Section 14.1.5. The best Fourier

A protein crystallographer desires to obtain a Fourier synthesis that can most readily be interpreted in terms of an atomic model of the structure. One synthesis which could be calculated is the `most probable Fourier', obtained by choosing the value of for each reflection which corresponds to the highest value of P(φ). Blow & Crick pointed out that although this Fourier is the most likely to be correct, it has certain disadvantages. In the first place, it might tend to give too much weight to uncertain or unreliable phases, and, in the second place, for cases where P(φ) is bimodal, there is a strong chance of making a large error in the phase angle. Blow & Crick suggested that in cases such as this, a compromise is needed, and that the centroid of the phase probability distribution provides just the required compromise. They showed that the corresponding synthesis is the `best Fourier', which is defined to be that Fourier transform which is expected to have the minimum mean-square difference from the Fourier transform of the true F's when averaged over the whole unit cell.

The centroid of the phase probability distribution may be defined as a point on the phase diagram with polar coordinates , where is the `best' phase angle. The quantity m, which acts as a weighting factor for , is called the `figure of merit' of the phase determination. Its magnitude, between 0 and 1, is a measure of the reliability of the phase determination.