Calculation and measurement of all (002) multiple diffraction peaks from a (001) silicon wafer

The geometry of the multiple diffraction of a (001) silicon wafer in an x-ray powder diffractometer is elucidated. In particular, multiple diffraction occurs only when the sample is rotated (along the sample normal) to one of the specific azimuthal angles. A derivation based on analytical geometry is performed to obtain a general but closed formula for these azimuthal angles of multiple diffraction. Subsequent calculation shows that multiple diffraction occurs at 128 different angles in a full 360 degrees scan to form a (002) 'forbidden' reflection. Among these 128 multiple diffractions, 64 are double diffractions and the others are combined double and triple diffractions. The results of measurement obtained by a powder diffractometer show that only 96 of the 128 multiple diffractions are observed in the x-ray diffraction pattern due to the overlapping of some closely spaced diffraction peaks.