An Intro to Covalent Lattices

Hello, everyone! It feels like it’s been ages since I’ve been back here, but I know that’s only because I’ve spent the past two days curled up in a fetal position and crying salty, salty tears over the shenanigans that friction plays with otherwise beautiful conservative systems. Yes, I’ve been studying physics. That’s the long and short of it.

Now that biochemistry is behind me (happily waving at it in the rear view mirror; I see you, trick-question-that-I-wasn’t-smart-enough-to-realize-was-tricksy-san!), it’s time for me to gear up for another test: inorganic. Yes, you know what that means, friends—if you’re my Facebook friend, your feed is about to get really clogged up with sciency stuff you don’t want to know.

Sorry about that. Turns out, this really helped me on my exam, so… I’mma keep doing it.

Kthxbai!


If you’ve taken much chemistry, you’ve probably done a lot of studying chemistry in aqueous solutions. If you’re super lucky (sarcasm punctuation), you may have even gotten an in-depth look at the gas phase. However, I doubt that you’ve done much with the state that we seem to notice the most—that is, the solid state.

Solid matter comes in a lot of different forms. There are atomic solids, which are solids of individual atoms held together by London forces (think really, really cold neon). Metalic solids are the ones we probably think of the most—they consist of metal atoms swimming in a sea of valence electrons (iron, copper). Ionic solids contain oppositely charged ions, and they form crystals wherein one ion forms the “framework” and the other falls into holes in the framework (salt). Molecular solids consist of individual molecules held together through noncovalent forces (sugar).

However, our favorite kind of solid is the covalent lattice, which is made up of atoms bonded to each other in such a way that you can’t separate out molecules. These can either be amorphous (having only short-range order), like glass, which is just a really, really viscous liquid [looks dubiously at window], or they can be crystalline (having long-range order), like quartz and diamond.

Although covalent lattices have no distinct molecules, crystalline covalent lattices can be defined in smaller terms. Turns out, every crystal is made through three-dimensional repetition of a basic unit, called a unit cell, such that there is no empty space. Unit cells are defined by three edge lengths and three angles, and they, in turn, define crystals.

All unit cells belong to one of the seven crystal systems. I’ll reproduce them here, but it’s really a bit tedious. Where a, b and c are lengths, and α, β and γ are angles:

Triclinic: a ≠ b ≠ c ; α ≠ β ≠ γ

Monoclinic: a ≠ b ≠ c; α = γ = 90º, β ≠ 90º

Orthorhombic: a ≠ b ≠ c; α = β = γ = 90º

Tetragonal: a = b ≠ c; α = β = γ = 90º

Cubic: a = b = c; α = β = γ = 90º

Trigonal: a = b = c; α = β = γ ≠ 90º

Hexagonal: a = b ≠ c; α = β = 90º,  γ = 120º

I told you it was tedious, didn’t I?

Well, if that isn’t complicated enough, we’re going to spice it up for you. Not only do unit cells each belong to one of these categories, but they can have different arrangements of what we call “lattice points.”

Now, before I go any further, I’ll tell you that every corner of a unit cell must contain the same kind of atom/ion/molecule, which we’ll call a lattice point. (I’m not just saying atom because these unit cells apply to other solids, too!) However, unit cells can contain more than this basic framework. There are four possible arrangements of lattice points:

Primitive (P)

Primitive unit cells only contain a lattice point at every corner. There are eight corners in a cube (or vaguely cube-like structure), each with a single lattice point. However, since each corner is shared by eight cells (three beside the cell in question, and four above it), we can really only think of an eighth of every point as belonging to the cell. Thus, each P cell contains 1/8 (8) = 1 lattice point.

End (C)-Centered (C)

C-Centered cells contain the eight lattice points that P cells have, but they also have two more—one on each of two opposite faces. Since each lattice point on a face is shared by two cells, C-Centered cells contain 1/8 (8) + 2(1/2) = 2 lattice points.

Body-Centered (I)

Body-centered cells contain the eight lattice points that P cells have, but they also have a single lattice point directly in their center. Since this body isn’t shared by any other cells, I cells have 1/8 (8) + 1 (1) = 2 lattice points.

Face-Centered (F)

Face-centered cells contain, in addition to the eight corner lattice points, a lattice point in the middle of each of their six faces. Since each face is shared by two cells, F cells contain 1/8 (8) + 1/2 (6) = 4 lattice points.

Mattaku… Okay, surely we’re at the end now, right? Crystals can’t be that complicated, right? … right?

Unfortunately, yes, they can be.

Now that we’ve defined the possible arrangements of lattice points, we have to apply those to the crystal systems. Theoretically speaking, for every crystal system, there should be a P, C, I and F unit cell.

As it turns out, however, we don’t get 28 different kinds of unit cells, as we’d expect. Some of them turn out to be repeats. (I’ll draw this when I have my tablet back.) Instead, we end up with a set of fourteen magical sets of unit cells called Bravais lattices.

(And yeah, that’s “brah-vay.” Like Auguste Bravais.)

Even though you don’t really need to know them, as far as I know, here they are:

Triclininc: P

Monoclinic: P, C

Orthorhombic: P, C, I, F

Tetragonal: P, I

Cubic: P, I, F

Trigonal: P

Hexagonal: P

And that, ladies and gentlemen, is just the beginning. We haven’t even started talking about closest packed structures yet.


Questions? Comments? Salty, salty tears? Put ’em down there!

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