Permissible domain walls in monoclinic M_AB ferroelectric phases

The concept of monoclinic ferroelectric phases has been extensively used over recent decades for the understanding of crystallographic structures of ferroelectric materials. Monoclinic phases have been actively invoked to describe the phase boundaries such as the so-called morphotropic phase boundary in functional perovskite oxides. These phases are believed to play a major role in the enhancement of such functional properties as dielectricity and electromechanical coupling through rotation of spontaneous polarization and/or modification of the rich domain microstructures. Unfortunately, such microstructures remain poorly understood due to the complexity of the subject. The goal of this work is to formulate the geometrical laws behind the monoclinic domain microstructures. Specifically, the result of previous work [Gorfman et al. (2022). Acta Cryst. A78, 158-171] is implemented to catalog and outline some properties of permissible domain walls that connect 'strain' domains with monoclinic (MA /MB type) symmetry, occurring in ferroelectric perovskite oxides. The term 'permissible' [Fousek & Janovec (1969). J. Appl. Phys. 40, 135-142] pertains to the domain walls connecting a pair of 'strain' domains without a lattice mismatch. It was found that 12 monoclinic domains may form pairs connected along 84 types of permissible domain walls. These contain 48 domain walls with fixed Miller indices (known as W-walls) and 36 domain walls whose Miller indices may change when free lattice parameters change as well (known as S-walls). Simple and intuitive analytical expressions are provided that describe the orientation of these domain walls, the matrices of transformation between crystallographic basis vectors and, most importantly, the separation between Bragg peaks, diffracted from each of the 84 pairs of domains, connected along a permissible domain wall. It is shown that the orientation of a domain wall may be described by the specific combination of the monoclinic distortion parameters r = [2/(γ - α)][(c/a) - 1], f = (π - 2γ)/(π - 2α) and p = [2/(π - α - γ)] [(c/a) - 1]. The results of this work will enhance understanding and facilitate investigation (e.g. using single-crystal X-ray diffraction) of complex monoclinic domain microstructures in both crystals and thin films.