CARMELO GIACOVAZZO 305 if Rp and Rq correspond to symmetry operators of order two, E 5 and E 6 are centrosymmetric reflexions. In fact, if R m is a rotation matrix for which Rp R m = Rq (then Rq R,n = Rp also), we have h(Rp-Ro)R m = h(R ° --Ru ) =--h(Rp--Rq), which in terms of phases gives, because of (4) (sp-aq). If E 5 and E 6 are centrosymmetric reflexions, (60) no longer holds; in fact, the modified Bessel function of zero order involving Z 5 and Z 6 has to be replaced by hyperbolic cosines of suitable arguments and a suitable B' value has to replace B. Furthermore, the problem of generalizing (60)to cases in which more (Rp, Rq) pairs exist, which give rise to the crystallographically independent generalized solutions (h~, h2) of system (5), needs to be solved. All these theoretical aspects are discussed elsewhere (Giacovazzo, 1979b) where a general distribution function is given, which in several cases can be considered a useful approximation of the 'true' distribution of O.
Concluding remarksA theory has been described which is capable of deriving for any space group the value of a two-phase seminvariant of first rank, • = tp= + ~0v, given all or some of the magnitudes belonging to the first phasing shell of O. The probabilistic formulae are derived both by using the exponential forms of the characteristic functions of the joint probability distributions studied and via their Gram-Charlier expansion. A general algebra for two-phase seminvariants of first rank has been developed which makes their estimation easier in the automatic procedures for phase solution.
References
DEBAERDEMAEKER, T. & WOOLFSON, M. M. (1972). ActaCryst. A28, 477--481. GIACOVAZZO, C. (1974). Acta Cryst. A30, 481-484. GIACOVAZZO, C. (1977a). Acta Cryst. A33, 531-538. GtACOVAZZO, C. (1977b). Acta Cryst. A33, 539-547. GIACOVAZZO, C. (1977c). Acta Cryst. A30, 933-944. GtACOVAZZO, C. (1978). Acta Cryst. A34, 27-30. GIACOVAZZO, C. (1979a). To be published. GIACOVAZZO, C. (1979b
AbstractWith the space-filling elongated dodecahedron or its truncated form as a coordination polyhedron for larger atoms, structures like BaAI4, CeMg2Si2, BaHgl~ and ThMnl2 can be accurately described.