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

International Tables for Crystallography (2006). Vol. C, ch. 2.5, pp. 87-88

## Section 2.5.2. White-beam and time-of-flight neutron diffraction

J. D. Jorgensen,c W. I. F. Davida and B. T. M. Willisd

### 2.5.2. White-beam and time-of-flight neutron diffraction

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#### 2.5.2.1. Neutron single-crystal Laue diffraction

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In traditional neutron-diffraction experiments, using a continuous source of neutrons from a nuclear reactor, a narrow wavelength band is selected from the wide spectrum of neutrons emerging from a moderator within the reactor. This monochromatization process is extremely inefficient in the utilization of the available neutron flux. If the requirement of discriminating between different orders of reflection is relaxed, then the entire white beam can be employed to contribute to the diffraction pattern and the count-rate may increase by several orders of magnitude. Further, by recording the scattered neutrons on photographic film or with a position-sensitive detector, it is possible to probe simultaneously many points in reciprocal space.

If the experiment is performed using a pulsed neutron beam, the different orders of a given reflection may be separated from one another by time-of-flight analysis. Consider a short polychromatic burst of neutrons produced within a moderator. The subsequent times-of-flight, t, of neutrons with differing wavelengths, λ, measured over a total flight path, L, may be discriminated one from another through the de Broglie relationship: where mn is the neutron mass and h is Planck's constant. Expressing t in microseconds, L in metres and λ in Å, equation (2.5.2.1) becomes Inserting Bragg's law, , for the nth order of a fundamental reflection with spacing d in Å gives Different orders may be measured simply by recording the time taken, following the release of the initial pulse from the moderator, for the neutron to travel to the sample and then to the detector.

The origins of pulsed neutron diffraction can be traced back to the work of Lowde (1956) and of Buras, Mikke, Lebech & Leciejewicz (1965). Later developments are described by Turberfield (1970) and Windsor (1981). Although a pulsed beam may be produced at a nuclear reactor using a chopper, the major developments in pulsed neutron diffraction have been associated with pulsed sources derived from particle accelerators. Spallation neutron sources, which are based on proton synchrotrons, allow optimal use of the Laue method because the pulse duration and pulse repetition rate can be matched to the experimental requirements. The neutron Laue method is particularly useful for examining crystals in special environments, where the incident and scattered radiations must penetrate heat shields or other window materials. [A good example is the study of the incommensurate structure of α-uranium at low temperature (Marmeggi & Delapalme, 1980).]

A typical time-of-flight single-crystal instrument has a large area detector. For a given setting of detector and sample, a three-dimensional region is viewed in reciprocal space, as shown in Fig. 2.5.2.1. Thus, many Bragg reflections can be measured at the same time. For an ideally imperfect crystal, with volume Vs and unit-cell volume vc, the number of neutrons of wavelength λ reflected at Bragg angle by the planes with structure factor F is given by where is the number of incident neutrons per unit wavelength interval. In practice, the intensity in equation (2.5.2.3) must be corrected for wavelength-dependent factors, such as detector efficiency, sample absorption and extinction, and the contribution of thermal diffuse scattering. Jauch, Schultz & Schneider (1988) have shown that accurate structural data can be obtained using the single-crystal time-of-flight method despite the complexity of these wavelength-dependent corrections.

 Figure 2.5.2.1 | top | pdf |Construction in reciprocal space to illustrate the use of multi-wavelength radiation in single-crystal diffraction. The circles with radii kmax = 2π/λmin and kmin = 2π/λmax are drawn through the origin. All reciprocal-lattice points within the shaded area may be sampled by a linear position-sensitive detector spanning the scattering angles from 2θmin to 2θmax. With a position-sensitive area detector, a three-dimensional portion of reciprocal space may be examined (after Schultz, Srinivasan, Teller, Williams & Lukehart, 1984).

#### 2.5.2.2. Neutron time-of-flight powder diffraction

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This technique, first developed by Buras & Leciejewicz (1964), has made a unique impact in the study of powders in confined environments such as high-pressure cells (Jorgensen & Worlton, 1985). As in single-crystal Laue diffraction, the time of flight is measured as the elapsed time from the emergence of the neutron pulse at the moderator through to its scattering by the sample and to its subsequent detection. This time is given by equation (2.5.2.2). Many Bragg peaks, each separated by time of flight, can be observed at a single fixed scattering angle, since there is a wide range of wavelengths available in the incident beam.

A good approximation to the resolution function of a time-of-flight powder diffractometer is given by the second-moment relationship where , and are, respectively, the uncertainties in the d spacing, time of flight, and Bragg angle associated with a given reflection, and is the uncertainty in the total path length (Jorgensen & Rotella, 1982). Thus, the highest resolution is obtained in back scattering (large ) where cot is small. Time-of-flight instruments using this concept have been described by Steichele & Arnold (1975) and by Johnson & David (1985). With pulsed neutron sources a large source aperture can be viewed, as no chopper is required of the type used on reactor sources. Hence, long flight paths can be employed and this too [see equation (2.5.2.4)] leads to high resolution. For a well designed moderator the pulse width is approximately proportional to wavelength, so that the resolution is roughly constant across the whole of the diffraction pattern. For an ideal powder sample the number of neutrons diffracted into a complete Debye–Scherrer cone is proportional to (Buras & Gerward, 1975). j is the multiplicity of the reflection and the remaining symbols in equation (2.5.2.5) are the same as those in equation (2.5.2.3).

### References

Buras, B. & Gerward, L. (1975). Relations between integrated intensities in crystal diffraction methods for X-rays and neutrons. Acta Cryst. A31, 372–374.
Buras, B. & Leciejewicz, J. (1964). A new method for neutron diffraction crystal structure investigations. Phys. Status Solidi, 4, 349–355.
Buras, B., Mikke, K., Lebech, B. & Leciejewicz, J. (1965). The time-of-flight method for investigations of single-crystal structures. Phys. Status Solidi, 11, 567–573.
Jauch, W., Schultz, A. J. & Schneider, J. R. (1988). Accuracy of single crystal time-of-flight neutron diffraction: a comparative study of MnF2. J. Appl. Cryst. 21, 975–979.
Johnson, M. W. & David, W. I. F. (1985). HPRD: the high resolution powder diffractometer at the spallation neutron source. Report RAL-85-112. Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, UK.
Jorgensen, J. D. & Rotella, F. J. (1982). High-resolution time-of-flight powder diffractometer at the ZING-P′ pulsed neutron source. J. Appl. Cryst. 15, 27–34.
Jorgensen, J. D. & Worlton, T. G. (1985). Disordered structure of D2O ice VII from in situ neutron powder diffraction. J. Chem. Phys. 83, 329–333.
Lowde, R. D. (1956). A new rationale of structure-factor measurement in neutron-diffraction analysis. Acta Cryst. 9, 151–155.
Marmeggi, J. C. & Delapalme, A. (1980). Neutron Laue photographs of crystallographic satellite reflections in alpha-uranium. Physica (Utrecht), 102B, 309–312.
Schultz, A. J., Srinivasan, K., Teller, R. G., Williams, J. M. & Lukehart, C. M. (1984). Single-crystal time-of-flight neutron diffraction structure of hydrogen cis-diacetyltetracarbonyl­rhenate. J. Am. Chem. Soc. 106, 999–1003.
Steichele, E. & Arnold, P. (1975). A high-resolution neutron time-of-flight diffractometer. Phys. Lett. A44, 165–166.
Turberfield, K. C. (1970). Time-of-flight neutron diffractometry. Thermal neutron diffraction, edited by B. T. M. Willis, pp. 34–50. Oxford University Press.
Windsor, C. G. (1981). Pulsed neutron diffraction. London: Taylor & Francis.