International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 5.2, pp. 495496

Latticeparameter determination with synchrotron radiation has a number of advantages over focusing methods (Parrish, Hart, Huang & Bellotto, 1987; Parrish, 1988; Huang, 1988). The doublet problem does not arise; the symmetrical single profiles greatly simplify the accurate angular measurement of peaks. The higher intensity and low uniform background out to the highest θ values give a higher statistical counting precision, an important factor in accurate measurements. Short wavelengths (0.65 to 1.4 Å) can be used to increase greatly the number of reflections without compromising the accuracy of the peak measurements. If desired, the patterns can be recorded with two or more wavelengths of about the same intensity, instead of being confined to the and lines (Popović, 1973). The large specimensurfacedisplacement and flatspecimen errors associated with most other methods do not occur, so that systematic errors are small or absent. The wavelength can be selected to obtain the desired dispersion, to avoid fluorescence, and to reduce specimen transparency.
The reflections are virtually symmetrical narrow peaks (Subsection 2.3.2.1 ), with widths of the order of 0.02–0.04°(2θ) when an analysing crystal is used instead of a receiving slit, and of the order of 0.05° when a long Soller slit is used as a collimator. These increase with increasing because of wavelength dispersion and small particle size. The angular positions of the peaks can be determined with high precision by the use of profilefitting or peaksearch measurements, and the only significant geometrical aberration is axial divergence.
There are no lines in the synchrotronradiation spectrum, and this creates the problem of determining the wavelength selected by the monochromator. If a highly accurate diffractometer were used for the monochromator and the monochromator d spacing were known accurately, the wavelength could be determined directly from . The angular accuracy of the diffractometer would have to be 0.0002° to achieve an accuracy of one part in 10^{6} in the wavelength at λ = 1.54 Å.
In practice, the wavelengths can often be determined by scanning the absorption edges of elements in the specimen or a metal foil placed in the beam. There is no feature of the absorption edge that is accurately measurable, and the wavelengths are usually listed to one or two decimal places fewer than those for the emission lines.
The wavelength problem could be avoided by using the ratio of the lattice parameter of the specimen to that of an accurately known standard measured with the same experimental conditions (Parrish et al., 1987). The standard may be mixed with the specimen or measured separately, as there is no specimensurface displacement shift. Mixing reduces the intensity of both patterns and worsens the peaktobackground ratio. The limitation is the accuracy of the lattice parameter of the standard. The only widely available one is the National Institute of Standards and Technology [NIST, formerly National Bureau of Standards (NBS)] silicon powder 640b (see Section 5.2.10). This accuracy may not be sufficient for measuring doping levels, stoichiometry, and similar analyses now possible with synchrotronradiation methods and the wavelength is normally determined directly from data for a standard whose lattice parameter is known with a high degree of precision, such as NIST SRM silicon 640b.
The most promising method is to use a highquality singlecrystal plate of floatzoned oxygenfree silicon, now widely available. Its lattice parameter is known to about one part in 10^{7} (Hart, 1981), which is much higher accuracy than that of the published lists of Xray wavelengths. Several orders of reflection (for example 111, 333, 444) should be used to improve the accuracy of the measurement.
Data are usually collected by stepscanning with selected constant angular increments and count times. To avoid interruptions due to refilling of the synchrotron ring, it is better to make a number of short runs rather than one long one. The data can then be added together and treated as a single data set. A shift in the orbit may cause a change in the wavelength reflected by the monochromator, and it is important to be aware of this in accurate latticeparameter determination. The peaks are narrow, and the angle increments should be small enough to produce at least a dozen points in each peak. In practice, the scans may be made to cover a range of one to two halfwidths (full widths at half height) on both sides of the peak, with increments of about 0.1 to 0.2 of the halfwidth, in order to record a sufficient number of data points for accurate profile fitting. The count time, which depends on the intensity, should be checked by determining the goodnessoffit of the calculated profiles and the experimental points (Subsection 2.3.3.8 and Chapters 8.4 and 8.6 ).
The lowerangle peaks generally have higher intensities and are therefore preferred to the higherangle peaks because of the better counting statistics. If the diffractometer can scan to negative angles, the number of strong peaks can be doubled by measuring the reflections on both sides of the zero position. The specimen can be used in either reflection or transmission, but reflection generally gives higher intensity. The lattice parameters are determined by a leastsquares analysis of the peak angles determined by profile fitting, and it is therefore necessary to measure a sufficient number of reflections to give a statistically valid result. The zeroangle position should be included as a variable parameter in the leastsquares calculation.
A precision of a few parts per million in the lattice parameter of NIST silicon has been reached with the highprecision diffractometer in the Daresbury Laboratory (Hart, Cernik, Parrish & Toraya, 1990). This instrument has an accurate gear and an incremental encoder driven by a DC servomotor with a feedback servoloop capable of positioning the detector arm within 0.36′′. A large number of repeated measurements showed a statistical accuracy of 0.0001°(2θ), corresponding to 1 in the fifth decimal place of d for λ = 1 Å and 2θ = 20°.
References
Hart, M. (1981). Bragg angle measurement and mapping. J. Cryst. Growth, 55, 409–427.Hart, M., Cernik, R., Parrish, W. & Toraya, H. (1990). Lattice parameter determination for powders using synchrotron radiation. J. Appl. Cryst. 23, 286–291.
Huang, T. C. (1988). Precision peak determination in Xray powder diffractometry. Aust. J. Phys. 41, 201–212.
Parrish, W. (1988). Advances in synchrotron Xray polycrystalline diffraction. Aust. J. Phys. 41, 101–112.
Parrish, W., Hart, M., Huang, T. C. & Bellotto, M. (1987). Latticeparameter determination using synchrotron powder data. Adv. Xray Anal. 30, 373–382.
Popović, S. (1973). Unitcell dimension measurements from pairs of Xray diffraction lines. J. Appl. Cryst. 6, 122–128.