Accurate and precise lattice-parameter determinations

Gałdecka, E.

doi:10.1107/97809553602060000597

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

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International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 509-510

Section 5.3.2.3.4. Accurate and precise lattice-parameter determinations

E. Gałdecka^a

^a Institute of Low Temperature and Structure Research, Polish Academy of Sciences, PO Box 937, 50-950 Wrocław 2, Poland

5.3.2.3.4. Accurate and precise lattice-parameter determinations

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To measure with a precision and an accuracy better than is possible in routine photographic methods, additional work has to be performed. The first methods allowing precise measurement of lattice parameters were photographic powder methods (Parrish & Wilson, 1959). Special single-crystal methods with photographic recording to realize this task (earlier papers are reviewed by Woolfson, 1970, Chap. 9) combine elements of basic single-crystal methods (presented in §§5.3.2.3.1 and 5.3.2.3.2) with ideas more often met in powder methods (asymmetric film mounting). A similar treatment of some systematic errors (extrapolation ) is met in both powder and single-crystal methods.

(i) The relative accuracy ΔI/I of the identity period I in the rotating-crystal method, estimated by differentiation of formula (5.3.2.1), is given by $[\Delta I/I=-\cot \nu\Delta\nu. \eqno (5.3.2.3)]$ This formula shows that the highest accuracy is obtained for ν tending to 90°. Since reflections with large values of ν are difficult to record in commonly used cameras, a special camera may be used for this task, in which a flat film is placed perpendicular to the rotation axis, or a different one, whose axis coincides with the primary beam (Umansky, 1960). The accuracy achieved with these improvements is still no better than 5 parts in 10³.
(ii) The asymmetric film mounting proposed by Straumanis & Ieviņš (1940) in the case of powder cameras can also be used in a simple oscillating camera (Farquhar & Lipson, 1946). In particular, this idea can be realized in a precision Debye–Scherrer camera adapted to single-crystal measurements by mounting in it a goniometer head (Popović, 1974). The Straumanis mounting allows the recording of the high-angle reflections close together on the film, thus reducing the effect of film shrinkage and making it possible to measure the effective camera radius.
(iii) Sometimes, to eliminate systematic errors (uncertainty of the camera radius), the separations resulting from the wavelength differences of the $[K\alpha_1]$ and $[K\alpha_2]$ doublet are measured rather than the absolute distances on the film (Main & Woolfson, 1963; Alcock & Sheldrick, 1967). The first reference related to the zero-layer normal-beam photograph, the second to higher layer lines (in the equi-inclination method also) and oscillation photographs.
(iv) Systematic errors connected with film shrinkage can also be eliminated by means of the ratio method, introduced by Černohorský (1960) for powder samples and adapted by Polcarová & Zůra (1977) for single crystals . In this method, pairs of reflections that differ from one another in wavelength and/or in hkl indices are used and the ratio of the two diameters of the diffraction rings corresponding to these reflections is taken into account. The accuracy of the method is about 1 part in 10⁴ if systematic errors due to absorption, refraction, Lp factor, temperature , changes of the camera radius , and misalignment of the sample and the goniometer are corrected. The ratio method was generalized by Horváth (1983) to the monoclinic crystal system.
(v) Graphical extrapolation, similar to that used in powder methods (Parrish & Wilson, 1959), can also be used for single crystals (Farquhar & Lipson, 1946; Weisz, Cochran & Cole, 1948), to reduce systematic errors proportional to $[\sin\theta]$ . Least-squares refinement , on the other hand, permits a reduction of the standard deviations of the results (Main & Woolfson, 1963; Clegg, 1981). Mathematical methods of processing the data obtained from oscillation photographs, including `eigenvalue filtering' and profile fitting (Rossmann, 1979; Reeke, 1984) have been applied to the refinement of unit-cell parameters, crystal orientation, and reflecting-range parameters needed to process oscillation photographs.
(vi) By measuring the angle between two reflecting crystal positions, symmetrical in relation to the primary beam [the idea used in the original Bragg spectrometer (Bragg & Bragg, 1915)], one can eliminate some sources of systematic errors. Such a spectrometer with photographic recording was used by Weisz, Cochran & Cole (1948). In spite of the great simplicity of the arrangement, the accuracy obtained was about 1 part in 10⁴. The authors indicated the need for introducing counter recording to the method. 12 years later, their idea was realized by Bond (1960) (cf. Subsection 5.3.3.4, in particular §5.3.3.4.3).
(vii) The other way of reducing some systematic errors is to introduce a reference crystal. Singh & Trigunayat (1988) adapted the idea to the oscillation method. By mounting the specimen crystal and the reference crystal, properly centred and set, on two identical goniometer heads with a screw-type base, they recorded layer lines of the two crystals simultaneously. The identity period I of the crystal was then determined from the formula that results from a combination of (5.3.2.1) and (5.3.2.2) for layer lines of the two crystals (notation of the present Section): $[I= n\lambda \left[{{{l^2_n}({I^2_r} - m^2\lambda^2)} \over {{l^2_{m,r}}m^2\lambda{^2}}}+1\right]^{1/2}, \eqno (5.3.2.4)]$ in which and $[l_{m,r}]$ are the measured distances between nth layer lines of the crystal and between mth layer lines of the specimen, respectively, and is the identity period of the reference crystal. The result is thus independent of the camera radius . When the differences between and $[l_{m.r}]$ are no greater than a few mm, the error due to film shrinkage is automatically taken care of, and the error due to a parallel shift of the axis of the cylindrical cassette in relation to the axis of rotation is negligible in practice. The other possible misalignments related to the cassette and the collimator can be readily detected beforehand by taking a complete rotation photograph.

Reference crystals are commonly used in multiple-crystal methods reviewed in Subsection 5.3.3.7.

References

Alcock, N. W. & Sheldrick, G. M. (1967). The determination of accurate unit-cell dimensions from inclined Weissenberg photographs. Acta Cryst. 23, 35–38.Google Scholar

Bond, W. L. (1960). Precision lattice constant determination. Acta Cryst. 13, 814–818.Google Scholar

Bragg, W. H. & Bragg, W. L. (1915). X-rays and crystal structure, Chap. 2. London: G. Bell and Sons.Google Scholar

Černohorský, M. (1960). The ratio method for absolute measurements of lattice parameters with cylindrical cameras. Acta Cryst. 13, 823–826.Google Scholar

Clegg, W. (1981). Least-squares refinement of unit-cell parameters from precession photographs. Acta Cryst. A37, 437–438.Google Scholar

Farquhar, M. C. M. & Lipson, H. (1946). The accurate determination of cell dimensions from single-crystal X-ray photographs. Proc. Phys. Soc. London, 58, 200–206.Google Scholar

Horváth, J. (1983). Lattice-parameter measurements of PbHPO₄ single crystals by the ratio method. J. Appl. Cryst. 16, 623–628.Google Scholar

Main, P. & Woolfson, M. M. (1963). Accurate lattice parameters from Weissenberg photographs. Acta Cryst. 16, 731–733.Google Scholar

Parrish, W. & Wilson, A. J. C. (1959). Precision measurements of lattice parameters of polycrystalline specimens. International tables for X-ray crystallography, Vol. II, Chap. 4.7, pp. 216–234. Birmingham: Kynoch Press.Google Scholar

Polcarová, M. & Zůra, J. (1977). A method for the determination of lattice parameters on single crystals. Czech. J. Phys. B27, 322–331.Google Scholar

Popović, S. (1974). Determination of unit-cell parameters of single crystals from rotation patterns. J. Appl. Cryst. 7, 291–292.Google Scholar

Reeke, G. N. J. (1984). Eigenvalue filtering in the refinement of crystal and orientation parameters for oscillation photography. J. Appl. Cryst. 17, 238–243.Google Scholar

Rossmann, M. G. (1979). Processing oscillation diffraction data for very large unit cells with an automatic convolution technique and profile fitting. J. Appl. Cryst. 12, 225–238.Google Scholar

Singh, K. & Trigunayat, G. C. (1988). Accurate determination of lattice parameters from XRD oscillation photographs. J. Appl. Cryst. 21, 991.Google Scholar

Straumanis, M. & Ieviņš, A. (1940). Die Präzizionsbestimmung von Gitterkonstanten nach der asymmetrischen Methode. Berlin: Springer. [Reprinted by Edwards Brothers Inc., Ann Arbor, Michigan (1948).]Google Scholar

Umansky, M. M. (1960). Apparatura rentgenostrukturnykh issledovanij. Moscow: Fizmatgiz.Google Scholar

Weisz, O., Cochran, W. & Cole, W. F. (1948). The accurate determination of cell dimensions from single-crystal X-ray photographs. Acta Cryst. 1, 83–88.Google Scholar

Woolfson, M. M. (1970). An introduction to X-ray crystallography. Cambridge University Press.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 509-510