The Infrared Spectra of Minerals
The principal concern of this book is the use of vibrational spectroscopy as a tool in identifying mineral species and in deriving information concerning the structure, composition and reactions of minerals and mineral products. This does not mean that the approach is purely empirical; some theoretical understanding of the vibrational spectra of solids is essential to an assessment of the significance of the variations in the spectra that can be found within what is nominally a single mineral species, but which usually includes a range of compositions and defect structures. Theory alone, however, can give only limited support to the mineral spectroscopist, and careful studies of well-characterized families of natural and synthetic minerals have played an essential role in giving concrete structural significance to spectral features. The publication of this book represents a belief that theory and practice have now reached a state of maturitity and of mutual support which justifies a more widespread application of vibrational spectroscopy to the study of minerals and inorganic materials. The wide area of theory and practice that deserves to be covered has required a careful selection of the subject matter to be incorporated in this book. Since elementary vibrational spectroscopy is now regularly included in basic chemistry courses, and since so many books cover the theory and practice of molecular spectroscopy, it has been decided to assume the very basic level of knowledge which will be found, for example, in the elementary introduction of Cross and Jones (1969). With this assumption, it has been possible to concentrate on those aspects that are peculiar to or of particular significance for mineral spectroscopy.
We know that a crystal containing N atoms constitutes a mechanical oscillator with 3N degrees of freedom, which give the 3N frequencies of the normal mode vibrations. These frequencies are excited by thermal agitation. Under the action of infrared monochromatic radiation one forced oscillation can be excited in a crystal. Such a vibration has a large amplitude when the frequency of the radiation resonates with one eigen frequency of the oscillator. Knowing the deformation of the crystal lattice under the action of infrared radiation, its polarization P can be predicted. Now the relative dielectric constant εR can be determined from the polarization. In the case of an isotropic medium, the relation is particularly simple: P=ε0(εR– l)E. Thus the dielectric constant can be obtained and so also the complex index of refraction n = n – jk of the crystal, where n is the index of refraction, and k the index of absorption. They respectively give the phase velocity v = cjn for the radiation of angular frequency ω, and the absorption coefficient K = 2kw/c, related to Beer's law which gives the transmission T for a plate of thickness x: T = e−Kx. The aim of this chapter is to show how to calculate the optical constants of the crystal from data concerning both its structure and its dynamics.
Let us consider a chain (Fig. 3.1) formed by a regular array of two ions of masses M and m respectively at a