The elongated rhombic dodecahedron in alloy structures

Abstract

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.

A new structural principle in anion-excess fluorite-related superlattices

“…The transformation from the fluorite to the tveitite grouping can then be formulated as M6X32 (fluorite) + 5X ~ M6X37 (tveitite); the origin of the series MnX2n+5 is thus immediately apparent. This same grouping of six square antiprisms around a filled cuboctahedron has been described by Nyman & Andersson (1979) in the alloy Cr23C 6.The juxtaposition of these entities in the tveitite structure is illustrated in Fig. 2, and Fig.…”

Tetrahedral Stars as Flexible Basis Clusters in sp-Bonded Intermetallic Frameworks and the Compound BaLi7Al6 with the NaZn13 Structure

“…In the cubic Th 6 Mn 23 structure ( Fd 3̄ m ) TS units are connected via type D atoms in such a way that one D atom belong to two TS clusters, and the resulting three-dimensional array of clusters is of the simple cubic type. This structural description has already been made by Nyman and Andersson . However, for comparing the Th 6 Mn 23 structure with related ones the rhombohedral representation is preferable.…”

“…Nyman and Andersson established TS as a valuable building unit for the description of numerous inorganic structure types. Ionic as well as intermetallic compounds are covered, ranging from subunits in Rh 7 Mg 44 to extended frameworks in W 3 Fe 3 C, Th 6 Mn 23 , γ-brass, SiF 4 , CaWO 4 , NaZn 13 , Ru 7 B 3 , Tl 3 VS 4 , and NaPb . Sometimes TS units are not linked directly via T or C atoms, but a third type of atom is introduced, bridging the edges of the central tetrahedron (type D atoms).…”

Sn8.7(Ni0.5,Zn0.4,Cu0.1)10.4, a new tin-based alloy from a complex metallic mixture

“…l c). This representation clearly shows the relationship between this structure and that of BaHgl~ (Peyronel, 1952), which has been described by means of the von Fedorov polyhedron (Nyman & Andersson, 1979). The networks of TRD and von Fedorov polyhedra are similar in the two structures, but the contents of the polyhedra are quite different.…”