Symmetry plays a crucial role in most formalisms of physics, and provides an elegant way to get a variety of results starting from a fully general mathematical basis. Such a principle of course applies to the realm of vibrational modes too. The structural symmetry of crystals, due to the ordered arrangement of atoms, causes particular properties of the vibrational modes, in terms of both the eigenvectors and eigenvalues of the dynamic matrix. The mathematical link that relates a space group, describing a geometrical symmetry, to the dynamics of a crystal, i.e. to a physical phenomenon, is provided by the theory of representations. Vibrational spectroscopy involved with lattice dynamics heavily benefits from the inferences attainable by combining the dynamics of the atomic motion in crystals with symmetry. Such a synergy has led to results of great practical use: selection rules, for instance, are a well-known powerful tool in interpreting experimental spectroscopic data.
This presentation aims at the following objectives:
illustrating the basic mathematical notions which underlie a rigorous approach to the relationships between lattice dynamics and symmetry, through the concept of group representations;
showing how the dynamic equations can be re-cast into a form leading straight to group representations;
providing the main links between lattice dynamics and crystal spectroscopy, in terms of theoretical inferences stemming from a combination of symmetry and dynamics, and experimental issues.
The points above have been developed with a particular care to the formal aspects, for three reasons:
Figures & Tables
The present book shows the arguments which have been considered in the EMU School (No. 4), dealing with Energy Modelling in Minerals; these arguments have been selected in order to provide examples of application of the most advanced theories to several cases. It should be pointed out that although the ultimate solution of our problems should involve “ab initio” quantum-mechanical calculations, at present such sophisticated procedures are far from being routine. Therefore, although “ab initio” approaches will play an ever-increasing role in the future and some important and most recent examples of such approaches are illustrated here, the greatest part of the contributions is dealing with empirical atom-atom calculations. Remarkably enough, such “semi-empirical” applications are often quite successful, providing excellent (or comparatively excellent) results in spite of their more or less approximate nature. It often happens that the methods here illustrated are some steps ahead of the current level of empirical treatment, thereby indicating a possible way of improvement by figuring out routines to be adopted in practice. If some methods seem to be too speculative to be actually usable, here they also are shown, in view of their possible discussion, or just to indicate a way to obtain promising developments. Among the descriptions of practical methods and results, some purely theoretical arguments have been inserted; these arguments — although abstract — according to our opinion are fundamental for earth scientists. Owing to the present status of the art, in a number of arguments there is no unique opinion with respect to their theoretical treatment as it is explained by different authors. Instead of having all of them discarded except the one which looks to be the most appropriate to the Editor (who might sometimes be personally involved in the question), most of such controversial points have been left just as they are, in the original draft of their advocates. Accordingly, the reader might find some discrepancies between some articles and others, which may lead to some obscurity; there are, however, several good reasons in favour of our behaviour. First of all, with a few exceptions we apologize about, our attention in inviting the contributors has been extended to all the principal authors in the world, with no limitation to a group of particular friends; moreover, the presence of different opinions in the context might give rise to interesting debates and critical objections; a further point is that the validity of the different treatments is shown per se by either the level of the theory and most of all by the agreement with the corresponding experimental data. Since we have to do with an advanced school, and in line with what should be a scientific procedure, it is important to provide the user with the possibility of choosing what seems to be the most appropriate method among a number of selected possibilities, rather than yielding to the assertion that something is indeed the ultimate and unquestionable “truth”.