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.