°, (c) ϕ = −6°, (d) ϕ = −6.5°, and R = 7 m.
IV. SUMMARY
MBRs related to forbidden Si(002) primary reflection and realized in the bent-perfect crystal were investigated in symmetric transmission geometry (with respect to the primary 002 reflection) for two different crystal cuts. As can be seen from Figures 3–7 and 9–11, several strong MBR effects were observed which could be alternatively used for very high-resolution diffraction or transmission experiments. Inclination of the pattern of the monochromatic beam can be avoided by a special cut of the crystal slab having the main face at a suitable chosen ϕ-angle with respect to the 100 or 110 planes. However, not all possible MBR effects which could be expected from Figure 3 were realized and strengthened by the elastic bending. The appearance of the MBR effects and their reflectivity power strongly depend on the mutual orientation of the reciprocal lattice vectors of the secondary and tertiary reflections g 2i and g 3i with respect to the deformation vector which is in this case of cylindrical bending perpendicular to the g 1 lattice vector of the primary reflection. Moreover, it also depends on the values of Miller indexes h,k,l. For large values of (h 2 + k 2 + l 2)1/2, the corresponding Bragg angle increases while the reflectivity power decreases (e.g., for Si(353) reflection planes, the Bragg angle is θ = 78°).
The MBR effect in elastically deformed perfect crystal is realized by the double-diffraction process on pairs of secondary and tertiary lattice planes which are mutually in dispersive setting. Therefore, as expected, the obtained output beam is naturally highly monochromatic and highly collimated (namely in the horizontal plane) without the use of any collimator. However, there are still open questions which should be solved in the search of strong MBR effects, Monte Carlo (MC) simulations of the MBR processes would be desirable. First attempts of MC simulations of an Si(111) primary reflection accompanied by an MBR reflection have been done (Šaroun et al., Reference Šaroun, Mikula and Kulda2011).
ACKNOWLEDGEMENTS
Measurements were carried out at the CANAM infrastructure of the NPI CAS Řež. The presented results were also supported by the infrastructural MŠMT project LM2018120 (Experimental nuclear reactors LVR-15 and LR-0). P. Mikula acknowledges support from ESS participation of the Czech Republic – OP (CZ.02.1.01/0.0/0.0/16_013/0001794) and J. Šaroun acknowledges support from the project ESS Scandinavia-CZ II (LM2018111). Furthermore V. Ryukhtin acknowledges support from the Czech Academy of Sciences in the frame of the program “Strategie AV21, No. 23”. The authors thank Ms. B. Michalcová from NPI CAS for a significant help with the measurements and basic elaboration of the data.
