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.
The quantity usually measured in relation to each X-ray reflection is the intensity, which is proportional to |F(hkl)2| and hence it is |F(hkl)| that is determined experimentally. This quantity may be called the 'geometrical structure factor' since it depends only on the positions of atoms and not on any differences in their scattering behaviour.
Quite a number of crystal structures contain parts (e.g. heavy atoms or building units) with a higher symmetry (e.g. with additional translation or pseudotranslation) compared with the whole structure. In those cases, standard methods may not lead to correct results, this is why special methods may have to be applied.
On définit habituellement le cristal parfait comme un objet infini engendré par la répétition triplement périodique d'un motif central remplissant tout le volume convexe (0, a, b, c) appelé alors maille centrale.
On a défini ici un cristal parfait à trois dimensions si a, b, c sont trois vecteurs linéairement indépendants (c'est-à-dire non contenus dans un même plan) de l'espace à 3 dimensions. Les vecteurs qui répètent infiniment ce motif sont les vecteurs
The crystal structures of a large number of metals, alloys and inoganic compounds can be described geometrically in terms of a close-packing of equal spheres, held together by interatomic forces. Frequently, the positions of one kind of atoms of ions in inorganic structures correspond approximately to those of equal spheres in a close-packing with the other atoms distributed among the voids. All such structures will be referred to as close-packed structures though they may not be ideally close-packed. The close-packed arrangement of equal spheres in a plane is shown in Fig
The fundamental property of a crystal is its triple periodicity and a crystal may be generated by repeating a certain unit of pattern through the translations of a certain lattice called the direct lattice. The macroscopic geometric properties of a crystal are a direct consequence of the existence of this lattice on a microscopic scale. Let us for instance consider the natural faces of a crystal. These faces are parallel to sets of lattice planes.
In X-ray crystallography the structure factor F(hkl) of any X-ray reflection (diffracted beam) hkl is the quantity that expresses both the amplitude and the phase of that reflection. It plays a central role in the solution and refinement of crystal structures because it represents the quantity related to the intensity of the reflection which depends on the structure giving rise to that reflection and is independent of the method and conditions of observation of the reflection.
In my view the basic problem in presenting X-ray diffraction to non-specialist audiences is to remove some of the atmosphere of mathematical difficulty and mysticism and to show, first of all, that the processes involved are, in principle, identical with those of microscopy.