ibility for a larger suite of 2-gram samples. Each replicate determination was obtained from the average of a t least two integrations of X-ray intensity. Notice that in every case of great differences between replicates, the opposite member of the La, Ce pair changes in value also, and in the direction of the first member. It would seem that the precision limit of the method is the intentionally introduced inhomogeneity of the rare earth sulfide precipitate. The absence of Pr determinations in steel reflects the low concentration of Pr found in the starting materials. F o r both the rare earth-silicon alloy and Mischmetall (a mixture of the metallic rare earths), Pr normally occurs at one-tenth the concentration of Ce. It is expected, however, that the
arguments advanced for the La and Ce cases would also hold true for Pr. In conclusion, the method developed can determine the three rare earths a t concentration levels of 0.002-0.05%. This range can be enlarged by appropriate adjustments t o the sample size and solution volume. Estimates of the precision, derived from elemental additions of the rare earths to steel solutions, are 8-14 pg. Accuracy cannot be determined a t this time because of a lack of suitable standards. Detection limits are 0.002-0.004 %, based on 2-gram steel samples. RECEIVEDfor review March 30, 1971. 1971.
Accepted July 19,
Theoretical Analysis of Quantitative X-Ray Emission Data: Glasses, Rocks, and Metals Donald A. Stephenson Research and Decelopmmt Laboratories, Corning Glass Works, Corning, N . Y. 14830 A new approach to the theoretical treatment of quantitative X-ray emission (fluorescence) data gives results for test cases involving glasses, rocks, and metal alloys whose accuracy is comparable with that of good wet-chemical analysis. The method is independent of the spectral distribution of the X-ray source and requires only one standard per element analyzed (one standard containing all the elements is also sufficient). The standards can be either pure elements, simple oxides, or up to 20 component mixtures. A computer program, CORSET, applies the necessary corrections for absorption and secondary fluorescence and should permit complete analysis of any homogeneous sample provided the measured X-ray intensities of the analyzed elements are obtained with sufficient accuracy.
edge of mass absorption coefficients, so the complete integral cannot be very accurately evaluated. The concept of a n effective energy (wavelength) defined in various ways by many of the aforementioned authors may help somewhat, but it is often slightly sensitive to compositional variation and again requires knowledge of the spectral distribution of the X-ray source. An effective energy defined by invoking the mean value theorem is probably unique only in the case where no characteristic X-ray lines from the tube possess energy greater than the absorption edge of the analyzed element-i.e., all fluorescence is due t o Bremsstrahlung. The mean value theorem guarantees that for a continuous function = .f(x), over an interval a 5 x 5 b that possesses a n antiderivative a t each x, there is at least one number c
IN RECENT PUBLICATIONS, Sherman ( I ) , Shiraiwa and Fujino (2), and Criss and Birks (3)have described general approaches to the solution of absorption and secondary fluorescence (enhancement) effects in quantitative X-ray emission spectrometry from first physical principles. Ebel, Meizer, Dirschmid, and Wagendirstel ( 4 ) , Ebel (9,and Ebel, Meizer, Wagendirstel, and Dirschmid (6) have approached the problem using variable take-off angles, and Tertain (7) has employed a multiple dilution technique. The general approaches, although offering a n exact solution, are seriously handicapped by the necessity of integrating over the spectral distribution of the X-ray source. While the spectral distribution of any X-ray source can now, in principle, be easily determined using energy dispersive X-ray detectors, integration of the higher energy (greater than about 18 keV) portion of the spectrum suffers from the absence of a precise knowl-
between a and h such that
(1) J.
Sherman. Spccttvrliim. Actci. 1. 283 (1955).
(2) T.Shiraiwa and N. Fujino, J c q w J . Appl. Phys., 5, 886 (1966). (3) J . W. Criss and L. S. Birks, ANAL.CHEM., 40, 1080 (1968). (4) H. Ebel. W. Meizer. H. Dirschmid, and A. Wagendirstel. Spcctrocliini. Ac/N..24B, 351 (1969).
( 5 ) H . €bel, At/t’cc/i.X - M A/Io/., ~ 13, 68 (1970). ( 6 ) H. Ebel. W. Meizer, A. Wagendirstel. and H. Dirschmid. Sprc/rocliini. Actcr, 25B, 83 ( 1 970). (7) R . Tertain, ibid.. 26B, 71 (1971).
~3
J’(x) d x
=
f (c)
.
( h - a).
There can indeed be more than one c, as may be the case where characteristic lines as well as Bremsstrahlung are capable of causing a n element t o fluoresce. Methods requiring specialized instrumentation for varying take-off angles and multiple dilution techniques, although they are applicable in specialized analytical situations, are neither sufficiently general nor easily enough implemented t o be of immediate practical benefit. The multiple regression techniques and semitheoretical approaches reviewed and briefly discussed by Stephenson (8) either require large numbers of well defined standards or involve a trade-off between theory and empiricism that renders the assessment of systematic errors very difficult. What is needed is a new approach to the theoretical reduction of X-ray emission data that accounts for absorption and secondary fluorescence effects, is independent of the spectral distribution of the X-ray source, and requires only one standard per element determined (one standard containing all the elements or several standards, each containing some of the elements, should also be acceptable). The procedures described herein meet these requirements and appear to be sufficiently accurate, about 1 to 2z relative, over large (8) D. A. Stephenson, ANAL. CHEM., 43, 310 (1971).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1971
1761
compositional variations. A Fortran computer program, CORSET, that performs all the necessary calculations will be made available in either batch or time-sharing versions. The only analytical requirements made by CORSET are that standards and unknowns are completely homogeneous or can be made so by a n appropriate procedure like flux-fusion, as described by Stephenson (9), and that the X-ray intensity data are free of line interferences and can be corrected for background and dead time. The principles briefly described here will be more fully documented in a future paper. DESCRIF'TION OF THE THEORETICAL FORMULAS
Iicalcd
el e2
CLiE
Wi Ki Oi
Ri
is the original intensity of the most efficient excitation energy for the ith element is the contribution of secondary fluorescence to the X-ray intensity of the ith element is the observed characteristic intensity of the ith element is the calculated X-ray intensity of the ith element is the angle of incidence of the excitation energy, measured with respect to the sample normal is the angle at which characteristic X-rays are observed, measured with respect t o the sample normal is the mass absorption coefficient of a sample for the most efficient excitation energy of the ith element is the mass absorption coefficient of the sample for the ith characteristic X-ray energy is the mass absorption coefficient of the ith element for its most efficient excitation energy is the weight fraction of element i is the absorption edge jump ratio of element i is the fluorescent yield of element i is the fraction of the ith element's characteristic X-rays in the analyzed line's series, e.g., Kw,?/(Ka1,2
+
KP1,3)
For incident monochromatic X-rays of sufficient energy t o generate characteristic X-rays within a pure element, the observed intensity, neglecting all instrumental factors, of a characteristic X-ray emission line from that element is given by
Equation 1 contains two intrinsic unknowns, I ~ i g the , original intensity of incident monochromatic X-radiation, and the energy of that incident radiation. If we are able t o specify a unique energy of incident radiation, Equation 1 can be solved for Zprig, thus
incident X-radiation is concentrated (monochromatized) at an energy somewhat greater than that of the absorption edge of the analyzed line and then ask: What is the original intensity of that assumed energy? In fact, the seemingly naive guess a t the value of the effective energy may turn out t o be fairly good in many cases; see, for example, LeRoux, Mahmud, and Davey (IO). The fact that the value of the effective energy varies with composition may in many instances be completely offset by the uncertainty in the mass absorption coefficients and the derived jump ratios. I n essence, this procedure asks each element in the standard(s) what it thinks is the original intensity of its most efficient excitation energy via Equation 2. What we have done, without any loss of generality, is t o switch the uncertainties involved from mass absorption coefficients and spectral distributions t o derived X-ray intensities, which are much easier t o handle. Any error made in assigning the most efficient excitation energy relative t o the effective energy is reflected in the calculated value of the original intensity as are all the random and systematic errors associated with fluorescence yields, jump ratios, etc. Moreover, all errors made in the calculation of the original intensities for the standard(s) will also be made in the analysis of unknowns, so that this procedure is at least self-consistent. The most damaging error likely t o arise is a n imprecise knowledge of the composition of the standard(s). Neglecting secondary fluorescence effects for the moment, we may now analyze unknowns based upon our assumed most efficient excitation energies and calculated original intensities by employing Equation 1 with Itcalcd substituted for Iiobsd for each analyzed element. This is accomplished by iteratively adjusting the weight fraction W , of each element in the unknown until observed and calculated X-ray intensities agree within a preset value. Errors due to counting statistics may in many instances be equivalent t o an error of around 1 or 2x relative in the actual weight fractions, since the answers can certainly be no better than the data from which they were derived. What this has accomplished is really a first approximation t o the true composition of the unknown, since secondary fluorescence effects have been ignored. Where secondary fluorescence poses little threat t o the desired analytical accuracy, the absorption corrections alone will often sufficee.g., where the unknowns deviate less than around *lox relative from a composite standard. This is often the case where heavier elements are determined in a predominently light matrix such as the analysis of rocks, glass, certain refractories, and other high-oxygen-content material. However, secondary fluorescence is usually significant, so it is wise t o account for it from the onset. Fortunately, under the assumption of a most efficient excitation energy for each analyzed element, the fluorescence correction reduces t o a n expression that can be easily evaluated, namely,
Since we assume no knowledge of the physical (real) energy distribution of the X-ray source, the various definitions of effective energy are of little assistance, but we can attempt to make a reasonable estimate of what it might be. Now the energy that is capable of generating the greatest fluorescent radiation (the most efficient incident energy for the element under consideration) is that energy slightly greater than the energy of the absorption edge associated with the measured X-ray line. I n effect, we shall assume that all (9) D. A. Stephenson, ANAL.CHEM., 41, 966 (1969). 1762
{ T l n [ l p s+2 sec T O2 Ps COS
81
1
+
In
(10) J. Leroux, M. Mahmud, and A. B. C. Davey, Carl. Spectrosc., 15, 169 (1970).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1971
Table I. Accepted Compositions, Relative X-Ray Intensities and CORSET Analysis of the Five Glasses Glass 2 Glass 3 Glass 5 Glass 1 Glass 4 Corn- Accept. X-ray Accept. X-ray CORSET Accept. X-ray CORSET Accept. X-ray CORSET Accept. X-ray CORSET value inten. value value inten. value value inten. value value inten. value ponent value inten. Si02 A1203
Fez03
CaO MgO
Na20 K20
so
3
BaO AS203
TiOz SrO ZrOl
where
72.10 1.80 0.047 7.60 3.73 14.20 0.25 0.25 0.01 0.05 0.03 0.06 0.02
Q5E
= prE
0.958 0.512 0.283 0.670 1.OOo 0.858 0.109 0.840 0.128 1.000 0.293 0.834 0.509
W5
72.92 1.50 0.045 4.93 3.47 16.54 0.32 0.20 0.01 0.00 0.02 0.03 0.04
0.965 0.443 0.273 0.443 0.901 1.000 0.140 0.654 0.030 0.060 0,219 0.445 1.000
72.88 1.59 0.042 4.97 3.41 16.49 0.33 0.20 0.003 0.003 0.02 0.03 0.04
73.30 1.OOO 1.20 0.357 0.115 0.668 8.20 0.722 3.40 0.911 13.40 0.816 0.15 0.060 0.25 0.783 0.01 0.070 0.00 0.011 0.02 0.178 0.08 1.000 0.01 0.210
Kj - 1 wjRj Kj
and the summation extends over all n significant fluorescent interactions due t o the j element lines fluorescing element i. Equation 3 may be found in various forms in Sherman ( I ) , Shiraiwa and Fujino (2), and Criss and Birks (3). Moderate caution should be exercised ;n the use of Equation 3, since the error associated with small corrections can exceed the correction itself. The full expression for the calculated X-ray intensity of each element then becomes Equation 1 plus Equation 2 or I,calcd =
sec Bs Ztorig .p _ b EW , (Kt- l/KJw,R, sec BIksE sec B2 ps'
+
~~
+
The weight fraction of each element in a n unknown can, as before, be iteratively adjusted until observed and calculated intensities agree or until no further convergence is realized. At this point the adjusted weight fractions are the best estimates of the composition and in this case secondary fluorescence has been taken into account. So far, the standards were assumed to be pure elements so that Equation 2 could be employed t o solve for the original intensity of the most efficient excitation energy for each analyzed element. In the case where the standards are composites, Equation 2 can no longer be used, because of the influence of secondary fluorescence upon the calculation of original intensities. Instead, Equation 4 is written with Itohsdsubstituted for ItcaLcd for each analyzed element, and the matrix equation
I
[Izobsd] = [C][I,orig
(5)
is solved for [Z,nrig], the column vector or original intensities associated with each element. The quantity [C] is the i X i matrix of coefficients indicated by Equation 4. Once the original intensities are obtained, the weight fraction of each analyzed element in an unknown is iteratively adjusted using Equation 4 until the observed and calculated intensities agree.
73.46 1.22 0.111 8.19 3.27 13.28 0.14 0.23 0.01 0.00 0.02 0.08 0.01
72.03 0.35 0.17 11.40 2.35 13.35 0.03 0.33 0.01 0.00 0.10 0.02 0.02
0.997 0.100 1.000 1,000 0.626 0.801 0.011 1,000 0.051 0.102 1.000 0.160 0.495
72.05 0.34 0.18 11.39 2.28 13.31 0.02 0.29 0.005 0.00 0.10 0.01 0.02
70.51 3.14 0.04 11.02 0.34 12.40 2.23 0.19 0.14 0.00 0.13 0.03 0.01
0.965 1.000 0.198 0.916 0.097 0.743 1.000 0.642 1.000 0.047 0.119 0.218 0.155
70.26 3,32 0.04 10.96 0.35 12.45 2.29 0.19 0.11 0.00 0.13 0.02 0.01
Any discrepancy between the calculated and accepted values for a test sample whose composition is regarded as accurate as the standard(s) can probably be traced directly t o the assumed value of the most efficient excitation energy for each element, so we now a t least know where t o look for any systematic error, providing, of course, the necessary constants such as fluorescence yields, etc., are accurate. EXPERIMENTAL VERIFICATION
To test the validity of the theory just described in actual analytical situations, five well analyzed glass standards were chosen for the initial test case. The accepted compositions, relative X-ray intensities, and values calculated by CORSET are listed in Table I. Glass 1 was selected as the only standard for the analysis of the remaining four. In every instance, the accepted and calculated values agree within the error associated with the two methods. Oxygen is in all cases determined by difference from weight fraction 1.0. CORSET also allows for the inclusion of any number of elements determined by independent methods into the equations. For example, had the glasses contained lithium and boron which ordinarily cannot be determined by X-ray, the weight fraction of these elements would have been entered, but not allowed t o vary during iterative adjustment of the other elements. These data on glasses may be compared with those given in General Electric Company's Laboratory Application Report No. 8 issued in 1966. Realize, however, that no actual analysis of unknowns is performed in the GE report, but rather the data are fitted by least-squares techniques, after which accepted and calculated values are compared. A second test case is taken from the work of Holland and Brindle (11). Table I1 gives their observed intensities for the Rock W-1and their five test unknowns along with the accepted compositions, values obtained by their self-consistent method, and values calculated by the CORSET program, using W-1 as the only standard. Overall agreement between the Holland-Brindle and CORSET methods is good when one considers the errors likely t o be inherent in the data. Two obvious systematic discrepancies do, however, exist. The CORSET values for K 2 0are much too high and the MgO values somewhat low. The potassium discrepancy might be attributed t o the fact that W-1, the standard, contains roughly (11) J. G. Holland and D. W. Brindle, Spectrocliim. Acta., 22, 2083 (1966).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1971
1763
Component SiOz A1203 Fez03
MgO CaO NazO KzO TiOn MnO
Component SiOz A1203 Fez03 MgO CaO NazO Kz0
TiOz MnO
Table 11. Accepted Values, X-Ray Intensities, Holland-Brindle Values, and CORSET Values for W-1 and the Five Test Unknowns w-1 Unknown 2 Unknown 1 Accept. X-ray Accept. X-ray H-B CORSET Accept. X-ray H- B CORSET value value value inten. value value inten. value inten. value 60.59 204 62.01 52.51 170 60.80 62.14 62.02 201 60.90 14.96 316 19.70 416 17.50 479 19.43 19.78 17.98 17.68 6.46 1588 7.83 6.65 7.18 7.69 11.07 2076 1419 6.60 1.78 911 2.13 6.59 2886 775 1.77 1.58 2.12 1.92 3.60 346 2.31 572 3.58 2.35 3.88 10.91 1800 2.28 3.07 3.43 205 2.06 2.06 206 341 3.55 2.09 1.94 408 3.00 3.36 4.94 3.09 623 4.96 0.64 82 5.28 1.11 0.98 1.05 1.08 277 293 1.06 1.04 282 1.07 0.09 0.12 0.09 440 0.125 0.18 614 0.09 340 0.093 Unknown 3 Unknown 4 Unknown 5 H-B CORSET Accept. X-ray H-B CORSET Accept. X-ray H-B CORSET Accept. X-ray value inten. value value value inten. value value value inten. value value 59.29 58.86 192 59.27 64.08 64.52 59.41 194 59.05 64.83 218 58.78 16.35 20.05 490 20.08 16.47 16.23 17.83 18.01 422 18.16 397 20.64 8.65 6.64 7.29 7.21 8.33 1713 8.17 8.02 1684 8.50 1518 8.24 2.44 2.46 2.23 1.75 1.91 1.76 1.56 2.11 929 2.12 1047 758 0.60 0.92 91 0.68 2.37 2.35 2.99 3.15 3.00 2.49 361 447 1.97 2.13 214 2.10 2.50 2.27 1.49 1.54 150 1.44 2.50 247 6.31 747 5.99 5.99 6.03 6.30 6.06 4.40 4.74 748 4.38 564 1.38 1.43 385 1.44 1.21 1.29 311 1.18 1.02 1.00 0.98 276 0.16 585 0.161 0.15 0.104 0.10 0.126 0.09 431 0.12 37 I 0.08
Table 111. Comparison of Accepted and Calculated Values for Inconel and 310 Alloy CORSET a Is Pure Element Standards, CORSET b Results Obtained by Using Each Alloy to Analyze the Other Inconel 3 10 Alloy CORSET a CORSET b CORSET a CORSET b Component X-ray inten. Accept. value value value X-ray inten. Accept. value value value 20.4 20.6 0.095 20.5 Ni 0,597 79.0 78.7 79.1 53.9 54.0 0.384 55.0 6.5 6.5 6.7 Fe 0.083 25.7 25.4 0.365 24.5 14.5 14.8 14.2 Cr 0.227
ten times less potassium than the test unknowns, so that CORSET is unable t o apply the proper corrections. However, the same argument would have t o apply t o CaO in W-1 (10.91 wt %) and CaO in unknown 5 (accepted value 0.92 wt %) where CORSET and the Holland-Brindle method essentially agree. Holland and Brindle also experienced some difficulty in the analysis of potassium in the Rock G-1 using W-1 as a standard, so inaccurate intensities, sample inhomogeneity, or the mineralogical constitution of the samples may be responsible for the bias. To test the equations in an extreme case of secondary fluorescence using pure element standards, the data of Sherman ( I ) for inconel and 310 alloy were processed by the CORSET program. Table I11 lists Sherman’s observed X-ray intensities (assumed t o be K a , , 2 relative t o the pure element), accepted compositions of the two alloys, the results obtained by the CORSET analysis using pure element standards (CORSET a), and also results obtained using each alloy t o analyze the other (CORSET b.) The worst agreement obtained is for chromium in 310 alloy, about 6 . 5 % relative too high. Both pure-element standards and inconel high for chromium in 310 alloy, so predict about 1.0 wt this consistency probably implies that either the sample contains more chromium than assumed or that the X-ray intensities are slightly in error. In any case, the theory seems adequately justified. All these data point to the need for accurate, well documented information on X-ray intensities and compositions for readily available standards t o test various theoretical and statistical methods of quantitative analysis. 1764
SUMMARY Application of the theoretical method briefly described here yields quite acceptable results for the analysis of certain glasses, rocks, and metal alloys. The data imply that equally acceptable results may be obtained for other kinds of materials if sufficient care is taken t o obtain good X-ray intensity data, providing the standards and unknowns are sufficiently homogeneous. Particle size and phase (mineralogical) effects, background, dead time, and line interferences can be absorbed into regression coefficients without much concern on the analyst’s part about what is really happening. However, confidence in the results of regression analysis usually requires many well known standards. Theoretical approaches should give equally good results using only one standard per element, provided the input information is of high quality. The choice between the two depends really upon the nature of the samples and a willingness t o obtain accurate intensity data. The advantage of this theoretical treatment over those described by Shiraiwa and Fujino ( 2 ) and Criss and Birks (3) lies in its ability t o apply the necessary corrections without any knowledge of the spectral distribution of the X-ray source. Evaluation of the complete integral over the spectral distribution or the determination of the true effective energy, if a unique value exists, should give equally good results. Until this can be routinely established for individual instruments, either in the field or a t the factory, the method proposed here will hopefully serve as a successful interim measure. RECEIVED for review May 26,1971. Accepted August 2,1971.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 13, NOVEMBER 1971
