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A high-dimensional description has been applied to a new structural model of decagonal A1-Mn, which is constructed on the basis of the structure of the A13Mn phase [Li (1995). Acta Cryst. B51, 265-270]. The structure of the A1-Mn decagonal quasicrystal consists of a sequence of six layers, involving two types of layers. Acceptance domains in a five-dimensional structure description are proposed for the structures of the layers. An ideal quasilattice of the A1-Mn decagonal quasicrystal can be referred to as a periodic stack of two-colour Penrose patterns in the sequence TTTT .... in which 7" is T in reverse colour.

Orthorhombic (O-1, O-2 and O-3) and monoclinic (M(1) and M(2)) phases are the crystalline approximants of the Al-Cu-Fe-Cr decagonal quasicrystal. The structures of these crystalline phases can be described using three subunits, which are derived from the O-3 structure. As an example, the structural model of the O-1 phase is presented in detail. This model agrees with the X-ray powder diffraction pattern and the high-resolution electron image of the O-1 phase.

Monoclinic tau(2)-Al13Co4, tau(3)-Al13Co4 and tau(4)-Al13Co4 phases with a and c parameters about tau(2), tau(3) and tau(4) times [tau = (1 + 5(1/2))/2 = 1.61803, which is the golden number], respectively, greater than their counterparts in Al13Co4 (space group Cm, a = 1.5183, b = 0.8122, c = 1.2340 nm, beta = 107 degrees 54' [Hudd & Taylor (1962). Acta Cryst. 15, 441-442]) have been found by selected-area electron diffraction and high-resolution electron microscopy (HREM). The [010] HREM images showed that these tau-inflated phases have a similar pentagonal layer structure as Al13Co4, but the edge length of the pentagons is tau(2), tau(3) and tau(4) times larger, respectively, than that in Al13Co4. This explains why these tau-inflated phases have about the same beta angle (similar or equal to 108 degrees) and b representing a repeat unit of the stacking of these pentagonal layers. The tau(2) pentagon (1.23 nm) consists of six smaller pentagons (0.47 nm) existing in Al13Co4, one inverted and five of the same orientation as the tau(2) pentagon. Similarly, the tau(3) pentagon (2.00 nm) consists of six tau pentagons (0.76 nm), and the tau(4) pentagon (3.23 nm) of six tau(2) pentagons which in turn contain six 0.47 nm pentagons in each of them. The greater the inflation, the larger the unit-cell parameters, and the better the approximation of its structure to that of a decagonal quasicrystal (DQC). In other words, a DQC is considered to be the end member of this family of tau-inflated phases.

Al3Mn (with the stoichiometric composition Al28.1Mn10.9), M(r) = 1356.5, orthorhombic, Pn2(1)a, a = 14.837 (4), b = 12.457 (2), c = 12.505 (2) angstrom, V = 2311.2 (8) angstrom3, atoms/cell = 4 x 39, D(x) = 3.90 g cm-3, lambda(Mo Kalpha) = 0.71069 angstrom, mu = 65.82 cm-1, F(000) = 2550.15, room temperature, R = 0.068 for 1289 reflections. About two-thirds of Mn atoms have icosahedral coordination. In a repeat unit b, there are four interpenetrated icosahedra with two Mn and two Al atoms at their centers. This phase has a layer structure consisting of an almost 'flat' layer F sandwiched between two puckered P and p layers in the sequence PFpP'F'p'..., where P'F'p' and PFp are related by the 2(1) axes parallel to [010]. A similar layer structure has also been found in the pi-AlMnM (M = Ni, Cu, Zn) phases. Both these two crystalline structures can be obtained from that of the Al-Mn decagonal quasicrystal by substituting a rational ratio of two consecutive Fibonacci numbers for the irrational tau = [1 + (5)1/2]/2 in the two quasiperiodic directions.

A new hexagonal aluminium–chromium–nickel phase, Al14.4Cr3.4Ni1.1, is described. The structure includes 12 icosahedral clusters in a unit cell, each of which consists of a transition metal (TM) atom at the center and 11 aluminium and one TM atom on the fivefold vertices of the icosahedron. Pairs of the clusters are linked along the c axis by sharing a triangular face. Two pairs of the clusters are joined by two trigonal antiprisms, constructing a column along the c axis. Six such columns are included in a unit cell and form the crystal structure. The second neighboring atoms connecting the columns are on the threefold axes of the icosahedral cluster.