Let us consider a plane monochromatic wave incident on a crystal and let
**k**_{o} = s be its wave vector. Each scatterer will diffuse
this wave in every direction with the same wavelength (coherent scattering).
The total amplitude scattered in a particular direction **s**_{h} will be
obtained by summing the amplitudes scattered in this particular direction by all
scatterers, taking into account their phase relations. Let *A* and *B* be two
homologous points in the structure, that is **AB** = **r** is a direct
lattice vector. The phase differences between the waves scattered by *A* and
*B* is equal to:

(5.1) |

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:

(5.2) |

The modulus of the diffusion vector has the dimension of the reciprocal of a
length. **R** can therefore be expanded in reciprocal space:

The position vector **r** can in the same way be expressed in terms of its
coordinates *u*, *v*, *w* in direct space. Applying relations (2.3), we may
therefore write the phase difference in the following way:

(5.3) |

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

where*A reciprocal lattice node may thus be associated with each Bragg
reflection* .

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:

(5.4) |

(5.5) |

**Copyright © 1981, 1998 International Union of
Crystallography**