Diffractive point sets with entropy

After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of lattice gases. We show that taking the site occupation of a model set stochastically results, with probabilistic certainty, in well-defined diffractive properties augmented by a constant diffuse background. We discuss both the case of independent, but identically distributed (i.i.d.) random variables and that of independent, but different (i.e., site dependent) random variables. Several examples are shown.Comment: 25 pages; dedicated to Hans-Ude Nissen on the occasion of his 65th birthday; final version, some minor addition

Urotropin azelate: a rather unwilling co-crystal

Urotropin (U) and azelaic acid (AA) form 1:1 co-crystals (UA) that give rise to a rather complex diffraction pattern, the main features of which are diffuse rods and bands in addition to the Bragg reflections. UA is characterized by solvent inclusions, parasite phases, and high vacancy and dislocation densities. These defects compounded with the pronounced tendency of U to escape from the crystal edifice lead to at least seven exotic phase transitions (many of which barely manifest themselves in a differential scanning calorimetry trace). These involve different incommensurate phases and a peritectoid reaction in the recrystallization regime (T-h >0.6). The system may be understood as an OD (order-disorder) structure based on a layer with layer group P(c)c2 and cell a(o) similar or equal to 4.7, b similar or equal to 26.1 and c similar or equal to 14.4 Angstrom. At 338 K the layer stacking is random, but with decreasing temperature the build-up of an orthorhombic MDO (maximal degree of order) structure with cell a(1) = 2a(o), b(1) = b, c(1) = c and space group Pcc2 is begun (at similar to 301 K). The superposition structure of the OD system at T = 286 (1) K with space group Bmmb and cell (a) over cap = 2a(o), (b) over cap = b and (c) over cap = c/2 owes its cohesion to van der Waals interactions between the AA chains and to three types of hydrogen bonds of varied strength between U-U and U-AA. Before reaching completion, this MDO structure is transformed, at 282 K, into a monoclinic one with cell a(m) = a(o) + c/4, b(m) = b, c(m) = -2(a(o) + c/2), space group P2(1)/c, spontaneous deformation similar to2degrees, and ferroelastic domains. This transformation is achieved in two steps: first a furtive triggering transition, which is not yet fully understood, and second an improper ferroelastic transition. At similar to 233 K, the system reaches its ground state (cell a(M) = a(m), bM = b, c(M) = c(m) and space group P2(1)/c) via an irreversible transition. The phase transitions below 338 K are described by a model based on the interaction of two thermally activated slip systems. The OD structure is described in terms of a three-dimensional Monte Carlo model that involves first- and second-neighbour interactions along the a axis and first-neighbour interactions along the b and c axes. This model includes random shifts of the chains along their axes and satisfactorily accounts for most features that are seen in the observed diffraction pattern

Stacking faults and superstructures in a layered brownmillerite

Stacking faults in Ca4Fe2Mn0.5Ti0.5O9 have been examined using X-ray diffraction and high-resolution transmission electron microscopy. Electron diffraction revealed two superstructures with ordered stacking sequences

A neutron diffuse scattering study of PbZrO₃ and Zr-rich PbZr_1-xTi_xO₃

A combined neutron diffuse scattering study and model analysis of the antiferroelectric crystal PbZrO3is described. Following on from earlier X-ray diffuse scattering studies, supporting evidence for disordering of oxygen octahedral tilts and Pb displacements is shown in the high-temperature cubic phase. Excess diffuse scattering intensity is found at theMandRpoints in the Brillouin zone. A shell-model molecular dynamics simulation closely reproduces the neutron diffuse scattering pattern. Both in-phase and antiphase tilts are found in the structural model, with in-phase tilts predominating. The transition between disordered and ordered structure is discussed and compared with that seen in Zr-rich PbZr1−xTixO3.</jats:p