The Unit Cell

What's the V in the electron density equation?

It's the the unit cell volume, the smallest volume that can be used to generate the entire crystal using translation operations only; translations along the unit cell edges.

In general, the unit cell is a parallelopiped with non-orthogonal axes. The non- orthogonality causes some problems, i.e., Pythagorus' theorem doesn't hold for distances anymore, but it's the most natural set of axes to deal with often.

So, x,y,z are the coordinates along the unit cell axes. What are the units in which these are measured?

Well, for much of what we'll talk about, the coordinates are fractional coordinates.

They are fractions of the unit cell translations along the axes. They range from 0 to 1.0. So if an atom is on the x axis a quarter of the way across the cell, its coordinates are 0.25, 0.0, 0.0. And because of the symmetry of the crystal, if there's an atom at 0.25, 0.0, 0.0, there's also one at 1.25, 0.0, 0.0 and -.75, 0.0, 0.0. These coordinates are a pain when it comes to determining distances between atoms, because the distances also depend on the lengths of the unit cell edges, but they are quite handy for crystallographic calculations and for understanding symmetry.

Here are examples of how you can choose unit cells in two-dimensional wallpaper patterns.

But remember, symmetry makes the asymmetric unit the basic building block of a crystal.

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copyright © Ron Stenkamp stenkamp@u.washington.edu Most recent update 4/1/03