Computer modeling and description of nonstoichiometric apatites Cd5-η/2(VO4)3I1-η and Cd5-η/2(PO4)3Br1-η as modified chimney-ladder structures with ladder-ladder and chimney-ladder coupling

Diffraction patterns from apatite-structure compounds Cd5-η/2(TO4)3X1-η with T=P, V and X=Br, I show sheets of diffuse scattering normal to c* at incommensurate l=q (q=1.63 for Cd-V-I apatite and q=1.78 for Cd-P-Br apatite), because the c repeat of the average unit cell is shorter than two X diameters. The equilibrium X..X spacing along c defines the incommensurate periodicity c/q and stoichiometry (1-η)=q/2. The layers show a honeycomb texture for the Cd-V-I apatite, which is condensed into discrete spots for the Cd-P-Br compound. In both phases, X..X repulsions along 〈100〉 force neighboring rods of X atoms out of phase. In the Cd-P-Br phase, additional 〈210〉 attractions drive incipient formation of a rhombohedral superstructure. Average structure site occupancies and the observation of second-order diffuse layers at both l=2q and l=q+2 imply the existence of strong Cd..X in addition to X..X interactions. A three-dimensional computer model was used to produce finite-temperature structure simulations as a function of X..X interactions along 〈001〉, 〈100〉, and 〈210〉, and X..Cd interactions, from which diffraction patterns were calculated. The experimental patterns were fit and approximate values for the interaction energies obtained (hundreds to thousands of joules per mole). It was apparent that lock-in to commensurability caused by the X..Cd term and the formation of nonprimitive incommensurate modulated structures driven by X..X interactions were mutually antagonistic, and the actual structures are compromises between the two