Systematic errors

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. 523-524

Section 5.3.3.4.3.2. Systematic errors

E. Gałdecka^a

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

5.3.3.4.3.2. Systematic errors

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As mentioned above (§5.3.3.4.1), some systematic errors that affect the asymmetric diffractometer are experimentally eliminated in the Bond (1960) arrangement. According to Beu (1967), who has supplemented the list of errors given by Bond, the following systematic errors are eliminated at the $[0.001^\circ\theta]$ level:

(a) absorption, source profile, radial divergence and surface flatness; removed since the detectors are used only to measure intensities and not angular positions;
(b) zero, eccentricity, misalignment and diffractometer radius; eliminated since $[\theta]$ depends only on the difference in the crystal-angle positions and not on these geometrical factors;
(c) ratemeter recording does not affect the measurements since the detectors are used only for point-by-point counting;
(d) 2:1 tracking error is eliminated because the 2:1 tracking used in most commercial asymmetric diffractometers is not used;
(e) dispersion, if the peak position of the profile is determined rather than the centroid or the median, and the wavelength has been determined for the peak position also.

As well as these errors there are other systematic errors, due to both physical and apparatus factors, which should be eliminated by suitable corrections.

(1) Lorentz–polarization error. The Lorentz–polarization factor Lp influences the shape of the profile as follows: $[v_1(x)={\rm Lp} v(x), \eqno (5.3.3.23)]$ where v(x) and are the shape functions [see equations (5.3.3.10), (5.3.3.10a,b)] of the undistorted and distorted profiles, respectively. It therefore produces a shift $[(\Delta\theta_{\rm Lp})]$ in the peak position.

The correction for the Lp factor was estimated, assuming that is the Cauchy function [equation (5.3.3.19)], by Bond (1960, 1975), Segmüller (1970), and Okazaki & Ohama (1979) for two cases. For perfect crystals, when the Lp factor has the form (James, 1967, p. 59; Segmüller, 1970; Okazaki & Ohama, 1979) $[{\rm Lp}=(1+|\cos2\theta|)/\sin2\theta, \eqno (5.3.3.24)]$ the correction is given by $[\theta-\theta_p=(\omega_h/2){}^2[\cot2\theta_p+\sin2\theta_p/(1+|\cos2\theta_p|)], \eqno (5.3.3.25)]$ where $[\omega_h]$ is the half-width of the profile, $[\theta_p]$ is the Bragg angle related to the distorted profile, and $[\theta]$ is the corrected Bragg angle. In contrast, the following formulae are valid for mosaic crystals: $[{\rm Lp}=(1+\cos^22\theta)/(2\sin2\theta), \eqno (5.3.3.26)]$ and $[\theta-\theta_p=(\omega_h/2){^2}\cot2\theta_p(2+\sin{^2}2\theta_p)/(2-\sin{^2}\theta_p). \eqno (5.3.3.27)]$ Because of a notable difference between the values calculated from (5.3.3.25) and (5.3.3.27), the problem is to choose the formulae to be used in practice. However, the Lorentz–polarization error is usually smaller than the rest.
(2) Refraction. In the general case, when the crystal surface is not parallel to the reflecting planes but is rotated from the atomic planes around the measuring axis by the angle $[\varepsilon]$ , the correction, which relates directly to the determined interplanar distance, has the form (Bond, 1960; Cooper, 1962; Lisoivan, 1974, 1982) $[d=d_p\left[1+{\delta\cos^2\varepsilon \over \sin(\theta+\varepsilon)\sin(\theta-\varepsilon)}\right], \eqno (5.3.3.28)]$ where δ is unity minus the refractive index of the crystal for the X-ray wavelength used, and and d are the uncorrected and corrected interplanar distances, respectively.
(3) Errors due to axial and horizontal (in-plane) divergence. The axial divergence of the primary beam, given by an angle $[2\Delta_p]$ depending on the source and collimator dimensions, causes the angle $[\theta']$ , formed by a separate ray of the beam with a given set of crystallographic planes, to differ from the proper Bragg angle. In general, if the plane of diffraction is not sufficiently perpendicular to the axis of rotation but lacks perpendicularity by an angle $[\Delta]$ , the measured Bragg angle $[\theta']$ can be described, according to Bond (1960), as $[\sin\theta'=\sec\Delta\sin\theta. \eqno (5.3.3.29)]$ Let us assume that both the crystal and the collimator have been accurately adjusted so that the lack of perpendicularity results from axial divergence only. By averaging the expression (5.3.3.29) over the limits $[\pm\Delta_p]$ , the mean value of $[\sin\theta']$ can be found and, as a consequence, the following formula describing the correct d spacing can be obtained: $[d=d'(1+\Delta^2_p/6), \eqno (5.3.3.30)]$ where is the apparent d spacing.

According to Berger (1984), this correction is valid only for the case of infinitely small focus, when all rays have the same intensity. Taking into consideration the shift of the centroid caused by vertical divergence when the focus emits uniformly within the axial limits (−F, F), he proposes an alternative correction for $[\theta]$ : $[\Delta\theta_d=\textstyle{1\over6}\tan\theta(P^2+F^2), \eqno (5.3.3.31)]$ where 2P is the sample height.

As tested using computer modelling (Urbanowicz, 1981b) and estimated analytically (Härtwig & Grosswig, 1989), the effect of the horizontal divergence on the peak position of the recorded profiles cannot be neglected, contrary to suggestions of Bond (1960). The respective systematic error is dependent on asymmetries of both the focus-tube emissivity and the spectral line, and so it is difficult to express it with a simple formula [cf. point (7) below]. In practice (Härtwig, Grosswig, Becker & Windisch, 1991), it proves to be the second largest error. (The first is the one caused by refraction.)

(4) Specimen-tilt and beam-tilt error. Since the three main sources of systematic error in diffractometer measurements, i.e. zero, eccentricity, and absorption, have been eliminated in the Bond method, two errors due to misalignment of the crystal and the collimator can strongly influence results of lattice-parameter determination. They are difficult to control because of the random character; numerous authors analysing the Bond method have tried to cope with them. A review is given by Nemiroff (1982).

Bond (1960) considered the crystal-tilt error separately from the collimator tilt. However, in subsequent papers on this subject it was shown that the errors connected with the crystal tilt and the collimator tilt, i.e. with the angles that the normals to the crystal and collimator make, respectively, with the plane of angular measurement, are dependent and should be treated jointly.

Foreman (in Baker, George, Bellamy & Causer, 1968) derived a formula for the real value of the angle between two reflecting positions [i.e. $[\omega_1]$ and $[\omega_2]$ in equation (5.3.3.22)] when affected by both tilts. Burke & Tomkeieff (1968, 1969), in contrast, have found a dependence between the crystal tilt α and the beam tilt β and the relative error Δa/a in lattice parameter a in the form $[\Delta a/a=\alpha\beta/\sin\theta-(\alpha^2+\beta^2)/2. \eqno (5.3.3.32)]$ A separate analysis is given by Gruber & Black (1970) and by Filscher & Unangst (1980).

Two approaches are used to eliminate the systematic errors considered, based on the above formula:

(i) The error resulting from the crystal tilt and the collimator tilt can be reduced experimentally. Baker, George, Bellamy & Causer (1968) have given a simple procedure that allows a collimator tilt of small but unknown magnitude to be tolerated and, at the same time, the tilt of the crystal to be adjusted to its optimum value. Burke & Tomkeieff (1968, 1969) propose a method for setting the crystal so that α = β, since, as is obvious from (5.3.3.32), the error has then its minimum value; α and β have to be of the same sign. Then the influence of crystal tilt and beam tilt on the accuracy of lattice-parameter determination is negligible at the level of 1 part in 10⁶.
(ii) Equation (5.3.3.32) permits calculation of the exact correction due to both crystal and collimator tilts, if the respective values of α and β are known. Halliwell (1970) proposed a method for determining the beam and the crystal tilt that requires measuring reflections from both the front and back surfaces of the crystal. In a method described by Nemiroff (1982), the two tilts are measured and adjusted independently within ±0.5 mrad.

(5) Errors connected with angle reading and setting. Errors in angle reading and angle setting depend both on the class of the device and on the experimenter's technique. Some practical details are discussed by Baker, George, Bellamy & Causer (1968). Since the angles are measured by counting pulses to a stepping motor connected to a gear and worm, the errors due to angle setting and reading depend on the fidelity with which the gear follows the worm. To diminish errors affected by the gearwheel (notably eccentricity), the authors propose a closed error-loop method, which involves using each part of the gear in turn to measure the angle and averaging the results. In the diffractometer reported in the above paper, there was, originally, an angular error of about +15′′ around the gearwheel, and this can be corrected by means of a cam so that the residual error is reduced to about ±5′′.

Another example of a high-precision drive mechanism is given by Pick, Bickmann, Pofahl, Zwoll & Wenzl (1977). In the diffractometer described in their paper (see also §5.3.3.7.2), the gear was shown to follow the worm with fidelity even down to 0.01′′ steps, and a drift of ±10% per step was traced to insufficient stability of temperature (±0.15 K).
(6) Temperature correction. An error $[\Delta d_T]$ in the lattice parameter d owing to the uncertainty ΔT of the temperature T can be estimated from the formula (Łukaszewicz, Pietraszko, Kucharczyk, Malinowski, Stępień-Damm & Urbanowicz, 1976): $[\Delta d_T=d\,\alpha_d\,\Delta T, \eqno (5.3.3.33)]$ if the thermal-expansion coefficient α_d in the required direction is known.

In the case of the 111 reflection of silicon, for which $[\alpha_d\approx2.33\times10^{-6}]$ , to obtain a relative accuracy (precision) of 1 part in 10⁶, the temperature has to be controlled with accuracy (precision) not worse than ±0.05 K if the temperature correction is to be neglected (Segmüller, 1970; Hubbard & Mauer, 1976; Łukaszewicz et al., 1976).
(7) Remarks. The above list of corrections, sufficient when the Bond (1960) method is applied under the conditions similar to those described by him (large, perfect, specially cut single crystal; well collimated primary beam; large open detector window) has to be sometimes complemented in the case of different specimens and/or different measurement conditions (§5.3.3.4.3.3). When an asymmetric diffractometer is used, all the systematic errors listed in this section (see also §5.3.3.4.1) must be taken into account.

Using a complete convolution model of the diffraction profile, Härtwig & Grosswig (1989) were able to derive all known aberrations (and so respective corrections) in a rigorous, analytical way. The analytical expressions given by the authors, though based on some simplifying assumptions, are usually much more complex than the ones shown in points (1)–(6) above. Some coefficients in their equations depend on physical parameters characterizing the particular device and experiment. So, to follow the idea of Härtwig & Grosswig, one must individually consider all preliminary assumptions. As shown by the authors, to achieve the accuracy of 1 part in 10⁷, all aberrations mentioned by them must be taken into account. The most important aberrations prove to be those related to refraction and to horizontal divergence.

References

Baker, T. W., George, J. D., Bellamy, B. A. & Causer, R. (1968). Fully automated high-precision X-ray diffraction. Adv. X-ray Anal. 11, 359–375.Google Scholar

Berger, H. (1984). A method for precision lattice-parameter measurement of single crystals. J. Appl. Cryst. 17, 451–455.Google Scholar

Beu, K. E. (1967). The precise and accurate determination of lattice parameters. Handbook of X-rays, edited by E. F. Kaelble, Chap. 10. New York: McGraw-Hill.Google Scholar

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

Bond, W. L. (1975). Precision lattice constant determination. Erratum. Acta Cryst. A31, 698.Google Scholar

Burke, J. & Tomkeieff, M. V. (1968). Specimen and beam tilt errors in Bond's method of lattice parameter determination. Acta Cryst. A24, 683–685.Google Scholar

Burke, J. & Tomkeieff, M. V. (1969). Errors in the Bond method of lattice parameter determinations. Further considerations. J. Appl. Cryst. 2, 247–248.Google Scholar

Cooper, A. S. (1962). Precise lattice constants of germanium, aluminium, gallium arsenide, uranium, sulphur, quartz and sapphire. Acta Cryst. 15, 578–582.Google Scholar

Filscher, G. & Unangst, D. (1980). Bond-method for precision lattice constant determination. Dependence of lattice constant error on sample adjustment and collimator tilt. Krist. Tech. 15, 955–960.Google Scholar

Gruber, E. E. & Black, R. E. (1970). Analysis of the axial misalignment error in precision lattice parameter measurement by the Bond technique. J. Appl. Cryst. 3, 354–357.Google Scholar

Halliwell, M. A. G. (1970). Measurement of specimen tilt and beam tilt in the Bond method. J. Appl. Cryst. 3, 418–419.Google Scholar

Härtwig, J. & Grosswig, S. (1989). Measurement of X-ray diffraction angles of perfect monocrystals with high accuracy using a single crystal diffractometer. Phys. Status Solidi A, 115, 369–382.Google Scholar

Härtwig, J., Grosswig, S., Becker, P. & Windisch, D. (1991). Remeasurement of the Cu Kα₁ emission X-ray wavelength in the metrical system (present stage). Phys. Status Solidi A, 125, 79–89.Google Scholar

Hubbard, C. R. & Mauer, F. A. (1976). Precision and accuracy of the Bond method as applied to small spherical crystals. J. Appl. Cryst. 9, 1–8.Google Scholar

James, R. W. (1967). The optical principles of the diffraction of X-rays. London: Bell.Google Scholar

Lisoivan, V. I. (1974). Local determination of all the lattice parameters of single crystals. (In Russian.) Appar. Methody Rentgenovskogo Anal. 14, 151–157.Google Scholar

Lisoivan, V. I. (1982). Measurements of unit-cell parameters on one-crystal spectrometer. (In Russian.) Novosibirsk: Nauka.Google Scholar

Łukaszewicz, K., Pietraszko, A., Kucharczyk, D., Malinowski, M., Stępień-Damm, J. & Urbanowicz, E. (1976). Precyzyjne pomiary stałych sieciowych kryształów metoda Bonda (Precision measurements of lattice constants of crystals by the Bond method). Wrocław: Instytut Niskich Temperatur i Badań Strukturalnych PAN.Google Scholar

Nemiroff, M. (1982). Precise lattice-constant determinations using measured beam and crystal tilts. J. Appl. Cryst. 15, 375–377.Google Scholar

Okazaki, A. & Ohama, N. (1979). Improvement of high-angle double-crystal X-ray diffractometry (HADOX) for measuring temperature dependence of lattice constants. I. Theory. J. Appl. Cryst. 12, 450–454.Google Scholar

Pick, M. A., Bickmann, K., Pofahl, E., Zwoll, K. & Wenzl, H. (1977). A new automatic triple-crystal X-ray diffractometer for the precision measurement of intensity distribution of Bragg diffraction and Huang scattering. J. Appl. Cryst. 10, 450–457.Google Scholar

Segmüller, A. (1970). Automated lattice parameter determination on single crystals. Adv. X-ray Anal. 13, 455–467.Google Scholar

Urbanowicz, E. (1981b). The influence of in-plane collimation on the precision and accuracy of lattice-constant determination by the Bond method. II. Verification of the mathematical model. Systematic errors. Acta Cryst. A37, 369–373.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 523-524