X-ray scattering

X-ray scattering is one of the most widely used method for solving crystal structures. The basic technique was developed by W. L. Bragg and W. H. Bragg in 1914. Recall that X-rays can promote electrons to excited states. When the excited electrons decay back down to the ground state, they emit photons of the same frequency. This is the phenomenon of X-ray scattering. If two neighboring atoms emit photons, these photons will be quantized spherical waves, which can add constructively if they are in phase (an integer number of wavelengths apart) or destructively if they are 180$^{\circ}$ out of phase. Now, we apply this idea to two layers of atoms in the crystal. We use X-ray scattering to determine the distance between the layers. Referring to the figure below, the angle

Figure: Illustration of Bragg (X-ray) scattering from planes of atoms in a crystal

between the direction of propagation of the X-ray and the crystal plane is $\theta$. $\theta$ is also the angle shown between the line joining the two planes, and the segment that shows the path difference between the X-ray that scatters off the top layer and the one that scatters off the second layer. If $l$ is the path difference between the two incoming X-rays, then we have the condition $l=d\sin\theta$, hence the total path difference is $2l = 2d\sin\theta$. If $2l$ is an integer number of wavelengths $n\lambda$ apart, then the condition for constructive interference and the appearnce of a bright spot in the interference pattern is

$$2d\sin\theta = n\lambda$$

so that the distance $d$ between the layers is

$$d = {n\lambda \over 2\sin\theta}$$

We can tell the value of $n$ from the interference pattern. The brighest spot at the center of the pattern will correspond to $n=1$. Spots that fan out from there are due to X-rays that scatter off more distant layers, e.g. $n=2$ are from layers a distance $2d$ apart, $n=3$ are from layers that are $3d$ apart, etc.