Components, end-members and mineral species

Abstract I consider the conditions for defining a mineral by its dominant end-member formula. One can calculate the end-member proportions of the end-members of a mineral provided that the end-members are linearly independent (i.e. they are phase components of the mineral); the result includes the dominant end-member of the mineral. If the end-members used in this calculation are not linearly independent, the corresponding set of simultaneous equations is indeterminate. One may remove an end-member from the system, removing the linear dependence; however, any end-member formula may be removed, leaving various sets of end-members that function as phase components. Each set of end-members produces a different solution for the end-member proportions. Each set of positive end-member proportions may (or may not) result in a different dominant end-member; however, within the compositional limits of the species, the same end-member is dominant over all others calculated with different combinations of component end-members. Problems previously encountered in attempting to calculate the dominant end-member formula were due to (1) using mineral formulae that do not accord with the requirements of stoichiometry, and (2) using end-members that are not components of the system. Where the set of end-members chosen to relate mineral composition to end-member proportions contains an end-member that is a linear combination of the other end-members, one must calculate the end-member proportions for all distinct subsets of linearly independent end-members. The dominant end-member over all sets of end-member proportions with all proportions positive is the dominant end-member. Thus for any mineral formula, the dominant end-member formula may be identified and serves to uniquely characterize and identify the mineral. The arguments used here are illustrated by reference to the minerals of the garnet supergroup.