Let to be the primitive vectors of the crystal lattice , whose atoms are located at the points that are integer linear combinations of the primitive vectors.
Let be the wavevector of the incoming (incident) beam, and let be the wavevector of the outgoing (diffracted) beam. Then the vector is called the scattering vector (also called transferred wavevector) and measures the change between the two wavevectors.
The three conditions that the scattering vector must satisfy, called the Laue equations, are the following: the numbers determined by the equations
must be integer numbers. Each choice of the integers , called Miller indices, determines a scattering vector . Hence there are infinitely many scattering vectors that satisfy the Laue equations. They form a lattice , called the reciprocal lattice of the crystal lattice. This condition allows a single incident beam to be diffracted in infinitely many directions. However, the beams that correspond to high Miller indices are very weak and can't be observed. Anyway, it is enough to find a basis of the reciprocal lattice, from which the crystal lattice can be determined. This is the principle of x-ray crystallography.
The incident and diffracted beams are planar wave excitations
of a field that for simplicity we take as scalar, even though the main case of interest is the electromagnetic field, which is vectorial.
The two waves propagate through space independently, except at the points of the lattice, where they resonate with the oscillators, so their phase must coincide. Hence for each point of the lattice we have
or equivalently, we must have
for some integer , that depends on the point . Simplifying we get
Now, it is enough to check that this condition is satisfied at the primitive vectors (which is exactly what the Laue equations say), because then for the other points we have
where is the integer .
This ensures that if the Laue equations are satisfied, then the incoming and outgoing wave have the same phase at all points of the crystal lattice, so the oscillation of the atoms, that follows the incoming wave, can at the same time generate the outgoing wave.
Kittel, C. (1976). Introduction to Solid State Physics, New York: John Wiley & Sons. ISBN 0-471-49024-5
^More realistically, the oscillators of the lattice should lag behind the incoming wave, and the outcoming wave should lag behind the oscillator. But since the lag is the same at all point of the lattice, the only effect of this correction would be global shift of phase of the outcoming wave, which we are not taking into consideration.