Sunday, January 29, 2017

What is a crystal?

(I'm bringing this up because I want to write about "time crystals", and to do that....)

A crystal is a larger whole comprising a spatially periodic arrangement of identical building blocks.   The set of points that delineates the locations of those building blocks is called the lattice, and the minimal building block is called a basis.  In something like table salt, the lattice is cubic, and the basis is a sodium ion and a chloride ion.  This much you can find in a few seconds on wikipedia.  You can also have molecular crystals, where the building blocks are individual covalently bonded molecules, and the molecules are bound to each other via van der Waals forces.   Recently there has been a ton of excitement about graphene, transition metal dichalcogenides, and other van der Waals layered materials, where a 3d crystal is built up out of 2d covalently bonded crystals stacked periodically in the vertical direction.

The key physics points:   When placed together under the right conditions, the building blocks of a crystal spontaneously join together and assemble into the crystal structure.  While space has the same properties in every location ("invariance under continuous translation") and in every orientation ("invariance under continuous orientation"), the crystal environment doesn't.  Instead, the crystal has discrete translational symmetry (each lattice site is equivalent), and other discrete symmetries (e.g., mirror symmetry about some planes, or discrete rotational symmetries around some axes).   This kind of spontaneous symmetry breaking is so general that it happens in all kinds of systems, like plastic balls floating on reservoirs.  The spatial periodicity has all kinds of consequences, like band structure and phonon dispersion relations (how lattice vibration frequencies depend on vibration wavelengths and directions).


Anonymous said...

A crystal necessary has to have a discrete diffraction spectrum, but does it necessarily have to have discrete translational symmetry? I'm thinking about quasicrystals in particular here.

Anonymous said...

The crystalline structure and your comparison of it to space-time symmetries are really interesting. You mentioned that space-time is invariant under continuous translation and rotation, while the crystal is not, but it isn't really the case if we assume that space-time is also quantized with the frequency of the Planck length. Could a crystal be the background of our space-time? Or could we study the structure of space-time in a crystal model?

Anonymous said...

^^ This is the strategy of lattice field, which simulates quantum field theories on a discretized spacetime lattice, then takes the limit as the lattice spacing goes to 0 go recover the continuum theory. It is currently an open question as to whether a true theory of quantum gravity implies that spacetime is fundamentally discrete and grainy.

Douglas Natelson said...

Anon@12:53, I was thinking of mentioning quasicrystals (see here). I believe in that case the atoms are arranged quasiperiodically, in a projection into 3d of an arrangement that would have discrete translational symmetry in four spatial dimensions. As a non-expert on quasicrystals, I don't know what that means for things like wavefunctions - whether or not there is some variant of the Bloch Theorem for quasiperiodic systems.

Anon@10:30, the answer by Anon@15:26 is right on. Conceivably spacetime could have some underlying lattice structure, and it would be fun to write a sci-fi novel about the Umklapp Drive, where your spacecraft scatters off the fundamental periodicity of the universe and acquires momentum in units of the Planck momentum. A serious challenge of lattice theories is to construct versions that don't break Lorentz invariance (and thus special relativity).