Properties of the dispersive double bent crystal monochromator setting with a multicrystal sandwich at the place of the second axis

1. Introduction

It is well known that nondispersive as well as dispersive DC settings could be often found as beam optical elements at synchrotron sources as e.g. [5]. However, due to a lower neutron monochromatic current, some nondispersive ( 1 , − 1 ) DC settings exploiting the same crystals are sometimes used for neutron scattering instruments e.g. see ref. [6,8,10,11,23]. The luminosity problem can occur, namely, for mosaic crystals having the peak reflectivity r < 1 and in the case of the DC system the neutron monochromatic current is proportional roughly to r 2 . The situation is even much worse in the case of a dispersive double-crystal setting. However, the DC monochromator with two parallel identical crystals in ( 1 , − 1 ) setting has some advantages: The extracted monochromatic beam is parallel to the incident white beam even when changing the neutron wavelength and the scattering instrument is generally characterized by a lower background detector signal. Sometimes, this allows a smaller space for the instrument installed at the reactor [4]. Nevertheless, in the case of mosaic crystals, there is a commonly accepted opinion that the DC setting is not convenient due to a lower monochromatic neutron current and thus a low luminosity of the scattering instrument. It is not known whether some dispersive DC monochromator arrangements could be permanently employed even though some attempts have already been carried out [20,21]. On the other hand, with respect to neutron scattering, in the case of bent perfect crystals (e.g. Si or Ge) the situation is rather different from that of mosaic crystals and some advantages of the dispersive DC setting can be found. First, one should consider that when the curvature of the crystals is not drastic, the reflection probability r 1 , 2 of both bent perfect crystals can be close to 1 and there is a small and acceptable loss of neutrons due to the double-reflection process [7,9]. Namely, it is valid when neutrons of λ > 0.18  nm are used [9]. First particular experimental results related to the dispersive DC setting and its possible use in residual strain/stress measurements were published in ref. [13,14,16–19] in which its excellent properties fully confirmed a promising future of the unconventional DC setting. In this paper we present extensive experimental results obtained by testing the properties of the dispersive DC ( n , − m ) setting of Si(111) in combination with the Si(220), Ge(111), Si(311) or Si(400) BPC slabs in relation to possible use in neutron diffractometry.

2. Experimental arrangement

For this experiment, we used a testing neutron diffractometer installed at the 10 MW medium power research reactor LWR-15 situated in Řež operating at the fixed neutron wavelength of λ = 0.162  nm which is provided by a cylindrically BPC Si(111) slab of the dimensions of 200 × 40 × 4 mm³ (length × width × thickness) and the fixed radius of curvature of R I ≈ 12  m. Its curvature was not changed during the experiment. A schematic sketch of the diffractometer performance is drawn in Fig. 1. The testing diffractometer can operate in two or three axis mode. The sandwich, situated on the second axis, consisted of four thin (1.3 mm) BPC slabs Si(220), Ge(111), Si(400) and Ge(311) (which we had at a disposal) which were all cut for symmetric reflection geometry. When using the individual slabs of the sandwich for the experiment in the DC setting, their curvatures were easily comparable and it avoided some possible errors in determining the curvature coming from the construction of the bending device. The dimension of each slab in the sandwich was 200 × 40 × 1.3 mm³ (length × width × thickness). By using a sandwich of such thin slabs, a large range of curvatures, without a danger of breaking them, could be used. For neutron detection either a point detector or neutron Imaging Plate (IP) with a 50 μm × 50 μm spatial resolution was used. The polycrystalline sample was put on the third axis of the instrument which was at a distance of 50 cm from the sandwich. The distance between the Si(111) and the sandwich of slabs was fixed and equal to 170 cm. No Soller collimators were installed before and after the Si(111) crystal. During these extensive experimental studies the cross-section of the incident beam coming from the bent Si(111) crystal of 20 × 20 mm² was used (see Fig. 1). As to the imaging, we focused our attention mainly on α-Fe(211) diffraction profiles taken at a scattering angle 2θ of about 88° when a well annealed standard sample of the diameter ϕ = 2  mm was used.

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Fig. 1.

Schematic sketch of the diffraction performance as used for the experimental investigations.

Generally, the BPC elements are desirable to use from the following reasons: predictability and reproducibility of the effective mosaicity, predictability and reproducibility of a rather high peak reflectivity and its uniformity over large areas of the crystal, rather high peak reflectivity for asymmetric or transmission geometry and predictability of the integrated reflectivity. When a neutron passing the BPC crystal meets the Bragg condition, it is reflected with the probability r ( R ) often called peak reflectivity, which is of course dependent on the bending radius R. For the asymmetry cut of the crystal and asymmetry diffraction geometry, r ( R ) is equal for the both geometries, output beam compression as well as for output beam expansion. Then, the parameter r ( R ) was derived for cylindrical BPC slab in a simple general form [7,9] (1) r ( R ) = { 1 − exp [ − Q h k l ( ∂ Δ θ / ∂ s o ) − 1 ] } , where Q h k l = ( F h k l ) 2 · λ 3 / ( Ω 2 · sin 2 θ o ) is the kinematical reflectivity of the crystal unit volume ( F h k l , λ and Ω are the structure factor, neutron wavelength and the unit-cell volume, respectively [1]). As can be seen in Fig. 2, the rate of change of the Bragg angle on the flight path Δ s 0 between points A and B in the crystal (in the incident beam direction) for general asymmetric diffraction geometry is (2) ∂ Δ θ / ∂ s 0 = ( 1 / R M ) ( cos ψ / cos θ o ) [ 1 − ( 1 + σ ) sin ( θ o + ψ ) sin ( θ o − ψ ) ] , where σ is the Poisson constant and ψ is the angle of the crystal asymmetry cut. The difference δ θ = θ 1 − θ 2 (see Fig. 2) is called effective mosaicity as an alternative to the case of mosaic crystals. A strong dependence of r ( R ) on λ is evident. Figure 3 shows the calculated values and the dependence of r ( R ) on the curvature 1/R for individual crystal slabs from the sandwich [7,9]. Inspection of Fig. 3 reveals that for small curvatures the peak reflectivity is very close to 1 as in the case of a perfect crystal, but for stronger curvatures it decreases, however, still being comparable or higher to that of mosaic crystals. More details about the other properties of BPC crystals (e.g. integrated reflectivity) can be found in [9].

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Fig. 2.

Detail of the incident beam passing through the BPC slab in the general asymmetric diffraction geometry.

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Fig. 3.

Peak reflectivity vs crystal curvature for individual crystal slabs as calculated according to the formulae (1) and (2) for symmetric diffraction geometry ( ψ = 0 °).

First, it is necessary to point out that in order to be able to use the DC setting as a neutron monochromator, only neutrons extracted from the polychromatic beam for Δ ω = 0 o of the rocking curve (peak intensity) are usable. The peak intensity is proportional to the area resulting from mutually overlapping of phase space elements of the individual BPC slabs. A comparison of the peak reflectivities of the individual BPC slabs of the sandwich (see Fig. 3) shows that thanks to a higher scattering amplitude of Ge with respect to Si, it would be more advantageous to use the Ge-slabs instead of the Si-slabs, namely, for stronger curvatures 1 / R . As the peak reflectivity of the double-crystal setting is proportional to r 1 ( R 1 ) × r 2 ( R 2 ) , the loss of intensity could be small if the curvature is not drastic.

An inspection of Figs 8, 14, 19 and 24 reveals that the best resolution in the powder diffraction tests on a α-Fe(211) standard sample is obtained at smaller curvatures below 0.125 m⁻¹. There was also a little difference between the parallel ( 1 , − 1 ) and antiparallel ( 1 , + 1 ) settings that was observed. This property can be used successfully in the case of powder diffraction, namely, strain/stress measurements when two strain components can be measured simultaneously with an equivalent resolution as it is usually done in the case of the TOF method. This result can be expected, because for smaller curvatures of the second bent crystal the extracted monochromatized beam has a spread Δ λ / λ as well as a Δ θ collimation smaller (a self-collimation effect). The excellent resolution property of the double diffracted beam can be used effectively in studies of microstrains as well as the grain size distribution in polycrystalline materials after applying shape analysis of the neutron diffraction peak profiles [2,3].

As to the spatial distributions of the double diffracted beams (see Figs 5, 11, 16 and 21 or Figs 6, 12, 17 and 22)) minimum FWHMs are observed at larger curvatures above 0.2 m⁻¹, namely, for Si(220), Ge(311) and Si(400) slabs of the sandwich. As can be seen from the related figures, the focusing of the double diffracted beam is evident, while the peak intensity of the double crystal setting (for Δ ω = 0 o ) does not change significantly. Furthermore, focusing can be easily manipulated in a large range of curvatures. For strong curvatures, a high resolution in the powder diffraction experiment was achieved only for a parallel ( 1 , − 1 ) setting for Si(220), Si(400) and Ge(311) slabs from the sandwich, which means that the BPC slabs allowing for the ( 1 , − 1 ) setting to be sufficiently dispersive. However, it was not the case of the Ge(111) one having the value of the lattice constant close to that of Si(111). In the practical experiment a compromise in the choice of curvature should be made. As was demonstrated by the results in Figs 9 and 27 the double diffracted beams have very low divergence Δ ( 2 θ S ) of about 1 × 10 − 3 rad which makes it possible to use them for investigations e.g. of edge refraction effects. Particular studies with the dispersive double-crystal Si(220) + Si(311) and Si(111) + Si(311) settings containing the BPC Si(311) slab in strongly or fully asymmetric diffraction geometry can also be found in [12,13,19].

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Fig. 28.

Rocking curves of the DC Si(111) + Si(220) setting with the sandwich containing 1, 2 or 4 slabs all of the thickness of 1.3 mm and bent with the radius of curvature R = 6 m – (a) and FWHM of monochromatic beam spatial distribution from the bent sandwich as a function of curvature taken at a distance of 70 cm from the sandwich [15].

In order to find the optimum performance of the diffractometer using a dispersive DC focusing monochromator, there are still some open questions which could be answered by means of Monte Carlo simulations [22]. Namely, the use of a thinner Si(111) crystal and optimization of the distance between the first and second crystal slab can open the door to obtaining a highly focused beam of submillimeter width at the focal point. On the other hand, with the use of thicker slabs some compromise between the slightly relaxed focusing as well as resolution and the instrument luminosity can be found as was pointed out in [15] (see an example in Fig. 28). Thus, the simulations can provide best possible resolution and luminosity properties of the optimized DC setting.

Finally, it can be stated that the presented dispersive DBC settings offer high-resolution monochromator alternatives to the conventional one, namely, for measuring the finer effects in powder diffractometry which can be successfully exploited at the high-flux neutron sources.