The determination of unit-cell parameters from Bragg reflection data using a standard reference material but without a calibration curve

The procedure for determining unit-cell parameters from Bragg reflection data, using a standard reference material but without a calibration curve, is described. In this procedure, the observation equations for both the sample to be measured (SMP) and a standard reference material (SRM) are solved simultaneously and the unit-cell parameters and a form of error function are determined during the least-squares calculation. The theory, which was first proposed as a linear least-squares procedure [Toraya & Kitamura (1990). J. Appl. Cryst. 23, 282-285], has been extended to create a nonlinear least-squares procedure. The procedure can be used in two different ways. In one approach, the unit-cell parameters of the SMP and the parameters in the error function are refined while the unit-cell parameters of the SRM are fixed during the least-squares calculation. This procedure requires knowledge of the wavelength but it gives a stable solution and the tangent term in the error function gives a perfect correction for the error in wavelength. In the other approach, the unit-cell parameters of both the SMP and the SRM are refined, together with the parameters in the error function. The procedure does not require knowledge of the wavelength. The solution, however, became unstable when the correlation was strong between the unit-cell parameters and the error function and careful selection of the error function was required. The first approach gave the same result as the second and is, therefore, more practical to use. Since the form of the angle-dependent error function is determined using reflection data from both the SMP and the SRM, just one or two reflections from the SRM were enough for the correction of systematic error. The procedure, coupled with high-precision reflection data, can determine accurately the unit-cell parameters in a routine analysis.