Formulas forn-tet phase invariants and embedded seminvariants for all space groups

Abstract

Formulas are presented for the calculation of the cosines of n-tet phase invariants and embedded seminvariants in all the space groups. They are shown to be of the form of a particular type of expected value formula that is derivable from the joint probability distribution. In the recent literature, formulas for phase invariants and seminvariants have been given in the form of conditional probability distributions. A detailed comparison of the relative merits of the two types of formulas, expected value and conditional distribution, has not yet been made. The variety of potential applications is quite vast and therefore it may require much effort to make evaluations of current theories. Should it seem worthwhile, the determinantal joint probability distributions employed in this paper could provide the basis for the derivation of additional conditional probability distributions. They are likely to be much more complex, however, than the expected value formulas. Some simple calculations with triplet and quartet invariants involving random structures in space group P1 show a considerable decrease in the reliability of the expected value formulas as the complexity of the structure increases. A comparable observation had been made in the past for conditional probability distributions for triplet phase invariants. Current theories present the possibility of obtaining information in special circumstances, for example, with respect to selected embedded seminvariants. How extensive and how useful such information might be, particularly with respect to the truly difficult structures that occur among the essentially equal-atom, noncentrosymmetric crystals with 100 or more nonhydrogen atoms in the asymmetric unit, remains to be seen.