This invention describes a general class of new focusing crystal surfaces for the x-ray Bragg spectroscopy of high-density plasmas and small (point-like) x-ray sources that makes it possible to maximize the photon throughput of a Bragg crystal spectrometer for a specified spectral range. These surfaces are the proper three-dimensional expansions of two-dimensional curves, which are solutions of a general differential equation and which are defined by the value of a single parameter. Before this invention, a few standard crystal forms, such as flat crystals, spherically, cylindrically, or toroidally bent crystals and crystals of a logarithmic-spiral form, were used in x-ray Bragg spectroscopy. These standard crystal forms offered only limited choices for the optimization of the photon throughput for a desired spectral range by varying the postion of the crystal relative to the x-ray source.
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