The approximate maximum and minimum amounts of any phase in a complex mineral mixture can be determined by solving a linear programming problem involving chemical mass balance and X-ray powder diffraction (XRD) data. The chemical information necessary is the bulk composition of the mixture and an estimation of the compositional range of each of the minerals in the mixture. Stoichiometric constraints for the minerals may be used to reduce their compositional variation. If only a partial chemical analysis for the mixture is available, the maximum amounts of the phases may still be estimated; however, some or all of the stoichiometric constraints may not apply. XRD measurements (scaled using an internal standard) may be incorporated into the linear programming problem using concentration-intensity relations between pairs of minerals. Each XRD constraint added to the linear programming problem, in general, reduces the difference between the calculated maximum and minimum amounts of each phase. Because it is necessary to define weights in the objective function of the linear programming problem, the proposed method must be considered a model. For many mixtures, however, the solution is relatively insensitive to the objective function weights.

An example consisting of a mixture of montmorillonite, plagioclase feldspar, quartz, and opal-cristobalite illustrates the linear programming approach. Chemical information alone was used to estimate the mineral abundances. Because quartz and opal-cristobalite are not chemically distinct, it was only possible to determine the sum, quartz + opal-cristobalite, present in the mixture.