Classifying crystals of rounded tetrahedra and determining their order parameters using dimensionality reduction

Using simulations we study the phase behavior of a family of hard spherotetrahedra, a shape that interpolates between tetrahedra and spheres. We identify 13 close-packed structures, some with packings that are significantly denser than previously reported. Twelve of these are crystals with unit cells of N = 2 or N = 4 particles, but in the shape regime of slightly rounded tetrahedra we find that the densest structure is a quasicrystal approximant with a unit cell of N = 82 particles. All 13 structures are also stable below close packing, together with an additional 14th plastic crystal phase at the sphere side of the phase diagram, and upon sufficient dilution to packing fractions below 50-60% all structures melt. Interestingly, however, upon compressing the fluid phase, self-assembly takes place spontaneously only at the tetrahedron and the sphere side of the family but not in an intermediate regime of tetrahedra with rounded edges. We describe the local environment of each particle by a set of l-fold bond orientational order parameters q_l, which we use in an extensive principal component analysis. We find that the total packing fraction as well as several particular linear combinations of q_l rather than individual q_l’s are optimally distinctive, specifically the differences q₄ - q₆ for separating tetragonal from hexagonal structures and q₄-q₈ for distinguishing tetragonal structures. We argue that these characteristic combinations are also useful as reliable order parameters in nucleation studies, enhanced sampling techniques, or inverse-design methods involving odd-shaped particles in general.