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

International Tables for Crystallography (2006). Vol. C, ch. 5.2, pp. 501-504

Section 5.2.13. Factors determining accuracy

W. Parrish,a A. J. C. Wilsonb and J. I. Langfordc

aIBM Almaden Research Center, San Jose, CA, USA,bSt John's College, Cambridge CB2 1TP, England, and cSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England

5.2.13. Factors determining accuracy

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Many factors influencing accuracy in lattice-parameter determination have been mentioned in passing or discussed at length in this and previous chapters. This section attempts to summarize them and put them into perspective. Accuracy in the range of 1 to 0.1% can now be achieved routinely with average care. Increasing the accuracy to 0.01% requires considerable care in specimen preparation, data collection, instrument alignment, and calibration. The range 0.001 to 0.0001% is rarely reached and each determination is virtually a research project. The more important factors are:

  • (1) Differentiation of the Bragg equation, as in (5.2.1.4)[link], shows the advantage of using the highest-angle reflections; because of the [\cot\theta] term, the error in Δd is smaller for a given angular accuracy [\Delta\theta]. The gain is not as great as one might expect at first, as the experimental accuracy of the back reflections is lowered because of (i) their lower intensity, (ii) their lower peak-to-background ratio, (iii) their broadening by wavelength dispersion and crystallite imperfection, and (iv) problems of overlapping.

  • (2) The lower-angle reflections show the converse effects of (i) higher intensity, (ii) higher peak-to-background ratio, (iii) less broadening, and (iv) fewer problems of overlapping. In any particular case, a balance of advantage must be sought.

    The forward reflections have been used in parallel-beam synchrotron-radiation lattice-parameter studies (Parrish et al., 1987[link]).

  • (3) The profile shape has a strong influence on the accuracy of the angle measurement. The geometrical aberrations produce asymmetries that reduce the accuracy; the effects can be minimized by a proper selection of slit sizes. In most cases, it is inadvisable to use [K\beta] radiation to avoid [K\alpha]-doublet splitting, as the intensity is reduced by a factor of seven. Symmetrical profiles are obtained with parallel-beam optics, but it is usually necessary to use synchrotron radiation to achieve sufficient intensity.

  • (4) The largest and commonest source of systematic error in focusing geometry is the specimen-surface displacement. Several remountings of the specimen in the diffractometer and measurement of some low-angle reflections may be helpful in determining and minimizing the error. This aberration does not occur in parallel-beam geometry unless a receiving slit is used.

  • (5) The precision of the diffractometer gears (or the equivalent) may be the limiting factor in high-precision measurements. The use of an electromagnetic encoder mounted on the 2θ-output shaft can increase the precision considerably. It is not normally included in commercial diffractometers because of its cost, but it is essential for adequate accuracy when the 2θ angles must be determined to better than 0.001°. The various types of mechanical error have been described by Jenkins & Schreiner (1986[link]).

    The diffractometer must be carefully adjusted to avoid mechanical problems. The effect of backlash can be minimized by slewing beyond and then returning to the starting angle, and by always scanning in the same direction. It is essential to avoid over-tight worm-and-gear meshing, as it causes jerky rather than smooth movement.

  • (6) The beam must be precisely centred, the slits and monochromator (if used) must be parallel to the line focus of the X-ray tube, and the scanning plane must be perpendicular to the line focus.

  • (7) The use of standard specimens with accurately known lattice parameters (Section 5.2.10[link]) and ideally free of line broadening is strongly recommended as a test of the overall precision of the instrumentation and method.

  • (8) For a given total time available for an experiment, it is necessary to strike a balance between numerous short steps with short counting times and fewer longer steps with longer counting times. The former alternative may give a better definition of the line shape; the latter may give lower calculated standard uncertainties (formerly called estimated standard deviations) in any derived parameters. Obviously, the step length must be considerably shorter than the width of any feature of the profile that is considered to be of importance.

  • (9) Least-squares refinement is discussed in Subsection 5.2.3.2[link]. The programs and the methods of handling the data should be carefully checked, as various programs have been found to give slightly different values from the same experimental data (see, for example, JCPDS – International Centre for Diffraction Data, 1986[link]; Kelly, 1988[link]).

  • (10) Specimen preparation is very important; the particle size should preferably be less than 10 µm, and a flat smooth surface normal to the diffraction vector is essential. The linearity of the detector and the temperature of the specimen must be properly controlled during the collection of the experimental data.

  • (11) The accuracy of the 2θ measurements is directly dependent on the individual step-scanned points. The counting statistical accuracy is determined by the intensity and the background level, and is a major factor in lattice-parameter precision. Preliminary tests on typical profiles ensure that fullest advantage can be taken of the experimental conditions.

  • (12) At present (2003), the best approach to precision lattice-parameter determination is to follow the suggestions listed above, and to use peak search or profile fitting to calculate the observed 2θ positions. All the well determined peaks are used in the least-squares refinement against 2θ to obtain the zero-angle calibration correction, and in the case of focusing methods the specimen-surface displacement is added. The use of standards is recommended.

References

JCPDS–International Centre for Diffraction Data (1986). Task group on cell parameter refinement. Powder Diffr. 1, 66–76.
Jenkins, R. & Schreiner, W. N. (1986). Considerations in the design of goniometers for use in X-ray powder diffraction. Powder Diffr. 1, 305–319.
Kelly, E. H. (1988). A summary of a `round-robin' exercise comparing the output of computer programs for lattice-parameter refinement and calculations. British Crystallographic Association.
Parrish, W., Hart, M., Huang, T. C. & Bellotto, M. (1987). Lattice-parameter determination using synchrotron powder data. Adv. X-ray Anal. 30, 373–382.








































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