Satellite Reflections from a Displacive Modulation

The effect of modulation, such as has been discussed in incommensurate crystal phases, may also be incorporated into the scattering equation (equation 2.10) in a similar manner to the disorder effects. Only the fluctuations from the basic structure will now possess strong correlations between unit cells. A simple but useful example is that of a single harmonic displacement wave. If the direction of the wave is given by the vector ${\bf q}$, then the displacements, $\Delta{\bf r}$, may be represented as a sinusoidal function with an amplitude ${\bf D}$ (the polarization of the modulation will be encapsulated by this vector). The position of the $n$th atom in the $m$th unit cell, equation 2.12, now becomes

$${\bf R} = {\bf m} + {\bf r}_n + {\bf D}sin({\bf q}.({\bf m} + {\bf r}_n))$$ (4.18)

The substitution of this new expression for R into the scattering equation, making the simplification again of a monatomic lattice, results in

$$I({\bf k}) = \sum_{m}^N \sum_{m'}^{N'} f^2 \exp(i{\bf k}.({\... ...{\bf k}.{\bf D}( sin({\bf q}.{\bf m}) - sin({\bf q}.{\bf m'}))$$ (4.19)

This equation may again be expanded, in a similar way to the case of static disorder (equation 2.14), only now it is complicated by the presence of the $\exp(i z sin\theta)$ term. Using the Jacobi-Anger generating function for the Bessel functions $J_n(z)$, these terms maybe transformed into $\exp(i z sin\theta)=\sum_{n} exp(i n \theta)J_n(z)$, and the resulting intensity expression into a series of terms such that:

$$\begin{eqnarray*} I({\bf k}) = \sum_{m}^{N} \sum_{m'}^{N'} &f^2& J_0({\bf k.D})^... ...p(i({\bf k \pm 2q}).({\bf m - m'})) + .... \hspace{0.5cm}(2.20) \end{eqnarray*}$$

The first of these intensity components describes the fundamental reflections, which are dependent upon the average structure and also the amplitude of the displacement wave through the Bessel function $J_0({\bf k.D})$. The subsequent components, the diffuse intensity, can now be seen to be a function of $({\bf m - m'}).({\bf k \pm q})$ So, in addition to the fundamental reflections at ${\bf k}={\bf G}$ there will be an accompanying series of reflections which exist with ${\bf k}= {\bf G} \pm {\bf q}$, ${\bf k}= {\bf G} \pm 2{\bf q}$, and so on. These are, of course, the satellite reflections. More rigorous derivation of the terms in equation 2.20 are given by [33,34], and discussion of their application in [30,9,32].

In the experiments which will follow, making up the bulk of this thesis, extensive application of the modulation wave approach to interpreting diffuse scattering will be made. A measurement of the position of a satellite allows the modulation vector q to be established. While the measurement of satellite intensity can, in part at least, be related to the amplitude of the modulation. Equation 2.20 shows that, for a single harmonic modulation, the intensity of satellites decrease as their order $m$ increases, in accordance with $J_m^2({\bf k.D})$. Non-harmonic modulation components will be responsible for abnormally increasing the intensities of higher order satellites.