Fractional coordinates

Fractional coordinates are numbers (between 0 and 1) that indicate the positions of lattice points in a unit cell.

The edges of a unit cell, which may be primitive or non-primitive, are defined by the basis vectors a, b and c, and any lattice point within the unit cell is described by the position vector


where x’, y’, z‘ are fractions with 0 ≤ x’ ≤ 10 ≤ y’ ≤ 10 ≤ z’ ≤ 1. Therefore, (x’, y’, z’) are called fractional coordinates (see diagram below).

With the different combinations of the positions of lattice points and angles between primitive translation vectors, one may think that there are possibly an infinite number of types of unit cell. However, the number is finite as explained in the next section.


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