Crystal Size Distribution in Magmatic Rocks: Proposition of a Synthetic Theoretical Model
Résumé
The crystal size distribution (CSD) corresponds to the number of crystals of a mineral per unit volume within a series of defined size intervals. Many crystal size distributions in igneous systems are straight lines when plotted on the ‘classical' diagram of ln(population density) versus size. Other common CSDs are concave-down on such graphs for the low size values. The effect of growth rate on CSDs seems to be small and nucleation apparently increases exponentially with time. Although magmatic systems are always multiphase, only a few CSD studies deal with several phases. On the basis of the few available published examples, it seems that the parallelism (defined as y-spacing constancy) of the normalized CSDs (i.e. where the crystal lengths are normalized to the greatest length for each phase) is a common feature in igneous systems. The aim of our modelling is to propose a mathematical framework to unify the CSD typology, to explain the common occurrence of straight lines on the ‘classical' CSD diagram and the y-spacing constancy of the normalized CSDs for multiphase rocks, and also to provide a convenient tool to easily test petrogenetic scenarios through efficient computer simulations. The bulk balance modelling is based on a square matrix having for coefficients the number of nuclei of one phase by the crystallized volume of another phase. This interaction matrix is the mainspring of our CSD modelling, which requires straightforward assumptions about the uniformity of crystal shape and growth: (1) at each time, every crystal of a phase has the same shape and (2) the one-dimensional growth rate of a crystal is independent of its size. Consequently, the values of the normalized CSD are linked together by means of an integral equation. For the most part, the behaviour of the CSDs theoretically obtained with a constant matrix may also be observed in computer simulations of more general cases. This extended approach makes feasible the application of the model to natural cases. Concave-down shapes for the lowest size are classically interpreted as a consequence of the ending of crystallization. By roughly taking into account crystal interactions during the growth process through the concept of mean clutter, we successfully simulate such concave-down CSDs.