Structure determination without Fourier inversion. Part IV: Using quasi-normalized data

For a centric crystal structure represented by m equal point scatterers at rest, absolute scaling of a small number n of reflection data reduced to relative geometrical structure amplitudes g'(h) = K · |Σ cos (2πihr j )|, = 1, ..., m; K = scaling factor) is obtained by dividing each amplitude through the r.m.s. average of the amplitudes to be considered. For the same batch of reflections, the resulting values e(h, n) are proportional to the well known normalized structure amplitudes |E(h)| in Direct Methods. Choosing a set of n harmonic reflections of a central reciprocal lattice row, the e(h, n) serve to determine the m independent coordinates of the point scatterers projected onto the corresponding direct space direction, e.g. h00-reflections for coordinates x j , hh0-reflections for (x + y) j (j = 1,..., m), etc. This is achieved by applying the concept of an m-dimensional parameter space P m with asymmetric part A m containing (m ― 1)-dimensional iso-surfaces E(h, n; e) determined by the values e(h, n), which define boundaries between forbidden and permitted solution regions (the latter containing test structure vectors Xt) based on observed inequalities, e.g. e(h, n) m data in order to obtain the "best" solution of the considered one-dimensional structure projection. The refined coordinates of various different projections can then be combined for reconstructing the three-dimensional structure. Properties of the e(h, n) and their iso-surfaces E(h, n; e) are discussed and determinations of two very small structures (centric and acentric) as well as of a centric 15-atom structure are presented as examples for the applicability of the method.