DISCRETE HILBERT TRANSFORMS AND THE PHASE PROBLEM IN CRYSTALLOGRAPHY. A.F.Mishnev. S.V.Belyakov, Latvian Institute of Organic Synthesis, Riga, Latvia.

On the analogy to the optical image reconstruction theory the role of in-between structure factors (having half-integral Miller indices) has been investigated in application to the crystallographic phase problem. Relation between normal (having integral Miller indices) and in-between structure factors (SF) is given by the discrete Hilbert transforms (DHT)

where the prime on the summation sign indicates that terms with h=h', k=k' and l=l' are omitted, j= When h',k',l' in the left-hand side of(1) are odd, than in the right-hand side terms with h, k, l odd will cancel and (1) will give the values of in-between SF in terms of normal ones. In-between SF can be incorporated into the phasing process by means of the autocorrelation function (ACF, Patterson for a single unit cell), which has the following form

Formula (2) contains both normal and in-between intensities, which can be calculated from normal SF using (1). When the causal Fourier transform condition (Fourier transform vanishes for negative arguments) is satisfied for projections of the ACF, DHT applied to intensity function provide intensity oversampling at in-between points as follows

Properties of the ACF (non-negativity, boundedness in space and magnitude) and its relation to the Patterson function give rise to new criterions for the correctness of phase sets. Phase refinement procedures based on minimum-negativity constraint and fitting of ACF to Patterson are suggested.