Symmetry-inconsistent Three-phase Structure Invariants Space Groups of Quasicrystallographic Tilings

Since again no reference is given by HL, the reader might assume that no previous work has been done on the subject. On the contrary, several references must be quoted" the problem of the influence of the space-group symmetry in the quartet relationships was first treated in paper G3 from both the algebraic and the probabilistic points of view and the implementation of the theory in a procedure for phase solution was described by Busetta, Giacovazzo, Burla, Nunzi, Polidori & Viterbo (1980). (d) An effective implementation in the MULTAN package of the results previously quoted for triplets has been described by Main (1985). The correct space-group weight for a triplet relationship is given by Wh, k : E-hEkEh-k ~ ¢~p,q exp [27ri(-hTp +kTq)] P,q where 8p.q= 1 when h(I-Rp) =k(I-Rq)-0 otherwise. The summations are over all the space-group symmetry operations. Main's algorithm is clearly able to single out symmetry-consistent and-inconsistent triplets and to provide relative weights for their use in the phasing process. The last consideration introduces a final remark. Tables 1-3 in HL's paper are of limited use in direct-methods practice because: (1) the method used by HL to derive the list of equivalent or inconsistent triplets can fail to recognize some special combinations of indices producing multiple solutions for (2). The supplementary rules derivable by means of the algorithm described in the present paper and those concerning triplets with restricted phase values are only two examples, but others could exist in principle. (2) the use of large tables in routine programs is not advisable. Main's algorithm is an effective example of how relatively simple in practice the use of symmetry in such types of problems may be. Thanks are due to a referee for useful suggestions. Abstract I. Introduction A method is described for producing tilings with various quasicrystallographic space groups, paying particular attention to the two-dimensional space groups pnml and pnlm that can exist as distinct possibilities when the order of rotational symmetry n is a power of an odd prime number. Rokhsar, Wright and Mermin have discussed the definition and classification of lattices and space groups with crystallographically forbidden point-group symmetries, taking the view that such quasi-crystallographic concepts are best formulated in Fourier space. For any material whose diffraction