International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 4.4, pp. 435-438
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The refractive index, n, for neutrons of wavelength λ propagating in a nonmagnetic material of atomic density N is given by the expression where is the mean coherent scattering length. Values of the scattering-length density for some common materials are listed in Table 4.4.2.2, from which it can be seen that the refractive index for most materials is slightly less than unity, so that total external reflection can take place. Thus, neutrons can be reflected from a smooth surface, but the critical angle of reflection, given by is small, so that reflection can only take place at grazing incidence. The critical angle for nickel, for example, is 0.1° Å−1.
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Because of the shallowness of the critical angle, reflective optics are traditionally bulky, and focusing devices tend to have long focal lengths. In some cases, however, depending on the beam divergence, a long mirror can be replaced by an equivalent stack of shorter mirrors.
The principle of mirror reflection is the basis of neutron guides, which are used to transmit neutron beams to instruments that may be situated up to 100 m away from the source (Christ & Springer, 1962; Maier-Leibnitz & Springer, 1963). A standard neutron guide is constructed from boron glass plates assembled to form a rectangular tube, the dimensions of which may be up to 200 mm high by 50 mm wide. The inner surface of the guide is coated with approximately 1200 Å of either nickel, 58Ni ( = 0.12° Å−1), or a `supermirror' (described below). The guide is usually evacuated to reduce losses due to absorption and scattering of neutrons in air.
Theoretically, a neutron guide that is fully illuminated by the source will transmit a beam with a square divergence of full width in both the horizontal and vertical directions, so that the transmitted solid angle is proportional to . In practice, owing to imperfections in the assembly of the guide system, the divergence profile is closer to Gaussian than square at the end of a long guide. Since the neutrons may undergo a large number of reflections in the guide, it is important to achieve a high reflectivity. The specular reflectivity is determined by the surface roughness, and typically values in the range 98.5 to 99% are achieved. Further transmission losses occur due to imperfections in the alignment of the sections that make up the guide.
The great advantage of neutron guides, in addition to the transport of neutrons to areas of low background, is that they can be multiplexed, i.e. one guide can serve many instruments. This is achieved either by deflecting only a part of the total cross section to a given instrument or by selecting a small wavelength range from the guide spectrum. In the latter case, the selection device (usually a crystal monochromator) must have a high transmission at other wavelengths.
If the neutron guide is curved, the transmission becomes wavelength dependent, as illustrated in Fig. 4.4.2.4 . In this case, one can define a characteristic wavelength, , given by the relation , so that (where a is the guide width and ρ the radius of curvature), for which the theoretical transmission drops to 67%. For wavelengths less than , neutrons can only be transmitted by `garland' reflections along the concave wall of the curved guide. Thus, the guide acts as a low-pass energy filter as long as its length is longer than the direct line-of-sight length . For example, a 3 cm wide nickel-coated guide whose characteristic wavelength is 4 Å (radius of curvature 1300 m) must be at least 18 m long to act as a filter. The line-of-sight length can be reduced by subdividing the guide into a number of narrower channels, each of which acts as a miniguide. The resulting device, often referred to as a neutron bender, since deviation of the beam is achieved more rapidly, is used in beam deviators (Alefeld et al., 1988) or polarizers (Hayter, Penfold & Williams, 1978). A microbender was devised by Marx (1971) in which the channels were made by evaporating alternate layers of aluminium (transmission layer) and nickel (mirror layer) onto a flexible smooth substrate.
Tapered guides can be used to reduce the beam size in one or two dimensions (Rossbach et al., 1988), although, since mirror reflection obeys Liouville's theorem, focusing in real space is achieved at the expense of an increase in divergence. This fact can be used to calculate analytically the expected gain in neutron flux at the end of a tapered guide (Anderson, 1988). Alternatively, focusing can be achieved in one dimension using a bender in which the individual channel lengths are adjusted to create a focus (Freund & Forsyth, 1979).
Optical imaging of neutrons can be achieved using ellipsoidal or torroidal mirrors, but, owing to the small critical angle of reflection, the dimensions of the mirrors themselves and the radii of curvature must be large. For example, a 4 m long toroidal mirror has been installed at the IN15 neutron spin echo spectrometer at the Institut Laue–Langevin, Grenoble (Hayes et al., 1996), to focus neutrons with wavelengths greater than 15 Å. The mirror has an in-plane radius of curvature of 408.75 m, and the sagittal radius is 280 mm. A coating of 65Cu is used to obtain a high critical angle of reflection while maintaining a low surface roughness. Slope errors of less than 2.5 × 10−5 rad (r.m.s.) combined with a surface roughness of less than 3 Å allow a minimum resolvable scattering vector of about 5 × 10−4 Å−1 to be reached.
For best results, the slope errors and the surface roughness must be low, in particular in small-angle scattering applications, since diffuse scattering from surface roughness gives rise to a halo around the image point. Owing to its low thermal expansion coefficient, highly polished Zerodur is often chosen as substrate.
Schoenborn, Caspar & Kammerer (1974) first pointed out that multibilayers, comprising alternating thin films of different scattering-length densities () act like two-dimensional crystals with a d spacing given by the bilayer period. With modern deposition techniques (usually sputtering), uniform films of thickness ranging from about twenty to a few hundred ångströms can be deposited over large surface areas of the order of 1 m2. Owing to the rather large d spacings involved, the Bragg reflection from multilayers is generally at grazing incidence, so that long devices are required to cover a typical beam width, or a stacked device must be used. However, with judicious choice of the scattering-length contrast, the surface and interface roughness, and the number of layers, reflectivities close to 100% can be reached.
Fig. 4.4.2.5 illustrates how variation in the bilayer period can be used to produce a monochromator (the minimum Δλ/λ that can be achieved is of the order of 0.5%), a broad-band device, or a `supermirror', so called because it is composed of a particular sequence of bilayer thicknesses that in effect extends the region of total mirror reflection beyond the ordinary critical angle (Turchin, 1967; Mezei, 1976; Hayter & Mook, 1989). Supermirrors have been produced that extend the critical angle of nickel by a factor, m, of between three and four with reflectivities better than 90%. Such high reflectivities enable supermirror neutron guides to be constructed with flux gains, compared with nickel guides, close to the theoretical value of m2.
Illustration of how a variation in the bilayer period can be used to produce a monochromator, a broad-band device, or a supermirror. |
The choice of the layer pairs depends on the application. For non-polarizing supermirrors and broad-band devices (Høghøj, Anderson, Ebisawa & Takeda, 1996), the Ni/Ti pair is commonly used, either pure or with some additions to relieve strain and stabilize interfaces (Elsenhans et al., 1994) or alter the magnetism (Anderson & Høghøj, 1996), owing to the high contrast in scattering density, while for narrow-band monochromators a low contrast pair such as W/Si is more suitable.
Capillary neutron optics, in which hollow glass capillaries act as waveguides, are also based on the concept of total external reflection of neutrons from a smooth surface. The advantage of capillaries, compared with neutron guides, is that the channel sizes are of the order of a few tens of micrometres, so that the radius of curvature can be significantly decreased for a given characteristic wavelength [see equation (4.4.2.6)]. Thus, neutrons can be efficiently deflected through large angles, and the device can be more compact.
Two basic types of capillary optics exist, and the choice depends on the beam characteristics required. Polycapillary fibres are manufactured from hollow glass tubes several centimetres in diameter, which are heated, fused and drawn multiple times until bundles of thousands of micrometre-sized channels are formed having an open area of up to 70% of the cross section. Fibre outer diameters range from 300 to 600 µm and contain hundreds or thousands of individual channels with inner diameters between 3 and 50 µm. The channel cross section is usually hexagonal, though square channels have been produced, and the inner channel wall surface roughness is typically less than 10 Å r.m.s., giving rise to very high reflectivities. The principal limitations on transmission efficiency are the open area, the acceptable divergence (note that the critical angle for glass is 1 mrad Å−1) and reflection losses due to absorption and scattering. A typical optical device will comprise hundreds or thousands of fibres threaded through thin screens to produce the required shape.
Fig. 4.4.2.6 shows typical applications of polycapillary devices. In Fig. 4.4.2.6(a), a polycapillary lens is used to refocus neutrons collected from a divergent source. The half lens depicted in Fig. 4.4.2.6(b) can be used either to produce a nearly parallel (divergence = ) beam from a divergent source or (in the reverse sense) to focus a nearly parallel beam, e.g. from a neutron guide. The size of the focal point depends on the channel size, the beam divergence, and the focal length of the lens. For example, a polycapillary lens used in a prompt γ-activation analysis instrument at the National Institute of Standards and Technology to focus a cold neutron beam from a neutron guide results in a current density gain of 80 averaged over the focused beam size of 0.53 mm (Chen et al., 1995).
Typical applications of polycapillary devices: (a) lens used to refocus a divergent beam; (b) half-lens to produce a nearly parallel beam or to focus a nearly parallel beam; (c) a compact bender. |
Fig. 4.4.2.6(c) shows another simple application of polycapillaries as a compact beam bender. In this case, such a bender may be more compact than an equivalent multichannel guide bender, although the accepted divergence will be less. Furthermore, as with curved neutron guides, owing to the wavelength dependence of the critical angle the capillary curvature can be used to filter out thermal or high-energy neutrons.
It should be emphasized that the applications depicted in Fig. 4.4.2.6 obey Liouville's theorem, in that the density of neutrons in phase space is not changed, but the shape of the phase-space volume is altered to meet the requirements of the experiment, i.e. there is a simple trade off between beam dimension and divergence.
The second type of capillary optic is a monolithic configuration. The individual capillaries in monolithic optics are tapered and fused together, so that no external frame assembly is necessary (Chen-Mayer et al., 1996). Unlike the multifibre devices, the inner diameters of the channels that make up the monolithic optics vary along the length of the component, resulting in a smaller more compact design.
Further applications of capillary optics include small-angle scattering (Mildner, 1994) and lenses for high-spatial-resolution area detection.
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