54
ANALYTICAL CHEMISTRY, VOL. 51, NO. 1, JANUARY 1979
in powder form directly from the original container. Figure 8 illustrates the decrease in signal-to-noise ratio beyond 2500 nm. Similarly, Figure 9 shows the spectrum of praseodymium oxide (Pr60,,) in the near-infrared. The ultraviolet and visible spectra are saturated for this substance. Photoacoustic spectra have been obtained for a wide range of substances including textiles. paint, semiconductors, asbestos, fungi spores, paper, blood smears, and catalysts to name a very few. The number of potential applications is large, and instrumentation is becoming more readily available to make possible the exploration of these essential investigations.
LITERATURE CITED
---+ c--
300
1220
t-+*I 1540
18M1
2180
25w
2820
WAVELENGTH inml
Figure 9. Near-infrared photoacoustic spectrum of praseodymium oxide, Pr,Oll,
f r o m 900 t o 2820 n m made with a modulation frequency of
48 Hz, a time constant of 2.5 s, and a bandpass of 26 4 n m
The
UV-visible spectrum IS saturated
well beyond the usual 2500-nm limit. T o illustrate the simplicity of operation, additional rare earth spectra were run. Figures 7 and 8 show the PAS spectrum of neodymium oxide again placed in a sample holder
(1) M. J. Adams, J. G. Highfield, and G. F. Kirkbright, Anal. Chem., 49, 1850 (1977). (2) M. J. Adams and G. F. Kirkbright, Analyst(London), 102, 678 (1977). (3) A. Hordvik and H. Schlossberg, Appl. Opt., 16, 101 (1977). (4) A. A . King and G. F. Kirkbright, Lab. Pract., 25, 377 (1976). (5) J. F. McClelland and R. N. Kniseley, Appl. Opt., 15, 2658 (1976). (6) Yoh-Han Pao, Ed., "Optacoustic Spectroscopy and Detection", Academic Press, New York, 1977. ( 7 ) A. Rosencwaig, Opt. Commun., 7, 305 (1973). (8) A. Rosencwaig, Science, 181, 657 (1973). (9) A. Rosencwaig, Anal. Chem.. 47, 592A (1975). (10) A. Rosencwaig, Phys. Today, 28 (9), 23 (1975). (11) R. B. Somoano, Angew. Chem.. 90. 250 (1978). (12) A. Rosencwaig, and A . Gersho, J . Appl. Phys., 47, 64 (1976). (13) J. F. McClelland and R. Kniseiey. Appl. Phys. Lett., 28. 467 (1976). (14) R. Blank and H. Pawlowski. to be published.
RECEIL'ED for review July 28, 1978. Accepted October 6, 1978.
Calibration of Photographic Emulsions by Cubic Spline Functions and Application in Spectrochemical Analysis Paolo F. Frigieri' and Ferruccio B. Rossi' Centro Informazioni Studi ed Esperienze, Via Reggio €milia, 20090 Segrate, Milan, Italy
A method of computation for calibration of photographic emulsion and conversion of microphotometer readings to relative intensities on a digital computer with special application to spectrochemical analysis is descrlbed. A numerical method using cubic spline functions for the treatment of experimental data is discussed. The method extends the operational range of the quantitative determinations from the gross fog level of the emulsion to very high absorbance values approaching saturation. A program is described, that includes an emulsion calibration curve, corrections for the background, and the internal reference ratio calculations. The program provides the automatic selection of the required application through a preliminary analysis of the input data. Various algorithms are described, and some results to test the procedure are given.
The use of the photographic plate as a detector has several advantages. A large portion of the spectrum can be recorded simultaneously, and a permanent record is obtained. Since the plate must be exposed for a given time, the radiation intensity is integrated over the exposure time, and random fluctuations typical of discharges are averaged and smoothed out. The outstanding limitation in the use of photographic plates, however, is the complicated relationship between the incident intensity and the consequent plate blackening, which 0003-2700/79/0351-0054$01 O O / O
is the information needed for quantitative analysis. The plot of absorbance as a function of the logarithm of the exposure (commonly known as a Hurter-Driffied curve) is linear for a limited region where the intensity calculation is easy so that the possibility of a further extension of this region would be extremely advantageous. In an attempt a t linearizing the calibration curve and thus also at extending the operational range, mathematical transformations of the absorbance values were made by various researchers (1-23). The results, however, were not fully satisfactory, This can be clearly understood since even the most complicated transformation models do not rigorously reflect the physical phenomena involved but they are based on some simplifications of the phenomena. Hence, it seems extremely difficult to know the exact functional relationship among the physical quantities involved which would simplify the analysis. The numerical analysis has recently made use of some techniques which enable us to solve the problem of the approximation of a set of data, even in the absence of an exact model (24-28). Such techniques provide algorithms for the construction of the so called "spline functions" which are briefly described hereafter.
THEORY Let us consider a set of N pairs of data (XI, Y J ...(X,,YN) to be interpolated. Instead of using various classical inter1978 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 1, JANUARY 1979
55
if one knows the order of the measurement random errors, it is possible to define a semi-amplitude band (Figure 1)which certainly contains the true values. Among all the splines which may be selected inside the band, it is again possible to select the most regular one, that is the spline which minimizes a suitable square sum of the differences between the functional and experimental values. This estimation enables one LO adjust the so-called “smoothing coefficient” of the spline function. The “dy” coefficient is selected considering that the spline function S ( X ) also satisfies the following relation:
Figure 1. Example of interpolating S ( X )and smoothing spline functions
polating polynomials, a search is made of a function that may be defined as follows: all intervals between two adjacent points ( X I ,Xz)...(XN-l,X,) are covered by third degree polynomials, that are generally one different from the other; however, they all satisfy the general condition that the overall function is continuous with continuous first and second derivatives over the whole range from X 1 to X N . Among all the functions f ( X ) which interpolate the points (XI, Y J...(XN,YN)and which are continuous along with their first and second derivatives over the interval ( X I ,X N ) ,it is necessary to have S ( X ) ,such that:
Such a function S ( X ) is called a spline function and has the important property of being the most “regular” interpolating function with the lowest number of oscillations. The experimental data are generally affected by various errors for which the best approximation curve is not the interpolating spline, but an approximation one. In practice,
where PI are weights which, for simplicity‘s sake, have all been taken equal t o (O.E1/3~/’).dy. In this way the inequality (3) becomes: ,v 1 C(g(x,) - Y,)’ 5 -ds3 (3) /=1 12 The ‘‘dy” coefficient is selected in order that the smoothing of the spline function results more or less “exact”. Figure 2 points out how the smoothing spline changes with variations of the smoothing coefficient “ d j “ for a generic set of experimental points. Previous experience suggests that, in the absence of a good analytical model, the use of the spline function is one of the best approaches to the treatment of experimental data. This technique has found a useful application in obtaining the emulsion calibration curves, which are described in the following. Emulsion Calibration Curve. There are several techniques for relating the absorbance to relative exposure values.
Figure 2. Spline functions with different smoothing coefficients for the spectral region 3600 f 50 A
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 1, JANUARY 1979
L.r> START
/17 REA0 CARDS
FROM PERMANENT F I L E
OFOATAiN h WAVELENGTH
I
TRANSFORMATION OF TRASMITTANCE OATA INTO ABSORBANCE VALUES
I
I
TRANSFORMATION OF THE PHOTOMETRIC TRANSMITTANCE DATA INTO ABSORBANCE VALUES
1
DETERMINATION OF PRELIMINARY CURVES BY CUBIC SPLINES FLTTING
FINDING K PAIRS OF ABSORBANCE VALUES WITH CONSTANT EXPOSURE
STRAIGHT.LINES
AN-,
Figure 3. Preliminary curve of density of a less absorbed step A, vs. absorbance of a more absorbed step A N + ,
I-
A rotating seven-step sector has been used in this case, but the procedures to obtain data for this calibration program are similar to those of the conventional two-step method. This allows a good adaptability of the program to other measuring techniques by using filters or rotating disks with a different number of steps. The spectral range is divided into regions (100 A wide) where the emulsion calibration curves are supposed to have nearly the same behavior. Transmittance values for different relative exposures of selected lines in the region of interest are read by a microdensitometer and used as an input for the program. For every N readings of blackening measured for each line in N different relative exposures, it is possible to obtain N - 1 pairs of absorbance values for the same exposure ratio. It is then possible to plot a preliminary curve showing the variations of the experimental density values with the blackening of the photographic emulsion (Figure 3). The abscissa reports the absorbance values AN+1whose ordinates are the absorbance values AN of the next more exposed step. In such a way a discrete set of points to be approximated by a spline function, which is directly plotted by the program (Figure 4),is obtained. The graphical representation shows the outline of the function with respect to the experimental points. The goodness of fit can be judged on the basis of such graphs, and can be improved-if necessary-by varying the smoothing coefficient “dy’’, keeping in mind that the more “dy” is decreased, the more the fit is exact (see relation 3). From this preliminary curve, the emulsion calibration curve A vs. log E can be obtained by a step-by-step procedure. By the AN vs. A.\r+l curve, starting from a very small value of the abscissa, say A,, the values A1, A z , AB, . . . assumed by the absorbance for constant exposure increments are determined. The zero value of log E is then associated with A, and with increasing values of log E to AI, AP,AB, . . . so that the difference between two successive values is constant. The value 0.3 has been used as the increment corresponding to the logarithm of the relative exposure ratio of the various steps of the rotating chopper. In such a way, a first set of data has been obtained (Ao,O), (Al, 0.3), ( A 2 ,0.6), . . . for the deter-
PLOTTING OF PRELIMINARY AN0 CALIBRATION CURVES
STORAGE OF CALIBRATION CURVES ON A PERMANENT FILE
Figure 4.
PLOTTING OF ANALYTICAL STRAIGHT.LINES ANOEXPERIMENTA 1
I
Computer programs flow chart
mination of the curve A vs. log E. Analyzing the preliminary curve again, giving a small increment A, to A , and repeating the above procedure, some increasing values Ai‘,A*’, A3’, . . . have been found which correspond to equal relative exposure increments. T o find the values of log E related to the absorbance, the program seeks the log E l value from the previous A vs. log E curve which is assigned to ( A , + Ao);the program relates (log E ) , + 0.3, (log E ) , + 0.6, (log E),+ 0.9, . . . to AI’, AZ’, A i . A new set of points comes out, which together with the points already calculated and those to be deduced by further iterations, are used to obtain the spline emulsion calibration curve. The corresponding values of A and log E are stored for use in the second program. Analytical Curve and Quantitative Analysis. The quantitative spectral analysis is based on the relationship between the intensity of the spectral lines and the concentration of the elements emitting these lines. The relative intensities of the emitted lines are obtained from the emulsion calibration curve discussed above. The analytical curve is obtained by a linear least-square fit among log E values and the corresponding concentrations. The following data are used as input for a second program: (a) analytical line of the element; (b) reference line, if any; (c) photometric transmittance values of the line (and the possible background) and the information on their exposure step; and (d) concentration values of the elements. After a preliminary analysis of the input data, the program converts the photometric transmittance into log E values, using the emulsion calibration curve peculiar to that wavelength.
ANALYTICAL CHEMISTRY, VOL. 51,
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Table I. Cumulated Standard Deviation Values of Relative Intensities Obtained from the Same Set of Data cumulated standard deviation, reference % analyt- calibraical line, tion spline graphic Seidel A curve, A method method method 2580 2600 2.09 6.38 5.20 3273 3300 7.50 12.24 13.26 3413 3400 4.75 11.39 8.04 3730 3700 4.45 7.01 7.31
e I
1.60
I
1
I
l
2. YO
I
log E
Figure 5. Calibration curve obtained with S A 1 Kodak plate
If, together with the analytical line, a reference line is taken into account and their absorbance values are determined by means of a microphotometer in different regions at different exposure times, the program gives the logarithm of the exposure ratio as the average value of the values of all the possible combinations among the relative exposure data of the analytical and reference lines. It is now possible to obtain the analytical straight-line. The subroutines provide different analytical straight-lines, according to whether an internal standard is used, and the background of the plate for one or both lines is considered or not. Finally, by feeding the program with an input similar to that previously mentioned, the unknown concentration of the element in the analytical sample is obtained. RESULTS AND CONCLUSIONS Both programs are in FORTRAN IV and have been used in an I.B.M. 370/125 computer equipped with a 1627 plotter. A flow chart is shown in Figure 4. The programs, easy to use, provide the graphic representation of all functions. Figure 6 shows a calibration curve obtained for SA-1 Kodak plates, making use of data of spectral lines ranging from 3550 to 3650 A. Since the cubic spline method for the calibration of photographic emulsions disregards any analytical model representing the physical phenomenon and, moreover, the calculation code, by means of the smoothing coefficients, enables the user to choose the approximation level between the experimental points and their trend curve, it is, therefore. meaningless to calculate the fitting between the experimental points and the calibration curve. T o test the precision of the method, calculations have been performed to evaluate the dispersion of the intensity data, obtained from different blackening values coming out of different exposure times of the same emission source. As known, the poor precision of the analytical spectrographic methods often depends on the poor reproducibility of the microphotometric measurements, so that the calculation of the dispersion of the intensity data has been carried out by calculations of the percentage of the cumulated standard deviation of five sets of data obtained from different spectrophotometric measurements of the same photographic plate, where the lines recorded by seven different exposure times show, of course, seven different blackening values. T o allow us a better evaluation of the results, the same statistical
estimation has been performed on the same series of experimental data by using in the intensity calculations either the standard graphic procedure (23) or a method (20) based on the Seidel transformation. Table I shows the results of such a comparison together with the information on the analytical line under analysis and the calibration curve used for the intensity calculations. As can be seen in the table, the method allows one to obtain a precision, which, though not so high, is, however, significantly better than that provided by the other methods. The use of the cubic spline functions for the treatment of spectrographic data permits the elaboration of all the blackening data in the whole range from the gross fog level of the emulsion to 3.0 absorbance units with a sufficient precision. Moreover, the described method can be codified in a speedy execution and easy to use program, which, at any time, gives the user the best choice of the function fitting the experimental data. Copies of the FORTRAN computer programs are available upon request.
ACKNOWLEDGMENT The authors thank R. Trucco for helpful suggestions and a critical review of the method and R. Anzani and M. Cambiaghi for technical assistance during the experimental work. LITERATURE CITED W. Seidei. "Sitzung des spektralanalytischen Ausschusses der Gesellschafi-Metal und Erz" (1939). H. Kaiser, Spectrochim. Acta, 2, 1 (1941). M. Honerjager-Sohm and H. Kaiser, Spectrochim. Acta, 2, 396 (1944). H. Kaiser, Spectrochim. Acta, 3, 159 (1947). T. Torok and K. Zimmer. Acta Chim. Hung., 29, 273 (1961). K. Zimmer and T. Torok, Acta Chim. Hung., 28, 59 (1961). T. Torok, K . Zimmer, and S. %ti, Microchim. Acta, 611 (1962). T. Torok and P. Zentai, Acta Chim. Hung., 30, 11 (1962). T. Torok, Emissionspektroskopie, Berlin, 141 (1962). T. Torok, Chem. Anal., (Warsaw), 7, 47 (1962). T. Torok and K. Zimmer, "Quantitative Evaluation of Spectrograms by t-Transformation, Heyden, London, 1972. K. Zimmer, T. Torok. and I . Asztalos, Chem. Anal. (Warsaw), 11, 1065 (1966). C. S. Joyce, Can. Spectrosc., 10, 33 (1965). C . R. Bosweil, S. S. Berman and D. S. Russell, Appl. Spectrosc., 23, 268 (1969). D.M. Shaw, Can. Spectrosc., 10, 3 (1965). A. Carnevaleand A. J. Lincoln, Spectrochim. Acta, Part B,24, 313 (1969). M. Margoshes and S. D. Rasberry, Spectrochim. Acta, Par7 B , 24, 497 (1969). R. J. Heemstra, U . S . Bur. Mines Rept., R I 7447 (1970). R. J. Decker and D. J. Eve, Spectrochim. Acta, Part E , 25, 479 (1970). J. A. Holcombe, D. W. Brinkman, and R. D. Sacks, Anal. Chem., 47, 441 (1975). W. A. Gordon and A. K. Gallagher. TMX-1220, April 1966. D. R. Blevins and W. R. O'Neill Appl. Spectrosc., 30, 190 (1976). ASTM Designation E 115-71 and E 116-70 Recommended Practice for Photographic Photometry in Spectrochemical Analysis, American Society for Testing and Materials, Philadelphia, Pa. C. H. Reinsch, Numer. Math., 10 177 (1967). T. N. E. Grevilie Math. Methods Digital Comput., 2, 156 (1967). T. N. E. Greville, MRC Tech. Summ. Rpt, 893 (1968). C. de Boor and J. R. Rice, Rpt. CSD TR 20, Computer Science Department, 1968. C. de Boor and J. R. Rice, Rpt. CSD TR 21, Computer Science Department. 1968.
RECEIVED for review November 29, 1976. Resubmitted July 24, 1978. Accepted September 11, 1978.
