Let  be primitive translation vectors (shortly called primitive vectors) of a crystal lattice , where atoms are located at lattice points described by  with , , and  as any integers. (So  indicating each lattice point is an integer linear combination of the primitive vectors.)
Let  be the wave vector of an incoming (incident) beam or wave toward the crystal lattice , and let  be the wave vector of an outgoing (diffracted) beam or wave from . Then the vector , called the scattering vector or transferred wave vector, measures the difference between the incoming and outgoing wave vectors.
The three conditions that the scattering vector  must satisfy, called the Laue equations, are the following:
where numbers  are integer numbers. Each choice of integers , called Miller indices, determines a scattering vector . Hence there are infinitely many scattering vectors that satisfy the Laue equations as there are infinitely many choices of Miller indices . Allowed scattering vectors  form a lattice , called the reciprocal lattice of the crystal lattice , as each  indicates a point of . (This is the meaning of the Laue equations as shown below.) This condition allows a single incident beam to be diffracted in infinitely many directions. However, the beams corresponding to high Miller indices are very weak and can't be observed. These equations are enough to find a basis of the reciprocal lattice (since each observed  indicates a point of the reciprocal lattice of the crystal under the measurement), from which the crystal lattice can be determined. This is the principle of x-ray crystallography.