Global least-squares determination of Eulerian angles from single electron diffraction patterns of tilted crystals

Three-dimensional structure determination using electron diffraction of crystalline samples necessitates the determination of the Eulerian angles of tilted samples. For experimental tilt series, even with approximately known tilt, the resolution of the final three-dimensional reconstructions is reduced as a result of the large errors of the refined tilt angles and crystal axes positions. The presented new least-squares procedure determines the orientation of the crystal with very high accuracy from a single electron diffraction pattern. Instead of evaluating the averaged pattern geometry, each diffraction spot position is individually included in an analytical non-linear fit. This procedure is very stable against potential experimental errors, as demonstrated by Monte Carlo simulations. As a test sample, a three-dimensional microcrystal of an organic crystal compound was used. Contrary to the conventional method, which produced erroneous Miller indices for some reflections, the indexing obtained with the new algorithm was more consistent for each individual pattern. Preliminary data from frozen hydrated protein crystals, the samples of which are beam sensitive and for which only a few patterns can be recorded from a single crystal, indicate that the new angle determination promises to be particularly beneficial under such conditions.