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In special relativity, spacetime is $\mathbb{R}^{1,3}$ with isometries $\text{ISO}(1,3)^{[1]}$, i.e. the Poincaré group. Wigner's classification postulates that fundamental particles are in correspondence to irreducible projective representations of the restricted Poincaré group. In this sense, the isometries of spacetime tell a lot about what can happen in spacetime; it gives an exhaustive list of all possible fundamental particles.

Consider some Bravais lattice $\mathcal{L}$ with space group $G$. Does the space group $G$ give me similar information about "fundamental" particles on the Bravais lattice $\mathcal{L}$?

In a sense, a Bravais lattice is not quite spacetime. Imagining that a Bravais lattice is of ion cores (comprised of ion + bound electrons), then at the least conduction electrons are not restricted to be even localized (in fact, Bloch's theorem says they are utterly delocalized) around the lattice structure. This is very different from fundamental particles in special relativity, which must exist in spacetime.

[1] Note that "$\text{ISO}$" does not stand for isometry.

Qmechanic
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Silly Goose
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1 Answers1

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What constrains crystal lattices is energy. An atom will try to surround itself with neighbors in the lowest possible energy configuration. But so will the neighbors. In a crystal, all atoms find the same lowest energy configuration.

The lowest energy configuration for one atom might be to surround itself with neighbors at the faces of a dodecahedron. But it cannot because dodecahedrons cannot tile space. This would force neighbors into higher than optimal configurations.

To understand why crystals form, you need to ask not only what are the crystal groups, but why do all atoms find the same environment? Why don't they form a non repeating glass like state?

mmesser314
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