Resolution revisited: limit of detail in electron density maps

Application of the Rayleigh criterion for the limit of resolution of a simple lens with axial illumination leads to the value 0.61λ. In two-dimensional electron density maps based on X-ray data, the limit of resolution has been considered to be 0.61 dmin, the counterpart of the optical case, and in three-dimensional maps 0.715 dmin. It is shown here that point atoms separated by these distances are not resolved in two- and three-dimensional electron density maps. Such maps are amplitude functions rather than intensity functions as in the optical case. Application of the Rayleigh criterion to the three-dimensional amplitude function for point atoms leads to a value of 0.917 dmin, for the limit of resolution in three-dimensional electron density maps. This result is confirmed by superposing both analytic and numeric functions for point atoms and numeric functions for real atoms with B = 0 and 10 A2. Finally, some implications of diffraction effects in X-ray structure analysis are considered.