LRL_WEB, A Website for a More Thorough Exploration of Bravais Lattice Types of Your Unit Cells

Even today, the standard programs for determining likely lattice types still sometimes fail to recognize the correct symmetry of a crystal (Le Trong & Stenkamp, 2007). Experience has shown that the programs BGAOL (Andrews & Bernstein, 2014) and the newer program Sella (in preparation) produce more complete results. Sella produces an especially useful map of the hierarchy of lattice types (Grimmer, 2016, described the hierarchy). LRL_WEB allows access to Sella, BGAOL, SAUC (McGill et al., 2014), PlotC3, Delaunay and Niggli reductions. An example of the hierarchy maps is a case cited by Le Trong & Stenkamp, PDB entry 1G2X, as C-centered monoclinic, C 80.95 80.57 57.1 90 90.35 90. Sella produces Figure 1, which of course shows a perfect match to mS (side centered monoclinic), but also a relatively good match to hR (rhombohedral). The example is one of a group of probably identical structures of Krait toxin phospolipase A2. Figure 2 shows two Bravais lattice types, as defined by Delaunay. We can use PlotC3 to learn more about these two orthorhombic types. O3 is described as body-centered, and O4 is side-centered. In Figure 3, the upper panel was produced using O3 data and the lower using O4 data. For each case, CmdGen (on the web site) was used to create 200 random cells of the selected type. Considering the upper panel (O3), there are two panes that show the projections as lines, and the third shows a point. That tells us that the plot is described by a plane, even though we are examining a 6-dimensional plot. The lower panel also shows two projections as lines, but the third is random. In this case we are looking at a 3-space object in the 6-dimensional space S6, one representation of unit cells. So Figure 3 has told us that O3 is a 2-dimensional object, but we know that orthorhombic unit cells have to have 3 parameters. O4 being a 3-space object is what is expected. We learn that O3 is a degenerate type, and we can determine that in a different way using Figure 2. The characteristic of S6 vectors for O3 is (rs0 rs0), and for O4 it is (00r sst). O3 has only 2 available parameters, and O4 has 3, which we expect for orthorhombic. In fact, some additional research shows that O3 is the boundary between two other types.

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