Most monographs on physics discussing the physical properties of matter usually proceed from isotropic materials and as a generalization include a more or less limited description of the behaviour of crystalline bodies. This way of presentation is doubtless advantageous, however, it implies a separate discussion of the various properties, inevitably obscuring the general principles and methods applicable in the theory of crystal properties.
The term 'direct methods' is applied to that class of methods which seek directly to solve the phase problem by the use of phase relationships based on the observed intensities.
The object of this pamphlet is to familiarize the reader with the phase relationships used in Direct Methods, and to explain why they work and how they are used in practice. Some prior knowledge of the phase problem, the structure-factor equation and the application of Fourier theory in crystal-structure analysis is assumed.
Classically, the two main ways of studying metals and alloys were metallography (the examination of polished and etched surfaces) and cooling curves (looking for discontinuities that indicated some sort of phase change). Both these methods involved considerable skill and experience, and the results were not always unambiguous. The introduction of X-ray diffraction provided a much clearer, simpler and more objective way of investigation.
The results of X-ray diffraction studies of both small and large molecules of biological interest have greatly enhanced our understanding of many biochemical processes such as enzyme mechanisms, nucleic acid flexibility and virus assembly. Since biologists are primarily interested in biology, if a physical method is described to them, it is important that it be carefully explained how the results can be of use in understanding some aspect of biology.
Two-dimensional periodic patterns provide a simple means for teaching the principles of crystallographic symmetry. In particular they help to develop some desirable skill in recognizing space groups at an elementary level.
The general idea of symmetry is familiar to almost everyone. Formally it can be defined in various ways. The Concise Oxford Dictionary says '1. (Beauty resulting from) right proportion between the parts of the body or any whole, balance, harmony, keeping. 2. Such structure allows of an object's being divided by a point or line or plane or radiating lines or planes into two or more parts exactly similar in size and shape and in position relative to the dividing point, etc., repetition of exactly similar parts facing each other or a centre, ... '.
Crystals are three-dimensional objects and are represented on paper by suitable projections. The use to which the resulting picture is to be put determines the choice of projection. Clinographic, orthographic and perspective projections are briefly described here, with examples taken from the cubic crystal system.
The stereographic projection is a projection of points from the surface of a sphere on to its equatorial plane. The projection is defined as shown in Fig.
In the first part of this monograph the concepts of symmetry operations, symmetry elements and symmetry groups based on the metric tensor invariance are introduced.
In the second part the crystallographic point groups are derived: first the enantiomorphic groups using all possible combinations of the rotation axes; secondly, the centrosymmetric groups; and, finally, the non-enantiomorphic, non-centrosymmetric groups.
Rotation matrices (R) and translation vectors (t) are very powerful descriptions of the symmetry within the crystal and give aid in origin specification, in determining phase restrictions, systematic absences, systematic enhancement and seminvariants, in distinguishing centric and acentric reflections, general and spatial reflections and are helpful in making correct space group determinations.