International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 2.6, p. 106

## Section 2.6.2.1.2. Geometry

R. Mayb

#### 2.6.2.1.2. Geometry

| top | pdf |

With typical neutron wavelengths, low Q need not necessarily mean small angles: The interesting Q range for an inhomogeneity of dimension D can be estimated as 1/D Q 10/D. The scattering angle corresponding to the upper Q limit for D = 10 nm is 1.4° for Cu Kα radiation, but amounts to 9.1° for neutrons of 10 nm wavelength. Consequently, it is preferable to speak of low-Q rather than of small-angle neutron scattering.

Pin-hole'-type cameras are the most frequently used SANS instruments; an example is the SANS camera D11 at the Institut Max von Laue–Paul Langevin in Grenoble, France (Ibel, 1976; Lindner, May & Timmins, 1992), from which some of the numbers below are quoted. Since the cross section of the primary beam is usually chosen to be rather large (e.g.  cm) for intensity reasons, pin-hole instruments tend to be large. The smallest Q value that can be measured at a given distance is just outside the image of the direct beam on the detector (which either has to be attenuated or is hidden behind a beamstop, a neutron-absorbing plate of several 10 cm2, e.g. of cadmium). Very small Q values thus require long sample-to-detector distances. The area detector of D11, with a surface of  cm and resolution elements of 1 cm2, moves within an evacuated tube of 1.6 m diameter and a length of 40 m. Thus, a Q range of to 5 nm−1 is covered. The geometrical resolution is determined by the length of the free neutron flight path in front of the sample, moving sections of neutron guide into or out of the beam (collimation'). In general, the collimation length is chosen roughly equal to the sample-to-detector distance. Thus, the geometrical and wavelength contributions to the Q resolution match at a certain distance of the scattered beam from the direct-beam position in the detector plane. In order to resolve scattering patterns with very detailed features, e.g. of particles with high symmetry, longer collimation lengths are sometimes required at the expense of intensity.

Much more compact double-crystal neutron diffractometers [described for X-rays by Bonse & Hart (1966)] are being used to reach the very small Q values of some 10−4 nm−1 typical of static light scattering. The sample is placed between two crystals. The first crystal defines the wavelength and the direction of the incoming beam. The other crystal scans the scattered intensity. The resolution of such an instrument is mainly determined by the Darwin widths of the ideal crystals. This fact is reflected in the low neutron yield. Slit geometry can be used, but not 2D detectors.

A recent development is the ellipsoidal-mirror SANS camera. The mirror, which needs to be of very high surface quality, focuses the divergent beam from a small (several mm2) source through the sample onto a detector with a resolution of the order of 1 × 1 mm. Owing to the more compact beam image, all other dimensions of the SANS camera can be reduced drastically (Alefeld, Schwahn & Springer, 1989). Whether or not there is a gain in intensity as compared with pin-hole geometry is strongly determined by the maximal sample dimensions. Long mirror with cameras (e.g. 20 m) are always superior to double-crystal instruments in this respect (Alefeld, Schwahn & Springer, 1989), and can also reach the light-scattering Q domain (Qmin of some 10−4 nm−1, corresponding to particles of several µm dimension).

### References

Alefeld, B., Schwahn, D. & Springer, T. (1989). New developments of small angle neutron scattering instruments with focusing. Nucl. Instrum. Methods, A274, 210–216.
Bonse, U. & Hart, M. (1966). Small-angle X-ray scattering by spherical particles of polystyrene and polyvinyltoluene. Z. Phys. 189, 151–162.
Ibel, K. (1976). The neutron small-angle camera D11 at the high-flux reactor, Grenoble. J. Appl. Cryst. 9, 296–309.
Lindner, P., May, R. P. & Timmins, P. A. (1992). Upgrading the SANS instrument D11 at the ILL. Physica (Utrecht), B180–181, 967–972.