specular direction (reflected angle equal to incident angle). It has been shown that a reflection off a parabolic mirror followed by a reflection off a hyperbolic mirror leads to the focusing of X-rays. Since the incoming X-rays must strike the tilted surface of the mirror, the collecting area is small. It can, however, be increased by nesting arrangements of mirrors inside each other. The ratio of reflected intensity to incident intensity is the X-ray reflectivity for the surface. If the interface is not perfectly sharp and smooth, the reflected intensity will deviate from that predicted by the Fresnel reflectivity law. The deviations can then be analyzed to obtain the density profile of the interface normal to the surface. For films with multiple layers, X-ray reflectivity may show oscillations with wavelength, analogous to the Fabry–Pérot effect. These oscillations can be used to infer layer thicknesses and other properties. Diffraction In X-ray diffraction a beam strikes a crystal and diffracts into many specific directions. The angles and intensities of the diffracted beams indicate a three-dimensional density of electrons within the crystal. X-rays produce a diffraction pattern because their wavelength typically has the same order of magnitude (0.1–10.0 nm) as the spacing between the atomic planes in the crystal. Each atom re-radiates a small portion of an incoming beam's intensity as a spherical wave. If the atoms are arranged symmetrically (as is found in a crystal) with a separation d, these spherical waves will be in phase (add constructively) only in directions where their path-length difference 2d sin θ is equal to an integer multiple of the wavelength λ. The incoming beam therefore appears to have been deflected by an angle 2θ, producing a reflection spot in the diffraction pattern. X-ray diffraction is a form of elastic scattering in the forward direction; the outgoing X-rays have the same energy, and thus the same wavelength, as the incoming X-rays, only with altered direction. By contrast, inelastic scattering occurs when energy is transferred from the incoming X-ray to an inner-shell electron, exciting it to a higher energy level. Such inelastic scattering reduces the energy (or increases the wavelength) of the outgoing beam. Inelastic scattering is useful for probing such electron excitation, but not in determining the distribution of atoms within the crystal. Longer-wavelength photons (such as ultraviolet radiation) would not have sufficient resolution to determine the atomic positions. At the other extreme, shorter-wavelength photons such