3.5.1. Diffraction of X-rays

Wavelength-dispersive spectrometers are used to select the X-ray of interest for analysis. This selection is made by Rayleigh scattering of the X-rays from a systematic crystal located between the sample and X-ray detector. By changing the angle of incidence, an analytical crystal can be made to constructively diffract X-rays of different wavelengths. The resulting X-rays are counted using X-ray detectors that must be moved to accommodate the changing incident angles on the crystal.

 

Like all electromagnetic radiation, X-rays have dual characteristics acting as particles (photons) and waves simultaneously. Photons have discrete energies and their corpuscular (particle) characteristics produce the phenomena of ionization, scattering, and visible fluorescence in some materials. The wave characteristics produce the diffractive properties of X-rays.

 

The diffraction of X-rays by a systematic crystal was first confirmed by Max von Laue, a junior colleague of Rontgen, and his associates W. Friedrich and E.P. Knipping at the University of Munich in 1912. They showed that a diffracting crystal intercepts X-rays of all wavelengths, but only those that undergo constructive interference are transmitted efficiently to the detector (Figure 3.5.1a). Von Laue got the Nobel Prize in 1914.

Max von Laue (1879-1960)

 

Figure 3.5.1a. A representation of first-order X-ray diffraction. The path-length difference between X-rays diffracted from adjacent atomic layers corresponds to an integral number of wavelengths.

 

Von Laue sent a copy of the paper reporting the results to William Henry Bragg, whose son, William Lawrence Bragg confirmed that X-rays produced ionization and also could be diffracted by a regular crystal (the wave-particle duality),  extending von Laue's work. In 1913, they built the first X-ray spectrometer and observed the first X-ray spectrum, examining the L lines produced from platinum using an NaCl crystal. TThe older Bragg developed an X-ray detector that when coupled with the younger Bragg’s diffracting crystal is the basis of all X-ray spectrometry. The Braggs received the Nobel Prize in 1915 for their work.

 

W. H Bragg (1862-1942) and W. L. Bragg (1890-1971) on Swedish postage stamp

 

The conditions necessary for constructive diffraction of X-rays (and light) by a crystal are described by Bragg's Law (Figure 3.5.1b):

Figure 3.5.1b. Scattering and diffraction of a coherent beam of X-rays from parallel planes of atoms, spaced a distance d apart. The diffraction condition is again that of coherence of the diffracted beam. The path lengths of the two rays shown differ by the distance CBD, which must therefore be equal to a whole (integral) number of wavelengths (nl) for the rays scattered at the angle f to be in phase with each other (after Williams 1987).

X-rays that are in-phase (those with wavelengths of 1l, 2l, 3l...) produce constructive interference, while out-of-phase waves (those with wavelengths like 4/3l, 3/2l, 5/2l, etc.) produce destructive interference. Note that wavelengths of l/2, l/3, etc. are also diffracted constructively. These are termed high-order wavelengths, because n must be a number higher than 1 for constructive diffraction to occur.

One may use monochromatic X-rays to determine the d-spacing of an unknown crystal as in X-ray diffraction (XRD), or, conversely, use a crystal of known d-spacing to produce monochromatic X-rays and reject all except a specific wavelength. Crystals are not perfect, so the resulting X-rays are not perfectly monochromatic either. An excellent web site covering Bragg's Law and X-ray diffraction is http://www.journey.sunysb.edu/ProjectJava/Bragg/home.html.