Crystal Structure Complexity and Approximate Limits of Possible Crystal Structures Based on Symmetry-Normalized Volumes.

Rules that control the arrangement of chemical species within crystalline arrays of different symmetry and structural complexity are of fundamental importance in geoscience, material science, physics, and chemistry. Here, the volume of crystal phases is normalized by their ionic volume and an algebraic index that is based on their space-group and crystal site symmetries. In correlation with the number of chemical formula units Z, the normalized volumes exhibit upper and lower limits of possible structures. A bottleneck of narrowing limits occurs for Z around 80 to 100, but the field of allowed crystalline configurations widens above 100 due to a change in the slope of the lower limit. For small Z, the highest count of structures is closer to the upper limit, but at large Z, most materials assume structures close to the lower limit. In particular, for large Z, the normalized volume provides rather narrow constraints for the prediction of novel crystalline phases. In addition, an index of higher and lower complexity of crystalline phases is derived from the normalized volume and tested against key criteria.