A defining characteristic of crystalline solids is that their constituent atoms are arranged in a spatially periodic way. In fancy lingo, the atomic configuration breaks continuous translational and rotational invariance (that is, it picks out certain positions and orientations in space from an infinite variety of possible choices), but preserves discrete translational invariance (and other possible symmetries).

The introduction of a characteristic spatial length scale, or equivalently a spatial frequency, is a big deal, because when other spatial length scales in the physical system coincide with that one, there can be big consequences. For example, when the wavelength of x-rays or electrons or neutrons is some integer harmonic of the (projected) lattice spacing, then waves scattered from subsequent (or every second or every third, etc.) plane of atoms will interfere constructively - this is called the Bragg condition, is what gives diffraction patterns that have proven so useful in characterizing material structures. Another way to think about this: The spatial periodicity of the lattice is what forces the momentum of scattered x-rays (or electrons or neutrons) to change only by specified amounts.

It gets better. When the wavelength of electrons bound in a crystalline solid corresponds to some integer multiple of the lattice spacing, this implies that the electrons strongly "feel" any interaction with the lattice atoms - in the nearly-free-electron picture, this matching of spatial frequencies is what opens up band gaps at particular wavevectors (and hence energies). Similar physics happens with lattice vibrations. Similar physics happens when we consider electromagnetic waves in spatially periodic dielectrics. Similar physics happens when looking at electrons in a "superlattice" made by layering different semiconductors or a periodic modulation of surface relief.

One other important point. The idea of a true spatial periodicity really only applies to infinitely large periodic systems. If discrete translational invariance is broken (by a defect, or an interface), then some of the rules "enforced" by the periodicity can be evaded. For example, momentum changes forbidden for elastic scattering in a perfect infinite crystal can take place at some rate at interfaces or in defective crystals. Similarly, the optical selection rules that must be rigidly applied in perfect crystals can be bent a bit in nanocrystals, where lattice periodicity is not infinite.

Commensurate spatial periodicities between wave-like entities and lattices are responsible for electronic and optical bandgaps, phonon dispersion relations, x-ray/electron/neutron crystallography, (crystal) momentum conservation and its violation in defective and nanoscale structures, and optical selection rules and their violations in crystalline solids. Rather far reaching consequences!

## 2 comments:

And incommensurate spatial periodicities between individually periodic entities carry their own 'hydrodynamic modes', the phasons, with them.

what about "periodicity" because periodicity in time also often occurs (such as lately the "time crystals", but also e.g. surface adsorbate phases in heterogeneous catalysis, see e.g. Ertl).

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