Solving the crystal structures of zeolites using electron diffraction data. II. Density-building functions

A density-building function is used to solve the crystal structures of zeolites from electron diffraction data using both two-and three-dimensional data sets. The observed data are normalized to give unitary structure factors vertical bar U-h vertical bar(obs). An origin is defined using one to three reflections and a corresponding maximum-entropy map, q(ME)(x), is calculated in which the constraints are the amplitudes and phases of the origin-defining reflections. Eight strong reflections are then given permuted phases and each phase combination is used to compute P(delta q) = integral(V)delta q (x)(2)/q(ME)(x) dx, where delta q(x) is the Fourier transform of vertical bar U-h vertical bar(obs) exp(i phi(perm)(h)) - vertical bar U-h vertical bar(ME) exp(i phi(ME)(h)), phi(perm)(h) is the permuted phase for reflection h and phi(ME)(h) is the phase angle for reflection h predicted from the Fourier transform of q(ME)(x). The 64 phase sets with minimum values of P(delta q) are subjected to entropy maximization and, following this procedure, those with the five highest log-likelihood gains are examined. Sometimes auxiliary potential histogram information is also used. The method worked routinely with seven zeolite structures of varying complexity and data quality, but failed with an eighth structure.