22.1 CRYSTAL MONOCHROMATORS

For X-ray energies higher than 2 keV or so, gratings become extremely inefficient, and it becomes necessary to utilize the periodicity naturally occuring in a crystal to provide the dispersion. Since the periodicity in a crystal is 3-dimensional, the normal single grating equation must be replaced by the three grating equations, one for each dimension, called the Laue equations,1 as follows:

where the a s are the repeat vectors in the three dimensions, the k s are the wave vectors for the incident and scattered beams, and the H s are integers denoting the diffraction order in the three dimensions. All of them must be simultaneously satisfied in order to have an interference maximum (commonly called a Bragg reflection). One can combine these equations into the well-known Bragg s law2 for one component of the crystalline periodicity (usually referred to as a set of Bragg planes),

where n is an integer indicating the order of diffraction from planes of spacing d, and ? is the angle between the incident beam and the Bragg planes. This equation is the basis for using crystals as X-ray monochromators. By choosing one such set of Bragg planes and setting the crystal so that the incident X rays fall on these planes, the wavelength of the light reflected depends on the angle of incidence of the light. Table 1 shows some commonly used crystals and the spacings...