The use of phase relationships between quartets of reflexions

Abstract

Likewise, the elimination of the noise of the refine-'ment of the mean structure from the A matrix may be performed in a similar manner as before. The pseudosymmetry operator which gives high covariance between parameters need not always allow a sensible selection of combination of parameters. An example is the pseudosymmetry operators (-}+x, y, k-z), (2-+x, y, ~--z). A refinement procedure for a crystal structure with these pseudosymmetry operators is currently being investigated, and will be published elsewhere.We have seen earlier that a full-matrix solution tends to overweight the changes in the 'difference' structure whereas a block diagonal approach overweights the changes in the 'mean' structure but underweights the 'difference' structure, and so other refinement procedures may be considered.As .,tated in the Introduction, the refinement of the 'difference' structure is only possible as (sin 0)/2 increases. Thus the simplest method to reduce covariance between highly covarying parameters is to use highangle data or to weight data according to (sin 0)/2. Similarly, if there is an index condition so that only the 'difference' structure contributes to certain reflexions then the weight of these reflexions can be increased.A combination of a full-matrix approach and a diagonal approach can be considered. This is simply brought about by multiplying the diagonal elements au of the A matrix by a constant, kl, greatei than 1, and the elements of the column matrix B by a constant k2. The constant kz should be less than kl, otherwise the refinement of the 'mean' structure will be overweighted. A down weighting of the refinement of the 'difference' structure is brought about by a reduction of the apparent covariance between parameters. The shifts of highly covarying parameters are reduced more than the shifts in parameters which are less covariant. A read-out/read-in interruption to the leastsquares refinement program prior to the inversion of the A matrix allows any number of modifications to be tested without re-evaluating the A matrix each time.This final method has a lot to recommend it. Consider our simple two-variable example again. The equation~'/~ varies from eo for 6 = 0 to 1 for 6 = 1. Also (2-6)1~ varies from ,2 for ~ = 0 to 1 for ~ = 1, whereas o)zlo)l = (2-3')/6' varies from (kl + 1)/(kl-1) for ~ = 0 to 1 for 6=1.Thus we have a built-in discrimination against those parameters of the 'difference' structure which are least accurately determined. If we choose kl as 1.4 say, then COz/O91 varies between 6 and 1. This procedure will also work when a larger set of covarying variables is considered.Acta Cryst. (1973 Phase relations between quartets of reflexions can be found either directly from a generalized Hughes expression or by elimination of the phase of a reflexion common to two Y2 relationships. By combining the information from both types, strengthened quartet relationships (SQR) can be constructed, which are comparable to the Y z relations in reliability and number. Most of the reliable SQR's involve ...

The negative-quartet relation from electron-density considerations

“…The electron-density will be found near the points of intersection of the three sets of planes H, K and L. clusion, large values of E 4, IEH÷KI, IEK+LI and IEH÷LI indicate that the positive quartet (1) is likely to be true. This positive-quartet relationship proved to be reliable (Schenk, 1973a).…”

“…They therefore will be particularly helpful at an early stage of the phase determination process. This conclusion was drawn in the first paper on seven-magnitude quartets (Schenk, 1973a) from experimental evidence. Since then quartets have been used with success in the program system SIMPEL (Overbeek & Schenk, 1978) for the determination of a large and reliable starting set in terms of the smallest possible number of symbols.…”

“…which holds for large E 4 and large IEH+KI (Schenk, 1973a). This can be understood by drawing the H + K reflection into Fig.…”

“…This relationship, now commonly referred to as a positive quartet, was already known as the ~5 formula (Hauptman & Karle, 1953) and independently it was derived by Simerska (1956). Later it was shown (Schenk, 1973a) that, for a successful application, it is of paramount importance to include the cross terms H + K, K + L and H + L. Only when these terms are strong does the positive quartet give a reliable indication.…”

Improved treatment of severely or exactly overlapping Bragg reflections for the application of direct methods to powder data

“…= 0) for phase determination (Cochran, 1955) are selected if x = 2o3o'23/21EhEkE|[ > Kmin, and ~o triplets (Cochran & Douglas, 1957) are selected if I Ehl is close to zero and IEkl, IEtl > IEl,~j,,. Negative quartets (h + k + 1 + m = 0) (Schenk, 1973;Hauptmann, 1975) a is defined as 0-, = ~= l(Zs) ", where Zs is the atomic number for the N atoms in the unit cell. Comparison of perfectly partitioned (PP) powder data and equipartitioned (EQ) powder data shows that (a) there are more generating reflections in the EQ data set since more [El values are larger than I Elmin, (b) there are fewer triplets and quartets since there are fewer large [El values and therefore fewer products larger than K'mi n and Kmin"quartet, respectively, and (c) there are fewer negative quartets because there are fewer quartets and fewer cross-term sums with X < Xma x (Table 3).…”