Problems of resolution and bias in the experimental determination of the distribution of electron density and other densities in crystals

Abstract

The limited 'window' in reciprocal space through which it is possible to observe diffraction phenomena sets a reciprocal limit to the resolution of detail in density distributions. The limit of resolution depends on whether the detail is one-, two-, or three-dimensional, and to some extent it is possible to choose between (i) lack of resolution associated with a large central maximum representing a point-object and (ii) false details associated with a smaller central maximum and more pronounced diffraction troughs. In any case, however, the limit of resolution is about one-quarter to one-half of the wavelength of the radiation used. Intensities measured by photon or particle counting are unbiased estimates of the true intensities, but their square roots are not unbiased estimates of the structure factors, and this bias may carry over into parameters based on structure factors rather than intensities. A satisfactory correction can be made for the strong reflexions, but weak reflexions (which are required if the theoretical limit of resolution is to be reached) remain a problem.