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! Continue reading →


Evolution & Student-centered Teaching Practices

Why does education "work"? How should we think about its best practices as a whole?I’ve been reading The Righteous Mind by Jonathan Haidt, and lately it’s gotten me thinking about the role of morality in education. If education is a garden, morality is the soil. What implicit moralities best cultivate learning? What keeps thirty students itching for A’s from cornering the teacher in his/her office and demanding that grade?

That’s a little far-fetched, but you see where I’m going. The classroom is bound by certain ethical principles, but what keeps students (or instructors) from violating them? Part of that can be explained by student self-interest: “this content will improve me, so I have incentive to follow the rules,” or “I want the grade, so I’ll go along with what the instructor says.” But there’s good reason to believe that’s not the whole story. For example, many instructors take an arbitrary approach to assigning grades, and for these teachers doing that is in their self-interest: it keeps students off their backs and frees up more time for [writing grants|lab work|time with family|anything else]. Of course, the best instructors know better. They understand that arbitrary grades (e.g. curves) are demotivating and encourage cutthroat behavior in students. They know that students must have a reason to buy into the morality of education, and that many practices in the classroom undercut education’s lofty foundations. What’s the core reason to buy into education, and what practices have evolved to promote that buying in? Consider an evolutionary perspective. Continue reading →