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ANALYTICAL CHEMISTRY, VOL. 50, NO. 9, AUGUST 1978
spectrum Rz(xl, a,), using Houston's notation, where the subscript 2 indicates that the second derivative of the original data is first taken, and then this second derivative is integrated twice to recover the original spectrum minus the constant and linear components lost during differentiation. Using a least-squares-fit smoothed-derivative array, the smoothing operation occurs automatically in the process of determining the second derivative. x , is the independent variable used in the integration, and a, is the lower limit of integration. Then calculating RZ(x,,a ) ) and R&, a J f l )where , a, and alfl are two x , values in the spectrum below and above the threshold feature to be extracted, respectively, the extracted threshold signal is given by Rz(x,,a,) - R2(x1,a,+1). The locations of the threshold signals are indicated by arrows at the top of Figure 1. The values of x, that were chosen for the a,'s in the decomposition of the original spectrum were cy1 = l; a 2 = 118; a3 = 165; a4 = 210; a5 = 280; a6 = 349; and ai = 400. These ai's were chosen to lie in regions of the spectrum assumed to contain only background, and no threshold-signal, information. There were 512 data points in the spectrum and a 19-point convolution array (Equation VI1 of Table I) was used to determine the second derivative. The seven spectral components resulting from the decomposition are plotted as curves (a) through (g) in Figure 1. In the original data, three thresholds are clearly discernible while the other three are less so. In the component spectra (curves a through g), all six thresholds are clearly seen. As this example calculation was performed for illustrative purposes only, linear extrapolation of the signals below the threshold energy (in the direction of increasing x , values) was made instead of using the more correct reciprocal-energy dependence (14). Note that the decrease in counts-per-channel below threshold for the molybdenum signal indicates an enhancement of the molybdenum concentration at the solid surface. This result was corroborated by backscattering data taken with 1.4-MeV protons incident on the sample (14). Curve h gives the sum of curves a through g, all with unit scaling, and is seen to be essentially a smoothed version of the original data. Thus, the differentiation techniques demonstrated by Peisach (13) as useful in analyzing backscattering spectra can be simply extended to give component threshold signals. Least-squares-fit smoothing is beneficial in this operation in ameliorating the noise buildup in the differentiation. I t provides a simple one-operation method
of obtaining the smoothed derivative. The Savitzky-Golay convolution method of least-squares-fit smoothing and differentiation of digital data is versatile and extremely simple to use. Using only the coefficients given in the Savitzky-Golay tables ( I ) , one is limited to a 25-point smooth, however. Wider smoothing arrays are sometimes desirable and may be simply generated using the eleven equations given in Table I of this letter. Relaxing the condition that integer numerator and denominator values for the ps(q)be calculated separately leads to further simplification in the computer program. Smoothing has been shown in data processing involving deconvolution (7)and/or differentiation ( I 1 , 1 2 )to be beneficial. Peak position, height, and width-of lines of known profile-are not the only information that may be sought in some spectroscopies. The smoothing techniques of Savitzky and Golay offer an extremely simple aid in extracting additional lineshape information.
LITERATURE CITED (1) (2) (3) 14) , . (5) (6)
(7) (8)
(9) IO) 11)
12) 13) 14) (15)
A. Savitzky and M. J. E. Golay. Anal. Chem., 36, 1627 (1964). "Citations Index" (1965-1976). J. Steinier, Y. Termonia, and J. DeRour, Anal. Chem., 44, 1906 (1972). R. R. Nernst. Adv. Maon. Reson.. 2. 39 (1966). P. D. Willson and T. H. Edwards, App/. Spectrosc. Rev., 12, 1 (1976). (a) E. Whittaker and G. Robinson, "The Calculations of Observations", 4th ed.,Bbcke and Son, Ltd.. London, 1944, pp 291-296. (b) P. G. Guest, "Numerical Methods of Curve Fitting", Cambridge University Press, 1961, pp 349-353. (c) M. G. Kendall and A . Stuart, "The Advanced Theory of Statistics", Hafner Publishing, Inc., New York, N.Y., 1966, Vol. 3, "Design and Analysis, and Time-Series", pp 366-375. H. H. Madden and J. E. Houston, Adv. X-Ray Anal., 19. 657 (1976): J Appl. f h y s . , 47, 3071 (1976). T. H. Edwards and P. D. Willson, Appl. Spectrosc., 28, 541 (1974). C. G. Enke and T. A. Niernan. Anal. Chem., 48, 705A (1976). M. G. Kendall and A. Stuart, ref 6c. pp 375-379. H. H. Madden and D. G. Schreiner, "Convolution Method for LeastSquares-Fit Smoothing and Differentiation of Digital Data", Report No. SAND76-0283, Sandi Labs., Albuquerque, N.M., 1976, unpublishedwork. S. Rubin, T. 0. Passell, and L. E. Bailey, Anal. Chem.. 29, 736 (1957). M. Peisach, Thin Solid Films, 19, 297 (1973). J. A. Borders and G. D. Peterson, "Ion Backscanering Studies of Synthoil Catatyst Materials", Report No. Sand77-0479, Sandia Labs., Albuquerque, N.M., 1977, unpublished work. J. E. Houston, Rev. Sci. Instrum., 45, 897 (1974).
H a n n i b a l H. M a d d e n Sandia Laboratories, Albuquerque, New Mexico 87185
RECEIVED for review April 18,1977. Accepted May 22, 1978. This work was supported by the U S . Department of Energy under Contract AT(29-11789.
Extension of the Ras ber ry- He inric h Equat ion for X-ray FIuorescence Analysis Sir: For an "infinitely" thick sample, the x-ray fluorescence intensity is shown to be given by the following expression:
where K, is a factor to take care of the geometrical arrangement of the sample-detector and of fundamental parameters of element j . C, is weight fraction of element j . ~ Q ( A ) is mass absorption coefficient of the sample (denoted by Q ) a t the incident wavelength, A. p Q ( A J ) is mass absorption coefficient of the sample for the primary fluorescence radiation 0003-2700/78/0350-1386$01 OO/O
of element j . k j ( X ) is mass absorption coefficient of element j for the incident wavelength, A. Ami, is minimum wavelength of the generator tube as given by the excitation voltage. XJKab is wavelength of the K-absorption edge of element j . Equation 1 does not include higher order fluorescence effects (secondary, tertiary, etc.) that occur when the sample contains more than one chemical element. Furthermore, we consider monochromatic incident radiation in order to make the physical processes therein easier to understand. This would lead to a better understanding of what happens when the incident spectra is polychromatic, as it is in most of the practical cases. C 1978 American Chemlcal Society
ANALYTICAL CHEMISTRY, VOL. 50, NO. 9, AUGUST 1978
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CONCENTRAT ION
Figure 1. Possible cases of calibration curves for binary samples, I: A,, = 0 , BA, = 0; 11: AB, > 0 , BE, < 0, (JBBAI< AB,); 111: A,, > 0, BAB= 0; Iv: A,, > O r < 0, 86, < 0, (-1 < A,, + BBA< 0); V: A,, < 0 , B,, = 0
For a binary sample containing two elements, denoted by
A and B, expression 1 reduces to:
CA ) ( P B (1)+
PB
(1, )) (2)
Dividing this intensity by the one from a pure sample of element A we get:
LLIILlilI
‘0
0.1
0.3
0.5
0.7
09
CONCENTRATION
Figure 2. C I R as a
function of C for binary samples. The numbering has the same meaning as in Figure 1 When the characteristic radiation intensity from the element of interest is enhanced by fluorescence from the other element, an additional term shoud be included in Equation 5 to represent such enhancement; in this case the shape of the curve will no longer be hyperbolic. The semiempirical formula proposed by Rasberry and Heinrich ( 1 ) is:
R A =I G Q = CA IA CA + (1- C A b A B where:
in the case in which both the incident and take off angles are equal (cosecant factors are cancelled out). Defining:
In this expression the coefficients A l krepresent absorption, while the Bikrepresent enhancement effects. Only the Alk’s have a theoretical justification and they could be determined this way as well as experimentally. The Blk’s can only be determined experimentally since a closed theoretical expression for them does not exist. Denoting the components of a binary sample by A and B, expression 6 reduces to:
we arrive a t a more generally used expression:
RA =
-C A
1 + A A B (-~ C A )
(5)
ANALYSIS OF THE EMPIRICAL EXPRESSION Equation 5 has been used by many authors (1-6) as a relationship between the relative fluorescence intensity of one element and its concentration. I t is valid whenever the element of interest is not affected by characteristic radiation emitted by the other element in the sample. In Figure 1, curves I11 and V illustrate the behavior of RA as a function of CA for different values of the coefficients, Am. The corresponding range of values of Am are shown for each case, correcting those ranges initially given by Rasberry and Heinrich ( I ) . These authors gave the values 0 < A < 1 (“negative absorption”) and A > 1 (“positive absorption”) when it should be -1 < A < 0 and A > 0, respectively. However, this error does not affect the method proposed by them, neither was the interpretation given to their results, because the samples of Cr-Fe-Ni they studied do not show “negative absorption” effects. Negative absorption occurs, for example, in an A1-Cd sample, when it is irradiated with the Fe K a line and the emitted Cd Lal line is observed.
and
Element A is the one capable of exciting with its characteristic line the absorption edge of interest in element €3. Figure 2 shows curves of C I R vs. C. Five cases are now possible: I: A A B = 0, RAB = 0 (linear calibration curve); 11: A B A > 0, R B A < 0 (absorption effects overcome enhancement); 111: Am > 0, BAB= 0 (preferential absorption); IV: ABA> or < 0, BBA< 0, (-1 ABA+ BBA< 0) (enhancement effects overcome absorption); V: A A B < 0, R A B = 0 (“negative absorption” effects dominate). Case I1 was not considered by Rasberry and Heinrich even though it is possible; while case V is not completely dealt with. Cases I, 111, and IV are discussed by them but the ranges of the coefficients were erroneous. A few comments about Equation 6 are in order at this point. First, this formula does not take into account tertiary and higher order fluorescence effects, always present in multicomponent system. Thus, if one tries to apply it to these
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ANALYTICAL CHEMISTRY, VOL. 50, NO. 9, AUGUST 1978
systems, the BIk’s will be slightly dependent on the concentration of t h e elements. T o overcome this, Hawthorne and Gardner ( 7 ) ,found an additional empirical term to take into account tertiary contributions, while keeping the coefficients B , k independent of the concentrations. Secondly, Alkcannot be zero when enhancement is the predominant effect, as was proposed by Rasberry and Heinrich ( I ) , mainly because absorption effects are always present, as noted by Budesinsky (8),Tertian (9),and Hawthorne and Gardner (7). If one takes Aik = 0, then the BIk’swould again be dependent (linearly) on t h e concentration of the elements.
DISCUSSION We wish to point out that the importance of Equation 6 in the analysis by x-ray fluorescence of multicomponent samples, resides in the fact that coefficients A and B should be independent of the concentrations CA and CB. This is achieved when the only effects present in the spectral line of interest are of secondary order. When higher order effects (tertiary, quaternary, etc.) of relative importance are present, an additional term should be added for each of them. Suppose now that in the analysis of a binary sample Equation 6 is used and BAB= 0, then two cases are possible: (i) Am > 0, in this case A U C B represents a loss of fluorescent x-rays since t h e matrix absorbs more incident and/or fluorescent radiation than a standard of element A. (ii) Am < 0, in this case A A B C B represents an excess of fluorescent radiation as compared with a standard of element A. This effect could be thought of as a case of “negative absorption” since its net result is similar to an enhancement. Let us further suppose that in Equation 8 the element B be such that the terms A B A C A are negligible compared with B B A C A / ( l + C B ) . Then the predominant effect is enhancement of fluorescent radiation. This last term takes into account the net excess of x-rays in the characteristic radiation of element B, due to the fact that radiation emitted by element A is energetic enough to produce secondary fluorescence in element B (note that BBAis always negative). Finally, if the coefficient BBAis such that: JBBAI < ilBA/2 we have curves above the line (C/R)B = 1 in Figure 2. For this relation the curve ( C / R ) B vs. CB will show a maximum in the range 0 < C B < 1. This fact is of no physical significance, since the curve we always use to determine the un-
known element concentration is
RB
vs.
CB.
APPENDIX Consider a binary sample composed of chemical elements denoted by indices A and B such t h a t Z A > Z B and hAK < XBKab, then we can write: ,n
r
,
Hyperbolic curves will lie above (below) the line ( C / R ) = 1, if t,he factor between brackets in Equation A-1 is greater than zero (less than zero). Curves of type I1 in Figure 2 occur when the second term in Equation A-1 is positive. Since A B * is a positive constant, 1+ BBA/ABA(l C B ) must be positive too. Because B B A is always negative, the relation between BBAand A B * is A B * ( 1 + CB) > - B B A or A B * > lBBAl since (1 + C,) 1 1. Curves of type V in Figure 2 are given by Equation 5, namely, ( C / R ) A = 1 + A A B - A A B . CA with A A B < 0.
+
LITERATURE CITED (1) S.D. Rasberry and K. F. Heinrich, Anal. Chem.. 46, 81 (1974). (2) J. W. Criss and L. S. Birks, Anal. Chem., 40, 1080 (1968). (3) R. Tertian, Spectrochim. Acta, Part 6,24, 447 (1969). (4) A. Guinier, Rev. Univers. Mines, 17, 143 (1961). (5) H. J. Beanie and R. M. Brissey, Anal. Chem., 26, 980 (1954). (6) G. R. Lachance and R. J. Traill, Can. Spectrosc., 11, 43 (1966). (7) A. R. Hawthorne and R. P. Gardner, Anal. Chem., 48, 14 (1976). (8) B. W. Budesinsky, X-Ray Spectrom., 4, 166 (1975). (9) R. Tertian, X-Ray Spectrom., 2, 95 (1973).
Jose A Riveros Rita D. Bonetto Raul T. .Mainardi* Instituto de Matemfitica, Astronomia y Fbica, IMAF, Universidad Nacional de Cbrdoba 5000 Cdrdoba, Argentina
R E C E I ~ Efor D review January 11, 1978. Accepted April 28, 1978. The authors acknowledge financial assistance from the ComisiBn Nacional de Estudios Geoheliofisicos (CNEGH) and the Universidad Nacional de CBrdoba of Argentina.
