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

Abstract

Abstract 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 · |∑ cos (2πi hr j )|, 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 xj , 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) < e( k , n). Due to the spatial resolution potential of the concept even less than m data suffice to yield in A m tractable amounts of such test structure vectors ready for conventional least-squares refinement based on 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.