When plane waves scatter off two point scatterers in three dimensions, the amplitude of the scattered waves at position $\vec{r}$ is,

$$F(\vec{r})=\frac{F_1}{R}\cos\left(\vec{k}'\cdot\vec{r}-\omega t + (\vec{k}-\vec{k}')\cdot\vec{r}_1\right)+\frac{F_2}{R}\cos\left(\vec{k}'\cdot\vec{r} -\omega t +(\vec{k}-\vec{k}')\cdot\vec{r}_2\right).$$

If you focus on any one point in the interference pattern, it executes simple harmonic motion at angular frequency $\omega$. Harmonic motion is related to circular motion. This is illustrated in the simulation below.

Your browser does not support the canvas element.

\[ \begin{equation} \large e^{i\omega t} = \cos\omega t + i \sin\omega t \end{equation} \]

The red ball represents the position of the complex number $e^{i\omega t}$ as it moves through the complex plane with real numbers plotted horizontally and imaginary numbers plotted vertically. The black circle is the unit circle. The blue ball represents the position of $\cos \omega t$, and the green ball represents the position of $i\sin \omega t$. The simulation on the left is a graphical representation of Euler's formula on the right.

Oscillations that can be described by $\sin \omega t$ or $\cos \omega t$ are called harmonic oscillations. From the simulation, it is clear that there is a relationship between circular motion and harmonic oscillations. If you look at the motion of the red ball from above, it moves in a circle, but if you look at the motion of the red ball from within the plane of the circle, it executes harmonic motion.

The relationship between circular motion and harmonic oscillations is described easily using complex numbers. Sometimes, when we observe a harmonic oscillation, it is convenient to imagine that we are looking at circular motion. We can't measure the component of the motion in the imaginary direction, we just imagine it.

For our case of the two scatterers, both terms are exhibiting harmonic motion $\cos\left(\vec{k}'\cdot\vec{r}-\omega t + (\vec{k}-\vec{k}')\cdot\vec{r}_1\right)$. We imagine that this is really the real part of circular motion in the complex plane and write it as, $\exp\left(i\left(\vec{k}'\cdot\vec{r}-\omega t + (\vec{k}-\vec{k}')\cdot\vec{r}_1\right)\right)$. Written in complex form, the amplitude of the scattered waves is,

$$F(\vec{r})=\frac{F_1}{R}\exp\left(i\left(\vec{k}'\cdot\vec{r}-\omega t + (\vec{k}-\vec{k}')\cdot\vec{r}_1\right)\right)+\frac{F_2}{R}\exp\left(i\left(\vec{k}'\cdot\vec{r}-\omega t + (\vec{k}-\vec{k}')\cdot\vec{r}_2\right)\right).$$

This equation can be represented graphically in the complex plane. The waves scattered from point $\vec{r}_1$ will result in a harmonic oscillation at point $\vec{r}$ represented by the phasor labeled $F1$. Similarly, the waves scattered from point $\vec{r}_2$ will result in a harmonic oscillation at point $\vec{r}$ represented by the phasor labeled $F2$. The vector sum of these two components is the phasor with the solid red circle. The blue circle is the real part of the sum phasor, which represents the real oscillations observed at point $\vec{r}$.

Your browser does not support the canvas element.

The terms $\vec{k}'\cdot\vec{r}$ just rotate the whole pattern by a fixed phase. It does not change the length of the phasors or the angles between them. Since we are only interested in the length of the phasors, these terms can be ignored. The magic of the complex representation is that $F(\vec{r})$ can be factored into a time-independent part $\mathcal{A}(\vec{r})$ and a time-dependent factor $e^{-i\omega t}$,

$$F(\vec{r})=\left(\frac{F_1}{R}\exp\left(i\left( (\vec{k}-\vec{k}')\cdot\vec{r}_1\right)\right)+\frac{F_2}{R}\exp\left(i\left((\vec{k}-\vec{k}')\cdot\vec{r}_2\right)\right)\right)e^{-i\omega t}=\mathcal{A}(\vec{r})e^{-i\omega t}.$$

The factorization is not possible in the real form with the cosines. The time-independent part is a complex number, and the magnitude of this complex number $|\mathcal{A}(\vec{r})|$ is the amplitude of the oscillations at point $\vec{r}$. The intensity $I$ is proportional to the amplitude squared,

$$I\propto \mathcal{A}^*\mathcal{A}.$$

If there are more scatterers, this generalizes to,

$$\mathcal{A}=\sum\limits_{i=j}^N\frac{F_j}{R}\exp\left(i\left( (\vec{k}-\vec{k}')\cdot\vec{r}_j\right)\right).$$

In elastic x-ray scattering, the amplitude of the scattering from a specific point is proportional to the electron density. The higher the electron density is at a point, the more scattering at that point. Since the electron density is a continuous function, the amplitude of the scattered waves is written as a volume integral over the crystal.

$$\mathcal{A}\propto\int n(\vec{r}')\exp\left(i\left( (\vec{k}-\vec{k}')\cdot\vec{r}'\right)\right)d^3r'.$$

Since the electron density is a periodic function, it can be written as a Fourier series,

$$n(\vec{r}') = \sum\limits_{\vec{G}}n_{\vec{G}}\exp\left(i\vec{G}\cdot\vec{r}'\right).$$

The amplitude at $\vec{r}$ can then be written,

$$\mathcal{A}\propto\int \sum\limits_{\vec{G}}n_{\vec{G}}\exp\left(i\left( (\vec{G}+\vec{k}-\vec{k}')\cdot\vec{r}'\right)\right)d^3r'.$$

As we integrate over the volume of the crystal, the factor $\exp\left(i\left( (\vec{G}+\vec{k}-\vec{k}')\cdot\vec{r}'\right)\right)$ will take on many values as $\vec{r}'$ changes. In the representation in the complex plane, this corresponds to phasors pointing in all directions so that they mostly cancel each other out. However, if $\vec{G}+\vec{k}-\vec{k}'=0$, the exponential factor will be $e^0=1$ and the integral is,

$$\mathcal{A}\propto n_{\vec{G}}V,$$

where $V$ is the volume of the crystal. This results in a sharp peak in the intensity of the measured x-rays at this point. These peaks occur when the diffraction condition,

$$\bbox[10px, border:1px solid black]{\vec{k}'-\vec{k} = \vec{G}}$$

The diffraction condition is also sometimes called the Laue condition and is an important formula in solid-state physics. The momentum of an incident x-ray photon is $\hbar\vec{k}$ and the momentum of a scattered photon is $\hbar\vec{k}$, so the diffraction condition can be seen as a statement of the conservation of momentum in the scattering process, $\hbar\vec{k}'-\hbar\vec{k} = \hbar\vec{G}$. All of the 14 reciprocal lattices have inversion symmetry, so if there is a reciprocal lattice vector at $\vec{G}$, there is also one at $-\vec{G}$. For this reason, the diffraction condition can be written $\vec{k}'-\vec{k} = \vec{G}$, or $\vec{k}-\vec{k}' = \vec{G}$, or $\Delta\vec{k} = \vec{G}$.

In an experiment, an x-ray source is used to generate a beam of x-rays, with a certain wavelength, moving in a certain direction. This specifies $\vec{k}$. Most of the x-rays scatter elastically which means that the wavelength of scattered x-rays is the same as the wavelength of the incident x-rays, $|\vec{k}'|=|\vec{k}|$. The direction of $\vec{k}'$ is the direction from the crystal to the detector. This means that both $\vec{k}$ and $\vec{k}'$ are known in the experiment. Typically, $\vec{k}$ is held fixed because the x-ray source is large and heavy. The orientation of the crystal and the position of the detector are swept through all possible positions to search for diffraction peaks. Every time a peak is encountered, the three components of the reciprocal lattice vector, $G_x = k_x'-k_x$, $G_y = k_y'-k_y$, $G_z = k_z'-k_z$, and the intensity of the peak are recorded. The intensity of the peak is proportional to the Fourier component $|n_{\vec{G}}|^2$. Often, hundreds or thousands of diffraction peaks are measured.

The next step is to try to fit the measured reciprocal lattice vectors to just three primitive reciprocal lattice vectors.

$$\vec{G}_{hkl}= h\vec{b}_1 +k\vec{b}_2 +l\vec{b}_3$$

This is often visualized using an Ewald sphere.