Sphere packings continued – space groups P2_1 and C2/c

In this post, we’re diving back into the world of sphere packings, focusing on the space groups P2_1 and C2/c. Feel free to check out an earlier discussion on Densest P1, P-1 and P21/c packings of spheres. Both of these groups reach the packing density of \frac{\pi}{\sqrt{18}}.

The densest P2_1 packing almost identical to the P\overline{1} group. Even though they belong to different crystal systems—triclinic for P\overline{1} and monoclinic for P2_1—they share the same unit cell parameters: a = c = 2, b = 2\sqrt{2}, and \beta = \frac{\pi}{2}. Here’s a snapshot of what this packing looks like:

Next up is the C2/c space group, which also falls under the monoclinic category. This group is a bit more complex, with eight symmetry operations. Its densest packing has unit cell parameters of a = b = 4, c = 2\sqrt{3}, and \beta = \pi - \text{arcsin} \left( \frac{\sqrt{2}}{\sqrt{3}} \right). Check out the illustration below of the densest packing configuration:

So far, we’ve seen that the optimal packing density for the space groups P1, P\overline{1}, P21/c, P2_1, and C2/c matches the general optimal sphere packing density. This isn’t too surprising since these groups are all related to the Hexagonal Closed Packed structure, via group – subgroup relations. Interestingly, these space groups are among the top ten of the most frequently occurring in the Cambridge Structural Database, making up 74% of its entries.