these the critical reflection angle is energy dependent. For gold at 1 keV, the critical reflection angle is 2.4°. The use of X-ray mirrors simultaneously requires: the ability to determine the location of the arrival of an X-ray photon in two dimensions, a reasonable detection efficiency. Multilayers for X-Rays No material has substantial reflection for X-rays, except at very small grazing angles. Multilayers enhance the small reflectivity from a single boundary by adding the small reflected amplitudes from many boundaries coherently in phase. For example, if a single boundary has a reflectivity of R = 10−4 (amplitude r = 10−2), then the addition of 100 amplitudes from 100 boundaries can give reflectivity R close to one. The period Λ of the multilayer that provides the in-phase addition is that of the standing wave produced by the input and output beam, Λ = λ/2 sin θ, where λ is the wavelength, and 2θ the half angle between the two beams. For θ = 90°, or reflection at normal incidence, the period of the multilayer is Λ = λ/2. The shortest period that can be used in a multilayer is limited by the size of the atoms to about 2 nm, corresponding to wavelengths above 4 nm. For shorter wavelength a reduction of the incidence angle θ toward more grazing has to be used. The materials for multilayers are selected to give the highest possible reflection at each boundary and the smallest absorption or the propagation through the structure. This is usually achieved by light, low-density materials for the spacer layer and a heavier material that produces high contrast. The absorption in the heavier material can be reduced by positioning it close to the nodes of the standing-wave field inside the structure. Good low-absorption spacer materials are Be, C, B, B4C and Si. Some examples of the heavier materials with good contrast are W, Rh, Ru and Mo. Applications include: normal and grazing-incidence optics for telescopes from EUV to hard X-rays, microscopes, beam lines at synchrotron and FEL facilities, EUV lithography. Mo/Si is the material selection used for the near-normal incidence reflectors for EUV lithography. Hard X-ray mirrors An X-ray mirror optic for NuStar space telescope working up 79 keV was made using multilayered coatings, computer-aided manufacturing, and other techniques. The mirrors use a tungsten/silicon (W/Si) or platinum/silicon-carbide (Pt/SiC) multicoating on slumped glass, allowing a Wolter telescope design. See also Kirkpatrick–Baez mirror