Cu TARGET DIFFRACTION TUBE 45 KVCP.)
tron probe microanalysis (16) if he assumes that continuous radiation is generated as the same function of depth. There is really no valid reason for the assumption because depth distribution of the continuum has not been measured. However, for a Cu target diffraction tube the calculated and measured jump ratios were 6.0 and 4.5, respectively. For the Cr OEG-50, the ratios were 2.8 and 3.0. Secondary peaking was observed fo: each of the diFerent target tubes. In W it occurred a t 1.5 A, in Cu a t 0.9 A, and Its cause is unknown but it cannot be ascribed Cr at 0.8 t o the spectrometer characteristics because it occurs at different wavelengths for different targets. It is not a function of voltage because it remains at the same wavelength as the voltage is changed in Figure 5. It is not a tube window phenomenon because all the tubes have Be windows. Characteristic lines d o contribute a significant fraction of the total intensity, as shown above. This is significant in the fluorescent excitation of elements whose X-ray emission wavelengths are just longer than the characteristic lines from the tube target material. It points out the danger of assuming a single “equivalent” wavelength for exciting the specimen as is suggested in some of the regression methods. It seems appropriate that one should settle on a few standard operating conditions such as 15, 25, 35, or 45 kV, as it is obviously not possible to determine the spectral distribution for all tubes at all imaginable operating situations. Certainly, there are some circumstances which might make other than “standard” operating conditions desirable but these would be considered the exception.
A.
Figure 6. X-ray continuum from a Cu target X-ray diffraction tube DISCUSSION
The spectral distributions which have been measured represent the actual spectra available to excite the specimen in X-ray fluorescence analysis-that is, the spectra outside the tube window, not the original spectra generated in the targets. They show several features of significance in quantitative X-ray fluorescence analysis. Absorption jump has been generally neglected in the literature although it was recognized by Kuhlenkampff and must always be present in the spectra emerging from the target. The increased intensity on the long wavelength side of the absorption edge certainly contributes to increased fluorescent excitation of elements lower in atomic number than the target. One can calculate the approximate jump ratio from the depth distribution of characteristic radiation used in elec-
RECEIVED for review November 15, 1967. Accepted March 28, 1968.
(16) J. Philibert, “X-Ray Optics and X-Ray Microanalysis,” Academic Press, New York, 1963, p 379.
Calculation Methods for Fluorescent X-Ray Spectrometry Empirical Coefficients vs. Fundamental Parameters J. W. Criss and L. S . Birks U . S. Nacal Research Laboratory, Washington, D.C. 20390 Two mathematical approaches for correcting matrix effects i n quantitative X-ray spectrochemical analysis are compared. The empirical regression method, which has been i n use for many years, represents absorption and enhancement effects of each element on each other element by parameters independent of mass concentrations. The formulation used in this paper applies to measured intensities that already have been corrected for background and detector response, and so the number of standards required equals the number of components to be determined. Application to 15 iron and nickel based alloys produced deviations of 0.1% to 3.8% from wet chemistry values, and demonstrates that the method is valid only for specific X-ray equipment and for the analysis of specimens that are similar in nature to the standards used. The fundamental parameter method used is a modification of equations developed many years ago, but only recently used successfully in quantitative analysis. The present method uses measured primary spectral distributions, rather than a theoretical expression, and accounts for matrix effects by means of
1080
ANALYTICAL CHEMISTRY
measured mass absorption coefficients p and fluorescent yields W. A method of iteration is proposed to facilitate rapid computer analysis. The fundamental parameter method is more general than the empirical techniques, but at present is limited by uncertainties i n g and W . Application to the same set of 15 alloys as above produced deviations of 0.1% to 1.7% f r o m wet chemistry values.
THEPAST 15 YEARS have brought many suggestions for calculating chemical composition from measured line intensities in fluorescent X-ray spectrometry. In this paper, two approaches are described. One of them, the empirical coefficient method, is essentially the same regression method rediscovered many times in recent years but now simplified, updated, and in optimum form for computer evaluation. The other, the fundamental parameter method, is a more general approach suitable for any combination of elements and limited in precision only by uncertainties in the parameters themselves.
The purpose of all calculation methods is to reduce the need for comparison standards in quantitative analysis and thus to save time (ergo, money). Both of the methods described here satisfy the criterion but only a variety of industrial applications will determine which is better suited for specific problems. EMPIRICAL COEFFICIENT METHOD
Background. X-ray intensity from some element in a matrix of other elements is not linear with composition because of selective absorption or secondary fluorescence within the specimen (matrix effects). In general, measured characteristic intensity is a function of the mass concentrations of all elements present. Symbolically,
where Rr may be the intensity from element i measured directly, or expressed as a ratio to the intensity from a standard specimen of known composition, often the pure element i.
Of the several forms proposed for Equation 1, perhaps the most common functional relationship is / n
which is used to take into account the absorption of incident radiation and of emerging fluorescent radiation. Equation 2 is the form used for intensities R i that have already been corrected for background and detector response. Other treatments (1, 2) include additional parameters to account for these effects, but this paper will consider only the simpler form (Equation 2). Similar equations have been proposed many times in the past 15 years, both for X-ray fluorescence (1-10) and electron probe analysis (11-15). The simple form of this equation results from assumptions that are physically unrealistic, yet often useful for analyzing specimens that fall into a single well-defined type. For instance, iron-andnickel base alloys are one single type of specimen; copper ore is another type; phosphate rock and cement are still other types, etc. The basic assumptions made in the derivation of Equation 2 are that the specimen is homogeneous, thick, and has a flat surface, that the radiation used to excite the specimen is (1) W. Marti, Spectrochim. Acta, 18, 1499 (1962). (2) H. J. Lucas-Tooth and C. Pyne, “Advances in X-Ray Anal-
ysis,” Volume 7, W. M. Meuller, G. R. Mallett, and M. J. Fay, Eds., Plenum Press, New York, 1964, p 523. (3) L. von Hamos, Arkiu. Math. Asrron. Fys., 31a, 25 (1945). 26, 981 (1954). (4) H. J. Beattie and R. M. Brissey, ANAL.CHEM., (5) L. S. Birks, “X-Ray Spectrochemical Analysis,” Interscience, New York, 1959, p 60. (6) A. Guinier, Rev. Uniuerselle Mines, 17, 143 (1961). (7) H. J. Lucas-Tooth and B. J. Price, Metallurgia, 64, 149 (1961). (8) A. H. Gillieson, D. J. Reed, K. S. Milliken, and M. J. Young, Am. SOC.Testing Mater., Spec. Tech. Publ. 376, 1964, p 3. (9) R. J. Trail and G . R. Lachance, Geol. Suru. Can., Paper 64-57 (1964). (IO) R. 0. Muller, “Spektrochemische Analysen mit Rontgenfluoreszenz,” R. Oldenbourg Munich, Vienna, 1967, p 174. (1 1) R. Castaing, “Advances in Electronics and Electron Physics,” Academic Press, New York, 1960, p 317. (12) J. Philibert and H. Bizouard, “X-Ray Microscopy and Microanalysis,” Elsevier, 1960, p 419. (13) D. M. Poole and P. M. Thomas, “X-Ray Optics and X-Ray Microanalysis,” Academic Press, 1963, p 41 1. 36, 322 (1964). (14) T. 0. Ziebold and R. E. Ogilvie, ANAL.CHEM., (15) D. B. Wittry, Am. SOC.Testing Mater., Spec. Tech. Publ., 349, 128 (1964).
monochromatic, or, equivalently, a unique wavelength would produce the same effects as the actual spectrum used, for all specimen compositions to be considered, and that enhancement (secondary fluorescence) has the same effect as low matrix absorption. As a consequence of these assumptions, each a f jin the resulting Equation 2 is represented as the effect of only one element on another element, independent of whatever other elements are present, and therefore is assumed to be a constant. Actually, each ai5is nearly constant only in a narrow range of compositions corresponding to the standards used to evaluate the coefficients. Determining the Coefficients a I j . Previous discussions of empirical coefficients have suggested that they should be determined for each pair of elements in a system by preparing binary standards of those elements. That is, the effect of nickel on iron and of iron on nickel should be determined in a binary sample of Fe-Ni. We know now that it is better to determine all the coefficients simultaneously from multicomponent standards containing all the elements of interest. Not only is the number of initial standards reduced to a number equal to the number of elements to be analyzed, but the shortcomings of the assumptions are minimized because second order effects are automatically included in the measured intensities. The coefficients are determined by solving sets of simultaneous equations. For instance, in a three-component system, there are nine coefficients and they are found three at a time. Let A, B, C be the elements and 1 , 2 , 3 be the three known standards (variation in composition of the standards should encompass the full range of composition expected in the unknowns). First, consider the three coefficients LYAA, o ~ A B , a A c . The relative X-ray intensity for element A is measured in each of the three standards and Equation 3, rewritten from Equation 2, shows the relationships. The reference standard used in defining relative intensity may be any one of these three standards or it may be a separate, pure element standard.
+
+ + aACCC2 + aACCC3
CAI/RAI= ~ A A C A I~ A B C B I~ A C C C I CAZ/RA =~a A A C A 2
f aABcB2
C A ~ / R= A ~a A A C A 3 f
aABCB3
(3)
CAImeans the composition of element A in standard sample 1, etc. A similar set of equations is used to find CYBA, LYBB, aBC by measuring the intensities for element B in the three , are found by measuring standards; likewise LYCA, L Y ~ B LYCC intensities for element C. It may be possible to compensate somewhat for errors in measuring intensities from the standards by using more than the minimum number of standards (2, 8). In that case, a least squares solution would give a “best fit” set of values for the LY parameters. With these nine coefficients, an unknown sample is analyzed by rewriting Equation 2 in the form of Equation 4.
+ R A ~ A B C+BR A ~ A C C C0
(4a)
RBLYBACA f ( R B ~ B-B 1 ) c B f R B ~ B C C=C0
(4b)
( R A ~ AA 1)cA
+ +
=
+ +
RCCWACA R C ~ C B C B(Rcacc CA
CB
CC
-
1)Cc = 0
1
(4~) (4d)
Equation 4d is necessary because the other three equations are not mutually independent. In practical analysis if minor constituents are known to be present but are not of interest, Equation 4d may be set equal to the sum of the major constituents4.e. 0.98 for example. VOL 40, NO. 7, JUNE 1968
1081
Table I. Empirical Coefficients Ai5 Used in Test 1 for Matrix Effect on Element i by Presence of Element j j
i Cr Fe
co Ni Mo
Fe
Cr 1.559 2.419 1.ooo 15.428 0.991
Co 0.971 1.149 2.184 -7.221 0.110
0.516 1.023 1.OOO -1.110 1.001
Ni 0.670 0.850 1.ooo -0.990 1.001
Mo 1.731 1.286
1.ooo 5.336 1.325
Examples of Analyses. The method above was tested with a set of alloys containing the five elements Cr, Fe, Co, Ni, and Mo. The set included 300 series and 400 series stainless steels, an Inconel, and other high-temperature alloys. Each of the specimens had been analyzed by wet chemistry so that all compositions were known and could be compared with X-ray analysis. The five specimens selected as standards, for calculating the 25 a f j coefficients from equations like 3, were No. 301, No. 410, No. 1187, Hastelloy B, and Inconel X. Table I shows the values of the a f j coefficients determined. It is interesting to note that, if the reference standard used in defining relative intensity for each measurement is a pure element, then the idealized Equation 2 demands that ai(= 1 for each element i. In practice this is not true, for the computer selects those values for all the coefficients that best fit the measured intensities from the standards. These coefficients were used to determine the compositions of No. 303, No. 304, No. 316, No. 321, No. 347, and No. 430 stainless steels, from measured X-ray intensities. The results are given in Table 11, under the heading “By Empirical Coefficients, Test 1”. Since the coefficients were determined from alloys covering a wide range of compositions (listed in Appendix 2), these coefficients should be useful in analyzing high-temperature alloys as well. Next, a new set of atj’s was determined from standards all of the 300 series steels. This might be the procedure used if
Alloy 303
Element Cr
Mn Fe Ni 304
Cr
Mn Fe Ni 316
Cr
Mn Fe Ni 321
Cr Mn
Fe Ni 347
Cr
Mn Fe Ni 430
Cr
Fe
1082
one were analyzing only stainless steels. Mn was neglected, as before, so only three standards were required to describe the interelement effects among Cr, Fe, and Ni. The values of ail calculated using No. 301, No. 303, and No. 316 as standards are shown in Table 111. A comparison of Tables I and I11 shows that the values of the coefficients are quite dependent on the nature of the standards, and might therefore be very sensitive to errors in measured intensities or in assumed compositions of the standards. However, this quite different set of coefficients led to approximately the same values of calculated weight per cent (Table 11, under the heading “By Empirical Coefficients, Test 2”). In both cases, the errors were small enough to be acceptable for many analyses. From the above observations, one must conclude that the coefficients selected in this way do not represent the simple effect of one element on another, as was hypothesized in the derivation of Equation 2. Rather, they are the best set of numbers to describe the intensities measured. Thus one has the reason for the earlier statement that the empirical coefficient method is most adequate when applied to repeated analyses of a single type of specimen, such as the iron- and-nickel base alloys. None of the coefficients obtained for this system of elements would be applicable to the analysis of these elements in ores such as chromite or olivine because the remainder of the matrix is completely different. FUNDAMENTAL PARAMETER METHOD
Background. In contrast to the empirical coefficient method, the fundamental parameter method to be described assumes only that the specimen is homogeneous, thick, and has a flat surface. Instead of assuming that the incident spectrum can be described by a single average wavelength, it uses the measured primary spectral distribution for a given target and operating voltage. The matrix absorption and secondary fluorescence enter explicitly for each specimen, and composition is calculated directly by iteration. The advantage is that no intermediate standards or empirical coefficients are needed for any matrix. The disadvantage is the
Table 11. Comparison of Chemical and X-Ray Analyses of Six Stainless Steels By chemical analysis By empirical coefficients, test 1 By empirical coefficients, test 2 By fundamental parameters Weight Difference Weight Difference Weight Difference Weight 17.2 17.8 0.6 Used as standard 18.5 1.3
z
1.3 71.2 8.7 18.6 1.4 69.5 9.4 17.7 1.8 64.8 12.8 17.8 1.6 68.2 10.8 17.7 1.6 67.9 10.7 17.5 81.3
ANALYTICAL CHEMISTRY
z
z
73.8 8.4 18.8
2.6 -0.3 0.2
71.0 9.7 17.8
1.5 0.3 0.1
66.0 13.2
1.2 0.4
18.5
0.7
18.4
0.6
71.7 10.9 18.5
3.5 0.1 0.8
70.6 11.0 18.4
2.4 0.2 0.7
71.7 11.7 17.6 79.3
3.8 1.0 0.1 -2.0
70.2 11.4 17.5 81.4
2.3 0.7 0.0 0.1
19.0
0.4
71.7 9.3
2.2 -0.1
Used as standard
1.3 71.5 8.7 19.8 1.5 69.6 9.1 19.0 1.8 66.5 12.7 19.0 1.7 68.7 10.6 18.9 1.7 68.5 10.9 18.8 79.6
0.0 0.3 0.0 1.2 0.1 0.1 -0.3 1.3 0.0 1.7 -0.1 1.2 0.1 0.5 -0.2 1.2 0.1 0.6 0.2 1.3 -1.7
1
I
W Target OEG-50
La1
5 0 KVP
/is
€
La,
3
Ldz Figure 2. Schematic of the production of primary fluorescent radiation
WAVELENGTH
(A)
Figure 1. Spectrum from tungsten target tube at 50 kvp as measured by Gilfrich and Birks present uncertainty in the mass absorption and fluorescent yield parameters. This shortcoming is being overcome rapidly however, as more laboratories devote themselves to remeasuring the parameters accurately. Derivation of Expressions. Formulas have been derived (16-18) to express measured characteristic intensities in terms of specimen composition and the fundamental parameters. The first parameter needed is the spectral distribution of the primary radiation. The present treatment differs from previous derivations (16-18) principally in the use of recently measured spectra rather than theoretical descriptions of the Xray tube output. Spectra have been measured for a tungsten target X-ray tube operated at 15,25,35, and 50 kilovolts peak potential with a full-wave-rectified power supply, and for tungsten, chromium, copper, and molybdenum target tubes operated at 45 kilovolts constant potential (19). Figure 1 shows the spectrum for tungsten at 50 kilovolts peak potential. Table IV lists the values from Figure l . It may be surprising to some readers that the characteristic lines contribute such a large share of the total. The discontinuity at the LIIIedge for W is due to the sudden change in target absorption for emerging radiation at that wavelength. The formulas for measured characteristic intensity assume that the incident radiation impinges at some angle E to the surface of a flat, homogeneous, thick specimen, and that the measured characteristic radiation emerges at some angle +, as shown in Figure 2. The penetration of radiation is assumed to obey the familiar exponential law
Z(z) = Z(0) exp
[-p(X)pz
cosec e]
if the incident spectrum is divided into a number of wavelength intervals AXk with corresponding intensities Z(Xa). Di(X,) is unity for all values of k for which XI, is small enough to excite element i , and is zero for all other values of k . Equation 7 and following formulas are essentially the same as those derived by other investigators (16-18), except that the integral over X is here replaced by a summation. The wavelength of the measured characteristic radiation is X f . The constant g i is related to the absolute intensity from element i, but cancels when one considers relative X-ray intensity
where Pi* refers to a standard specimen of any known composition (for example, a pure element). The expression for Pi* will be like that for PI,except that Cf, p M ( X k ) , and p M ( X i )will depend on the composition of the standard. In many specimens, some element j will produce characteristic radiation of wavelength short enough to excite the element
(5)
where the mass attenuation coefficient p depends on the X-ray wavelength and the absorbing medium. For an N-component specimen, p of the matrix for some radiation X is a massfraction average of the p’s of the individual elements: (16) (17) (18) (19)
The intensity of emergent primary characteristic radiation (fluoresced only by the incident spectrum) for some element i has been found to be
J. Sherman, Spectrochim. Acta, 7 , 283 (1955). J. Sherman, Ibid.,466 (1959). T. Shiraiwa and N. Fujino, Japan J. Appl. Phys., 5,886 (1966). J. V. Gilfrich and L. S. Birks, ANAL.CHEM., 40,1080 (1968).
Table 111. Empirical Coefficients Ai, Used in Test 2 for Matrix Effect on Element i by Presence of Element j
i i
Cr
Cr
1.801 3.391 -1.050
Fe Ni
Fe 1.608 1.997 3.651
VOL 40, NO. 7, JUNE 1968
Ni 2.616 2.215 5.611
1083
~~
Table IV.
Integrated Intensity in Spectrum of a Tungsten Target Tube a t 50 kVp Continuum
1,AX
A" (A)
~~
A" (A)
IxAX
A" (A)
IxAX
A" (A)
IxAX
1.03 30.0 1.79 58.6 95.8 2.55 14.1 1.05 1.81 56.6 53.2 93.8 2.57 13.8 1.07 1.83 76.8 91.8 2.59 13.4 54.6 1.09 1.85 52.4 104 90.0 1.11 1.87 50.0 124 88.4 1.13 1.89 47.6 148 87.0 Characteristic Lines* 1.15 1.91 169 85.8 45.4 1.17 1.93 43.3 184 84.6 L,2+3 1.06460 68.8 1.19 1.95 41.2 196 83.5 1,210 1.97 39.2 202 84.2 LY1 1.09855 155 1.23 1.99 37.3 205 90.2 1.25 2.01 35.6 206 89.8 L82 1.24460 532 1.27 2.03 205 34.0 89.3 1.29 2.05 32.4 202 88.8 L8l 1.281809 1464 1.31 2.07 31.0 195 88.3 0.55 1.33 2.09 187 87.8 29.7 0.57 La1 1.47639 2072 1.35 2.11 28.5 0.59 180 87.4 1.37 2.13 173 27.4 86.9 0.61 L, 1.6782 68.8 1.39 2.15 167 26.3 86.4 0.63 1.41 2.17 163 25.3 85.8 0.65 1.43 159 24.3 85.1 2.19 0.67 1.45 2.21 154 23.4 84.4 0.69 1.47 2.23 150 22.6 83.7 0.71 1.49 21.8 83.1 2.25 0.73 146 1.51 21.1 2.27 141 82.3 0.75 1.53 2.29 137 20.4 81.3 0.77 1.55 19.6 2.31 133 80.2 0.79 1.57 129 18.8 2.33 0.81 78.9 1.59 18.0 77.4 2.35 0.83 125 1.61 17.3 75.9 2.37 0.85 121 1.63 16.8 2.39 118 74.0 0.87 2.41 1.65 114 16.3 72.2 0.89 1.67 111 15.7 70.2 2.43 0.91 1.69 108 15.2 68.4 2.45 0.93 1.71 105 14.9 66.6 2.47 0.95 1.73 103 14.7 64.6 2.49 0.97 1.75 100 14.5 62.6 2.51 0.99 1.77 97.9 14.3 60.6 2.53 1.01 X for Continuum is the middle of the AX interval. * X for Lines is from Bearden (weighed average for Ly2+3). AX for Lines is natural line breadth (from Blokhin, "Physics of X-Rays," AEC translation 4502). c LIII edge occurs at 1.216 A. (1,AX is 66.1 from 1.200 to 1.216 and 18.1 from 1.216 to 1.220 A.) 0.27 0.29 0.31 0.33 0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49 0.51 0.53
Q
of interest i. Such secondary radiation can contribute as much as 30% to the total measured intensity from element i. The formula that includes both primary and secondary radiation is
pi + st
= gfct
F
Dr(Mpt(XdI(Xx)& . p w ( X k ) cosece p w ( X z ) cosec 4
+
All symbols except K j have been defined earlier.
K,
=
(1
-
:)
w
where J and w are the absorption-edge-jump ratio and fluorescent yield corresponding to the particular characteristic line of element j that excites element i. The summation over j indicates that the contributions of all such sufficiently energetic lines must be added together. When we express relative X-ray intensity for the care of secondary fluorescence, the constant gf cancels: 1084
0
ANALYTICAL CHEMISTRY
The denominator refers to a standard specimen of known composition. It is given by Equation 9, except that C t , C,, pIM(XL),pa&), and p M ( X ~will ) depend on the composition of the standard. If the standard is a pure element, then Sf* will be zero. These equations express emitted relative X-ray intensity in terms of specimen composition. However, it is relative intensity that is measured, whereas composition is the unknown. Unfortunately, algebraic manipulations cannot produce explicit formulas for the mass fractions C1, CZ,etc. Therefore, these mass fractions must be found by an iteration procedure that involves making successively better estimates of mass fractions until the corresponding calculated relative intensities agree with the measured relative intensities. We used a nonlinear (hyperbolic) interation equation (see appendix). In the computer program, the measured values of the R , are scaled to 100% and used as the first assumed values of the Ct's. From these Ct's a new set of Rt's calculated and compared with the measured values. The differences are used to adjust the assumed Cl's and the iteration is continued until, for some set of assumed Ct's, the calculated Rf's agree with the
Table V. Per Cent Composition of Hastelloy X Calculated in Successive Iterations, by Fundamental Parameters Method Mo Iteration co Ni Cr Fe 18.77 1.41 13.33 1 26.81 39.68 46.07 1.47 9.96 19.60 22.90 2 10.31 47.55 18.59 3 22.11 1.44 1.43 10.41 4 47.58 22.20 18.38 10.42 47.53 18.39 1.43 5 22.23 10.41 47.53 1.43 18.40 6 22.23 10.41 47.53 1.43 18.40 7 22.23
measured values. Table V shows the compositions calculated on successive iterations in an analysis of Hastelloy X. Even though seven iterations are shown, one can see that only three or four iterations would be required to achieve acceptable precision. The computer program ensures that the iteration will converge. However, the accuracy of the answer depends on the accuracy of the intensity measurements and the fundamental parameters involved. Examples of Analyses. It seemed appropriate to apply the fundamental parameter method to the same specimens analyzed with the empirical coefficients. The results are included in Table 11. The deviations from the chemical values for Cr are greater for the fundamental parameter method than for the empirical coefficient method. This definite bias in the results is probably traceable to uncertainties in p or o used in calculating Rc,. The bias could be corrected artificially but this would be inappropriate for the present. Elements such as Mn are included in Table I1 because any element present can be calculated by the fundamental parameters method. Mn as a minor constituent had been disregarded in setting up the empirical coefficients in Table I, so Mn could not be calculated by Equation 4. SUMMARY
The regression equations described here allow empirical coefficients representing the matrix effects to be determined with a minimum number of standards,-Le., the same number of standards as elements to be determined. Alternative empirical approaches would include additional parameters and would use more than the minimum number of standards. The number values of the parameters do not represent the direct effect of element A on element B but rather the best set of numbers for calculating composition from measured intensities. This limits application of a particular set of coefficients to the same general type of specimen for which the coefficients were determined. That is, coefficients determined for metal alloys cannot be used for minerals. The advantage is that the coefficients determined in this way give better analyses than coefficients determined on the assumptions that they truly represent the effect of one element on another or that they are universal constants and sufficient for all matrix combinations. A more direct but more complex mathematics is given for general analysis based only on tabulated values of the fundamental parameters I @ ) , p, and a. With present computer facilities the fundamental parameter method is just as easy as the empirical coefficient method. Application of both methods to a set of six iron-and-nickel base alloys shows deviations from wet chemical analysis of about 4 for the empirical coefficient method and 2 for the fundamental parameter method. The errors in the fundamental parameter method are caused by present uncertainties
in the mass absorption coefficients and fluorescent yields, but these are being remeasured and should be sufficiently accurate within a few years. In the meantime, bias in results can, if desired, be adjusted out by comparison with one or two standards. SYMBOLS USED
No one set of symbols satisfies all the international and national commissions. The symbols selected here conform to general acceptance to the best of our knowledge. R r is relative X-ray intensity for element i; that is, the intensity for element i in a matrix of other elements divided by the intensity from a reference standard containing element i. Cr is the weight fraction of element i. cyij is the empirical coefficient representing the matrix effect on element i by the presence of elementj. IAAAis the integrated intensity in the primary incident specAh, in unit of photons trum between wavelengths X and h per second on an arbitrary scale. p,(h) is the mass absorption coefficient of element i for primary radiation of wavelength h. p&) is the mass absorption coefficient of element i for characteristic radiation from element j . wK( is the K shell fluorescent yield for element i. J is the absorption edge jump-ratio corresponding to the characteristic line of interest.
+
APPENDIX 1. ITERATION METHOD
The iteration equation used at present in the fundamental parameters calculations was developed originally to treat intensities measured relative to a pure element standard (20). This is no limitation, however, because relative X-ray intensities obtained using a multicomponent comparison standard can be converted easily to intensities relative to a pure element by the formula
R = -
I (unknown)
I (unknown) I (standard)
The first term in brackets is the measured relative intensity, where the comparison standard may be any specimen of known composition, containing the element of interest. The second term in brackets is the intensity from the standard, relative to a pure element, calculated from the known composition. In the remainder of this section, relative intensity will mean the intensity of characteristic radiation of some element in the unknown, divided by the intensity from a standard of purely that element. In calculating composition from measured relative intensities, it is necessary to make some initial estimate of mass fractions and then compare the corresponding calculated relative intensities with those actually measured. Successively better estimates of composition are found by an iteration procedure. The most common method for finding the next estimate of concentration, C’t, from the present estimate, C t , for each element i, is to use the equation (20) J. W. Criss and L. S. Birks, “The Electron Microprobe,” T. D. McKinley, K. F. J. Heinrich, and D. B. Wittry, Eds., Wiley, New York, 1966, p 217. VOL. 40, NO. 7, JUNE 1968
1085
where Rt is the calculated relative intensity corresponding to the present estimates, and Rimis the measured relative intensity, On each iteration, this formula is applied to every element and then the new values, C’t, are scaled so that their sum is unity. This iteration equation is equivalent to a linear interpolation between the points (0, 0) and (Ct, Rt) on a graph of relative intensity us. concentration. A better iteration equation can be obtained by using a threepoint interpolation, among the points (O,O), (Ct, Rt), and (1, 1). The curve chosen to fit these three points was a rectangular hyperbola whose conjugate axis is parallel to the line joining (0, 0) and (1, 1). The resulting iteration equation is
This equation should cause more rapid convergence to a solution than a straight line. The steps in the iteration procedure are summarized as follows: (I) Make initial estimates of concentrations, such as the measured relative intensities, normalized to add to unity. (11) Calculate the corresponding expected relative intensities. (111) Scale the measured intensities to add to the sum of the calculated intensities. This step is not essential, but it guards against the possibility of bad measurements preventing a solu-
tion. (IV) Calculate the next estimates of concentrations by applying the iteration Equation A-3 to each element i. (V) Scale the new concentrations to add to unity. (VI) Decide whether these concentrations fit the measured relative intensities, within the desired precision. For example, one might test whether the per cent change between Cc and Clt, for every element i, were less than some arbitrary limit, say 0.1%. If more iterations were necessary, the sequence of steps would be repeated, starting with step 11, using these new concentrations.
APPENDIX 2. Per cent compositions of the standards used in the empirical coefficients, test one, are as follows. Hastelloy B No. 1187 InconelX No. 410 No. 301 Cr 0.00 21.62 14.00 12.60 17.90 Fe 6.00 27.40 8.50 86.09 72.67 co 0.00 20.80 0.00 0.00 0.00 Ni 62.00 20.26 78.10 0.16 7.23 Mo 32.00 3.41 0.00 0.05 0.21
RECEIVED for review November 15, 1967. Accepted March 28, 1968. Information presented in part at the Fifth National SAS Meeting, Chicago, Ill., June 1966, and in part at the Denver X-Ray Conference, Denver, Colo., August 1967.
Ultraviolet Spectrophotometric Study of the Determination of Cadmium, Cobalt ,Bismuth, and Molybdenum by the Pyrrolidinedithiocarbamate Method Melvyn B. Kalt’ and D. F. Boltz Department of Chemistry, Wayne State University, Detroit, Mich. 48202
Ammonium pyrrolidinedithiocarbamate has been investigated as a reagent for the spectrophotometric determination of certain metals. Pyrrolidinedithiocarbamic acid forms metal chelates which are soluble in chloroform, methylisobutyl ketone, and other organic solvents. The extracted pyrrolidinedithiocarbamates of cadmium, cobalt, bismuth, and molybdenum have absorbance maxima in the ultraviolet region. The main advantage of pyrrolidinedithiocarbamate is that it can be used in acidic solutions. Procedures have been developed for the ultraviolet spectrophotometric determination of these elements. The molar absorptivities range from 3.50 x 104 I/molecm for the cadmium chelate to 8.5 x 103 for the molybdenum system.
nickel (5). Bode (6-8) has studied the extraction of certain metal-DEDTC chelates by carbon tetrachloride and noted that several complexes exhibited ultraviolet absorptivity below 320 mp. Lacoste, Earing, and Wiberley (9) reported that DEDTC could be used for the colorimetric determination of bismuth, cobalt, chromium, iron, nickel, and uranium. Chilton (10) determined copper, cobalt, and nickel by a simultaneous spectrophotometric method using DEDTC and a carbon tetrachloride extraction; the wavelengths of measurement were 436, 367, 328 mp, respectively. An ultraviolet
THEREhave been many analytical applications of sodium diethyldithiocarbamate as a reagent since it was introduced in 1908 by Delepine (1) for the detection of copper and iron. Sodium diethyldithiocarbamate (DEDTC) has been used extensively as a chromogenic reagent in the determination of copper (2-4) and occasionally for the determination of
(2) P. Callan and J. A. R. Henderson, Analyst, 54,650 (1929). (3) T. P. Hoar, ibid., 62, 657 (1937). (4) E. B. Sandell, “Colorimetric Determination of Traces of Metals,” 3rd Ed., Interscience, New York 1959, p 444. (5) 0. R. Alexander, E. M. Godar, and N. J. Linde, IND. ENG. CHEM.,ANAL.ED., 18, 206 (1946). (6) H. Bode, 2.Anal. Chem., 142, 414 (1954). (7) Zbid., 143, 182 (1954). (8) Zbid., 144,90, 165 (1955). (9) . _ R. J. Lacoste, M. H. Earing, - . and S. E. Wiberley, ANAL.CHEM., 23,871 (1951). (IO) J. M. Chilton, ibid., 25, 1274 (1953).
1
Present address, M&T Chemicals, Inc., Ferndale, Mich.
(1) M. Delepine, BUN. SOC.Chim, (3) 4, 652 (1908).
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