What is a Brillouin Zone?
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This is kind of hard to ELI5, especially without knowing the level of background you have
Basically, if you have a crystal, that crystal has some unit cell, the most basic block to build up the crystal lattice. For a cubic crystal, that crystal is going to be built up by a bunch of cubes. A hexagonal crystal will be built up from a bunch of repeating hexagons.
You can take the Fourier Transform of that shape, and you’ll get another lattice (physics-wise, you’re going from real space, to momentum space), which is also made up of a (different) unit cell. The Fourier Transform of a periodic structure gives you a periodic structure back.
For example, the Fourier transform of a simple cubic lattice is just a simple cubic lattice. The transform of a body-centered cubic (BCC) is a face-centered cubic (FCC).
You can think of it kind of like a function. FT(BCC)-> FCC. If you input a BCC into your function, you get out an FCC. The structure of the crystal in real space is going to have an impact on the properties of the Fourier transformed lattice.
Physics-wise, this would be like performing a diffraction experiment. If you do diffraction on a BCC crystal, your diffracted beam (could be photons, as in x-ray diffraction, or elections, in electron difraction, etc), will have an FCC structure to it.
Which will lead to restrictions in things such as how stuff diffracts. This is commonly used in e.g. crystal diffraction. Diffraction experiment results give you a map of the reciprocal lattice, so you can invert that to figure out the crystal structure in real space.
Mathematically, this generalizes to lattice theory (which shows up in geometry, group theory etc), and duals. It’s very analogous to dual vector spaces.
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