Cominciamo con un cristallo. Avete una cella fondamentale (diciamo un cubo per mantenerlo semplice) e questa cella si ripete SOPRATTUTTO nello spazio di posizione. Questi atomi continuano ad essere impilati. Si può pensare alla cella come all'unità spaziale fondamentale.

Ora, proprio come le funzioni periodiche possono essere decomposte nelle loro armoniche di Fourier, le strutture spaziali periodiche possono essere trasformate di Fourier nello spazio vettoriale delle onde con un fattore di [math]e^{i\vec{k}\dot\vec{r}}[/math]. Per un cristallo con quella struttura periodica ripetuta, la trasformazione di Fourier appare come una cella nello spazio vettoriale d'onda. A causa della meccanica quantistica, il vettore d'onda è proporzionale alla quantità di moto - quindi lo chiamiamo spazio della quantità di moto.

Questo ci porta all'unità fondamentale nello spazio VETTORE D'ONDA: codificare la periodicità in una singola cella. Così, se avete un cristallo cubico, con il suo cubo spaziale fondamentale, la sua trasformata di Fourier è di nuovo un cubo! And just like [math]e^{i\vec{k}\dot\vec{r}}e^{i\vec{k}\dot\vec{r}+2\pi n}=[/math] for any integer n, the edges of the Brillouin zone (this fundamental unit of the crystal in wave vector space) will repeat upon itself.

So let's recap:

• Brillouin zones encode the spatial periodicity of a structure (think CRYSTAL) in a fundamental unit in wave vector space / momentum space.
• Since energy usually depends on momentum, Brillouin zones serve as the fundamental unit of energy levels within the crystal. Due to periodicity, this is the formation of energy bands.

Also answered at Solid State Physics: What are Brillouin zones?
A Brillouin zone can be thought of as the Fourier Transform of the minimum unit cell of a periodic structure. It's an over-simplified view, but it gives it a fair amount of intuition.

In any periodic structure in which you want to solve a wave equation (say, Schrodinger's equation for an electron in a crystal, or a photon in a Bragg grating), then you want to discover the relationship of frequency to wavenumber. For an electron, frequency and wavenumber multiplied by h-bar are energy and momentum, respectively.

The most common approach to solving it for w-k relationship is to use Bloch waves. Bloch's theorem for waves in a periodic potential states that any wave travelling through a periodic potential can be expressed as the product of a periodic function and a plane wave.

This periodicity makes it so k is not unique, but rather every k is equivalent to k+ 2pi/a, where a is the period of the structure. Because of this, the solution wraps around the period boundary.

the same function centered at zero is replicated at k = 2pi/a and -2pi/a and so on. You can think of it instead just wrapping around the boundary.

This is where bands come from (but not necessarily band gaps). For every kvector describing a solution, there are multiple frequencies that can have that kvector because the relationship is periodic (i.e. k = 0 is equivalent to k = 2pi/a, where a is the lattice distance, or the equivalent point one unit cell over).

The Brillouin zone for a fiber Bragg grating is really simple since it is periodic in only one dimension.

Much like the lattice has a unit cell (its minimum unit cell is referred to as theWigner–Seitz cell), the periodic omega-k relationship also has a unit cell since it is periodic. That minimum cell is the Brillouin zone.

Explaining this in full detail is about a 4-week discussion in an undergraduate solid state physics class,