The double-crystal X-ray spectrometer: corrections, errors, and alignment procedure

The theory of the X-ray double-crystal spectrometer is outlined. For a vertical rotation axis, crystal planes should be vertical and the X-ray beam horizontal. Geometrical errors are introduced by crystal tilts (departure from vertical) and misalignment (departure of central ray from horizontal). Even with perfect alignment a vertical divergence correction is required. A typographical error in Schwarzschild's analysis of these geometrical effects led to erroneous results by Schnopper, later corrected in an erratum. The physical significance of this revision is considered here. The resulting expression for relative wavelength error indicates that, for zero tilt, misalignment produces no significant errors in relative wavelengths. Geometrical errors are shown to be negligible in recent measurements of Bearden and very small in earlier work of Bearden & Shaw. The Williams vertical divergence correction, used by Bearden, is proven valid; the Parratt form, preferred by Schnopper, is incorrect. An alignment procedure which can reduce geometrical errors below 1 part per million is described. It employs a precision engineer's level (with autocollimating telescope) and a simple technique for grinding crystal faces parallel to atomic planes.