has been observed that in most cases three iterations are sufficient to obtain the concentrations. A numerical application is given as an example in Table IV. This calculation was done in a few minutes with a pocket calculator. I t must be noted that, since the calculation is done on theoretical K’s, the accuracy obtained is better than the usual 1-2% relative accuracy of the ZAF method, but is only a comparison parameter. In the case of systems with more than three elements, the method is the same as for simpler systems. The number of a coefficients is larger, specially for ternary coefficients ( ~ A B C ~, A B D. . . , ~ A C N. . .I, but calculations can be simplified by introducing these ternary coefficients in the last iteration only or by omitting them completely. A comparison of these two methods is shown in Table V for a four-element hypothetical alloy. Obviously, the most accurate results are obtained when all the a coefficients are used. An example of application of the a coefficient method to real samples is given in Table VI. A comparison is made between results obtained by the ZAF method and by the computed a coefficients method. The two sets of results are in good agreement and confirm the potential of the computed a coefficients method which is simpler than the usual computer correction procedures used in microprobe analysis if continued work is to be carried out in a particular system.
In a recent publication (IO), Rasberry et al. discussed very extensively the interelement effects in X-ray fluorescence analysis and they emphasized the better accuracy of Equation 3 but also the difficulty of determining the greater number of coefficients it requires, due to the lack of standards. This latter objection is somewhat less valid since the a’s can now be computed.
ACKNOWLEDGMENT The authors are grateful to K. Heinrich for critical comments on the manuscript.
LITERATURE CITED T. 0. Ziebold and R. E. Ogilvie, Anal. Chern., 36,322 (1964). G. R. Lachance and R. J. Traill, Can. Spectrosc., 11, 43 (1966). F. Claisse and M. Quintin, Can. Spectrosc., 12, 129 (1967). R. Rousseau and F. Claisse, X-Ray Spectrorn.,3, 31 (1974). (5) A. L. Aibee and L. Ray, Anal. Chern., 42, 1408 (1970). (6) D. Laguitton, Doctoral Thesis, Universite Laval, Quebec, 101 (1973). (7) D. Laguitton, Y. Berube, and F. Claisse, Can. Spectrosc., 19, 100 (1974). (8) G. R. Lachance, Can. Spectrosc.,15, 64 (1970). (9) D. R. Beaman and J. A. Isasi, Anal. Chern.. 42, 1540 (1970). (10) S. D. Rasberry and K. F. J. Heinrich. Anal. Chern., 46, 81 (1974). (1) (2) (3) (4)
RECEIVEDfor review May 12, 1975. Accepted July 28, 1975. Work supported by the Canadian National Research Council (Grant No. A931).
Quantitative Chemical Analysis of Individual Microparticles Using the Electron Microprobe: Theoretical John T. Armstrong Division of Geochemistry, Department of Chemistry, Arizona State University, Tempe, Ariz. 8 5 2 8 1
Peter
R. Buseck
Departments of Geology and Chemistry, Arizona State University, Tempe, Ariz. 8 5 2 8 1
A correction procedure has been developed to enable the quantitative analysis of individual unpolished microparticles using the electron microprobe or related electron beam instruments. Equations expressing the amount of X-ray absorption have been derived for a variety of particle shapes. The effect of electron backscatter on observed X-ray intensities has been Considered. Analytical results are presented using these correction procedures on particles of known compositions. These results show that routine quantitative analysis of microparticles is both feasible and straightforward.
The ability to perform nondestructive chemical analyses of individual microparticles is important in a variety of areas. These include air pollution, corrosion, and pigment research, forensic chemistry, pathology, experimental petrology, and the study of terrestrial soils and lunar fines. To analyze such specimens, it is necessary to work with sample volumes of less than 1 pm3 and detect as little as gram of the element of interest. In the past, there were no practical techniques available for performing such analyses; consequently, the compositional determination of individual microparticles has been neglected. Electron beam instruments (e.g., the electron microprobe, the scanning electron microscope, and the transmis2178
sion electron microscope), with wavelength or energy dispersive spectrometers, are ideally suited for the analysis of such materials. These instruments bombard a sample with a focused beam of high energy electrons and thereby produce X-rays. These X-rays have energies and wavelengths characteristic of the elements which emitted them. A portion of the X-rays leave the specimen and are counted by a detector and can thus be used for nondestructive chemical analysis. However, the quantitative analysis of particulate material with electron beam instruments has generally been considered impossible because of the problems of determining the relation between X-ray intensity and element concentration (1). The characteristic X-ray intensity for a given element in a sample is only roughly proportional to the element’s concentration. The intensity is significantly affected by both instrumental and physical factors. The instrumental factors include the X-ray detection efficiency and the electron beam potential and current. The physical factors are dependent on the sample composition and include: (a) the efficiency of X-ray production by the various elements in the sample (the so-called atomic number factor), (b) the amount of absorption by the sample of the elements’ characteristic X-rays, and (c) the amount of secondary X-ray fluorescence (generated by the primary characteristic Xrays and the Bremsstrahlung radiation). The effect of instrumental variables can be minimized by
ANALYTICAL CHEMISTRY, VOL. 47, NO. 13, NOVEMBER 1975
ratioing the measured intensities of the elements in the sample to those in standards of known composition. These intensity ratios must still be corrected for differences in the physical factors between the sample and the standards. Corrections for the atomic number, absorption, and fluorescence effects have been developed and are widely used for thick polished specimens. (In terms of microprobe analysis, such thick polished specimens are generally >15-20 wm in diameter and >5-10 wm deep.) Complications arise, however, when attempting to apply the conventional corrections to the quantitative analysis of unpolished microparticles: (a) Some bombarding electrons, which would have generated characteristic X-rays in a thick polished specimen, may escape from the particle bottom (transmission) or sides (sidescatter) before they generate X-rays. The resulting decrease in the generated X-ray intensity will vary from element to element in the particle and depends on a variety of factors including particle shape and size. (b) The effect of matrix absorption on observed X-ray intensity depends on the distance the X-rays must travel from point of production until they leave the specimen in the direction of the detector. For a flat polished sample, the path length is a simple geometric function of the depth of X-ray production. For a particle, with its irregular shape, the X-ray path length is a far more complex function, dependent on the particle's shape and size (Figure 1). (c) Similarly, the amount of secondary X-ray fluorescence depends on X-ray path lengths (both of the fluorescing and fluoresced X-rays) and is a complex function of geometry for a particle. Two approaches have been used for semi-quantitative analysis of particulate materials. The first is to embed them in plastic, polish them, and treat them as flat materials. This procedure has several difficulties. Small embedded particles are diffncult to polish without severe rounding and preferential plucking. The surface morphology of the particle is often of interest; and this, of course, is destroyed by polishing. The particle may not be as deep as the effective depth of electron penetration, requiring it to be treated as a thin film. Finally, in order for the conventional matrix absorption correction to apply (in regard to X-ray path length), there is the geometric constraint that the particle have a minimum diameter, Dmin, expressed by the equation:
Dmin = D,
+
2(R, cot d~)
(For the definitions of the symbols used in this and subsequent equations, see the Nomenclature section.) Under ordinary experimental conditions for oxide or silicate materials, Dminis on the order of 10 Wm or greater. I t is also clearly important to be able to analyze particles in the
