research papers
Probability distributions associated with the third, fourth, fifth and sixth neighborhoods of the structure invariant φlmφh + φk + φl + φm are derived. These distributions yield estimates for φlm in terms of magnitudes |E| alone which are particularly good in the favorable case that the standard deviations of the distributions are small. The estimates for the φlm lead in turn to unique values for the individual phases φ even for very complex structures. Conditional distributions of φlm given the values of one or more structure invariants φpq, φrs, φtu and appropriate sets of magnitudes |E| are also described.