Standard diffractometers

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. 516-517

Section 5.3.3.2. Standard diffractometers

E. Gałdecka^a

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

5.3.3.2. Standard diffractometers

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The determination of lattice parameters by the use of a standard diffractometer is based, as in the case of photographic methods, on (5.3.1.1) and (5.3.1.2), and the main task is to measure a sufficient number of reflections (the $[\theta]$ values for various hkl indices) for determining and solving the equations and for calculating the unknown parameters. The reflections can be chosen arbitrarily or in a special way (high $[\theta]$ angle, axial or non-axial reflections).

The characteristic feature of measurements performed on a diffractometer is, however, that to satisfy the Ewald condition for a given reflection the crystal and the detector are rotated or, depending on the geometry (equatorial or inclination), shifted round their axes as well. Basic and more detailed information about the geometry of diffractometers is given elsewhere (Arndt & Willis, 1966, Chap. 3; Stout & Jensen, 1968, Section 6.3; Kheiker, 1973, Chap. 4; Luger, 1980, Chap. 4; Section 2.2.6 of this volume). For calculating the setting angles for given hkl reflections, the lattice parameters (at least preliminary values) have to be known, and conversely, if the setting angles are known, it is possible to calculate or to refine lattice parameters. Therefore, not only the $[\theta]$ values (given by the angle $[2\theta]$ of rotation of the detector about the goniometer axis) but also the values of the remaining setting angles (i.e. $[\omega]$ , $[\varphi]$ , and χ of the crystal rotation in equatorial geometry, or μ and $[\varphi]$ for the crystal and v for the detector in inclination geometry) can be used for lattice-parameter determination. This problem can be treated by a matrix analysis.

5.3.3.2.1. Four-circle diffractometer

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In the case of an automated four-circle (equatorial geometry) diffractometer, the setting angles are calculated by means of the orientation matrix U, i.e. a matrix such that $[{\bi A}^*={\bi UA}_G, \eqno (5.3.3.1)]$ where $[{\bi A}^*= \left[\matrix{ a^* \cr b^* \cr c^*}\right] \eqno (5.3.3.1a)]$ is the reciprocal-axis system with metric $[{\bi G}^{-1}=\left[\matrix{a^{*2}&a^*b^*\cos\gamma^*&a^*c^*\cos\beta^* \cr a^*b^*\cos\gamma^*&b^{*2}&b^*c^*\cos\alpha^* \cr a^*c^*\cos\beta^*&b^*c^*\cos\alpha^*&c^{*2}}\right] \eqno (5.3.3.1b)]$ and $[{\bi A}_G=\left[\matrix{a_G \cr b_G \cr c_G}\right] \eqno (5.3.3.1c)]$ is the crystal-fixed orthonormal system. As can be proved (Busing & Levy, 1967; Hamilton, 1974; Luger, 1980, Section 4.1.1; Gabe, 1980), the reciprocal-cell parameters are related to the orientation matrix by the following equation: $[{\bi A^*A^*{'}=U\cdot U'}, \eqno (5.3.3.2)]$ where $[{\bi A^*A^*{'}=G}^{-1}]$ is given by (5.3.3.1b). It is thus possible to calculate the lattice parameters from the terms of the orientation matrix.

The determination of the orientation matrix is usually the first step in measurements performed on the four-circle diffractometer. This task can be accomplished when the preliminary lattice-parameter values are known, and even when they are unknown. In the first case, the setting angles of two reflections, and, in the second, of three reflections, have to be determined. The procedure (Busing & Levy, 1967; Hamilton, 1974) is usually accomplished by the software of the four-circle diffractometer. Least-squares refinement of the lattice and orientation parameters may be performed when the setting angles of several reflections have been observed (Clegg, 1984). Appropriate constraints, resulting from the presence of symmetry elements in the given crystal structure, to be introduced during the refinement, are discussed by Bolotina (1989).

In a particular case, the four-circle diffractometer can be used for lattice-parameter measurements performed in the plane perpendicular to the main goniometer axis (say, the horizontal plane), for which χ = 0°, so that, in practice, only $[2\theta]$ and $[\omega]$ values are used for lattice-parameter determination (see also §5.3.3.4.1). The equations to be solved can be simplified if only axial reflections are taken into account. In an example described by Luger (1980, Section 4.2.2), the $[{\bf b}^*]$ axis of a monoclinic crystal is oriented in the direction of the main axis. Then each of the two axial lengths, [a^*] and [c^*] (see Fig. 5.3.3.1 ), can be obtained from only one measurement: $[\eqalignno{ a^*&={2\sin\theta \over|h|\lambda}, & (5.3.3.3a) \cr c^*&={2\sin\theta \over|l|\lambda}, &(5.3.3.3b)}%fd5.3.3.3b]$ whereas $[\varphi]$ values of two reflections are used to determine the $[\beta^*]$ angle between $[{\bf a}^*]$ and $[{\bf c}^*]$ axes, since $[\beta^*=\varphi_{h00}-\varphi_{00l}. \eqno (5.3.3.3c)]$ This method is more suitable for orthogonal systems than for non-orthogonal ones, because of the difficulties in obtaining the proper orientation in the case of the monoclinic and, particularly, the triclinic system. In the latter case, the crystal has to be set three times.

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Determination of reciprocal-lattice angles on the θ circle (after Luger, 1980).

5.3.3.2.2. Two-circle diffractometer

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Lattice-parameter determination by the use of the two-circle (inclination) diffractometer, the so-called `Weissenberg diffractometer', is more troublesome than by means of the four-circle one, because only two rotations [ $[\omega]$ (or $[\varphi]$ ) of the crystal, and $[2\theta]$ (or γ) of the detector] are motor-driven under computer control, while two inclination angles (μ for the crystal and ν for the detector) must be set by hand.

The problem of application of the popular two-circle (Eulerian-cradle) diffractometer for measurements similar to those presented in §5.3.3.2.1 was discussed by Clegg & Sheldrick (1984). The main idea of their paper was to introduce equations combining setting angles, obtained for selected reflections, with reciprocal-cell parameters, for calculating the latter. The authors started with zero-layer reflections for which, for a crystal mounted about the c axis, $[\eqalignno{ \sin\theta& = (x^2+y^2){}^{1/2}, &(5.3.3.4a) \cr \omega &= \omega_0+\theta-\tan^{-1} (\,y,x), &(5.3.3.4b)}%fd5.3.3.4b]$ where $[\eqalignno{ x &= \lambda(ha^*+kb^*\cos\gamma^*)/2, &(5.3.3.4c) \cr y&= (\lambda kb^*\sin\gamma^*)/2, &(5.3.3.4d)}%fd5.3.3.4d]$ and $[\omega_0]$ is a zero-point correction.

The remaining parameter c had to be determined from the inclination angle μ, measured by hand. The use of zero-layer reflections was advantageous, apart from the simplicity of the formulae (5.3.3.4a,b,c,d), because they were less affected by crystal misalignment than were upper-layer reflections. However, a zero-point correction $[\omega_0]$ for $[\omega]$ had to be performed. For this purpose, the $[\omega_0]$ value was treated as an additional parameter in off-line least-squares refinement.

As the next step, the authors introduced equations for a general crystal orientation instead of an aligned crystal (cf. §5.3.3.2.1) and derived equations defining the setting angles for an arbitrary reflection useful for data collection from a randomly oriented crystal if preliminary lattice-parameter values had been assumed. This made possible measurements of reflections on a range of layers; only one crystal mounting was required. The matrix formulae suitable for Eulerian-geometry diffractometers are also given by Kheiker (1973, Chap. 3, Section 9) and Gabe (1980).

In order to perform precise refinement of all six cell parameters, Clegg & Sheldrick (1984) used least squares with empirical weights: $[W_{hkl}=1/\sqrt {\omega_{hkl}}, \eqno (5.3.3.5)]$ where $[\omega_{hkl}]$ is the width of the hkl reflection. An additional (third) motor to control the μ circle was proposed.

The authors point out that the two-circle diffractometer, owing to its simpler construction in comparison with the four-circle one, is well suited to operations that require additional attachments; for example, for low-temperature operation.

References

Arndt, U. W. & Willis, B. T. M. (1966). Single crystal diffractometry. Cambridge University Press.Google Scholar

Bolotina, N. B. (1989). Refinement of unit-cell parameters and orientation of specimen in diffractometer, taking account of symmetry of single crystal. Kristallografiya, 34, 598–601. (English transl: Sov. Phys. Crystallogr. 34, 355–357.)Google Scholar

Busing, W. R. & Levy, H. A. (1967). Angle calculations for 3- and 4-circle X-ray and neutron diffractometers. Acta Cryst. 22, 457–464.Google Scholar

Clegg, W. (1984). Orientation matrix refinement during four-circle diffractometer data collection. Acta Cryst. A40, 703–704.Google Scholar

Clegg, W. & Sheldrick, G. M. (1984). The refinement of unit cell parameters from two-circle diffractometer measurements. Z. Kristallogr. 167, 23–27.Google Scholar

Gabe, E. J. (1980). Diffractometer control with minicomputers. Computing in crystallography, edited by R. Diamond, S. Ramaseshan & K. Venkatesan, pp. 1.01–1.18. Bangalore: Indian Academy of Sciences.Google Scholar

Hamilton, W. C. (1974). Angle settings for four-circle diffractometers. International tables for X-ray crystallography, Vol. IV, pp. 274–284. Birmingham: Kynoch Press.Google Scholar

Kheiker, D. M. (1973). Rentgenowskaya diffraktometriya monokristallow, Chaps. 3, 4, 5. Leningrad: Mashinostroyenie.Google Scholar

Luger, P. (1980). Modern X-ray analysis of single crystals. In particular, Chap. 4 and Section 4.2.2. Berlin: de Gruyter.Google Scholar

Stout, G. H. & Jensen, L. H. (1968). X-ray structure determination. London: Macmillan.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 516-517