Geometrical and wave-optical effects on the performance of a bent-crystal dispersive X-ray spectrometer

The X-ray focusing properties of a bent single crystal diffracting in Bragg geometry are discussed. First, it is assumed that a polychromatic point source is focused to a point image. The elliptical arc that the crystal must trace and the aberrations caused by bending the crystal cylindrically are derived from the ray paths. For a source of finite size, the magnification is found to vary over the crystal's length, so that rays of different wavelength produce images of different size. More realistic treatments of penetration and diffraction are performed with spherical monochromatic incident waves, using Takagi-Taupin calculations to create the diffracted wave and the Fresnel integral to trace the diffracted wave's evolution. Such “wave-optical” calculations on a symmetric Si (1 1 1) crystal with 7 keV X-rays predict beam sizes different from those found in ray traces. Optimal sample and detector placement therefore requires wave effects to be considered.