Moving-crystal methods

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. 508-510

Section 5.3.2.3. Moving-crystal methods

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. Moving-crystal methods

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Moving-crystal methods of lattice-parameter determination apply basic photographic techniques, such as:

(1) the rotating- or oscillating-crystal method;
(2) the Weissenberg method;
(3) the technique of de Jong–Bouman; or
(4) the Buerger precession method.

In the first of these methods, the film remains stationary, while in the others it is moved during the exposure. The principles and detailed descriptions of these techniques have been presented elsewhere (Buerger, 1942; Henry, Lipson & Wooster, 1960; Evans & Lonsdale, 1959; Stout & Jensen, 1968, Chapter 5; Sections 2.2.3 , 2.2.4 , and 2.2.5 of this volume) and only their use in lattice-parameter measurements will be considered here.

5.3.2.3.1. Rotating-crystal method

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The rotating-crystal method – the simplest of the moving-crystal methods – determines the identity period I along the axis of rotation (or oscillation), $[{\bf r}=u{\bf a}+v{\bf b}+w{\bf c}]$ , from the formula $[I(uvw)=n\lambda/\sin \nu, \eqno (5.3.2.1)]$ in which n is the number of the layer line and ν is the angle between the directions of the primary and diffracted beams.

The angle ν is determined from the measurement of the distance [l_n] between two lines corresponding to the same layer number n from the equation $[\tan \nu=l_n/R, \eqno \eqno (12)]$ where R is the camera radius.

All the lattice parameters may be determined from separate photographs made for rotations of the crystal along different rotation axes, i.e. the system axis, plane and spatial diagonals (Evans & Lonsdale, 1959), without indexing the photographs. In practice, however, this method is rarely used alone and is most often applied together with other photographic methods (for example, the Weissenberg method), but it is a useful preliminary stage for other methods. In particular, the length of a unit-cell vector may be directly determined if the rotation axis coincides with this vector.

Advantages of this method are:

(a) simple equipment (only rotation of the crystal is required, since the film is stationary);
(b) immediate determination of direct-cell parameters (photographs obtained with other cameras afford information about reciprocal-lattice parameters only);
(c) indexing of the photographs is unnecessary.

Drawbacks of the method are:

(a) poor precision and accuracy of the measurement $[(|\delta d|/d\approx10^{-2})]$ ;
(b) small amount of information from a single photograph (one parameter only);
(c) necessity of taking several photographs in the case of a lower-symmetry system if this method is the only one used.

5.3.2.3.2. Moving-film methods

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A two-dimensional picture of a reciprocal cell from one photograph can be obtained by the methods in which rotation of the crystal is accompanied by movement of the film, as in the Weissenberg, the de Jong–Bouman, and the Buerger precession techniques. These methods give greater precision $[(|\delta d|/d\approx10^{-4})]$ than the previous one (§5.3.2.3.1).

The advantages of the Weissenberg method in relation to the other two are:

(a) a simpler camera;
(b) a larger range of reciprocal-lattice points recorded on one photograph (larger range of $[\theta]$ angles, up to 90° for the zero layer).

On the other hand, the disadvantage, in contrast to the de Jong–Bouman and the Buerger precession methods, is that it gives deformed pictures of the reciprocal lattice. This is not a fundamental problem, especially now that computer programs that calculate lattice parameters and draw the lattice are available (Luger, 1980). In lattice-parameter measurements, both the zero-layer Weissenberg photographs and the higher-layer ones are used. The latter can be made both by the normal-beam method and by the preferable equi-inclination method. Photographs in the de Jong–Bouman and precession methods give undeformed pictures of the reciprocal lattice , but afford less information about it than do Weissenberg photographs.

5.3.2.3.3. Combined methods

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The most effective photographic method of lattice-parameter measurement is a combination of two techniques (Buerger, 1942; Luger, 1980), which makes it possible to obtain a three-dimensional picture of the reciprocal lattice ; for example: the rotation method with the Weissenberg (lower accuracy); or the precession (or the Weissenberg) method with the de Jong–Bouman (higher accuracy).

A suitable combination of the two methods will determine all the lattice parameters, even for monoclinic and triclinic systems, from one crystal mounting. This problem has been discussed and resolved by Buerger (1942, pp. 388–390), Hulme (1966), and Hebert (1978). Wölfel (1971) has constructed a special instrument for this task, being a combination of a de Jong–Bouman and a precession camera.

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.

5.3.2.3.5. Photographic cameras for investigation of small lattice-parameter changes

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Small changes of lattice parameters caused by thermal expansion or other factors can be investigated in multiple-exposure cameras.

Bearden & Henins (1965) used the double-crystal spectrometer with photographic detection to examine imperfections and stresses of large crystals. The technique allowed the detection of angle deviations as small as 0.5′′. A nearly perfect calcite crystal was used as the first crystal (monochromator), the sample was the second. The device distinguished itself with very good sensitivity. The use of the long distance (200 cm) between the focus and the second crystal made possible resolution of the doublet $[K\alpha_{1,2}]$ , and elimination of the $[K\alpha_2]$ radiation . An additional advantage was that the arrangement was less time-consuming, so that it was suitable for controlling the perfection of growing crystals and useful for choosing adequate samples for the wavelength measurements.

Kobayashi, Yamada & Azumi (1968) have described a special `strainmeter' for measuring small strains of the lattice. The strain [x_i] along an axis normal to the i plane results in a change $[\delta d_i]$ of the interplanar distance [d_i] : $[x_i=\delta d_i/d_i=-\cot\theta_i\delta_i. \eqno (5.3.2.5)]$ The use of a large camera radius R = 2639 mm makes it possible to obtain both high sensitivity and high precision (2 parts in 10⁶) even in the range of lower Bragg angles $[(\theta\simeq55^\circ)]$ . The device is suitable for the investigation of defects resulting from small strains and may be used in measurements of thermal expansion .

Glazer (1972) described an automatic arrangement, based on the Weissenberg goniometer, for the photographic recording of high-angle Bragg reflections as a function of temperature , pressure, time, etc. A careful choice of the oscillation axis and oscillation range makes it possible to obtain a distorted but recognizable phase diagram (Fig. 5.3.2.1 ) within several hours. The method had been applied by Glazer & Megaw (1973) in studies of the phase transitions of NaNbO₃.

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(a) Photographic recording of lattice-parameter changes. (b) Corresponding diagram of the variation of lattice parameters in pseudocubic NaNbO₃ (Glazer & Megaw, 1973).

Popović, Šljukić & Hanic (1974) used a Weissenberg camera equipped with a thermocouple mounted on the goniometer head for precise measurement of lattice parameters and thermal expansion in the high-temperature range.

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International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 508-510