While there is no shortage of excellent discussions about Bragg’s Law and the production of monochromatic x-rays, we present the following technical overview to explain the capability inherent in using various multilayer optics to create a range of monochromatic x-ray sources, and to illustrate the range of opportunities that exist for various applications.
Bragg's Law refers to the simple equation: nλ = 2d sinθ derived by the English physicists Sir W.H. Bragg and his son Sir W.L. Bragg in 1913 to explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence (theta, θ).
 The variable d is the distance between atomic layers in a crystal, and the variable lambda λ is the wavelength of the incident X-ray beam; n is an integer. This observation is an example of X-ray wave interference (Roentgen strahl interferenzen), commonly known as X-ray diffraction (XRD), and was direct evidence for the periodic atomic structure of crystals postulated for several centuries.
How graded multilayer coatings produce monochromatic x-rays
It is known that x-rays are diffracted by atoms. If the atoms concerned all lie in a single plane, then the diffracted rays reinforce one another. This phenomenon is widely used in the study of crystal structures by x-ray diffraction. X-rays can also be diffracted using thin films of alternating layers of high- and low-density elements, where the spacing between the layers is of the order of Angstroms. The effect of diffraction of x-rays at a multi-layer coating is shown below in Figure 1.
Figure 1. X-Ray Diffraction from Multiple Surfaces of a Multi-Layer Coating
X-rays readily penetrate the layers of the multi-layer coating. Strong reflections will only be made for x-rays that satisfy the Bragg Law:
nλ = 2d sin θ
Where λ is the x-ray wavelength, d is the interlayer spacing and θ is the grazing angle. The Bragg Law shows that the intensity of x-rays reflected by a multilayer coating is wavelength-selective. By selecting the spacing d and the grazing angle θ, only x-rays with wavelengths λ, 2λ, 3λ, 4λ, etc. will be reflected.
The coating shown above will produce monochromatic x-ray radiation only when all the incident beam rays have the same angle of incidence. Unfortunately this is not always the case.
Commonly, an x-ray tube is the source of x-rays (especially in imaging applications), which produces a divergent beam. This case is illustrated in the figure below. In this case, monochromatic x-rays can only be achieved using a graded multilayer coating, where the bi-layer spacing varies along the length of the filter such that the term d.sin θ (from the Braggs equation) remains constant. This is our area of focus and describes the behavior of our sources.
Figure2. Monochromatic x-rays from a point source with graded multilayer coating.
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