X-ray Bragg Diffraction

Shine X-rays of wavelength comparable to the atomic spacing at a crystal, and most angles give nothing, but at a few sharp angles a strong reflected beam appears. Bragg explained it with a picture so simple it earned a Nobel Prize: treat the crystal as a stack of parallel planes of atoms, each reflecting a little of the wave. A ray that reflects off the second plane travels an extra distance compared to one reflecting off the first, and a short construction shows that extra path is exactly $2d\sin\theta$, where $d$ is the plane spacing and $\theta$ the glancing angle. When that path difference is a whole number of wavelengths, the reflections from every plane arrive in step and add up; when it is not, they fall out of step and cancel. That is the Bragg condition, $n\lambda = 2d\sin\theta$. The scene draws two of those planes with the path-difference segments highlighted, and the reflected beam brightens as the angle passes through a value where the waves line up. The lower panel plots the reflected intensity against angle: a flat dark background broken by sharp peaks at the Bragg angles, labelled by their order $n$. Sweep the angle to walk through the peaks, or change the spacing and wavelength and watch the whole pattern shift, the inverse relationship that lets a diffraction pattern be read back as a map of the crystal.