Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T05:36:43.431Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantitative Analysis of Mineral Mixtures Using Linear Programming

Published online by Cambridge University Press:  02 April 2024

Gerald E. Braun
Gerald E. Braun*
Affiliation:
BJ Titan Services Company, 11211 FM 2920, Tomball, Texas 77375
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Chemical analysesLinear programmingMineral analysesMontmorilloniteOpal C-TX-ray powder diffraction
Type
Research Article
Information
Clays and Clay Minerals , Volume 34 , Issue 3 , June 1986 , pp. 330 - 337
Copyright
Copyright © 1986, The Clay Minerals Society

References

Albarede, F. and Provost, A., 1977 Petrological and geochemical mass-balance equations; an algorithm for leastsquares fitting and general error analysis Comput. Geosci. 3 309–326.CrossRefGoogle Scholar
Alexiades, C. A., Jackson, M. L. and Bailey, S. W., 1966 Quantitative clay mineralogical analysis of soils and sediments Clays and Clay Minerals, Proc. 14th Natl. Conf., Berkeley, California, 1965 New York Pergamon Press 35–52.Google Scholar
Chung, F. H., 1974 Quantitative interpretation of X-ray diffraction patterns of mixtures. I; matrix-flushing method for quantitative multicomponent analysis J. Appl. Crystallogr. 7 519–525.CrossRefGoogle Scholar
Chung, F. H., 1974 Quantitative interpretation of X-ray diffraction patterns of mixtures. II; adiabatic principle of X-ray diffraction analysis of mixtures J. Appl. Crystallogr 7 526–531.CrossRefGoogle Scholar
Gass, S. I., 1975 Linear Programming, Methods and Applications New York McGraw-Hill.Google Scholar
Hodgson, M. and Dudeney, A. W., 1984 Estimation of clay proportions in mixtures by X-ray diffraction and computerized chemical mass balance Clays & Clay Minerals 32 19–28.CrossRefGoogle Scholar
LeMaitre, R. W., 1981 GENMIX-a generalized petrological mixing program Comput. Geosci. 7 229–247.CrossRefGoogle Scholar
McClune, W. F., 1981 Inorganic Index to the Powder Diffraction File Pennsylvania Joint Committee on Powder Diffraction Standards.Google Scholar
Pearson, M. J., 1978 Quantitative clay mineralogical analses from the bulk chemistry of sedimentary rocks Clays & Clay Minerals 26 423–433.CrossRefGoogle Scholar
Reid, M. J., Gancarz, A. J. and Albee, A. L., 1973 Constrained least squares analysis of petrologic problems with an application to lunar sample 12040 Earth Planet. Sci. Lett. 17 433–445.CrossRefGoogle Scholar
Ross, C. S. and Hendricks, S. B., 1945 Minerals of the montmorillonite group U.S. Geol. Surv. Prof. Pap. 205–B 23–79.Google Scholar
Spivey, W. A. and Thrall, R. M., 1970 Linear Optimization New York Holt, Rinehart, and Winston.Google Scholar
Weaver, C. E. and Pollard, L. D., 1973 The Chemistry of Clay Minerals New York Elsevier.Google Scholar
Wright, T. L. and Doherty, P. C., 1970 Linear programming and least squares computer method for solving petrologic mixing problems Geol. Soc. Amer. Bull. 81 1995–2007.CrossRefGoogle Scholar
Zevin, L. S., 1977 A method of quantitative phase analysis without standards J. Appl. Crystallogr. 10 147–150.CrossRefGoogle Scholar