12.1.11: Interference of X-Ray Waves

Last updated
Save as PDF

Page ID: 18376

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

12.9.jpg — Figure 12.9: Wave interference

Just like visible light, X-rays propagate in all directions and may interact, or interfere, with each other (Figure 12.9). Two parallel X-rays of the same wavelength can exhibit constructive interference or destructive interference. If constructive, it means that the waves are in phase – that their wave peaks coincide. In such a case, wave energies add, increasing X-ray amplitude. Destructive interference occurs when waves are out of phase – one is moving up while the other is moving down. When destructive interference occurs, waves can appear to cancel. Often, waves are partially in phase and partially out of phase.

12.10.png — Figure 12.10: Interference of waves from two atoms

Figure 12.10 shows two atoms emitting monochromatic X-radiation. The solid lines show wavefronts moving away from each atom. If we could move an X-ray detector around the atoms, we would find that energy is intense in some directions (where wave peaks coincide and waves interact constructively). In other directions, the detector would register no X-rays due to destructive interference. This channeling of energy in specific directions is diffraction. The directions of diffraction depend on X-ray wavelength and distance between the two X-ray sources.

A narrow slit, a series of parallel grooves in a diffraction grating, regularly spaced atoms in a crystal, and many other things can cause diffraction. The main requirement is that two or more sources emit, or scatter, monochromatic waves.

In principle, all electromagnetic radiation can be diffracted, but unless the spacing of atoms, slits, or gratings is similar to the wavelength of the radiation, diffraction will not occur. Fortunately for mineralogists, atoms have radii on the order of angstroms. Because atoms pack closely together in crystals, space between atoms is of the same magnitude. So, atomic radii and spacing are about the same dimension as the wavelength of X-rays. Consequently, X-rays interacting with atoms in crystals can lead to intense X-ray diffraction. This is the understanding that led von Laue and his coworkers to their successful experiments.