Near coincidence site lattice misorientations in monoclinic zirconia

Zirconium dioxide, ZrO2, exists in three crystalline phases: monoclinic, tetragonal, and cubic. Calculations of the coincidence site lattice (CSL) misorientations for the last two lattices and for hexagonal ones using the methods developed represent little difficulty. However, no procedure for the determination of the CSL misorientations in the monoclinic system has been reported so far. Monoclinic zirconia has the crystallographic space group P21/c and the following parameters of the unit cell (e.g., 5, 6): a = 5.1490 angstrom, b = 5.2133 angstrom, c = 5.3161 angstrom, and β = 99.228 degree. Before discussing possible CSL misorientations in zirconia, consider a simple example based on geometric considerations. In any monoclinic crystal (with any lattice parameters) the two symmetrical boundaries along the (001) and (100) planes must have highly ordered atomic structure. The misorientation of the first boundary is descried as a rotation of either 180 degree around the [100] direction or 180 degree around the normal to the (001) plane. The misorientation of the second boundary is 180 degree [001] or 180 degree around the normal to the (100) plane. It can be shown that three-dimensional CSLs will exist in both cases if (c/a)cosβ is a rational number. This example justifies the following approximation of the unit cell in the monoclinic zirconia: a = b = c and cosβ = -1/6 (i.e., β = 99.594 degree). Consider the following prismatic cell in the monoclinic crystal structure: ([1 0 1], [bar 1 0 1], [0 1 0]). With the above approximation, this cell is orthogonal with the ratios of the squares of the edge lengths expressed as 5:7:3. Therefore, one can apply the algorithm for calculations of the CSL misorientations in orthorhombic lattices with rational ratios of squares of the lattice periods, which is based on the general vector-quaternion method of misorientation representation