Automated structure analysis in electron crystallography: phase determination with the tangent formula and least-squares refinement

The tangent formula (computer program QTAN) is used as an automated phasing technique to solve the crystal structure of 2,5-piperazinedione from published electron-diffraction intensity data. Unlike the case of the thiourea polymorphs studied previously, the correct phase set does not correspond to the lowest value of NQEST, so that four potential maps must be calculated before a chemically recognizable structure is seen. Even though dynamical scattering presents some difficulty in identifying the correct structure by changing some low-magnitude |E_h| `cross terms', hence affecting the NQEST figure of merit, it is still possible to refine the atomic coordinates found on the potential map by full-matrix least squares, leading to a structure similar to the earlier X-ray determination.