Evaluation of Correction Procedures Used in Electron Probe Microanalysis with Emphasis on Atomic Number Interval 13 to 33 D. R. Beaman Metallurgical Laboratory, Metal Products Department, The Dow Chemical Co., Midland, Mich.
An investigation was carried out to determine which of the many available absorption, atomic number, and characteristic fluorescence correction procedures would provide the most satisfactory results when correcting electron probe ratios obtained for elements in the atomic number interval 13 to 33. The effectiveness of the correction procedures was determined for low and intermediate atomic numbers and two probe ratio intervals, k > 0.1 and k < 0.1. All published absorption and characteristic fluorescence correction techniques were evaluated and several atomic number corrections were applied to sets of alloys in which atomic number effects were both large and small. It was found that absorption effects could be effectively removed in the intermediate atomic number interval using Green’s f(x) curves or the Duncumb and Shields technique. At low atomic number the techniques of Tong (modified), Thomas, and Philibert gave the most satisfactory results. Characteristic fluorescence effects were best treated using Reed’s approximate formula. The Thomas technique provided the most satisfactory atomic number correction.
*
SEVERAL EXCELLENT ARTICLES (1-5) have been published in which the correction procedures used in quantitative electron probe microanalysis have been evaluated. However, these articles have often been limited in scope in that some have considered only a few alloys or correction techniques while others have investigated restricted ranges of atomic numbers or mass absorption coefficients. Our particular interest is in the detection of elements of relatively low atomic number (Mg, Al), in which case the true chemical concentration can be more than four times that measured with the electron probe. Inasmuch as no comprehensive application of the many available correction techniques had been carried out in systems containing such elements, it was decided to initiate such a program and to include therein the following: (a) All known absorption correction techniques, (b) all known characteristic fluorescence correction techniques, (c) several atomic number correction techniques, and (d) analyzed elements of low (z = 13) and intermediate ( z = 22-33) atomic number (z). Because absorption is generally the most serious problem in this atomic number interval, the primary emphasis is on the absorption correction techniques. The purpose here was to use the many corrections advanced in the literature and see what sort of quantitative results (1) J. W. Colby, Fall Meeting of Electrochemical Society, Oct. 1115, 1964, Washington, D. C . , Extended Abstracts of Electrothermics and Metallurgy Division, Vol. 2, No. 2, p. 74. (2) J. W. Colby and D. K. Conley, Abstracts of “Fourth Inter-
national Congress on X-ray Optics and Microanalysis,” Sept. 7-10, 1965, Orsay, France, p. 16. (3) P. Duncumb and P. K. Shields, “X-ray Optics and X-ray Microanalysis,” H. Pattee, V. Cosslett, and A. Engstrom, Eds., Academic Press, New York and London, 1963, p. 329. (4) D. M. Poole and P. M. Thomas, UKAEK Report, AERE-R4796 (1964). (5) T. 0. Ziebold and R. E. Ogilvie, ANAL.CHEM., 35, 621 (1963). 418
ANALYTICAL CHEMISTRY
could be obtained in a variety of alloys. No attempt was made t o judge the theoretical validity of the techniques and the only criterion used in recommending techniques was the success of the method. Because many of the procedures have similar origins, none of the techniques have been rejected on a physical basis. This evaluation should offer a guide for selecting the best procedures and provide a method for assessing the reliability of computed concentrations. PROCEDURE
The procedure followed was straightforward. A fairly large number of alloys was carefully selected for analysis; while some were the outcome of measurements by the author, most were taken from the literature. Each correction technique was used to calculate the concentration from the x-ray intensity ratio, k , measured with the electron probe. These calculated concentrations were compared with the known chemical compositions and the corresponding errors were then grouped with respect t o atomic number, probe ratio, and predominant correction effect. This statistical approach was used in recognition of the fact that since good probe data are difficult t o obtain, some of the included data could be of dubious value. However, it is felt that the percentage of good data was sufficiently high to ensure that the determined ranges of expected accuracy and precision would be useful. Because there appears to be some confusion as to how the individual correction factors should be combined to establish a complete correction formula, the following derivation has been included. In the absence of secondary radiation effects the ratio of the intensity emitted (measured) by an alloy ( Z p f ) t o that emitted by a pure material (I,”) is given by k , where
J o
Cn is the concentration of the element being analyzed; the superscripts prime and zero refer to the alloy and pure materials, respectively; cp(pz) is the distribution in depth of primary x-radiation; x is ( p / p ) j i csc 4 where x is the spectrometer takeoff angle and ( ~ / pis) the ~ ~mass absorption coefficient of j for i radiation; z1 is the depth below the surface a t which the incident electron energy becomes equal to the excitation potential, E,, of the analyzed element; p is the density and F(x) is the defined integral. In the absence of absorption effects (X = 0) the ratio of generated intensities, k,, is given by:
Jo Combining Equations 1 and 2 we have: (3)
Procedure and ref. Thomas (8)
Table I. Absorption and Atomic Number Correction Procedures Symbol used in tables Quantity calculated Characteristics of procedure Kcalo, Ccalc absorption and atomic no.; h’ Thomas
Philibert (9) Philibert (9) Philibert (9)
Phil (A) Phil (B) Phil (C)
Kcalcr Ccalc
Philibert (9)
Phil (D)
Ccslo
Birks (10) Castaing (11) Ziebold (7)
Birks Castaing Ziebold
Dewey (12) Cameca (13) Green (14) Green (14)
Dewey Cameca Green -f(x) Green (Eo - Eo)
Koslc
Ilyine (15)
Ilyine
Koala
Theisen (16)
Th(o1d)
Kcale, Coalc
Theisen (17) Theisen (17) Belk (18) Belk (18) Belk (18) Belk (18)
Th - F ( x ) Th - f(xh R Belk (A) Belk (B) Belk (C) Belk (D)
Kcalc, Ccalc Kcalc, Ccalo Kealc, Coalc
Duncumb and Shie1.d~ (19)
D&S
Kcalc, C o a ~ o
Ziebold and Ogilvie (20)
Z&O
KoaIC
Tong (21) Tong (21) Archard and Mulvey (22) Criss and Birks (23) Trail1 and Lachancs ( 2 4
Tong - F(x) Tong - f ( x ) A&M not included not included
Tong (modified) (25)
Tong - M
=
ho = 1.2Ai/(2J2 absorption only; 11’ = Zcihi
absorption only; h‘ = Zaihi absorption and atomic no.;
KCdO
Koala
12’ = ZCihi =
1.2aiAJ
(Zaizi)Z
absorption only; F curves absorption only; f ( x ) curves absorption and atomic no.; Ziebold atomic no. tables and Philibert absorption correction absorption only absorption and atomic no. absorption only; f ( x ) curves absorption only; f ( x ) as fcn
Kcslc
Kale Kcalo
Kcslo Kcalc
KdC
(Eo - Ec) absorption only; not for multicomponent systems absorption and atomic no., h’ Kcale, C c s l c
Ccalc
CdC
KOalO
.....
..... Coal0
(4)
When characteristic fluorescence effects are present the measured probe ratio, k , is the ratio of the total characteristic x-ray intensity emitted by an alloy ( I t ’ ) to that emitted by a pure material (ZP). k is given by:
where Zp’is the primary (direct excitation by the electron beam) intensity emitted by an alloy and is given by the numerator of Equation 1; I,O is the primary intensity emitted by a pure material, is given by the denominator of Equation 1, and is equal to I? in the absence of excitation by the continuous x-ray spectrum; and 1,’ is the x-ray intensity emitted by an alloy due to characteristic fluorescence. Rewriting Equation 5 we obtain:
=
Zcihi
absorption and atomic no. absorption and backscatter absorption only absorption and atomic no. absorption only; no iteration absorption and atomic no.; no iteration absorption only; uc replaces Philibert u. absorption only; applicable if z
Kcalc, Cealc Coal0
where f(x) is the fraction of generated intensity that is emitted (Fh)/F(O)). Thus we see that k , and CA are related in the following manner:
Combining Equations 4 and 6 gives:
absorption only; 11’
> 20.
absorption and atomic no. absorption only absorption only; f(x) curves analytical expression for &z) similar to Z & 0; a(Trail1) = cy (Z & 0) - 1 for binary systems absorption and atomic number
The three terms in brackets represent in order the atomic number, absorption, and characteristic fluorescence correction terms. This expression is identical to the one derived by Duncumb and Shields (6) when continuous fluorescence effects are neglected. This type of expression has also been derived and used by Ziebold (7). The various forms in which Equation 7 is used are presented in the discussion of the individual correction effects to follow. (6) P. Duncumb and P. K. Shields, “The Encyclopedia of X-rays and Gamma Rays,” G. Clark, Ed., Reinhold, New York, 1963, p. 830. (7) T. 0. Ziebold, “The Electron Microanalyzer and its Applica-
tions,” Lecture notes for summer program at the Massachusetts Institute of Technology, Suppl. (1965). (8) P. M. Thomas, UKAEA Report, AERE-R-4593 (1964). (9) J. Philibert, “X-ray Optics and X-ray Microanalysis,” H. Pattee, V. Cosslett, and A. Engstrom, Eds. Academic Press, New York and London, 1963, p. 379. (10) L. S . Birks, “Electron Probe Microanalysis,” Interscience, New York, 1963, p. 107. (1 1) R. Castaing, “Advances in Electronics and Electron Physics,” L. Marton, Ed., Academic Press, New York and London, 1960, p. 317. (12) R. D. Dewey, R. S. Mapes, and T. W. Reynolds, “A Table of Coefficients for the Microprobe Analyst with Tables of X-ray Data,” Reynolds Metal Co., 1965. (13) C. Conty, private communication (1965), a technique de-
veloped at CAMECA, Courbevoie (Seine), France. (14) M. Green, in “X-ray Optics and X-ray Microanalysis,” H. Pattee, V. Cosslett, and A. Engstrom, Eds., Academic Press, New York and London, 1963, p. 361. VOL. 39, NO. 4, APRIL 1967
419
Absorption Correction. Table I lists the absorption correction procedures that have been proposed and the conditions under which each was applied herein. Some of the techniques [Ilyine (153, Cameca (13), Dewey (12)] have not been commonly used but are included to determine their relative merits while others, which d o not give good results, were included t o illustrate the consequences of improper technique selection. The means of applying the different techniques can be found in the indicated references. All of the procedures were utilized, with the exception of the Criss and Birks (23) and the Traill and Lachance (24) techniques. The former provides a n analytical form for the distribution in depth of primary x-radiation, cp(pz), while the latter is similar to that proposed by Ziebold and Ogilvie (20). The recently presented modified technique of Tong (25) has been applied only on a limited basis, as it was brought to the author’s attention after the bulk of the work discussed herein was completed. Because the relative merits of the graphical techniques established here depend upon the manner in which these curves are utilized, it is necessary to indicate how these curves, established for pure materials, are used in conjunction with alloys. In using the f ( x ) curves of Castaing and Descamps (11, 26) we have neglected the fact that the mass absorption coefficients used by these authors to correct for tracer radiation absorption in the pure metal overlay may differ from those used herein. The Heinrich (27) coefficients have been used throughout this investigation. The inherent difficulty associated with the tracer technique-the f(x) values represent a target with the scattering properties of the overlay and an ionization cross-section of the tracer-are also inherent in the Philibert (9) type techniques, a s the fitting parameters in the Philibert ( 9 ) technique were evaluated by fitting the Philibert (9) function to the experimental data of Castaing and Descamps (26). In order to use the Castaing and Descamps (26) curves for a variety of atomic numbers and acceleration potentials it is necessary to assume that the f(x) energy dependence established for aluminum holds for all other elements and that the atomic number dependence established at
29 kV holds for all other acceleration potentials-Le., d[f&)]jdE f fcn (z) and duh)l/dz # fcn (E). In using Green’s (14) or Archard and Mulvey’s (22)f(x) curves linear interpolation at constant x has been used in obtainingf(x) values for atomic numbers and voltages not falling on the experimental curves, For all of the graphical techniques the atomic number of the alloy, z’, was assumed to be given by: n
2’ =
CZZ{
i= 1
(8)
where ci is the concentration of component i in weight per cent, zi is the atomic number of component i and n is the number of components. f(x)’ was the value of f k ) for a chi value of x’ and an atomic number of z’ a t a given acceleration potential. The relative merits of the graphical techniques established here must be considered in view of the assumptions just listed. Alternative, and perhaps better, methods of using the f ( x ) curves will become available as more experimental data are obtained. In the absence of atomic number and characteristic fluorescence effects Equation 7 reduces to :
(9) However, to facilitate the evaluation of the large number of correction procedures, fk)’ was evaluated using the known chemical concentration, Ctrue,and a probe ratio, Kcslo,was calculated using:
The error is given by dkjk = (KCelc- k)/Kcslc. This error is generally different from the error: dc - -- Ctrue - Coalc __ c Ctrue
(1 1)
where Coalois the concentration obtained in an iterative process using Equation 9 in the form : (15) N. P. Ilyine, Bull. Acad. Sei., USSR, Vol. 25, No. 8, 940 (1961). (16) D. Quartaert and R. Theisen, “First International Conference on Electron and Ion Beam Science and Technology,” R. Bakish, Ed., Wiley, New York, 1965, p. 799. (17) R. Theisen, “Quantitative Electron Microprobe Analysis,” Springer-Verlag,New York, 1965, Chap. 3, p. 11. (18) J. A. Belk, Abstracts of “Fourth International Congress on X-ray Optics and Microanalysis,” Sept. 7-10, 1965, Orsay, France, p. 9. (19) P. Duncumb and P. K. Shields, Fall Meeting of the Electrochemical Society, Oct. 11-15, 1964, Washington, D. C., Extended Abstracts of Electrotherrnics and Metallurgy Dioision, Vol. 2, No. 2, p. 89. (20) T. 0 . Ziebold and R. E. Ogilvie, ANAL.CHEM.,36, 322 (1964). (21) C. I. Helgesson, Abstracts of “Fourth International Congress on X-ray Optics and Microanalysis,” Sept. 7-10, 1965, Orsay, France, p. 15. (22) G. D. Archard and T. Mulvey, in “X-ray Optics and X-ray Microanalysis,” H. Pattee, V. Cosslett, and A. Engstrom, E&., Academic Press, New York and London, 1963, p. 393. (23) J. W. Criss and L. S. Birks, Fall Meeting of the Electrochemical Society, Oct. 11-15, 1964, Washington, D. C., Extended Abstracts of Electrotherrnics and Metallurgy Division, Vol. 2, No. 2, p. 80. (24) R. J. Traill and G. R. Lachance, Geological Survey of Canada, Department of Mines and Technical Surveys, Paper No. 64-57 (1965). (25) J. Ruberol, M. Tong, and C. Conty, paper presented at the Groupement pour 1’ Avancement des Methodes Spectroscopiques (GAMS.) Conference in Paris, France, June 8, 1966. (26) R. Castaing and J. Descamps, J. Phys. Radium, 16, 304 (1955). (27) K. F. J. Heinrich, Fall meeting of the Electrochemical Society, Oct. 11-15, 1964, Washington, D. C., Extended Abstracts of Electrothermics and Metallurgy Division, ’Jol. 2, No. 2, p. 92. 420
ANALYTICAL CHEMISTRY
While Equation 12 represents the proper procedure for correcting probe data, iteration usingfk) curves is quite tedious unless the curves can be fitted to an analytical expression. Thus, Equation 10 greatly simplifies the evaluation of the graphical methods. Differentiation of Equation 9 reveals that dk/k and dcjc differ by the amount of df((x)’/fk)‘. Since this quantity can be significant in systems where f(x)’ varies rapidly with composition (AI-Mg), Equation 12 was also used in evaluating most of the techniques. In the column headed “Quantity calculated” in Table I, Kcalemeans that the technique was evaluated using Equation 10 and C,,I, means the technique was evaluated using Equation 12. While the various techniques have been used as directed by the different authors, some minor additions have also been included. Several methods (9, 16, 21, 28, 29) have been used to determine the value of h‘ for a n alloy where hi, the quantity appearing in the Philibert ( 9 ) expression, is given by:
where A i is the atomic weight of component i and s is a fitting parameter which Philibert (9) found to be 1.2 by fitting his (28) I. Adler and J. Goldstein, NASA Technical Note NASA TN D-2984,1965. (29) J. D. Brown, U. S. Department of the Interior, Bureau of Mines, Rep. 6648 (1965).
~~
Table II. Unpublished Alloy Data Take-off angle Probe Chemical Other Acceleration Determined ratio concentration Reference (degrees) component potential (kV) component 20 20 0.0190 0.08 (31) A1 Mg 0.051 0.20 (31) 20 20 A1 Mg 0.34 (31) 0. 100 20 20 AI Mg 0.50 (31) 0.175 20 20 A1 Mg 0.241 0.60 (31) A1 Mg 20 20 0.13 0.41 (31) 20 20 A1 Mg A1 Mg 20 20 0.03 0.12 (31) 0.49 0.80 (31) A1 Mg 20 20 0.15 0.472 20 20 A1 Mg 0.845 0.766 27 18b Fe Cr 0.516 (36) 0.441 26 18b As Ga 0.156 0.186 Cr Fe 27 18b 0.0034 27 18b 0.0042 Cr Fe 0.0097 0.0083 (13) 20 18b Cr Fe 0.0055 0.0048 (13) Cr Fe 20 18b 0.5725 0.481 (36) 26 18b Ga As 0.639 0.72 (45) Cd Mg 30 52.5 0.454 0.55 (45) 30 52.5 Cd Mg a The first eight alloys are taken from a calibration curve obtained by analyzing six AI-Mg alloys of known composition (composition from chemical analysis and phase diagram for single and two phase materials, respectively). b Data obtained using Cameca probe in which take-off angle depends upon the Bragg angle; in such cases the correct take off angle was used.
expression to the fk)data of Castaing and Descamps (26). To illustrate the effects of using expressions that give large differences in h’ Equations 14-16 have been used in conjunction with the Philibert ((9)technique. h’ =
cfhr.
h’ =
athi.
ai is the atomic concentration of component i. While the different approaches can lead to a considerable difference in h‘ values in a system in which ai and ci are significantly differis generally negligible because fk)is ent, the effect on fk)’ not very sensitive to atomic number. Belk (18) has developed a technique in which no iteration is required; this procedure has been evaluated with and without iteration. The Thomas (8) technique has been used as prescribed except in some multicomponent systems where divergence occurred upon iteration ; in such cases a modified expression was used. The alloys used in evaluating the absorption correction procedures were as follows with the detected element listed first: AI-Mg (31, 32); AI-Ni and AI-Co (32); Ni-Cr, Ni-Co-Cr, Co-Cr, Co-Ni, and Co-Ni-Cr (33); AI-Mg and Fe-Cr (present investigation); Fe-Cr (2, 34); Ni-FeCr-Si-Mn (1); Ti-Zr (35); and As-Ga (36). The alloy data not previously published are presented in Table 11. Space limitations prohi;bit inclusion of a complete listing of the alloys; however, this is available upon request. The probe ratio, k , ranged from 0.014 to 0.826, the electron accelerating
(30) D. B. Wittry, “Advances in X-ray Analysis,” Vol. 7, M. Mueller, G. Mallett, and 14. Fay, Eds., Plenum Press, New York, 1964, p. 395. (31) D. Adenis, Dow Ch’tmicalCo., Midland, Mich., unpublished data, 1963. (32) D. B. Clayton, Brit. J . Appl. Phys., 14, 117 (1963). (33) A. G. Guy and V. Leroy, “Journees Internationales des Applications due Cobalt,” Bruxelles, 1964. (34) R. Castaing, Thesis, University of Paris, O.N.E.R.A. Publ. No. 55 (1951). (35) L. S . Birks, J . Appl I’hys., 31, 1297 (1960). (36) J. R. Mihalisin and ;S. Wielgus, International Nickel Co., Research Rept., P.R.228.1. (1964).
voltage from 15 to 40 kV and the take-off angle from 6” to 52.5O. The techniques of Ilyine (15) and Dewey (12) could not be evaluated using the alloys just listed because these techniques require experimental calibration. Consequently, they were evaluated primarily using the alloys of Colby and Conley (2), Ziebold and Ogilvie (3,and Adenis (31) because these authors provide a large amount of data for a particular take-off angleacceleration potential combination. Invalid extrapolations would occur if each technique were used with every alloy--e.g., use of Archard and Mulvey (22) curves at 15 kV. Some extrapolation and interpolation have been used, but whenever the results indicated that such procedures might be invalid the results were disregarded in accumulating the averages discussed below. It was assumed that all probe data had been corrected for background (30), contamination (30), detector and electronic dead time (30), and intensity-induced pulse amplitude shifts (37, 38). Atomic Number Correction. Consideration here was restricted to the techniques which include atomic number corrections listed in Table I. Poole and Thomas (4) have applied several techniques to a large number of alloys in which atomic number effects were large. Only a few of the alloys included in the present study would be expected to give effects of such magnitude but it was of primary interest to ascertain if the application of an atomic number correction would have a deleterious effect in a system where absorption effects were expected to predominate. This was studied by applying the atomic number corrections to the alloys used in studying the absorption correction. In the absence of fluorescence effects Equation 7 reduces to the following expression which is used to correct the measured probe ratio when atomic number effects are expected.
Characteristic Fluorescence Correction. The following techniques were used to correct for characteristic fluores(37) D. R. Beaman, ANAL.CHEM., 38,599 (1966). (38) S.L. Bender and E. J. Rapperport, Fall Meeting of the Electrochemical Society, Oct. 11-15, 1964, Washington, D. C.. Extended Abstracts of Electrothermics and Metallurgy Division, Vol. 2, No. 2, p. 104. VOL 39, NO. 4, APRIL 1967
421
Table In. Example Data Calculation for a Single Correction Technique Alloy number 1 2 3 4 0.019 0.175 Measured probe ratio.. . . . . . .k 0.13 0.241 0.08 0.50 0.60 0.41 True chemical composition. . . Ct, - k)/Ct,, 76.2 65.0 39.9 Probe error.. . . . . . . . . . . . . . . 1O0(Ctru. . 68.3 0.0870 0.4872 0,5813 0.4057 Corrected concentration.. . . . .CCalo -0.0070 0.0128 0.0187 Ctru, - C c a l e . . . . . . . . . . . . . . ..dc 0.0043 2.56 3.12 -8.71 1.04 Error. ..................... 100(dc)/Ctru. 2.56 3.12 8.71 1.04 I Error I .................... 1 loO(dc)/Ct,,, I arthmetic mean = average of row 8 = 3.6% algebraic mean = average of row 7 = -0.9% standard deviation = ([Z(averageof row 7 - value in row 7)2]/4)1'2= &4.9% effectiveness = I dc/c 1 /( (Ct,, - k)/Ct,,,) = average of row 8/average of row 4 = 3.6/64.8 = 0.056
cence effects: Wittry (39), Castaing ( 3 4 , Reed (40), Birks (ZO), Ilyine (15), Ziebold and Ogilvie (20), and Dewey (12). Reed's (40) approximate formula and his general expression were both evaluated. Atomic number effects have been neglected in performing these corrections so that Equation 7 reduces to :
Average of indicated row
5
0.051 0.20 74.5 0.2047 -0.0047 -2.35 2.35
64.8x -0.87x 3.56z
In this investigation the effects of continuous fluorescence have been neglected because the atomic number of the analyzed element, z(A),the atomic number difference, z(A) - z(B), and the acceleration potential were generally low and thus minimized effects due to continuous fluorescence. RESULTS AND DISCUSSION
The ratio of the emitted intensity due to characteristic fluorescence in an alloy (1,') to the emitted intensity resulting from direct electron excitation in the alloy (&') is given by the expressions of Castaing and Reed. In the present investigation when reference is made to the Castaing ox Reed technique the expressions for I , '/Ip' derived by these two authors are used in Equation 18 along with the Philibert expression for f&)'/ f(x)O to calculate the concentration from the measured probe ratio. With the exception of the Ziebold and Ogilvie (20) and the Dewey (12) techniques which contain absorption parameters, the Philibert (9) expression was used to correct for absorption effects. The following expression has been used elsewhere ( 2 ) to convert measured probe ratios to chemical compositions :
Such a procedure is invalid because the measured intensity ratio is then not properly corrected for absorption; Equation 18 indicates that such a correction is essential. If atomic number effects are expected Equation 7 should be used. Iteration was used in calculating the corrected concentrations. The K fluorescence yields of Hagedoorn and Wapstra (41) and the absorption jump ratios of Lindstrom (42), which have been tabulated by Colby (4.9, were used. No attempt was made to be as comprehensive, with respect to the variety of alloy systems, as in the absorption correction evaluation, as fluorescence effects are relatively small at low atomic number. Nineteen alloys from the following systems were used: Cr-Fe (2, 5, 13, 34) (present investigation); Fe-Ni (3,Ga-As (36); and Cr-Mn-Si-Ni-Fe (1). The probe ratio, k , ranged from 0.004 to 0.5, the electron accelerating voltage from 20 to 40 kV and the take off angle from 15.5" to 52.5'. A complete listing of these alloys is available upon request. (39) D. B. Wittry, University of Southern California, USCEC Rep. 84-204 (1962). (40) S. J. B. Reed, Brit.J. Appl. Phys., 16,913 (1965). (41) H. L. Hagedoorn and A. H. Wapstra, Nucl. Phys., 15, 146 (1960). (42) B. Lindstrom, Acta Radiol. Suppl., 125,39 (1955). (43) J. W. Colby, USAEC Rept. NLCO-917 (1964). 422
e
ANALYTICAL CHEMISTRY
The calculated probe ratio ( Z L C in Equation IO) and/or the concentration (Coalcin Equation 12) are determined for every alloy in which absorption effects are predominant using each absorption correction procedure. CA (or Coalo)in Equation 18 is determined for every alloy in which characteristic fluorescence effects are predominant using each characteristic fluorescence correction technique. The errors corresponding to these calculated quantities are dkjk = (Kcalc- k ) / Koala and dcjc = (Ct,, - Cealo)/Ctrue where k is the experimentally determined x-ray _ intensity Overall averages _ _ratio. dkjkl or dc/cl) and algebraic of these errors, arithmetic - _ _ (dk/k or dc/c) are calculated for different alloy combinations representing a variety of conditions--e.g., all alloys in which the atomic number of the analyzed element lies between 22 and 33; a group of 11 A1-Mg alloys; a group of 10 Fe-Cr alloys; five A1 alloys with a measured probe ratio below 0.13 and several others. The algebraic mean so obtained represents the accuracy of a particular correction procedure under specific conditions. The standard deviation from the algebraic mean is calculated and is indicative of the precision obtainable from measured probe ratios using a particular technique under the specified conditions. Table I11 illustrates how these quantities are calculated for an alloy set consisting of five AI-Mg alloys using the Thomas correction procedure. The ratio of the arithmetic mean, __dcjcl , to the mean probe error, (C,,,, - k)/Ct,,,I, for the alloy set is representative of the fraction of probe error that remains after correction and thus is a good indication of a particular technique's effectiveness. In addition, this ratio may be used as a basis of comparison in cases where techniques are evaluated using different alloys. In the present investigation this ratio is used in the comparison of the Dewey and Ilyine techniques with the remaining techniques, as the mean probe error for the alloys used with the Dewey and Ilyine procedures is 19 as compared to 32 for the alloys used with the remaining procedures. The techniques are listed in tables in an order such that the technique having the lowest arithmetic mean for the alloy set is given first. In view of the number of alloys used in the present analysis and the precision of experimental probe intensity ratios, adjacent techniques in the tables would not generally be expected to give
I
(I
I
I
Table IVa. liesuits of Applying Absorption Corrections to 22 Alloys in the Intermediate Atomic Number Interval ( G l 0 calculated, complete listing of results) Algebraic Arithmetic Relative average - error -mean Standard E x 100 deviation Techniqu'e k 0.0 0.9 0.057 1.4 Green - f(xP 0.2 0.064 1.8 1 .o D & So 1.6 1.1 0.0 0.072 Green - f(x) -1.1 1.3 0.084 1.3 Phil (AP 0.1 1.4 0.090 1.8 Thomasa -0.4 1.5 0.096 1.9 D&S 2.0 0.096 -1.2 1.5 Phil (C) -1.5 0.106 1.7 1.7 Phil (B) -1.5 0.108 1.7 1.8 Phil (A) 0.109 0.0 2.3 1.7 Thomas -0.5 0.127 2.0 3.4 A&M 0.130 2.0 3.7 -0.2 Green (Eo - EJ 0.145 2.4 -2.2 2.3 Belk (A) 2.4 0.152 2.4 -2.3 Cameca 2.9 0.162 2.5 -2.2 Castaing 0.167 2.6 3.5 -0.4 Birks 2.9 -2.4 Ziebold 2.7 0.172 2.6 -2.5 Belk (B) 2.7 0.173 2.9 3.5 -3.5 0.228 Th (old) 5.4 0,230 3.6 0.2 Th - F ( x ) 3.2 0.236 3.7 -3.3 Z&O 5.7 0.256 4.0 -0.3 Tong F (XI 5.6 0.264 -0.8 4.1 Th - f(x), R Table IVb. (Cc.lo calculated, best 5 of 13 techniques)
Arithmetic average
Relative error Ctm,
0
Technique D&S Phil (A) Phil (D) Thomas Belk (B) Averages for 20 alloys-Le.,
- Coalo -k
Ctme
Algebraic
-mean
eC x 100
1.3 0.084 0.7 1.5 0.099 1.4 1.6 0.103 1.5 1.7 0.109 -0.4 0.120 1.9 1.2 the two systems giving the largest errors for each technique have been omitted.
significantly different r1:sults. The number and nature of the alloys used in each set are given in the table heading. Since the primary interest is in the most successful of the techniques, results are given for these only, except in three cases where the entire listing is given in order to illustrate the relative merits of all the techniques. Absorption Correction. When the techniques listed in Table I are used to correct the measured probe ratios of 22 elements with atomic numbers between 22 and 33 the results presented in Tables IVa and b are obtained. While the best results are obtained when using Green's f(x) curves or the Duncumb and Shields technique, satisfactory accuracy and precision are provided by several techniques. Preliminary results indicate that the modified Tong technique gives slightly better results than the Cameca technique. Four of the techniques appear twicr: in this table to illustrate the fact that the removal of the two largest errors results in only a slight improvement in precision. While iteration in these alloys did not significantly affect the results (compare Tables IVa and b), comparison of the results obtained in all cases indicates that, while iteration improves the results, no change in the relative order of the various techniques is observed. The effect of composition can be determined by using the different techniques to make corrections in groups of alloys
Standard deviation 1.5 1.7 1.9 2.2 3.1
with high and low probe ratios. When the evaluation is restricted to alloys where k < 0.1 in the intermediate atomic number (z = 22 to z = 33) interval, it is found that satisfactory results are provided by the techniques of Green (0.1, 2.0), Duncumb and Shields (-1.8, 1.8), Thomas (-0.4, 3.9) and Archard and Mulvey (0.5, 4.1). In the parentheses the symaccuracy is followed by the precision and the Z and bols are omitted. On the basis of these results the Green f(x) curves would be preferred over the Duncumb and Shields technique; note, however, that only five alloys are included in this category. The Green and Duncumb-Shields techniques give as good results when k < 0.1 as when the entire group of 22 alloys is considered; this is not true of the remaining procedures. In using an iterative procedure (Equation 12) the techniques of Thbmas (-0.1, 4.0) and Tong (0.5, 4.4) provide the most satisfactory accuracy, while the Duncumb and Shields technique (1.7, 1.5) gives the best precision. Considering only alloys with k > 0.1 is a selective procedure that enables almost all techniques to give acceptable results. While 70 of the techniques attain a given level of performance when k > 0.1, only 2 0 Z attain the same level if k < 0.1. Only under these particular circumstances (intermediate atomic number range, low probe error, k > 0.1) is it found that the choice of correction technique is inconsequential.
*
VOL 39, NO. 4, APRIL 1967
0
423
Table Va. Results of Correcting Aluminum Concentration in 11 AI-Mg Alloys (Kcalecalculated, complete listing of results) Arithmetic Relative Algebraic average
-
Technique Thomas Phil (B) Phil (A) Phil (C) Ziebold Castaing Th (old) Cameca Birks Belk (A) Belk (B) Green - f(x) Tong - F ( x ) Z & O
D&S Green (Eo- Ec) Th - f(x), R Th - F(x) A&M
Technique Tong - M Thomas Phil (A) Phil (D) Th (old) Belk (A) Belk (B) Tong - f(x) Tong - F(x) D&S Belk (D) Belk (C) Th - f(x), R Th - F(x)
x
100
5.2 5.7 5.7 5.9 8.0 8.3 10.2 10.6 14.9 14.9 15.3 20.0 22.5 22.9 25.1 31 .O 39.8 41.5 52.7
Table Vb. (Ccalccalculated, complete listing of results) Arithmetic Relative average error de Ctwe - Cca~o x 100 C Ctrw - k 2.5 0.041 3.6 0.060 3.9 0,065 4.0 0,066 7.4 0.123 10.9 0,181 11.4 0.189 13.4 0,222 14.2 0,235 16.0 0.265 18.5 0.306 22.0 0,364 23.2 0.384 24.1 0.399
-
In restricting the evaluation to 10 Fe-Cr alloys it is found, as noted by Colby (2), that the technique of Duncumb and Shields (0.8, 2.0) works well in this alloy system. Green’s f ( x ) curves (0.2,2.0) are equally effective. In restricting the evaluation to a set of 16 alloys with relatively small probe errors, 14 techniques give precision values of better than 1 1 . 7 % . Comparison of these results with others illustrates the importance of testing correction techniques under different conditions. Application of the correction techniques to the measured AI intensity ratio in 11 AI-Mg alloys yields the results of Tables Va and b. The techniques of Tong (modified), Thomas, and Philibert provide the best results while some of the techniques that work well in the intermediate atomic number range appear inadequate here. Even though the precision and accuracy are considerably reduced, the probe error is removed to the same extent as in the intermediate atomic region. Inclusion of the A1-Ni and A1-Co systems has a negligible effect on the results and generally causes a slight improvement-cg., as in the Thomas (2.5, 6.3) technique; this indicates that absorption effects predominate in 424
ANALYTICAL CHEMISTRY
0.079 0.094 0.096 0.098 0.132 0.137 0.169 0.176 0.247 0.248 0.253 0.331 0.373 0.379 0.416 0.513 0.659 0.687 0.873
3.0 3.6 3.6 4.1 7.3 -3.1 -10.2 -10.6 14.9 -11.1 -11.6 20.0 22.5 -22.9 25.1 31 .O 39.8 41.5 52.7
Standard deviation 6.3 6.5 6.5 6.7 7.9 9.3 4.3 5.2 8.5 11.4 11.4 15.3 16.5 10.1 11.9 20.6 23.1 23.9 18.0
Algebraic mean dc x 100 C
-1.4 -2.3 -2.8 -2.9 1.4 8.8 10.4 -13.4 -14.2 -16.0 -10.7 -16.0 -23.2 -24.1
Standard deviation 3.1 4.3 4.7 4.8 3.7 8.4 6.8 12.7 13.4 12.5 21.9 24.0 19.8 20.5
these alloys. The modified Tong technique is an exception; inclusion of the AI-Co and A1-Ni systems changes the precision to 1 4 . 3 %. Comparison of Tables Va and b shows that iteration provides a substantial improvement in the results. The position of the Castaing technique is misleading because some systems with high chi values are excluded from the overall averages. The AI-Mg tracer f(x) curves (44) provide unsatisfactory results (23.4, 13.6). Most techniques overcorrect (Ccale> C,,,,). The Dewey and Ilyine techniques could not be satisfactorily evaluated because of an insufficient number of alloy systems; however, it is noteworthy that in the three A1-Mg systems investigated the Dewey technique removes 9 6 z of the probe error with good precision (1.2 %). Ilyine’s procedure is much less satisfactory, removing only 67 of the probe error.
z
(44) R. Castaing and J. Henoc, Abstracts of “Fourth International
Congress on X-ray Optics and Microanalysis,” Sept. 7-10, 1965, Orsay, France, p. 11.
’
For k < 0.13 in five A1-Mg systems the techniques of Tong (modified) (-2.5 3.7), Thomas (-3.8, 4.8), and Philibert (-4.8, 5.3) are most effective. For k > 0.13 the Tong (modified) (-0.4, 2.91, Thomas (-0.8, 3.4), and Philibert (-0.6, 3.5) techniques are again the most useful. It is apparent that the tendency to overcorrect is greatest at low concentrations. The modified Tong technique may also be useful at very low atomic number. Ruberol, Tong, and Conty (25) have reported converting a measured carbon intensity ratio of 0.102 to a corrected composition of 29.3 % C in S i c using this technique. The question may arise as to why the Philibert type procedures work well at low atomic number when the Philibert parameters, s and U , were evaluated by fitting the derived function to the experimental tracer work of Castaing and Descamps (26). Two points are significant here: (1) The Philibert technique works well only on a relative basis-Le., it is considerably more effective than many others at low atomic numbers. One might say that the Philibert technique is one of the best of a series of not very good procedures. (2) We have preliminary data concerning the fitting of the Philibert expression to the experimental data of Castaing and Descamps which indicate that the s value of 1.2 in Equation 13 and the u values proposed by Philibert constitute almost the best fit to the five experimental curves. However, the best fit to the three aluminum curves is s E 0.2, u (29 kV) = 1476, u (9.7 kV) = 7860 and u (15.1 kV) = 4764. The fit with these values is twice as good as with the Philibert values for the three aluminum curves. The best fit to the aluminum curves at 9.7 and 15.1 kV is s < 0, u (9.7 kV) = 7370, and u (15.1 kV) = 4387. For s = 0 the fit with these values is 12 times better than with the Philibert values. The fact that the Philibert fit to some of the experimental aluminum data is not particularly good may explain in part why the Philibert expression gives better results at low atomic number than would be expected from the experimental f(x) data. Consideration of the entire group of 35 alloys is of value in that it shows what results to expect if a technique is applied indiscriminately-Le., without concern for atomic number or probe ratio. The techniques providing the most satisfactory combination of accuracy and precision under these conditions are those of Thomas (-1.0, 3.1) and Philibert (0.1, 3.5). The following is a list of general conclusions based on the preceding results concerning the absorption correction: 1. The techniques cif Tong (modified), Thomas, or Philibert should be used when large absorption effects are present at low atomic number; the former provides slightly better results than the latter two; all overcorrect; all yield the best results when k > 0.13. The present results indicate that an accuracy of about - 2 x (Coalo> Ct,) and a precision of rt 3 to f5 % are probable in this atomic number interval. 2. Green’s f(x) curves or the Duncumb and Shields technique should be used in the intermediate atomic number region; both provide sirnilar precision values but the former gives superior accuracy, particularly if k < 0.1. Errors in the systems investigated indicate that an accuracy of better than -1 and a precision of r t 2 x of the amount present are attainable with these techniques. 3. In the evaluation of a particular technique, meaningful ranges of probe ratios, atomic numbers, and mass absorption coefficients should be considered. This is illustrated by the fact that many techniques give excellent results in the intermediate atomic number interval when k > 0.1. In the present investigation only the jPhilibert or Thomas techniques give acceptable results on an overall basis. Indiscriminate application of the Thomas technique should yield accuracy in the
x
x
range of -0.3 % to -4 % and precision in the range of 2 to & 5 % . 4. Iteration improves the results, but the techniques can be evaluated by using Kcalo(Equation 10) because the relative order of the techniques is independent of the method of calculation. The results using Green’s f ( x ) curves will not be significantly altered by using iteration in the intermediate atomic number interval. 5. Iteration leads to improved precision and accuracy in the Belk procedure. 6. The means of calculating h’ are inconsequential except where wide differences between atomic and weight per cent exist; in such cases h’ should be calculated using Equation 16. Atomic Number Correction. The Philibert, Ziebold, Belk, Tong, and Ziebold and Ogilvie techniques can be applied with or without their respective atomic number correction factors. Application of these techniques, with the atomic number factor both included and excluded, to the 35 alloys used in evaluating the absorption correction factor indicates that atomic number corrections can not be made indiscriminately. The Philibert atomic number factor increases the error in only 37 of the alloys but it reduces slightly the overall precision and accuracy. Application of the Ziebold atomic number correction, in conjunction with the Philibert absorption correction, raises the standard deviation from 4.6 to 6.9 and increases the error in 73 % of the alloys. The Belk, Tong, and Ziebold and Ogilvie atomic number factors cause increased errors in 73, 73, and 87% of the alloys, respectively. Thus, in systems where the absorption effect is predominant, the calculated concentration will be closer to the true chemical concentration when the atomic number factor is excluded from these five correction formulas. When the techniques of Table I are applied to a small group of alloys from the AI-Co (32), AI-Mn (32), Cd-Mg ( 4 3 , Ti-Zr ( 3 9 , and AI-Ni (32) systems the Thomas (-1.0, 3.8) and Ziebold (- 4.1, 4.9) techniques, both of which contain atomic number correction factors, give the best results. Because of the importance of absorption effects in these alloys, it is not possible to select a preferred atomic number correction procedure. The author found, after completion of the present evaluation, that the backscatter factors used by Ziebold have been recently improved (46); thus better results than those indicated herein may be attainable with this procedure. In the Ziebold (46) formula, the first term in brackets in Equation 7 is evaluated using electron backscatter factors computed by Duncumb and Shields (46) and the electron range data of Nelms (46); the second term in brackets is evaluated using the Duncumb and Shields (19) modification of the Philibert (9) expression; and the last term is computed using Reeds (40) formula. The approximate (approximate because it was not possible to determine the exact value of the errors in all systems from the published data) results of an evaluation of atomic number correction procedures performed by Poole and Thomas (4) presented in Table VI indicate that the most effective technique is that of Thomas. The Ziebold technique was not included in the Poole and Thomas evaluation. It is concluded on the basis of the limited information provided herein that when atomic number effects are known to exist they can be most adequately treated using the Thomas or Ziebold tech(45) D. J. Schmatz, H. A. Domian, and H. I. Aaronson, Trans. AZME 236, in press (1966). (46) T.0. Ziebold, in “The Electron Microanalyzer and its Appli-
cations,” Lecture notes for summer program at the Massachusetts Institute of Technology,Suppl., 1966. VOL 39, NO. 4, APRIL 1967
425
Table VI. Poole and Thomas (4) Evaluation of Atomic Number Correction Procedures Arithmetic
average Technique Thomas. Thomas
/%I
iCtme
Theisen A&M A & Mb Birks
- C,io
Ctme
-k
0.251 0.284 0.413 0.436 0.514 0.647 0.689
4.0 4.6 6.7 7.0 8.3 10.4 11.1
Z&O
Relative error
Algebraic
-mean
l-
dc X 100
c 0.1 -1.3 -4.2 -2.2 0.0 -1.8 -3.7
Standard deviation 9.5 10.9 17.6 16.3 19.5 15.4 26.4
rich (27) and Norelco (47) coefficients, respectively, and the lower aluminum value leads to a total alloy concentration closer to 100%. It must be emphasized that since the true composition is unknown it is not possible to determine which, if either, of the coefficients are correct, notwithstanding the totals near 100%. If the true concentration is 54% Al, the Philibert technique would be expected to give a calculated concentration of 55.5 =k 3 % (from Table Vb) which nearly encompasses both calculated values. While it is not possible to say that either the absorption coefficients or the correction technique is the problem here, it is possible that the low Norelco (47) coefficients compensate for the tendency of the Philibert technique to overcorrect at low atomic number. CONCLUSIONS
Using Heinrich’s (27) mass absorption coefficients. * Using selected data in which interpolation was minimized.
4
niques. Further evaluation of the latter technique is being carried out. Characteristic Fluorescence Correction. The approximate formula of Reed (-1.8, 4.6) and the Castaing expression (-0.8, 5.1) give the best results when 19 alloys are corrected for characteristic fluorescence effects. The former removes over 84% of the probe error. These two techniques also give the best results when the evaluation is restricted to alloys with k > 0.1; to alloys with k < 0.1; to alloys with low probe errors; and to Fe-Cr alloys only. The Ziebold and Ogilvie, Ilyine, Birks,tWittry, and Dewey procedures all remove less than 70 of the probe error. Most techniques exhibit poor accuracy as the result of a large undercorrection; Le., if the absorption correction is assumed to be correct, the probe ratio corrected for characteristic fluorescence is too high. This directional effect makes empirical adjustment of some techniques feasible-e.g., the accuracy of the Reed expression is improved, for the alloys studied here, by replacing the voltage dependent factor, { (UB - l)/(U, - 1] in the Reed formula by ( UB - 1)/ (U, - 1) where U is the ratio of the acceleration potential to the excitation potential. This empirical adjustment is not theoretically justified and cannot be used on a general basis as it has been applied to only 19 alloys. Insertion of the Wittry (39) voltage dependent factor, as suggested by Wittry (N), in the Castaing formula gives results that are generally not quite as good as those listed above. The results of the present investigation indicate that characteristic fluorescence effects can be most effectively accounted for by using Reed’s approximate formula. This technique offers an accuracy range of - 1 to -2 % and a precision of & 3 to =k5% of the amount present in addition to providing a method of correction in the event of K L ( K exciting L), L K and LL fluorescence. The present results are too dependent on data from the Fe-Cr system. Further work needs to be carried out in other systems and also in systems involving other than KK fluorescence. Mass Absorption Coefficients. In attempting to correct data obtained in two aluminum base multicomponent alloys, it is found that the Heinrich (27) absorption coefficients lead to an “apparent” overcorrection. A probe ratio of 0.234 for aluminum is corrected to 0.606 and 0.506 using the Hein-
x
la6’,
x
426
ANALYTICAL CHEMISTRY
While uncertainty in the mass absorption coefficients can lead to a relatively large discrepancy between corrected and true concentrations, other factors are often at fault: apparent dilution or concentration of k often accompanies particle analysis as a result of electron penetration or a large x-ray beam size; the presence of undetectable elements and contamination effects can cause significant errors; and electronic and detector dead time errors can lead to low k values as can pulse amplitude shifts. Accurate quantitative results are not always easily obtained with an electron probe but by combining good experimental technique with the proper correction procedure, one can often obtain good composition data from volumes approaching one cubic micron in size. The proper technique for correcting for absorption at low atomic number is provided by Tong (modified), Thomas, or Philibert. In the intermediate atomic number range excellent results will be obtained with Green’s f(x) curves or the Duncumb and Shields technique. To simplify calculations the former should be fitted to an analytical model. Reed’s approximate formula should be used to correct for characteristic fluorescence. Atomic number effects will be best treated with the Thomas or Ziebold procedures. Utilization of these selected techniques should under most circumstances yield an accuracy of - 1 to -2 % and a precision in the range of =k2%to is%, the better results occurring in the intermediate atomic number interval and when k > 0.1. Only if large atomic number effects are present will less than 86-94% of the probe error be removed. In the relatively few cases where better accuracy and precision are essential, or when the mass absorption coefficients are questionable, calibration standards will be required. ACKNOWLEDGMENT
The author expresses his appreciation to L. Sturkey for his helpful comments on the manuscript. The author also extends his thanks to H. A. Diehl and L. F. Solosky for their assistance in performing the calculations and J. A. Koenig for supplying the computer program used in calculating standard deviations. RECEIVEDAUGUST1, 1966. Accepted January 10, 1967. First National Conference on Electron Probe Microanalysis, College Park, Md., May 1966. (47) Norelco Reporter, Vol. 9,No.3, (1962).
