Waves in crystals

Wave in crystals

A crystal can be considered to be a periodic structure built up by the regular repetition of a particular unit cell. To a very large extent solid state physics is concerned with the manner in which waves are propagated through such periodic structures. Under what conditions can the waves travel through the crystal without being scattered? Or if there is scattering in what direction will it occur? The waves may be of several different types – they may be x-rays or matter waves associated with the electrons in the material, or they may be lattice vibrational waves (phonons). Whatever type of wave we consider, the general principles involved are the same.

The Bragg construction

The Bragg construction is used to determine the angles at which Xray diffraction lines or spots should be observed. In this construction it is assumed (with no physical justification_ that he X rays which enter a crystal at an angle Q to a certain set of parallel planes are reflected at successive planes. For any arbitrary angle Q, however, the emergent rays which have been reflected by successive planes will not be in phase with one another and on average they will cancel out. In fact, there will only be a diffracted beam if the rays which have been reflected by each plane emerge in phase with one another.

To observe a diffracted beam at an angle Q we must have

2d sin Q = nλ

Where n is an integer called the order of the diffraction and λ is the wavelength.

....the wave vector k is defined as 2π/λ


If two waves with vectors k1 and k2 combine, then the resultant wave vector is k1+k2.

the general principle which is used to calculate the diffraction of all types of waves in crystals is that the amplitude of the waves which ar escattered or re-emitted by the individual atoms may be superimposed.

in a large perfect crystal the algebraic sum of these amplitudes in any arbitarary direction will be zero, and only in very special direcitons (those where the Bragg conditions hold) will there be a non zero amplitude, ie. there wil be a diffracted beam.......the larger the the crystal the more sharply will any diffracted beam be defined, since the destructive interference between the atoms will be more efficient, even for small deviations from the Bragg angles.

If, however, there are any irregularities in the crystal structure then the destructive interference will not be complete. The effect of the displaced atoms will be to give a non zero amplitude even in directions which do not precisely satisfy the Bragg condition.