Diffraction Condition in the Reciprocal Lattice

Let us consider a plane monochromatic wave incident on a crystal and let k_o = s $_o/\lambda$ 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:

$\begin{displaymath} \phi = 2\pi \frac{(\textbf{s}_h - \textbf{s}_o)}{\lambda}\cdot \textbf{r}\end{displaymath}$

(5.1)

(s_h and s_o are unit vectors in the reflected and incident directions, respectively).
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 $\phi$ is equal to an integer times 2 $\pi$ whatever the direct lattice vector r. The phase $\phi$ may also be written:

$\begin{displaymath} \phi = 2 \pi \textbf{R} \cdot \textbf{r}\end{displaymath}$

(5.2)

where R = (s_h - s_o)/ $\lambda$ is the so-called diffusion vector.
The modulus of the diffusion vector has the dimension of the reciprocal of a length. R can therefore be expanded in reciprocal space:

$\begin{displaymath} \textbf{R} = h\textbf{a*} + k\textbf{b*} + l\textbf{c*}\end{displaymath}$

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 $\phi$ in the following way:

$\begin{displaymath} \phi = 2 \pi (hu + kv + lw)\end{displaymath}$

(5.3)

We may note that u, v, w being the coordinates of a direct lattice vector are integers. If $\phi$ is to be equal to an integer times 2 $\pi$ 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 $_o/\lambda$ and s $_h/\lambda$ .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 $IO = 1/\lambda$ , then draw a sphere centered in I with radius 1/ $\lambda$ . 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 $\times$ sin $\theta$ ,calling $\theta$ 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

$\begin{displaymath} OH = \frac{n}{d}\end{displaymath}$

where d is the lattice spacing of the direct lattice planes and n the order of H along the reciprocal lattice row OH. We find thus that:

$\begin{displaymath} \frac{n}{2d} = \frac{\sin \theta}{\lambda}\end{displaymath}$

which is of course Bragg`s law. 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:

$\begin{displaymath} A = A_e \int\int\int \rho(\textbf{r})e^{-2{\pi}i\textbf{R}{\cdot}r}d{\tau}\end{displaymath}$

(5.4)

using (5.1) and (5.2). A_e is the amplitude diffracted by one scatterer and $\rho$ (r) the density of scatterers electrons if we consider X-ray diffraction for instance. The integral is extended over the volume of the crystal. We shall assume it here to be infinite. Expression (5.4) shows that the distribution of diffracted amplitudes is the Fourier transform of the electron density $\rho$ (r). If the diffracting medium is crystalline, it is triply periodic. The Fourier transform of $\rho$ (r) is then a distribution of Dirac masses at each reciprocal lattice node. The weight associated with each one of them is equal to the structure factor:

$\begin{displaymath} \begin{array} {rcl} F_{hkl} = & A_e \displaystyle \int\!\!\!... ...xtbf{R}{\cdot}\textbf{r}}dc \\ & \hbox{unit cell} &\end{array}\end{displaymath}$