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. 5.4, pp. 537-538
Section 5.4.1.1. Introduction
A. W. S. Johnsona
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This article treats the recovery of cell axes and angles from (a) a single pattern with suitable Laue zones and (b) two patterns with different zone axes. It is assumed that instrument distortions, if significant, are corrected and that the patterns are free of artefacts such as twinning, double diffraction etc. (Edington, 1975). The treatment is valid for convergent-beam, micro and selected-area electron-diffraction patterns and accelerating voltages above approximately 30 kV. Relevant papers are by LePage (1992) and Zuo (1993), and background reading is contained in Edington (1975), Gard (1976), and Hirsch, Howie, Nicholson, Pashley & Whelan (1965).
The basic requirement in the determination of the unit cell of a crystal is to find, from one or more diffraction patterns, the basis vector set, a*, b*, c*, of a primitive reciprocal unit cell. The Cartesian components of these vectors form an orientation matrixwhich, when inverted, gives the vector components of the corresponding real-space cell. The elements of UB can be measured directly from the diffraction pattern in millimetres. Define axes x and y to be in the recording plane and z in the beam direction. A point in the diffraction pattern x, y, z is then related to the indices h, k, l by Note that points with non-zero z are observed on the plane z = 0, see Fig. 5.4.1.1.
The metric M of UB−1 is used to find the unit-cell edges and angles as where T means the transpose. Then, gives and where L is the effective distance between the diffracting crystal and the recording plane and λ is the wavelength. These quantities are defined in Fig. 5.4.1.1 together with the nomenclature and geometrical relationships required in this article.
If necessary, the cell is reduced to the Bravais cell according to the procedures given in IT A (2005, Chapter 9.2 ), before calculating the metric.
In practice, there may be a difficulty in choosing a vector set that describes a primitive reciprocal cell. Although a record of any reasonably dense plane of reciprocal space immediately exposes two basis vectors of a cell, the third vector lies out of the plane of the diffraction pattern containing the first two vectors and may not be directly measurable. Hence, some care must be taken to ensure that the third vector chosen makes the cell primitive. This presents no difficulty in the non-zero-zone analysis given in Subsection 5.4.1.3 but needs consideration when two patterns are used as described in Subsection 5.4.1.2.
Unit-cell parameters may be partly or fully determined depending on the extent of knowledge of the effective camera length L and wavelength λ, and upon the type and number of patterns. The different situations are distinguished in Table 5.4.1.1 for photographic recording. Note that L is simply a magnification factor.
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The accuracy of the non-zero-zone analysis described in Subsection 5.4.1.3 will depend on the influence of spiral, radial, and elliptical distortions. The camera length and wavelength need to be known to better than 1%.
For measurements made using microscope deflector systems, a knowledge of L may not be required depending on the location of the deflectors in the microscope column. Instead, the calibration factor of the deflectors, in suitable units, will replace L.
References
Edington, J. W. (1975). Electron diffraction in the electron microscope. Monographs in practical electron microscopy in materials science, No. 2. Eindhoven: N. V. Philips Gloeilampenfabrieken.Google ScholarGard, J. A. (1976). Electron microscopy in mineralogy, p. 52. Berlin: Springer.Google Scholar
Hirsch, P. B., Howie, A., Nicholson, R. B., Pashley, D. W. & Whelan, M. J. (1965). Electron microscopy of thin crystals. London: Butterworth.Google Scholar
International Tables for Crystallography (2005). Vol. A. Heidelberg: Springer.Google Scholar
LePage, Y. (1992). Ab initio primitive cell parameters from single convergent beam patterns: a converse route to the identification of microcrystals with electrons. Microsc. Res. Tech. 21, 158–165.Google Scholar
Zuo, J. M. (1993). New method of Bravais lattice determination. Ultramicroscopy, 52, 459–464.Google Scholar