There will be diffraction of the incident wave by the crystal if the wavelets diffracted by all homologous points are in phase, that is if is equal to an integer times 2 whatever the direct lattice vector r. The phase may also be written:
The modulus of the diffusion vector has the dimension of the reciprocal of a length. R can therefore be expanded in reciprocal space:
We may note that u, v, w being the coordinates of a direct lattice vector are integers. If is to be equal to an integer times 2 whatever u, v, w, we conclude that h, k, l are necessarily also equal to integers; in other words, the diffusion vector is a reciprocal lattice vector . This is the diffraction condition in reciprocal space. Bragg`s law and the Ewald sphere construction are easily deduced from this result.
Let O be the origin of the reciprocal lattice and IO and IH vectors respectively equal to s and s.The vector OH is therefore equal to R (Fig. 7). If the diffraction condition is satisfied, H is a reciprocal lattice node. We have therefore the following construction: we draw through O a line parallel to the incident direction, let , then draw a sphere centered in I with radius 1/. If it passes through another reciprocal lattice node H, there is a reflected beam parallel to IH.
We may notice in the triangle IOH that OH/2 = IH sin ,calling the angle between IO or IH with the bissectrix of OIH, that is with the trace of the set of direct lattice planes associated with the node H.
We know from (2.8) that
This result can also be obtained directly through the properties of Fourier transforms. The basic assumption of the geometrical theory of diffraction is that the amplitude of the incident wave at each scatterer is constant. This assumption is acceptable if the interaction between the incident wave and the scatterers is small enough. The total diffracted amplitude in a given direction is therefore simply equal to the sum of the amplitudes scattered in this direction by every scatterer, taking into account their phase relationships. It is equal to: