International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 5.3, pp. 660-661   | 1 | 2 |

Section 5.3.7. Applications of the dynamical theory of neutron scattering

M. Schlenkera* and J.-P. Guigayb

aLaboratoire Louis Néel du CNRS, BP 166, F-38042 Grenoble Cedex 9, France, and  bEuropean Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France
Correspondence e-mail:  schlenk@grenoble.cnrs.fr

5.3.7. Applications of the dynamical theory of neutron scattering

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5.3.7.1. Neutron optics

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Most experiments in neutron scattering require an intensity-effective use of the available beam at the cost of relatively high divergence and wavelength spread. The monochromators must then be imperfect (`mosaic') crystals. In some cases, however, it is important to have a small divergence and wavelength band. One example is the search for small variations in neutron energy in inelastic scattering without the use of the neutron spin-echo principle. Perfect crystals must then be used as monochromators or analysers, and dynamical diffraction is directly involved. As in the X-ray case, special designs can lead to strong decrease in the intensity of harmonics, i.e. of contributions of [\lambda/2] or [\lambda/3] (Hart & Rodrigues, 1978[link]). The possibility of focusing neutron beams by the use of perfect crystals with the incident beam spatially modulated in amplitude through an absorber, or in phase through an appropriate patterning of the surface, in analogy with the Bragg–Fresnel lenses developed for X-rays, was suggested by Indenbom (1979)[link].

The use of two identical perfect crystals in nondispersive (+, −, ∥) setting provides a way of measuring the very narrow intrinsic rocking curves expected from the dynamical theory. Any divergence added between the two crystals can be sensitively measured. Thus perfect crystals provide interesting possibilities for measuring very small angle neutron scattering. This was performed by Takahashi et al. (1981[link], 1983)[link] and Tomimitsu et al. (1986)[link] on amorphous materials, and by Kvardakov et al. (1987)[link] for the investigation of ferromagnetic domains in bulk silicon–iron specimens under stress, both through the variations in transmission associated with refraction on the domain walls and through small-angle scattering. Imaging applications are described in Section 5.3.7.4[link]. Badurek et al. (1979)[link] used the different deflection of the two polarization states provided by a magnetic prism placed between two perfect silicon crystals to produce polarized beams.

Curved almost-perfect crystals or crystals with a gradient in the lattice spacing can provide focusing (Albertini, Boeuf, Lagomarsino et al., 1976)[link] and vibrating crystals can give the possibility of tailoring the reflectivity of crystals, as well as of modulating beams in time (Michalec et al., 1988[link]). A double-crystal arrangement with bent crystals was shown by Eichhorn (1988)[link] to be a flexible small-angle-neutron-scattering device.

5.3.7.2. Measurement of scattering lengths by Pendellösung effects

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As in X-ray diffraction, Pendellösung oscillations provide an accurate way of measuring structure factors, hence coherent neutron scattering lengths. The equal-thickness fringes expected from a wedge-shaped crystal were observed by Kikuta et al. (1971)[link]. Three kinds of measurements were made. Sippel et al. (1965)[link] measured as a function of thickness the integrated reflectivity from a perfect crystal of silicon, the thickness of which they varied by polishing after each measurement, obtaining a curve similar to Fig. 5.1.6.7[link] , corresponding to equation 5.1.6.8[link] . Shull (1968)[link] restricted the measurement to wavefields that propagated along the reflecting planes, hence at exact Bragg incidence, by setting fine slits on the entrance and exit faces of 3 to 10 mm-thick silicon crystals, and measured the oscillation in diffracted intensity as he varied the wavelength of the neutrons used by rotating the crystal. Shull & Oberteuffer (1972)[link] showed that a better interpretation of the data, when the beam is restricted to a fine slit, corresponds to the spherical wave approach (actually cylindrical wave), and the boundary conditions were discussed more generally by Arthur & Horne (1985)[link]. Somenkov et al. (1978)[link] developed the inclination method, in which the integrated reflectivity is measured as the effective crystal thickness is varied nondestructively, by rotating the crystal around the diffraction vector, and used it for germanium. Belova et al. (1983)[link] discuss this method in detail. The results obtained by this group for magnetic crystals are mentioned in Section 5.3.6[link]. Structure-factor values for magnetic reflections were obtained by Kvardakov et al. (1995)[link] for the weak ferromagnet FeBO3.

5.3.7.3. Neutron interferometry

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Because diffraction by perfect crystals provides a well defined distribution of the intensity and phase of the beam, interferometry with X-rays or neutrons is possible using ingeniously designed and carefully manufactured monolithic devices carved out of single crystals of silicon. The technical and scientific features of this family of techniques are well summarized by Bonse (1979,[link] 1988[link]), as well as other papers in the same volumes, and by Shull (1986)[link].

X-ray interferometry started with the Bonse–Hart interferometer (Bonse & Hart, 1965[link]). A typical device is the LLL skew-symmetric interferometer, where the L's stand for Laue, indicating transmission geometry in all crystal slabs. In these slabs, which can be called the splitter, the mirrors and the recombiner, the same pair of opposite reflections, in symmetrical Laue geometry, is used three times. In the first slab, the incident beam is coherently split into a transmitted and a diffracted beam. Each of these is then diffracted in the two mirrors, and the resulting beams interfere in the recombiner, again yielding a forward-diffracted and a diffracted beam, the intensities of both of which are measured. This version, the analogue of the Mach–Zehnder interferometer in optics, offers a sizeable space (several cm of path length) where two coherent parallel beams can be submitted to various external actions. Shifting the relative phase of these beams (e.g. by π, introducing an optical path-length difference of [\lambda /2]) results in the intensities of the outgoing beams changing from a maximum to a minimum.

Applications of neutron interferometry range from the very useful to the very exotic. The most useful one is probably the measurement of coherent neutron scattering lengths. Unlike the Pendellösung method described in Section 5.3.7.2[link], this method does not require the measured samples to be perfect single crystals, nor indeed crystals. Placing a slab of material across one of the beams and rotating it will induce an optical path-length difference of [{(1 - n)t}] if t is the effective thickness along the beam, hence a phase shift of [2\pi (1 - n)t / \lambda]. With the expression of the refractive index n as given in Section 5.3.2.2[link], it is clear that for an isotopically pure material the scattering length [b_{\rm coh}] can be deduced from the measurement of intensity versus the rotation angle of the phase shifter. This is a very versatile and much used method. The decrease in oscillation contrast can be used to obtain information of relevance to materials science, such as statistical properties of magnetic domain distributions (Korpiun, 1966[link]) or precipitates (Rauch & Seidl, 1987[link]); Rauch (1995)[link] analyses the effect in terms of the neutron coherence function.

Many elegant experiments have been performed with neutron interferometers in efforts to set an upper limit to effects than can be considered as nonexistent, or to test expectations of basic quantum physics. Many papers are found in the same volumes as Bonse (1979)[link] and Bonse (1988[link]); excellent reviews have been given by Klein & Werner (1983)[link], Klein (1988)[link] and Werner (1995)[link]. Among the topics investigated are the effect of gravity (Colella et al., 1975[link]), the Sagnac effect, i.e. the influence of the Earth's rotation (Werner et al., 1979[link]), the Fizeau effect, i.e. the effect of the movement of the material through which the neutrons are transmitted (Arif et al., 1988[link]) and the Aharonov–Casher effect, i.e. the dual of the Aharonov–Bohm effect for neutral particles having a magnetic moment (Cimmino et al., 1989[link]).

5.3.7.4. Neutron diffraction topography and other imaging methods

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These are the neutron form of the `topographic' or diffraction imaging techniques, in which an image of a single crystal is obtained through the local variations in Bragg-diffracted intensity due to inhomogeneities in the sample. It is briefly described in Chapter 2.8[link] of IT C. It was pioneered by Doi et al. (1971)[link] and by Ando & Hosoya (1972)[link]. Like its X-ray counterpart, neutron topography can reveal isolated defects, such as dislocations (Schlenker et al., 1974[link]; Malgrange et al., 1976[link]). Because of the small neutron fluxes available, it is not very convenient for this purpose, since the resolution is poor or the exposure times are very long. On the other hand, the very low absorption of neutrons in most materials makes it quite convenient for observing the gross defect structure in samples that would be too absorbing for X-rays (Tomimitsu & Doi, 1974[link]; Baruchel et al., 1978[link]; Tomimitsu et al., 1983[link]; Kvardakov et al., 1992[link]), or the spatial modulation of distortion due to vibration, for example in quartz (Michalec et al., 1975[link]), and resonant magnetoelastic effects (Kvardakov & Somenkov, 1991[link]). In particular, virtual slices of bulky as-grown samples can be investigated without cutting them using neutron section topography or neutron tomography (Schlenker et al., 1975[link]; Davidson & Case, 1976[link]).

Neutron topography also shows the salient dynamical interference effect, viz Pendellösung, visually, in the form of fringes (Kikuta et al., 1971[link]; Malgrange et al., 1976[link]; Tomimitsu & Zeyen, 1978[link]). Its unique feature, however, is the possibility of observing and directly characterizing inhomogeneities in the magnetic structure, i.e. magnetic domains of all kinds [ferromagnetic domains (Schlenker & Shull, 1973[link]) and antiferromagnetic domains of various sorts (Schlenker & Baruchel, [link]1978), including spin-density wave domains (Ando & Hosoya, 1972,[link] 1978[link]; Davidson et al., 1974[link]), 180° or time-reversed domains in some materials and helimagnetic or chirality domains (Baruchel et al., 1990[link])], or coexisting phases at a first-order phase transition (Baruchel, 1989[link]). In such cases, the contrast is primarily due to local variations in the structure factor, a situation very unusual in X-ray topography, and good crystal quality, leading to dynamical scattering behaviour, is essential in the observation process only in a few cases (Schlenker et al., 1978[link]). It is often crucial, however, for making the domain structure simple enough to be resolved, in particular in the case of antiferromagnetic domains.

Imaging can also be performed for samples that need be neither crystals nor perfect. Phase-contrast imaging of a specimen through which the neutrons are transmitted can be performed in a neutron interferometer. It has been shown to reveal thickness variations by Bauspiess et al. (1978)[link] and ferromagnetic domains by Schlenker et al. (1980)[link]. The same papers showed that phase edges show up as contrast when one of the interferometer paths is blocked, i.e. when the sample is placed effectively between perfect, identical crystals set for diffraction in a nondispersive setting. Under the name of neutron radiography with refraction contrast, this technique, essentially a form of Schlieren imaging, was further developed by Podurets, Somenkov & Shil'shtein (1989)[link], Podurets, Somenkov, Chistyakov & Shil'shtein (1989)[link], and Podurets et al. (1991)[link], who were able to image internal ferromagnetic domain walls in samples 10 mm thick.

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