Fourier transform of almost-periodic lattice?

For a perfectly periodic lattice of atoms, the density can be written as a Fourier series like any other periodic function. This Fourier representation is very useful, because it can be connected to peaks you can measure with diffraction.

Mathematically this is written as: $$\rho(\mathbf{r})=\sum_j \rho(\mathbf{r}-\mathbf{r}_j)=\sum_{\mathbf{G}} \rho_{\mathbf{G}} e^{i \mathbf{G}\cdot \mathbf{r}}$$

Where $$\mathbf{G}$$ are the reciprocal lattice vectors. In ordinary diffraction you measure diffraction peaks at the reciprocal lattice vectors with intensity $$\vert \rho_{\mathbf{G}} \vert^2$$. Coherent diffraction can further get the phase of these coefficients.

Now let's assume that the crystal has strain fields, causing atoms to deviate from their perfect crystal position by $$\delta \mathbf{r}_j$$. Now we cannot write the charge density as a simple Fourier series because it is no longer periodic.

$$\rho(\mathbf{r})=\sum_j \rho(\mathbf{r}-\mathbf{r}_j - \delta \mathbf{r}_j ) \neq\sum_{\mathbf{G}} \rho_{\mathbf{G}} e^{i \mathbf{G}\cdot \mathbf{r}}$$

My question is: is there a useful Fourier-like series of the density when strain is introduced?. At first thought I would assume $$\delta r_j \ll r_j$$ and do a Taylor series in $$\delta r_j/r_j$$ but in the case of a dislocation (very common type of strain) they become the same order of magnitude so that isn't quite right.

As an example, in the literature on strain field imaging with electron microscopy, a Fourier series-like form is suggested.

$$\rho(\mathbf{r})=\sum_{\mathbf{G}} \rho_{\mathbf{G}}(\mathbf{r}) e^{i \mathbf{G}\cdot \mathbf{r}}$$

Where in the paper above $$\rho_{\mathbf{G}}(\mathbf{r})\equiv A_{\mathbf{G}}(\mathbf{r}) e^{i \mathbf{G}\cdot \delta\mathbf{r}}$$ treating $$\mathbf{r}$$ and $$\delta \mathbf{r}$$ as continuous variables. This form for $$\rho(\mathbf{r})$$ looks conceptually appealing, but as far as I can tell it is ad-hoc, and non-rigorous in general. Any insights here?

• Related: Debye--Waller factor. – knzhou Dec 22 '19 at 23:42
• @knzhou thank you. The Debye Waller effect is a great example of dealing with the averaged behavior of a disordered lattice. Maybe there is a similar way to get a particular instance of disorder to extract its strain field. I'll have to think about this some more – KF Gauss Dec 23 '19 at 2:55
• You could add a modulation whose period is incommensurate with the lattice spacing. That would displace the minimum location by a "random" amount for each lattice site... – SuperCiocia Dec 28 '19 at 0:26