**First Brillouin Zone**

**Introduction**

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Before explaining what a Brillouin zone is, let’s make some definitions

- An infinite array of discrete points generated by a set of discrete translation operations described by*Bravais lattice*

where *n _{i}* are integers, and

- Cell that contains only a single lattice point(whole point, or two halves and so on), such that the whole lattice could be constructed by positioning the cell’s clones, which are distant by a lattice vector from each other.*Primitive unit cell*

- Primitive unit cell, which is built around a certain lattice point, and contains all space which is closer to this point than any other lattice point.*Wigner Seitz cell*

- Set of all points, described by vector*Reciprocal lattice (of a Bravais lattice)**K*, such that

where R is a vector on the Bravais lattice and n is an integer.

Note that a Reciprocal lattice of a Bravais lattice is also a Bravais lattice.

Now we finally can define a First Brillouin zone

- Wigner Seitz cell of a Reciprocal lattice.*First Brillouin zone*

**Software**

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**First Brillouin zone calculation method for Reciprocal lattice point P****-**

- Calculate Reciprocal lattice primitive vectors.
- Find the mid point, M, between P and all other lattice points.
- Calculate a plane which is located at M and perpendicular to P-M.
- First Brillouin zone is all points in space which could be reached by not crossing any plane calculated in 3.

My program generates first Brillouin zones, using steps 2-4 of this algorithm.

Link to program explanations can be found __here__.

This program also calculates Fermi surfaces.

Explanations on Fermi surfaces can be found __here__.

Other programs and explanations, prepared in the Physics Department of the Technion can be found __here__