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In a Laue diffraction pattern, 10–20% of the spots result from the exact superposition of two or more reflections that are `harmonics'; a high proportion of these are low-resolution reflections. For the solution of large or difficult structural problems, the intensities of the remaining 80–90% of the reflections, measurable as singles, may not be sufficient and thus the evaluation of the intensities of the components of the multiple spots is important. A new method for this deconvolution is presented that is based on maximizing the entropy of the Patterson function subject to the constraints imposed by the observed intensities of single and overlapping reflections. This method does not require data redundancy and therefore is of particular interest for time-resolved studies on a short time scale. A new computer program (ME) was implemented and tested with Laue diffraction data from hen egg white lysozyme. The R-factor between the deconvoluted reflection intensities from Laue multiple spots and observed intensities from monochromatic data was 0.116.

A general method for the model-free reconstruction of magnetization densities is presented. The technique is based upon the maximum-entropy principle and is valid for systems with any space-group symmetry, any physically reasonable ordered moment and allows for the incorporation of relevant corrections such as extinction. Thus, it is possible, for the first time, to reconstruct the real-space magnetization densities in essentially any physical system exhibiting a ferromagnetic spin alignment (either natural or induced) from polarized neutron diffraction experiments. The technique is illustrated for the purely organic ferromagnet, β-4,4,5,5-tetramethyl-2-p-(nitrophenyl)-3-oxido-4,5-dihydroimidazolium 1-oxyl.

The charge densities derived with the maximum-entropy method (MEM) may be influenced to some extent by the completeness of the data set. In order to examine the effects of the incompleteness, structure-factor data of Si measured by the Pendellosung method [Saka & Kato (1986). Acta Cryst. A42, 469–478] were re-analysed by the MEM. This data set is incomplete: it contains all space-group-allowed reflections with sinθ/λ = 0.86 A−1, and in addition 844 and 880 with sinθ/λ = 1.04 A−1. Results of a MEM analysis of the complete subset of data are compared with those from the full but incomplete set published previously [Sakata & Sato (1990). Acta Cryst. A46, 263–270]. The smaller but complete set was found to give a smooth charge-density distribution that is consistent with previous theoretical work. It is found that the sharp peak maximum at the bond midpoint reported previously is exaggerated owing to the highest-order reflection 880. The completeness of the data set appears to be one of the key factors for obtaining reliable charge densities with MEM. The incompleteness of the data set may cause non-physical fine features of the MEM density distribution.

A Bayesian approach to the treatment of uncertainty in maximum-entropy maps is developed based on a deeper hypothesis space that includes the prior map. The new approach is able to take into account uncertainty in the knowledge of the prior and provides a generalization of `classic MaxEnt'. Some examples of the new approach are presented for simulated SAXS data for cytochrome c.

An algorithm is described for calculating the maximumentropy (ME) electron density map that is constrained to satisfy each of the observed structure-factor amplitudes, i.e. [Fh I - [F~'b~[ = 0. The use of these phaseless constraints enables one to refine phases, and hence crystal structures, in terms of entropy. From the structure factors observed with a small molecule as well as values extrapolated from them, numerical calculations were done at various resolutions. The algorithm can be used to obtain atomic models even with 1.5 A data, indicating that the ME method is a significant improvement over currently available direct methods. For practical purposes, the present algorithm may be used as an alternative to successive Fourier refinements following the initial stage of approximate phase determination.

Several general properties of maximum-entropy maps are reviewed that substantiate previous results from selected applications. In particular, the maximum-entropy method (MEM) is depicted as a smoothing scheme and the intrinsic bias introduced by this procedure is pointed out. It is argued that the MEM is not well suited for accurate charge-density mapping.

The maximum-entropy method (MEM) was applied to accurate γ-ray diffraction data from MnF2 and NiF2 to explore details of the charge-density distribution. For a fair judgement of the results, Si Pendellōsung data [Saka & Kato (1986). Acta Cryst. A42, 469–478] were also treated. It is shown that conclusions drawn from MEM maps must be accepted with some reserve, particularly in the regions of interest in charge-density studies.

A maximum entropy method (MEM) map of electron density is obtained by maximizing S = -Σ p log p subject to whatever data are available. MEM is derived as the only reconstruction technique that is consistent with simple and general requirements. The method is very widely applicable, but, in this paper, attention is focused on the problem of producing electron density maps in crystallography. The entropy formula can also be derived by analogy with a thermodynamic system of quanta, but it is shown that this model can be misleading, and can break down in practice. MEM applied to a different problem related to quantum fluctuations in the thermodynamic model is shown to lead to formulae equivalent to the maximum determinant method. It is argued that direct MEM will produce superior maps.