Why is the crystallography restriction obeyed?

Question

The crystallography restriction states that any 2-dimensional lattice can have rotational symmetry of degree 1, 2, 3, 4 or 6 - and that's it. A simple proof of I've heard of this is: the magnitude of $rot_\theta(x)-x$ is less than the magnitude of $x$ if $\theta$ is less than $\pi / 3$. (So if there's a rotation by less than $\pi/3$, it will fail to be a lattice, since we can always get smaller and smaller elements.)

That sounds fine to me, and I get why real crystals would have the requirement that all atoms be some epsilon away from each other. I don't understand why nature requires the lattice to be closed though - obviously real crystals aren't completely closed under addition, or else they would be infinitely large.

So why does the crystallography restriction limit the symmetries of real crystals? (Or does it not, and this application of the theorem is made up by math teachers to convince their students that math is useful?)

Answer 1

It really is just a definition, as Misha said. A crystal is defined as a state of matter where translational invariance is broken to a discrete subgroup. It occurs when you have a simple molecule substance, where units are rigidly arranged in a packing. The densest packings of most non-contrived simple shapes repeat with a particular periodicity.

You can arrange for complex molecules to crystallize, but it requires special preparation in aquaous solution, and proper ionic concentrations, which surround the complex molecule in a cage of a certain type. If you get a protein to crystallize, you write a paper, because then you can take an atomic scale picture of it using X-ray diffraction.

If you just jam proteins together and cool them down you make amorphous non-translational things, like gels or rubbers. These are heavily tangled polymer chains with no translational subgroup preserved. You don't study them, because they are complicated. They have a ground state entropy, which is caused by the trapping of the gel in metastable states. These systems are the subject of active research, and their theory is not well understood. Whether glasses have a sharp liquification transition was debated for decades (current consensus is that they do).

Answer 2

Crystal lattice has translational invariance by definition. Period.

The thing you are interested in has no translational invariance. Thus, it is not what people usually call crystal lattice. You might be interested in reading about quasicrystals. Which are more or less like crystal, but have a bit relaxed approach to periodivity.

By the way, if you count the number of atoms in real crystal you might get an idea why it is not that far from infinity.