Precision lattice parameter determination from transmission diffraction of thick specimens with irregular cross sections

Accurate determination of lattice parameters from X-ray diffraction requires that the diffraction angles be measured very precisely, and significant errors result if the sample–detector separation differs from that assumed. Transmission diffraction from bones, which have a complex cross section and must be left intact, is a situation where this separation is difficult to measure and it may differ from position to position across the specimen. This article describes a method for eliminating the effect of variable sample cross section. Diffraction patterns for each position on the specimen are collected before and after 180° rotation about an axis normal to the cross section of interest. This places the centroid of the diffracting mass at the center of rotation and provides the absolute lattice parameters from the average apparent lattice parameters at the two rotation angles. Diffraction patterns were collected across the cross section of three specimens: a 3D-printed elliptical cylinder of Hyperelastic Bone (HB), which is composed primarily of synthetic hydroxyapatite (hAp), a 3D-printed HB model of the second metacarpal bone (Mc2), and a modern human Mc2 containing nanocrystalline carbonated apatite (cAp). Rietveld refinement was used to determine precise unit-cell parameters a apparent and c apparent for each pattern of each scan, and these values determined the actual average 〈a〉 and 〈c〉 for each sample. The results indicate that the 0°/180° rotation method works well enough to uncover variations approaching 1 × 10−3 Å in cAp unit-cell parameters in intact bones with irregular cross sections.