The crystal structure, its variation within the homogeneity range and some physical properties of the new zinc-rich, partly disordered phase Ir7+7deltaZn97-11delta (0.31< or =delta< or =0.58) are reported. The structures of three phases with distinct composition were determined by means of single crystal X-ray diffraction. Ir7+7deltaZn97-11delta exhibits a significant homogeneity range, adopts a complex gamma-brass related cubic structure (cF403-406), is stable up to 1201(2) K, and transforms sluggishly below 1048(4) K into a phase with 394 atoms in the monoclinic primitive unit cell. It is a diamagnetic, moderate metallic conductor. Six distinguishable clusters consisting of 22-29 atoms comprise the structure. The clusters are situated about the 16 high symmetry points of the cubic F lattice. The structure can be subdivided into two partial structures, one with constant composition IrZn5 and 192 atoms per unit cell and a second being significantly richer in zinc with variable composition and 211-214 atoms per unit cell. The meandering triply periodic minimal surface of two interpenetrating diamond-like nets separates the compositionally variable from its complementary invariant part. The phase width is coupled with substitutional and positional disorder. A comprehensive analysis of composition-dependent site occupancy factors reveals a linear correlation between the various types of disorder which can be conclusively interpreted in terms of an incoherent intergrowth of distinctive partial structures in variable proportions on a length scale comparable to the size of the approximately 2 nm large unit cell. On the basis of the structural findings we derive the structure chemically meaningful formula Ir7+7delta)Zn97-11delta which quantitatively accounts for the interrelation between substitutional and positional disorder and provides a measure for the homogeneity range in structural terms.

An extension is proposed of the Shannon entropy-based structural complexity measure introduced by Krivovichev, taking into account the geometric coordinational degrees of freedom a crystal structure has. This allows a discrimination to be made between crystal structures which share the same number of atoms in their reduced cells, yet differ in the number of their free parameters with respect to their fractional atomic coordinates. The strong additivity property of the Shannon entropy is used to shed light on the complexity measure of Krivovichev and how it gains complexity contributions due to single Wyckoff positions. Using the same property allows for combining the proposed coordinational complexity measure with Krivovichev's combinatorial one to give a unique quantitative descriptor of a crystal structure's configurational complexity. An additional contribution of chemical degrees of freedom is discussed, yielding an even more refined scheme of complexity measures which can be obtained from a crystal structure's description: the six C's of complexity.

An analysis of certain types of multiplicative congruential generators--otherwise known for their application to the sequential generation of pseudo-random numbers--reveals their relation to the coordinate description of lattice points in two-dimensional primitive sublattices. Taking the index of the lattice-sublattice transformation as the modulus of the multiplicative congruential generator, there are special choices for its multiplier which induce a symmetry-preserving permutation of lattice-point coordinates. From an analysis of similar sublattices with hexagonal and square symmetry it is conjectured that the cycle structure of the permutation has its crystallographic counterpart in the description of crystallographic orbits. Some applications of multiplicative congruential generators in structural chemistry and biology are discussed.

Special types of number-theoretic relations, termed multiplicative congruential generators (MCGs), exhibit an intrinsic sublattice structure. This has considerable implications within the crystallographic realm, namely for the coordinate description of crystal structures for which MCGs allow for a concise way of encoding the numerical structural information. Thus, a conceptual framework is established, with some focus on layered superstructures, which proposes the use of MCGs as a tool for the quantitative description of crystal structures. The multiplicative congruential method eventually affords an algorithmic generation of three-dimensional crystal structures with a near-uniform distribution of atoms, whereas a linearization procedure facilitates their combinatorial enumeration and classification. The outlook for homometric structures and dual-space crystallography is given. Some generalizations and extensions are formulated in addition, revealing the connections of MCGs with geometric algebra, discrete dynamical systems (iterative maps), as well as certain quasicrystal approximants.