niques such as factor analysis, pattern recognition, principal component analysis, nearest neighbor, and including numerical taxonomy are merely classification techniques. The relationship between numerical taxonomy and the other classification techniques has been discussed already in a study by Massart et al. (IO)in which a classification of 226 liquid phases was carried out. Among these techniques, numerical taxonomy and pattern cognition as described by Wold (11)are the only ones yielding formal classifications. Numerical taxonomy permits one to obtain a complete hierarchical classification which is not the case for pattern cognition. By the classification obtained with numerical taxonomy, the selection of columns is facilitated. This is considered to be one of the main advantages of numerical taxonomy, namely, that other factors such as stability of columns, availability, and price can be taken into account, which is not the case for the information theoretical approach. In the calculation of the amount of information, correlation coefficients have been used. Use of the same correlation coefficients as the classification parameter, in order to be able to make the optimal choice of columns, seems to be justified and yields better results than the Euclidean distance. Application of both procedures therefore necessitates (rather elaborate) calculations of correlation coefficients. The estimation of the amount of information for n columns requires an additional calculation of a series of-in our case-(',6) determinants, but the calculations can be reduced by using the selection procedure of Dupuis and Dijkstra (13). The application of numerical taxonomy requires a reduction of a 16 X 16 matrix. The mathematical background of the information theory procedure is more complex. The final conclusion of this article is twofold. As far as the methods are concerned, it is clear that the combined use of mathematical tools such as information theory, pattern-cognition, nearest neighbor calculations, principal components analysis, and numerical taxonomy permits the classification and combination of chromatographic techniques. They should therefore be of value in comparative physicochemical studies of these systems and in the selection of sets of preferred phases. As far as the results of the GLC problem which is treated here are concerned, this ap-
proach can be considered as intermediate between the two extreme positions of those who propose a restricted set of stationary phases for all GLC uses, and of those who do not want any such sets, stating that special separation problems do occur which cannot be solved with such a restricted set. It is the belief of the authors of the present article that both positions are somewhat exaggerated at the present stage of development of GLC. It cannot be denied that there is a large redundancy in GLC phases. At the same time, it is clear that it is not possible to achieve all GLC separations with a restricted set. When selecting phases for separation of restricted sets of, for example, alcohols, esters, steroids, trimethylsilylesters, etc., it should be possible to arrive at a reduced number of phases without excluding the possibility of some separations to be carried out further.
LITERATURE CITED (1) W. 0. McReynolds, J. Cbromatogr. Sci., 8, 685 (1970). (2) S. T. Preston, J. Cbromatogr. Sci., 8, (Dec). 18A (1970). (3) J. J. Leary, J. 6.Justice, S. Tsuge, S. R. Lowry, and T. L. Isenhour, J. Cbromafogr. Sci., 11, 201 (1973). (4) R. A. Keller, J. Cbromatogr. Sci., 11, 188 (1973). (5) J. R. Mann and S. T. Preston Jr., J. Chromafogr. Sci,, 11, 216 (1973). (6) R. S. Henly, J. Cbromatogr. Sci., 1 1 , 221 (1973). (7) L. Rohrschneider, J. Cbromafogr., 22, 6 (1966). (8) P. H. Weiner and J. F. Parcher, J. Cbromafogr. Sci., IO, 612 (1973). (9) S. Wold and K. Andersson. J. Cbromafogr., 80, 43 (1973). (10) D. L. Massart, M. Lauwereys, and P. Lenders, J. Chromafogr. Sci., 12, 617 (1974). (1 1) S. Wold, Technical Report No. 364, Department of Statistics, University of Wisconsin, Madison, Wis., 1974. (12) A. C. Moffat, A. H. Stead, and K. W. Smalldon, J. Cbromafogr.. 90, 19 (1974). (13) P. F. Dupuis and A. Dijkstra, Anal. Chem., 47, 379 (1975). (14) S. Hawkes, D. Grossman, A. Hartkopf, T. Isenhour, J. Leary. J. Parcher, S. Wold, and J. Yancey, J. Cbromatogr. Sci., 13, 115 (1975). (15) D. L. Massart and H. De Clercq, Anal. Chem., 46, 1988 (1974). (16) C. E. Shannon and W. Weaver, "The Mathematical Theory of Communication", The University of Illinois Press, Urbana, Ill., 1949. (17) P. H. A. Sneath and R. R . Sokal, "Numerical Taxonomy", W. H. Freeman, San Francisco, Calif.. 1973. (18) W. 0. McReynolds, "Gas Chromatographic Retention Data", Preston Technical Abstracts Company, Evanston, Ill., 1966. (19) T. D. Sterling and S. V. Polak, "Introduction to Statistical Data Processing", Prentice-HallInc., Englewood Cliffs, N.J., 1968.
RECEIVEDfor review March 27, 1975. Accepted July 10, 1975. The Belgian authors thank FKFO and FGWO for financial assistance.
Computed Alpha Coefficients for Electron Microprobe Analysis D. Laguitton, R. Rousseau, and F. Claisse Department
of Mining and Metallurgy, Universite Laval, Quebec, Canada
Computed a coefficients for the application of empirical equations in microprobe analysis are determined by the method developed by Rousseau and Claisse for X-ray fluorescence analysis. It is shown that the Claisse-Quintln relation with such coefflclents yields results of analysis Identical with those obtained by the ZAF method. Nearly as accurate results are obtained when the second-order terms relative to three elements are neglected.
The influence coefficient method ( a coefficients) can be either empirical as in the Ziebold-Ogilvie ( 1 ) method, semi-empirical as in the Lachance-Trail1 (2) and Claisse2174
Quintin ( 3 ) methods or theoretical as shown in the recent publication of Rousseau and Claisse (4). The empirical method of Ziebold and Ogilvie is based on the observation that curves of C A / K Aas a function of CA are nearly linear: C A K A= QAB
+ (1 - QABICA
(1)
where CA is the weight concentration of element A and K A is the ratio of the measured X-ray intensity of A in a binary solid solution of elements A and B and the measured intensity of A in a reference specimen of pure element A. The semi-empirical method of Lachance and Trail1 was first developed for X-ray fluorescence analysis but was ap-
ANALYTICAL CHEMISTRY, VOL. 47, NO. 13, NOVEMBER 1975
plied later to microprobe analysis with satisfactory results. I t is based on the principle that the ratio C A / K Amust be unity if no interelement effect exists in a binary system A-B and that, in other cases, a correction must be added to represent the interelement effect of element B on element A. This correction is characteristic of the system A-B and is proportional to the concentration of B, then:
~ABCB
CAIKA = 1
(2)
For X-ray fluorescence analysis, Claisse and Quintin ( 3 ) have shown that the polychromaticity of the primary radiat4on and the enhancement require that additional terms must be added to Equation 2. For a binary system, the relation becomes: CAIKA= 1
+ ~ A B C+B ~ A B B C B ~
and for a complex system:
For microprobe analysis it is possible to develop a relation similar to Equation 3, starting from the empirical approach of Ziebold and Ogilvie. As mentioned by these authors, the experimental data are often not linear and a second-order polynomial would provide a better approximation than Equation 1. We can write:
CAIKA= a0
+
UlCA
+ a2CA2
where ao, a l , a2 are constants. Replacing CA by 1 - CB in the right side of this equation: CA/KA =
(a0
+ a1 +
U2)
+ (-
U l
- 2a2)CB + a 2 C B 2 (4)
when CB = 0, CAIK.4 is unity; then Equation 4 takes the form
+
+
CAIKA = 1 ~ A B C B ~ A B B C B ’ (5) where CYABand ~ A B Bare constants characteristic of the system A-B. For a system with more than two elements, Equation 5 can be generalized by replacing Cg by CB Cc . . , and indices B by BC . . . For instance, in a ternary system, Equation 5 becomes:
+
cA/KA = 1 + ~ A B C ( C+Bc c ) + ~ A B C B C ( ~+BCc)* (6)
The cy coefficients being a measure of the influence of an element on element A, in the case of a ternary system, the effective value of a should be proportional to the ratio of the two elements which influence A. For the linear term we have:
and for the second-order term, we have
aABC
CBCC
(CB
+ cC)2
Then Equation 5 becomes identical to Equation 3 in which the coefficients with two different subscripts are “binary” coefficients and those with three different subscripts are “ternary” coefficients. When K A is known for element A in a specimen containing elements A, B, C . . . N, the concentration CA can be calculated if all the a coefficients are known: AB, ~ A C. . . CYAN, ~ A B B ,~ A C C. . . , ~ A N N ,~ A B C , ~ A B D . .. , ~ A B N (, Y A C D . . . , ~ A C N . .. . The calculation must be iterative since CI,, Cc . . , are not known; the intensity ratios K B , K c . . . are used as a first approximation of the concentrations.
COMPUTATION OF a COEFFICIENTS Albee and Ray (5) have developed a program for the calculation of a coefficients which uses intensity ratios calculated by the ZAF method. Recently, Rousseau and Claisse ( 4 ) have developed another method for the calculation of a coefficients which has given excellent results in X-ray fluorescence analysis. Because of the similar form of equations of C vs. K in the case of electron microprobe analysis, we have used the Rousseau-Claisse method for the calculation of a coefficients for microprobe analysis. The a coefficients relative to the calculation of the concentration CA of element A in any ternary system A-B-C are determined by calculating by the ZAF method the values of K A at five particular compositions (CA, CB, C c ) and solving the system of five equations obtained by substitution of these C and K values in Equation 3 as shown in Table I. I t has been found by experience ( 4 , 6) that best results are obtained when the five compositions (CA, CB, Cc) are: (0.25, 0.75, 0), (0.75, 0.25, 0), (0.25, 0, 0.75) (0.75, 0, 0.25), and (0.25, 0.375, 0.375). An APL program has been written to calculate the a coefficients for any given system of elements and experimental conditions. It uses as a subroutine a ZAF program by Laguitton et al. (7) for the calculation of intensity ratios in electron probe microanalysis. Details on these programs are available upon request. It must be emphasized that one difference between this method and others previously published ( I , 2, 5, 8) is the starting equation which is not the Ziebold-Ogilvie empirical relation but an extension of it with the addition of higher order terms. Another difference is the calculation procedure for determining the a coefficients. For instance, in published works, the intensity ratios that are needed to calculate the a’s were measured experimentally (8) or were calculated by existing computer programs but only in the case of the Ziebold-Ogilvie linear approximation ( I , 2 , 5 ) . RESULTS AND DISCUSSION In order to determine which relation between a , C, and K for the correction of interelement effects is the most accurate, the following three relations were compared: CA = K A ( + ~~ABC+ B ~ACCC)
(7)
CA = K A ( + ~~ABC+ B ~ A B B C B+’ ~ A C C + C QACCCC~) (8)
and Equation 3. For the sake of clarity we will describe the procedure of comparison in two steps. In a first step, a coefficients have to be computed using each of the three above equations, 7, 8, and 3; we therefore get three sets of coefficients. The first set is obtained by a least-squares fitting of the theoretical CIK vs. C curve calculated by the ZAF method for five binary concentrations, namely (0.1, 0.9), (0.2, 0.8), (0.5, 0.5), (0.8, 0.2), (0.9, 0.1). The second set is obtained by the same way except that a second-order polynomial (Equation 8) instead of a firstorder polynomial is used in the least-squares method. The third set of a’s is obtained by the Rousseau-Claisse method as described above. In a second step, the efficiency of these coefficients to recalculate concentrations from intensity ratios is then compared for a wide-range of ternary compositions. Ternary systems for testing these equations were chosen to represent common situations in microprobe analysis: these are Fe-Cr-Ni, Mg-Al-Fe, Cr-Co-Mo in which atomic number, absorption, and fluorescence effects are present. In each system, ten compositions were considered; they are shown in Figure 1.
ANALYTICAL CHEMISTRY, VOL. 47, NO. 13, NOVEMBER 1975
2175
Table I. The Five Equations Used to Calculate the a Coefficients" 0.25/KA, = 1 + C Y A BX 0.75 + Q A C X 0 + Q A B B X (0.75)' + ~ A
C C X
(0)' +
QABC X
0.75
0
Table IIA. a Coefficients for the System Fe-Cr-Ni (20 kV, 52.5" take-off angle) a AB
A
*
1
B -
Fe
1 0 2 0 3 0 Cr 1 -0.22 51 2 -0.0083 3 -0.0805 0.0904 Ni 1 2 0.0899 3 0.0900 *Method used: (1)Least squares values
Fe
a ABB
Cr
Ni
Fe
Cr
Ni
0.1459 0.1426 0.1427 0 0 0 0.0742 0.0728 0.0729
-0.2377 -0.0016 -0.0782 -0.0944 0.0073 -0.023 5 0 0 0
0 0 0 0 -0.2 167 -0.1279 0 0.0005 0.0004
0 0.0033 0.0032 0 0 0 0 0.0015 0.0013
0 -0.2361 -0.1419 0 -0.1017 -0.0639 0 0 0
ABC
0.0389
-0.0981
0.0019
(Eq. 7), ( 2 ) Least squares values (Eq. 8), (3) Method of Rousseau and Claisse (Eq. 3).
Table IIB. a Coefficients for the System Mg-Al-Fe (20 kV, 52.5' take-off angle) "ABB
u~~
A
1
*
Mg
1 2 3 1 2 3 1 2 3
A1
Fe
B-
Mg
0 0 0 2.0308 1.6004 1.6133 0.2107 0.2250 0.2265
A1
Fe
-0.0224 0.0483 0.0187 0 0 0 0.1796 0.1884 0.1899
1.8508 2.0310 2.0136 1.0838 1.2099 1.2087 0 0 0
Mg
0
0 0 0 0.4304 0.4151 0 -0.0143 -0.0162
AI
0 -0.0707 -0.0336 0 0 0 0 -0.0088 -0.0106
Fe
0 -0.1802 -0.1582 0 -0.1261 -0.1237 0 0 0
a~~~
-0.086 1
-0.1 104
-0.0263
* See Table IIA.
Table IIC. a Coefficients for the System Cr-Co-Mo (20 kV, 52.5" take-off angle) a~~~
"AB
+
A
*
1 2 3 co 1 2 3 Mo 1 2 3 * See Table IIA,
Cr
2176
B -
Cr
0 0 0 0.1373 0.1328 0.1330 0.2039 0.1992 0.2038
co
-0.1775 -0.0315 -0.0779 0 0 0 0.3291 0.2987 0.3041
Cr
co
MO
0 0 0 0 0.0045 0.0043 0 0.0047 -0.0009
0 -0.1460 -0.089 0 0 0 0 0.0304 0.0239
0 -0.0290 -0.0284 0 -0.013 7 -0.0134 0 0 0
Mo
-0.0051 0.0239 0.0235 -0.0523 -0.03 86 -0.0043 0 0
0
ANALYTICAL CHEMISTRY, VOL. 47, NO. 13, NOVEMBER 1975
"ABC
0.0035
-0.0091
0.0277
Table VI. Analysis of Real Samplesa System
Ti-Nb
Element
Figure 1. Distribution of ternary compositions used to compare the a coefficients methods
Absolute error A C b
Fe-Cr-Ni Mg-A1-Fe Cr-Co-Mo
2
10
3
5
4
0.0056 0.0031 0.0039 0.0009 0.0005 0.0044 0.0015 0.0030 0.0007 0.0003 0.0023 0,0009 0.0017 0.0003 0.0001 Relative --
Cr-Co-Mo Nb
Cr
Fe-Ni
Co
Mo
Fe
Ni
0.326 0.620 0.117 0.797 0.077 0.494 0.525 0.350 0.650 0,099 0.806 0.094 0.436 0.566 0.346 0.652 0.103 0.804 0,099 0.451 0.559 0.346 0.652 0.103 0.805 0.099 0.452 0.546
a Measured intensities are from Ref. 9; concentrations were calculated by the SAF method (Ref. 7) and the a coefficients method using computed coefficients.
Table 111. Mean Absolute a n d Relative Errors for the Five Different Methods of Application of a Coefficients Sistem
Measured K* C, True Calcd (ZAF) Calcd (a)
Ti
error AC I C
Fer-Ni Mg-A1-Fe Cr-Co-Mo
0.0180 0.0110 0.0283 0.0056 0.0036 0.0126 0.0056 0.0207 0.0037 0.0017 0.0075 0.0032 0.0115 0.0016 0,0009 a (1) a I jfrom Eq. 7 used in Eq. 7. ( 2 ) cyLl and aIllfrom Eq. 8 used in Eq. 8. (3) cyli from Eq. 3 used in Eq. 7. (4) aili and all, from Eq. 3 used in Eq. 8. ( 5 ) alIIcylIland a l i k from Eq. 3 used in Eq. 3. Each value is the arithmetic mean of 30 values: 10 concentrations of 3 elements.
Theoretical values of K A were used instead of experimental values which would have required a large amount of standards and would have suffered from experimental errors. These theoretical values were obtained by means of the PREVISONDE program (7) for the following experimental conditions: pure elements as reference samples, 20-keV beam energy, K a lines except for Mo for which La was used, 5 2 . 5 O take off angle. Values of CA were then recalculated from these theoretical intensity ratios, using coefficients from the three sets given in Table 11, in five different ways. 1) The first set of a's was used with Equation 7. 2) The second set of a's was used with Equation 8. 3) The first-order coefficients (@AB, ~ A C ,. . .) of the third set were used with Equation 7.
Table IV. Example of Calculation of Concentrations for a Ternary Sample Element: Ctrue: K
Fe 0.125 0.141
Cr
0.125 0.1351
Xi C.75
C8.735
( acoefficients from Table IIA)
1st Iteration
+
C F e= 0.141 X [l + 0.1427 X 0.1351 + (-0.0782) X 0.735 + 0.0032 X (0.1351)' + (-0.1419) X (0.735)' 0.0389 X 0.1351 x 0.7351 = 0.1254 Cc, = 0.1351 X [ 1 + (-0.0805) X 0.141 + (-0.0235) X 0.735 + (-0.1279) X (0.141)' + (-0.0639) X (0.735)' + (-0.098)X 0.141 x 0.7351 = 0.1248 C N i = 0.735 X [l + 0.09 X 0.141 + 0.0729 X 0.1351 + 0.0004 x (0.141)' + 0.0013 x (0,1351)' + 0.0019 X 0.141 x 0.13511 =
0.7516
2nd Iteration
0.141 X [l +. 0.1427X 0.1248 + (-0.0782) X 0.7516 + 0.0032 X (0.1248)' + (-0.1419)X (0.7516)' + 0.0389 X 0.1248X 0.75161= 0.1245 Cc, = 0.1351 X [l + (-0.0805)X 0.1254 + (-0.0235) X 0.7516 + (-0.1279) X (0.1254)' + (-0.0639) X (0.7516)'+ (-0.0981) X 0.1254 X 0.75161 = 0.1249 C,, = 0.735 x [l -t- 0.09 x 0.1254 + 0.0729 X 0.1248 + 0.0004 x (0.1254)' + 0.0013 x (0.1248)' + 0.0019 x 0.1254 X 0.12481 = 0.7500 A third iteration gives the final values of concentrations: Ccalcd= 0.1246 0.1249 0.7500. CFe =
Table V. Application of the a Coefficients Method to a Quaternary System Element Cr
co
Mo
Fe
True concentration 0.250 0.250 0.250 0.250 Intensity ratio K* 0.266 0.2433 0.2074 0.247 Calculated concentrations C" 0.2530 0.2502 0.2490 0.2474
Cb
* Theoretical
0.2496 0.2502 0.2501 0.2505
values (lines Cr K a , Co K a , Mo La, Fe K a , Eo = 20 kV,0 = 5 2 . 5 " ) . Keglecting ternary coefficients ARC, ~ A R D ,. . . . Introducing ternary coefficients in the last iteration only.
4) The first- and second-order binary coefficients of the third set were used with Equation 8. 5 ) The whole third set of a's was used with Equation 3. Average absolute and relative errors for each method of calculation are given in Table 111. I t is observed that the most accurate results are obtained when binary and ternary coefficients are used (column 5 ) . If quicker calculation is desired, results of column 4 indicate that the use of binary coefficients only in the Rousseau-Claisse method is quite satisfactory: considering the large number of coefficients in complex systems, this represents a great simplification without a significant decrease in accuracy. In addition, it
ANALYTICAL CHEMISTRY, VOL. 47, NO. 13, NOVEMBER 1975
2177
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. 8528 1
Peter R. Buseck Departments of Geology and Chemistry, Arizona State University, Tempe, Ariz. 8528 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
