A least-squares method for the determination of the orientation matrix in single-crystal diffractometry

Abstract

The lattice parameters of ammonium chloride have been measured at 31.5 and 54"0°C. The coefficient of linear expansion is found to be 58-5 × 10-6°C -1.The thermal expansion of ammonium chloride (CsC1 structure) has been measured by Fizeau (1867) and Sharma (1950). The values reported by them for the coefficient of linear expansion near room temperature are 62.5 and 59"9 (in units of 10-6°C -I) respectively. HaussiJhl (1960) has quoted a very much lower value of 48 x 10-6°C -1. In view of these differences, an accurate X-ray determination of the coefficient of thermal expansion was undertaken.A flat-faced powder sample was prepared with Analar grade B.D.H. material. X-ray powder photographs were taken with a flat-film back-reflexion camera. With filtered copper radiation, the camera could record three ~:t2 doublets at high angles. Photographs were taken at 31.5 and at 54"0°C. The procedure for the precision determination of the lattice parameter was the same as in our earlier work * Present address: Physics Department, Post-graduate Centre, Warangal (A.P.), India. Acta Cryst. (1970). A26, 295A least-squares method for the determination of the orientation matrix in single-crystal ditfractometry.By K. TICHq,* Physics Department, University College, Cardiff', Wales (Received 6 October 1969) An appropriate choice of the function minimized permits linearization of the least-squares determination of the matrix which transforms the diffraction indices into the components of the reciprocal vector in the diffractometer ~0-axis system of coordinates. The coefficients of the least-squares equations are based on diffraction indices and measured diffractometer angles of three or more non-coplanar setting reflexions.

Linear least-squares adjustment of UB matrix elements and the prediction of reflection positions

“…In common with the methods of Tichy (1970) and Shoemaker & Bassi (1970), it is difficult using this method to apply symmetry constraints to the unitcell parameters. This is not a serious limitation, however, as the purpose of the method is to warn of changes in the state or setting of the crystal in terms of immediately observable diffractometer angles, and to use these subsequently in predicting accurately the positions of the centres of other reflections.…”

“…The methods described by Tichy (1970) and by Shoemaker & Bassi (1970) are linear, but refine instead the nine independent coefficients of the UB matrix from the values of the elements of a column matrix R { = [AS2Xq~]-~Z of (1)}, which are derived from the measured diffractometer angles. Since the UB matrix is refined with respect to the components of R and not with respect to the measured angles, the weighting of the least-squares equations with the inverse variances of the observations is cumbersome.…”

Angle calculations for a five-circle diffractometer used for surface X-ray diffraction

“…The computed value will therefore be most accurate if small values for h and k are used. Once two or more diffraction peaks have been observed, the procedure to determine U is well known for a standard four-circle diffractometer (Busing & Levy, 1967;Tichy, 1970) and computer programs of varying complexity exist for this purpose. These programs basically determine and/or optimize U by using the relation UBH = Qq,.…”

Angle and index calculations for `z-axis' X-ray diffractometer

“…Thus, the right-hand side of (13) (vector %) depends only on the diffractometer setting and thereby is known. The orientation matrix U can be calculated using (13) by exactly the same procedures as in the fourcircle case (Tichy, 1970). For example, if the crystal unit cell is not known (B is not known) one can, in the conventional way, determine UB when three reflections are found.…”