Unit Cell Hell

I experienced a wake-up call recently when a student dropped by to ask about unit cells. Wow, I realized, I know nothing about crystal structures. Analyzing simple cubic on the fly is easy enough, but the close-packed structures require quite a bit of mental gymnastics. Working mostly on the lab side of things, I don’t often think about this topic (although some nice activities with solids as their focus have been developed).

While the visualization skills needed to understand unit cells inside and out can turn students off, they’re a classic example of how chemists use microscopic structure and properties to explain and predict macroscopic phenomena. Stripping away the messy details, there are relatively few properties of unit cells that general chemists care about:

  • Packing fraction (also interesting from a physical and mathematical point of view)
  • Density
  • Hole geometry and count
  • Dimensions and atomic/molecular radii

This video is a great introduction to the most important crystal structures from a materials science point of view. The best thing a student can do, in my opinion, is use a physical model to build up the structures herself—this is particularly true for the close-packed structures, which to me have a kind of magical allure. How can there be two ways to pack hard spheres as closely as possible? The answer becomes apparent after you’ve stacked up two planes of close-packed spheres…

Two layers of close-packed spheres stacked one on top of the other. Note the two types of pockets for the next layer!

Two layers of close-packed spheres. Note the two types of pockets for the next layer!

The next layer of spheres will sit down in the triangular “pockets” between the red spheres, but there are two inequivalent types of pockets: those above tetrahedral holes and those above octahedral holes. Because of the size of the spheres, both types of pockets cannot simultaneously be occupied. Ergo, there are two inequivalent close-packed structures!

Then, of course, there is the insanity of the face-centered cubic structure, which appears cubic only after a perverse transformation from a top view of four stacked close-packed planes to a diagonal view. This would make a good project for a “Fun with ChemDraw” class.

Seeing the cubic unit cell of the fcc structure, starting from four horizontal close-packed layers.

If you stare at that unit cell long enough, I’m pretty sure it starts to wave at you. It just looks like a mess of balls to me (or maybe two wrestlers in grappling formation—tilt your head a bit) until each face is highlighted individually.

Faces of the FCC unit cell.

Faces of the FCC unit cell.

Better! But still, this is one of those subjects that could benefit from a healthy dose of educational technology. I might reach for the Cambridge Structural Database and a recent copy of Jmol the next time I have to teach unit cells.

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