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In the context of discrete Fourier transforms the idea of aliasing as due to approximation errors in the integral defining Fourier coefficients is introduced and explained. This has the positive pedagogical effect of getting to the heart of sampling and the discrete Fourier transform without having to delve into effective, but otherwise long and structured, introductions to the topic, commonly met in advanced, specialized books.

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• What is x-ray crystallography? • Statistical applications in x-ray crystallography • Use of R in crystallographic research • Description of cRy Statistical applications in x-ray crystallography The probability distribution of x-ray intensities A. J. Wilson (1949) Traitment statistique des erreurs dans la determination des structures cristallines V. Luzzati (1952) Probability distribution connected with structure amplitudes of two related crystals G. N. Ramachandran, R. Srinivasan and R. Sarma (1963) On the application of phase relationships to complex structures. The additional use of statistical information in tangent-formula refinement. S. E. Hull and M. J. Irwin (1978) Maximum entropy and the foundations of direct methods G. Bricogne (1984)

The reciprocal lattice is derived through the Fourier transform of a generic crystal lattice, as done previously in the literature. A few key derivations are this time handled in detail, and the connection with x-ray diffraction is clearly pointed out. The Ewald sphere is subsequently thoroughly explained and a few comments on its representation in a mixed real-reciprocal space are made. In particular, it appears that the majority of textbooks or papers on the subject are limited in their way of picturing it. This paper will be useful to solid state and/or crystallography teachers. It is also suitable for graduate students researching these subjects and for talented undergraduate students.

An ab initio method is described for solving protein structures for which atomic resolution (better than 1.2 A Ê ) data are available. The problem is divided into two stages. Firstly, a substructure composed of a small percentage ($5%) of the scattering matter of the unit cell is positioned. This is used to generate a starting set of phases that are slightly better than random. Secondly, the full structure is developed from this phase set. The substructure can be a constellation of atoms that scatter anomalously, such as metal or S atoms. Alternatively, a structural fragment such as an idealized -helix or a motif from some distantly related protein can be orientated and sometimes positioned by an extensive molecular-replacement search, checking the correlation co-ef®cient between observed and calculated structure factors for the highest normalized structure-factor amplitudes |E|. The top solutions are further ranked on the correlation coef®cient for all E values. The phases generated from such fragments are improved using Patterson superposition maps and Sayreequation re®nement carried out with fast Fourier transforms. Phase re®nement is completed using a novel density-modi®cation process referred to as dynamic density modi®cation (DDM). The method is illustrated by the solution of a number of known proteins. It has proved fast and very effective, able in these tests to solve proteins of up to 5000 atoms. The resulting electron-density maps show the major part of the structures at atomic resolution and can readily be interpreted by automated procedures.

Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems. Being able to change all variables and expression involved in a given problem, when a different coordinate system is chosen, is one of those skills a physicist, and even more a theoretical physicist, needs to possess. In this lecture a general method to express any variable and expression in an arbitrary curvilinear coordinate system will be introduced and explained. We will be mainly interested to find out general expressions for the gradient, the divergence and the curl of scalar and vector fields. Specific applications to the widely used cylindrical and spherical systems will conclude this lecture.