Calibration for Interelement Effects in X-Ray Fluorescence Analysis S. D. Rasberry and K. F. J. Heinrich Institute
for Materials Research, National Bureau of Standards, Washington, D.C. 20234
A new empirical method for the calibration of X-ray fluorescence analysis in the presence of interelement effects is given. The effects of secondary fluorescence and of absorption are considered separately, with different expressions, in the calibration equation. The new approach is compared with empirical methods previously proposed by other authors, and is accurate and applicable over wide ranges of composition, even when the number of standards available is limited. An alloy system, in which the interelement effects are severe (Fe-Ni-Cr, over a range of composition from 0 to ~ O O Y O is ) , studied experimentally; also evaluated are data, previously presented by other authors, for the calcium carbonate-silica system. With reference to 23 chemical determinations, in the two systems, the relative error is f l % or less in 1 3 instances and f4% or less in all but two determinations.
Intensities of fluorescent X-rays can be measured very precisely. Relative standard deviations of less than 1%are common in routine analysis. However, a, comparable level of accuracy is seldom obtained without the use of a nonlinear calibration function; the measured X-ray intensity is usually not proportional to the concentration of the analyte. Such lack of proportionality may occur if the samples are inhomogeneous or particulate (particle effects), or when the X-ray emission by the analyte is significantly affected by concentration variations of the other elements in the specimen (interelement effects). In this paper we will treat only the calibration of X-ray intensities obtained from homogeneous specimens which are infinitely thick with respect to X-ray penetration, have smooth surfaces, and do not show effects of differences in particle size. Methods of calibration which aim to account for interelement effects fall into two categories. One uses equations theoretically derived from first principles and certain physical constants and parameters (including absorption coefficients, fluorescence yields, excitation spectrum, and geometric factors) to predict the X-ray intensity emitted by an element in any prescribed sample composition (1-19). An alternative is the empirical use of stan-
(1) E. Gillam and H. T. Heal, Brit. J. Appl. Phys.. 3,353 (1952). (2) P. K . Koh, B. Caugherty. and R. E. Burket, J. Appl. Phys., 23. 698 (1952). (3)J. Sherman, Spectrochim. Acta, 7,283 (1955). (4) J. Sherman, Spectrochim. Acta, 15,466 (1959) (5) F. R . Bareham and J. G. M. Fox, J. lnsf. Metals. 88, 344 (195960). (6) W.Marti, Spectrochim. Acta. 17,379 (1961). (7) R. Muller, Spectrochim. Acta, 18,1515 (1962). (8) A. P. Nikol'skii, Bull. Acad. Sci. USSR. Phys. Ser., 28,786 (1964). (9) A. V. Pivovarov, Zavod. Lab.. 31,1081 (1965). (10) T. ShiraiwaaneN. Fujino, Jap. J. Appl. Phys., 5,866 (1966). (11) G.Anderrnann,Anal. Chem., 38,82(1966). (12) C. J. Carman, Develop. Appl. Spectrosc., 5, 45 (1966). (13) H. Ebel, Z. Metalik., 57,454 (1966). (14)J. V. Gilfrich and L. S. Birks. Anal. Chem., 40,1077 (1968). (15)T. Shiraiwaand N. Fujino, Advan. X-RayAnal.. 1 1 , 63 (1968). (16) R. Tertian, Spectrochim. Acta, 268,71 (1971). (17) D. A. Stephenson,Anal, Chem., 43,1761 (1971). (18)J. W. Crtss and L. S. Birks, Anal. Chem., 40,1080 (1968).
dards to quantitatively define the extent of the interelement effect (19-42). Our discussion will be of this latter approach. We will show that the many empirical methods which have been published are variations of a few types of equations, most of which are useful over limited ranges of composition. In this paper, we propose an empirical method applicable over a wide range of compositions. It can be used with relatively 'few standards, or be overdetermined with additional standards to improve accuracy.
PREVIOUS EMPIRICAL PROCEDURES Interelement Effects. The intensities of fluorescent Xrays produced within a specimen depend on the mass fractions of the elements within the specimen, and of the relationships of their mass absorption coefficients for the primary radiation. Similarly, absorption of the fluorescence radiation within the specimen depends on the mass absorption coefficients for the fluorescence radiation. When low concentrations are measured in a set of nearly identical specimens, the relative X-ray intensity emitted by one element is virtually proportional to its concentration. But, a calibration curve which is linear (Figure 1,A) over a wide range of composition is obtained only when the absorption coefficients of the specimen for the primary and fluorescence X-rays are nearly constant over the same range of composition. The shape of curve B , in Figure 1, may arise because the absorption by the specimen of either the primary or fluorescence radiation, or both, is greater than that by the analyte; consequently, the measured relative intensity is below that given by a linear calibration curve. Theory and experiments indicate (20) that in this case the response between relative intensity and concentration for binaries can be approximated adequately by a hyperbolic function: n
R=
1
+
L
a(l
-
C)
(1)
(19) R. Tertian, Spectrochim. Acta, 238,305 (1968). (20) H. J. Beattie and R. M. Brissey, Anal. Chem., 26,980 (1954). (21) H. D. Burnham, J. Hower, and L. C. Jones, Anal. Chem., 29, 1827 (1957), (22) A. Guinier, Rev. Unlverselle Mines, 17,143 (1961). (23) B. J. Mitchell, Anal. Chem.. 33,917 (1961). (24) H. J. Lucas-Tooth and B. J. Price, Metallurgia, 64,149 (1961). (25) H. J. Lucas-Tooth and C. Pyne, Advan. X-Ray Anal., 7,523 (1964). (26) W.Marti, Spectrochim. Acta, 18,1499 (1962). (27) A. M. Gillieson, D. J. Reed, K . S. Milliken, and M. J. Young, Amer. SOC.Test. Mater. Spec. Tech. Pub/., 376,3 (1964). (28) G.R. Lachance, Geol. Surv. Can.. Pap., 64-50(1964). (29) R. J. Traill and G. R. Lachance, Geol. Surv. Can.. Pap.. 64-57 (1964). (30)J. Lerouxand M. Mahmud, Anal. Chem., 38,76 (1966). (31)G. R. Lachance and R. J. Traill, Can. Spectrosc., l1,43 (1966). (32) R. J. Traill and G. R. Lachance, Can. Spectrosc., 1 1 , 63 (1 966). (33) B. J. Alley and R. H. Myers, Anal. Chem., 37,1685 (1965). (34) B. J. Mitchell and F. N. Hopper.Appl. Spectrosc., 20, 172 (1966). (35) L. Backerud, Appl. Spectrosc., 21,315 (1967). (36) F. Claisse and M. Quintin, Can. Spectrosc., 12,129 (1967). (37) F. Claisse. Can. Spectrosc., 12,20 (1967). (38) R. Tertian.Advan. X-RayAnal., 12,546 (1969). (39) R. Tertian, Spectrochim. Acta. 248,447 (1969). (40) B. Thiele, Siemens-Z., 44,707 (1970). (41) G. R. Lachance, Can. Spectrosc., 15,3 (1970). (42) S. D. Rasberry and K. F. J. Heinrich, Colloquium Spectroscopicum Internationale X V I , Heidelberg, Germany, Oct. 1971.
5
0
b *.
r
+
M
*
c:
0
z
-.0
+
SM R E 0 II 0
9
E
I -I
0 \
a II
&
+
EM b
*
a.
0
=.0 a -.
II i
+
5M b
P 0
+ 5M -m
9 +(..0
z
E! Y 5
P
I
Y
P
where R is the measured relative intensity (ratio of counts on unknown to th,ose of the pure element standard, both corrected for background and counting deadtime), C is the concentration (mass fraction) of the measured element, and a is a constant which can be determined experimentally by means of standards. The magnitude of a is related to the displacement of the hyperbola from the linear calibration curve. The constant a is a positive real number; it is greater than one for cases of selective absorption in which the measured relative intensity is less than linear to concentration (mass fraction). The relation of Equation I to the general equation of a hyperbola is given in the Appendix. If the specimen absorbs the primary radiation or the X-rays emitted by the analyte, or both, less than does the pure element standard, an absorption effect in the opposite direction is usually obtained (C in Figure 1). Equation l also applies with 0 < a < l. A fourth possibility, D, is the presence of secondary fluorescence following photoelectric absorption by the analyte of a characteristic X-ray line from one of the other elements. We have observed experimentally that, for the case of secondary fluorescence, the function relating the concentration to the relative intensity in binary systems is not hyperbolic. Previous Procedures. We have examined twelve formulas given in 16 earlier papers which treat the empirical correction of interelement effects (see chronological listing in Table I). The formulas are quite different in appearance. If one extends them to include end points (pure elements), this extension determines the values of some of the constants used by the authors; for example, of reference 18 becomes equal to unity. Over restricted ranges of composition the constraint that a l , = 1 may be relaxed. Another reasonable assumption in considering these equations is that the sum of mass fractions equals unity. When these assumptions are observed and the notation of the authors is changed to a common set of symbols, striking similarities result. The first column of Table I gives the names of the authors and references. The second column shows the proposed equations in the form and notation of the original publications. In the third column, we have preserved the original form of each equation and have standardized the notation to that used in this paper: I and I’ are the measured X-ray intensities (corrected for background and counting deadtime) for the unknown or standard and for the pure element, R is the ratio I/Z’, C is the weight fraction. The subscripts i, k , and 1 are element designations, i denoting the analyte. In equations applicable to binaries only, the subscripts i and k are replaced by 1 and 2. The symbols a and b are used for constants and A, B, a , and p are interelement-effect coefficients. The definitions of symbols used by the various authors can be ascertained by comparing column two with column three. In the fourth column, the substitutions of constants are given which render the equation most nearly hyperbolic. Six of the proposed equations now have the hyperbolic form first given by Beattie and Brissey (20). Their method requires, as a minimum, one standard less than the number of elements present. Four of the equations-those by Lucas-Tooth and Price, by Lucas-Tooth and Pyne, by Gillieson, and by Thieleare algebraically equivalent, giving rise to a second model, first proposed by Lucas-Tooth and Price ( 2 4 ) . This model is similar to the hyperbolic model except for the substitution of X-ray intensities for mass fractions in the righthand side of the equation. The purpose of this approach
Figure 1. A. Linear calibration curve. 8. Preferential absorption by matrix. C. Preferential absorption by analyte. D. Secondary
fluorescence I
I
I
- Ni
“IFe
0
I
I
BINARY
I
I
I
I
0.2
0.4
0.6
0.0
CFe Figure 2. C / R as a function of C for iron
in Fe-Ni-Cr alloys.
The binary curves are the boundaries for the ternary system. The measurements were made at 45 kV and the other experimental conditions are shown in Table I I
was to simplify the manual solution of the equations for the mass fractions rather than to achieve higher analytical accuracy than previous models. In two of the equations, terms proportional to X-ray intensity are used in lieu of actual intensities (pap in Lucas-Tooth and Pyne and C E in Thiele’s work). Thiele used intensities corrected for background; thus, his equation does not contain the background constant, a,, found in the other three equations. The calibration of Thiele’s equation requires, as a minimum, one standard more than the number of elements present, while the other equations of this model require two standards more than the number of elements. The third model, given by Claisse and Quintin (36), is similar to the hyperbolic model, with addition of quadratic and cross-product terms for the elements other than the analyte. The additional terms improve the potential for
ANALYTICAL CHEMISTRY, VOL. 46, NO. 1, JANUARY 1974
83
these methods for analysis. Except in the method of Claisse and Quintin, the intended range of calibration is limited. Similarity of equation forms does not imply that complete methods are equivalent or will produce the same results. This stems from differences in the procedures the various authors give for determining interaction coefficients and the way they treat over-determined sets of calibration data. For comparison, the model discussed in this work has been added at the bottom of the table.
t s n
0
0.2
0.4
0.6
1.0
0.8
GI
as a function of C 1 for the equation: ( C / R ) I = 1 + (B12C2)/(1 C1); the value of i312for each curve, beginning with the lowest, is -1.0, -0.7, -0.5, -0.3, -0.1, and 0.0 Figure 3. (C/R)I
+
SAMPLE ID A s . 8's. R's
NEW PROCEDURE The calibration curves illustrated in Figure 1 can be studied more incisively if C / R is plotted as R function of C, as is illustrated in Figure 2. This plot has been described by Ziebold and Ogilvie (43) for use in interpreting electron microprobe data, and one similar to it has been used by Claisse and Quintin (36) for X-ray fluorescence data. ( C / R = 1) denotes a linear calibration curve. Where the absorption is dominant, the ratio C I R is greater than unity due to the reduction in relative intensity. Conversely, when secondary fluorescence is dominant, the values of C / R fall below the unity level. On the graph of C I R as a function of C, the hyperbolae plotted in Figure 1 are represented as straight lines going through (1,l) and ( 0 , l a ) . The experimental values shown in Figure 1 are from a determination of iron in alloys of the Fe-Ni-Cr system, in which strong secondary fluorescence is present. The upper trace represents the FeKa radiation which is strongly absorbed by chromium in Fe-Cr binaries, whereas the lower curve shows the effects of secondary fluorescence of FeKa by NiKcv in Fe-Ni binaries. All intermediate points (filled circles) cosrespond to iron measurements in Fe-Ni-Cr ternaries. A calibration equation which will permit the accurate determination of iron in the presence of nickel and chromium must also apply in the absence of either nickel or chromium-ie., it must fit both binary curves. Hence, it must take into account the nonhyperbolic nature of the calibration curve for secondary fluorescence of iron by nickel. These conditions can be met by fitting separate terms to the intensities obtained from binary specimens, and by treating multielement compositions by a summation of binary interference terms proportional to the concentration of each element. Form of the Equation. We propose the following equation which is based on the experimental studies described later:
+
SAMPLE
ID
PRINT
6 Figure 4.
Flowchart of the computation program
accurately calculating the interelement effects; however, their determination requires many more standards. For a system containing n elements, a minimum of 2(n - 1) Z p , where the summation ranges from p = 1 t o p = n - 2, standards are required (all elements present in each standard). For example, nine standards is the minimum needed to calibrate a four-element system. Claisse and Quintin have stated that the cross-product terms are less significant than the quadratic terms and, in normal usage, can be dropped. The simplification reduces the minimum requirement for standards to 2(n - 1). Tertian's formula constitutes a fourth model. The equation is close to the hyperbolic form except for a change in the interaction expression which is written in terms of the analyte, (1 - C Z ) , instead of in terms of the matrix element. This approach is feasible for the binary case only. Furthermore, the equation cannot be valid over the full range of C, because when C1 1, R1 1/(1 a),which is impossible unless the constant a is zero. In all, only four models are required to describe the formulas found in sixteen publications. In most cases, the references give examples which illustrate the utility of
+
- -
84
+
+
X -BG, J~, .
CA,hCh + (2) i, * d IC73 where each symbol has the same definitions as given before. The coefficients A L kare used when the significant effect of element k on the analyte i is absorption; in such cases the corresponding B l k are zero and Equation 2 is hyperbolic. The coefficients B i R are used when the predominant effect of element h on element i is secondary fluorescence; then the corresponding Ai, are zero. Usually, an analyst can easily predict which elements will cause fluorescence and thus require use of the B coefficients. Alternatively, this prediction can be made either by examining tables to assess locations of emitted lines and absorption edges, or by directly examining graphs of C I K as a function of C. The value of Bik affects not only the slope of the curve showing C , / K , as a function of C,, but also its curvature. This is shown in Figure 3, where C,/R, is plotted for sev-
C,/R,= 1
(43) T. 0. Ziebold and R. E. Ogilvie. Anal. Chem.. 36. 322 (1964).
ANALYTICAL CHEMISTRY, VOL. 46, NO. 1, JANUARY 1974
7
Table I I . Instrument Settings
3'0
EIem ent Parameter
X-Ray tube voltage (W X-Ray tube current (mA)
Detector voltage ( V ) Amplifier gain Pulse height analyzer base line ( V ) Pulse height analyzer window ( V ) Measurement time (set)
Fe
Ni
Cr
45 (20)Q
40 (20)
45 (20)
3 (9) 1100 80
2 (5) 1050 80
15 (25) 1200
80
1 .o
1 .o
1 .o
6.0
6.0
10.0
50
50
50
57.52
48.66
69.35
1
0.5
Spectrometer angle, 20 (")
a A second set of data, taken at alternate voltage and current settings, is presented in Table IV.
0 0
I
0.2
0.4
1
I
0.6
0.8
CNi
Figure 5. Binary curves for nickel in iron and in chromium
era1 possible values of B L h .The divisor 1 + C, in the secondary fluorescence term was selected on the basis of observing that the fluorescence curve had a slope which was about twice as large when Ci approached zero as when it was near unity. In some cases, where a combination of effects occurs, it may be necessary to apply in one equation both A and B coefficients to adequately treat the effects caused by a given element. However, we have not observed such a case. Determination of Coefficients. In the procedure just given, ( n - 1) coefficients are required for the determination of each element in a specimen containing n elements. Thus, the total number of nonzero coefficients required for calibration of the complete composition is n ( n - I ) . We have determined, by two methods, the empirical coefficients needed in Equation 2 for the analysis of the Fe-Ni-Cr alloy system, with nearly identical results. In the first method, we use a direct graphical procedure t o evaluate the intercept a t the limit C h 1 of binary curves such as shown in Figure 2 . In the case of fluorescence of iron excited by nickel, C J R , approaches 0.53 as C, goes to zero. This yields for B F ~ aNvalue ~ of -0.47. The same procedure can be used to obtain the A or B coefficients for the other elements. In the second method, binary standards are not required. A set of simultaneous equations (Equation 2) is solved for each element; in the set there is one equation for each standard and the only unknown variables are the empirical coefficients. The minimum number of standards required for a system of n elements can be obtained from the rule that for each element ( n - 1) calibration points, in addition to the pure element, are required. For example, if n = 4, each of the four elements must be present in a t least three standards; this requires a minimum of three quaternary standards. The requirement could also be fulfilled by the use of four ternary or six binary standards. Consideration of possible errors in the compositions or measurements makes it desirable to use a larger number of standards to overdetermine the system. This is especially important if some of the standards do not adequately bracket the concentration ranges of interest. In case of overdetermination, consideration should be given to weighting of calibration points. The coefficients used in Equation 2 are simply related to the analytical calibration curve and the shape of the
-
The measurements were made at 40 kV and the other experimental conditions are shown in Table I I
2'or---,,Cr-
(L
Ni
BINAPY
10
0.5 -
0
1
I
,
I
0.2
0.4
0.6
0.8
ccr Figure 6. Binary curves for chromium in nickel and in iron The measurements were made at 45 kV and the other experimental conditions are shown in Table I I
curve is predicted by them; they are reasonably insensitive to minor alterations in the measurement conditions or selection of standards. This is in contrast to the procedures patterned after that of Lucas-Tooth and Price ( 2 4 ) in which the coefficients cannot be obtained graphically, are highly interdependent, and may vary by as much as lo5 when a substitution is made between equally applicable standards. A simplification is permissible if several elements are present a t concentrations low enough that their contributions to the interelement effects are small or similar. The calibration of the system may be accomplished with improved efficiency by summing the mass fractions of these elements and treating them as a single entity. Fewer coefficients and, hence, fewer standards are required; the consequent overdetermination of the systems may lead to more accurate analysis with a given set of standards. Application to Unknowns. When the coefficients have been obtained, Equation 2 is used in the determination of each element in a given unknown specimen. Because sets of such equations (one equation for each element) usually do not permit a simple, explicit solution for the C's, we have employed an iterative technique. The flowchart of the computation is given in Figure 4. In the first iteration,
A N A L Y T I C A L CHEMISTRY, VOL. 46, NO. 1, J A N U A R Y 1974
85
Table I I I. Specimen Compositions and Relative Intensities (Measured at 45 kV) Weight fraction, Ca Relative intensity, R Specimen Fe- N i
Fe
Ni
Cr
Fe
Ni
Cr
Fe
Ni
Cr
0.0462 0.0659 0.1018 0.2263 0.3067 0.3431 0.5100 0.6315 0.9549
0.9516 0.9322 0.8964 0.771 1 0.6931 0.6552 0.4820 0.3599 0.0329
... ...
0.8782 0.8321 0.7595 0.5483 0.4515 0.4073 0.2553 0.1 720 0.0125
... ... ... ... ...
...
0.5856 0.5969 0.6280 0.7134 0.7654 0.7846 0.8634 0.9076 0.9886
1.0836 1,1203 1.1803 1.4063 1.5351 1.6086 1.8880 2.0924 2.6320
... ...
... ... ...
0.0789 0.1104 0.1621 0.31 72 0.4007 0.4373 0.5907 0.6958 0.9659
0.9627 0.9372 0.8766 0.8080 0.7477 0.6786 0.6322
... ...
0.8970 0.8270 0.6974 0.5739 0.4748 0.4048 0.3579
... ...
...
0.061 7 0.1004 0.1817 0.2587 0.3326 0.4023 0.4476
1.073 1.133 1.257 1.408 1.575 1.676 1.766
... ...
...
0.0353 0.0608 0.1214 0.1900 0.2503 0.3194 0.3658
...
0.572 0.606 0.668 0.734 0.753 0.794 0.81 7
0.1425 0.1897 0:2069 0.21 04 0.2621 0.2765 0.3360 0.3883
...
... ...
0.6998 0.6556 0.651 5 0.6224 0.5543 0.5392 0.4684 0.4119
0.1906 0.2240 0.2304 0.2492 0.3085 0.31 74 0.3840 0.4305
...
...
0.8446 0.8041 0.7858 0.7837 0.7343 0.7192 0.6591 0.6064
... ...
1.207 1.227 1.206 1.259 1.325 1.334 1.407 1.472
0.748 0.847 0.898 0.844 0.850 0.871 0.875 0.902
0.6838 0.6945 0.5280 0.5919 0.7159 0.1501 0.0140
0.0498 0.0996 0.1927 0.2002 0.0829 0.6429 0.7260
0.2525 0,1988 0.2696 0.1988 0.1879 0.1688 0.2030
0.451 1 0.4971 0.3529 0.4343 0.5298 0.1460 0.01 25
0.0203 0.0416 0.0821 0.0898 0.0343 0.4367 0.5630
0.3258 0.2651 0.331 1 0.2582 0.2536 0.2072 0.2263
1.516 1.397 1.496 1.363 1.351 1.028 1.120
2.450 2.394 2.347 2.230 2.41 7 1.472 1.290
0.775 0.750 0.814 0.770 0.741 0.815 0.897
0.3431 0.7250 0.6303 0.4721 0.0660
0.6552 0.001 5 0.1480 0.2357 0.7265
0.0000 0.2577 0.21 30 0.2784 0.1540
0.4377 0.4689 0.4480 0.31 79 0.0667
0.41 10 0.0006 0.0642 0.1115 0.5534
... 0.3348 0.2784 0.3361 0.1 740
0.784 1.546 1.407 1.483 0.990
1.594 2.500 2.305 2.114 1.313
0.770 0.765 0.828 0.885
binaries 971 972 974 983 986 987 1159 1268 8098
... ...
... ...
... ... ...
... ... ...
...
... ... ...
Fe-Cr binaries 4061 4062 4065 41 73 4181 4183 4184
...
... ...
...
... ...
... ...
...
... ... ...
Ni-Cr
binaries 3995 4002 4003 4004 401 1 401 2 401 3 4014
... ...
... ...
... ... ...
... ...
... ...
...
... ... ...
... ...
Fe-Ni-Cr ternaries 5074 5181 5324 5321 7271 161 1189
Computed as unknowns 3987 5054 5202 5364 1188 a
Given by chemical analysis.
the concentrations are approximated by the measured intensity ratios. After a new set of concentrations is obtained, they are compared with the previous approximation, if the resulting change in any C, value is smaller than 0.001, the results are accepted as the final estimate of composition. If any C, value changes by 0.001 or more, the C's are summed and normalized for use in the next iteration. After the last iteration, there is no normalization of the final results. Solution of the equations (Equation 2) is not possible by matrix inversion when B coefficients are present ( 4 4 ) . Even in the linear case, the formal solution using matrix inversion is less advantageous since it requires one additional equation defining the sum of the components being analyzed to be unity. Consequently, by matrix inversion, Criss, Naval Research Laboratory, Washington, D.C. 20390, personal communication (1973).
(44) J. W .
86
...
one always obtains unity as the sum of concentrations. This normalization is not desirable: if an element has been overlooked or if there has been a gross error in a measurement, our procedure will indicate a sum of concentrations significantly different from unity.
EXPERIMENTAL Instrumentation. The instrument used was a North American Philips Co., Inc. X-ray spectrometer, Type 52530, with a Type 12045, full-wave rectified generator and a Machlett Inc. Type OEG-50 tungsten-target tube. The mean angle between the incident primary beam and the specimen is approximately 63" and the angle of emergence of the fluorescent radiation with the specimen surface is 33". Primary collimation before the crystal was provided by a parallel-blade collimator having a spacing of 0.31 mm and a length of 102 mm. The dispersing crystal was lithium fluoride 200 (2d spacing of 4.027 A). Following the crystal, secondary collimation was provided by a parallel-blade collimator having a spacing 1.02 mm and a length of 38 mm.
A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 1, J A N U A R Y 1974
Table I V . Specimen Compositions and Relative Intensities (Measured at 20 kV) Weight fraction, Ca
Specimen
Relative intensity, R
CIR
Fe
Ni
Cr
Fe
Ni
Cr
Fe
Ni
Cr
0.0462 0.0659 0.1018 0.2263 0.3067 0.3431 0.5100 0.6315 0.9549
0.9516 0.9322 0.8964 0.7711 0.6931 0.6552 0.4820 0.3599 0.0329
...
0.8891 0.8505 0.7767 0.5695 0.4718 0.4287 0.2695 0.1824 0.0133
...
...
0.5614 0.5720 0.6045 0.6843 0.7389 0.7652 0.8448 0.8912 0.9953
1.0703 1.0961 1.1541 1.3540 1.4691 1.5283 1.7885 1.9731 2.4737
... ...
...
0.0823 0.1 152 0.1684 0.3307 0.4151 0.4484 0.6037 0.7086 0.9594
0.9627 0.9372 0.8766 0.8080 0.7477 0.6786 0.6322
...
0.0353 0.0608 0.1214 0.1900 0.2503 0.3194 0.3658
0.8953 0.8364 0.7106 0.5890 0.4936 0.4222 0.3751
... ...
1.0753 1.1205 1.2336 1.3718 1.5148 1.6073 1.6854
...
...
0.0638 0.1032 0.1872 0.2738 0.3430 0.41 16 0.4572
...
0.5533 0.5891 0.6485 0.6939 0.7297 0.7760 0.8001
... ...
0.1425 0.1897 0.2069 0.2104 0.2621 0.2765 0.3360 0.3883
...
,.. ...
0.7199 0.6771 0.6717 0.6468 0.5773 0.5560 0.4882 0.4319
0.1942 0.2248 0.2371 0.2558 0.3141 0.3297 0.3881 0.4392
... ...
... ...
0.8446 0.8041 0.7858 0.7837 0.7343 0.7192 0.6591 0.6064
1.1732 1.1876 1.1699 1.2117 1.2720 1.2935 1.3501 1.4040
0.7338 0.8439 0.8726 0.8225 0.8344 0.8386 0.8658 0.8841
0.6322 0.0000 0.6838 0.6945 0.5280 0.5919 0.7159 0.1501 0.0140
0.0000 0.6064 0.0498 0.0996 0.1927 0.2002 0.0829 0.6492 0.7260
0.3658 0.3883 0.2525 0.1988 0.2696 0.1988 0.1879 0.1688 0.2030
0.3751 0.0013 0.4663 0.5123 0.3678 0.4512 0.5378 0.1522 0.0135
0.0002 0.4319 0.0216 0.0443 0.0933 0.0949 0.0362 0.4535 0.5852
0.4572 0.4392 0.3336 0.2721 0.3393 0.2662 0.2599 0.2129 0.2348
1.
,..
1.4664 1.3557 1.4356 1.3118 1.3312 0.9862 1.0370
1.4040 2.3056 2.2483 2.0654 2.1096 2.2901 1.4176 1.2406
0.8001 0.8841 0.7569 0.7306 0.7946 0.7468 0.7230 0.7929 0.8646
0.3431 0.7250 0.6303 0.4721 0.0660
0.6552 0.0015 0.1480 0.2357 0.7265
0.0000 0.2577 0.2130 0.2784 0.1540
0.4484 0.4844 0.4649 0.3313 0.0706
0.4287 0.0006 0.0684 0,1175 0.5689
0.0007 0.3422 0.2839 0.3450 0.1831
0.7652 1.4967 1.3558 1.4250 0.9348
1.5283 2.5000 2.1637 2.0060 1.2770
0.7531 0.7503 0.8070 0.8411
Fe-Ni binaries 971 972 974 983 986 987 1159 126B 809B
...
... ... ... ...
...
... ... ... ...
...
...
... ... ...
...
...
Fe-Cr binaries 4061 4062 4065 41 73 4181 4183 41 84
... ... ... ...
.
.
I
... ...
... ...
... ... ...
Ni-Cr
binaries 3995 4002 4003 4004 401 1 401 2 401 3 401 4
...
...
... ...
...
... ...
...
... ...
Fe- N i-Cr ternaries 41 84 401 4 5074 5181 5324 5321 7271 161 1189
Computed as unknowns 3987 5054 5202 5364 1188
...
Given by chemical analysis.
The detector and electronics for signal conditioning and pulse counting were built by Hamner Electronics Co., Inc. and were as follows: scintillation detector and preamplifier, Type NB-18-A; linear amplifier, Type NA-11, pulse height analyzer, Type NC-11; scaler, Type SS-11; timer, Type NT-11; and detector voltage Supply, KV-13. The instrument settings for the experiment are listed in Table 11. Specimens. The compositions of the 38 specimens measured in this experiment are given in Table 111. They were selected from NBS Standard Reference Materials and industrial standards for which chemical analyses were available. Each specimen was a disk 3.1 cm in diameter and approximately 2.5 cm high. One of the round surfaces was polished with a 6-pm diamond finishing wheel. Five of the specimens were not used in the calibration pro-
cess; rather, they were reserved to be measured and computed as unknowns, and are so noted in Table 111.
RESULTS AND DISCUSSION Iron-Nickel-Chromium System. The measurements, a t two operating voltages, for the Fe-Ni-Cr system (tabulated in Tables 111 and IV and shown graphically for one voltage in Figures 2, 5 , and 6) were used in developing Equation 2. The coefficients for this system and equation were calculated by both procedures described above and are presented in Table V. For a given operating voltage, the coefficients are virtually the same by either procedure. At the lower operating voltage, the B-coefficients (fluores-
ANALYTICAL CHEMISTRY, VOL. 46, NO. 1, J A N U A R Y 1974
87
2.0
.Ca Ka
trm
I
COCO,
I
- SiO,
B
A X-ray line
Fe
Cr Fe 45 kV-From graphs
FeKa NiKa CrKa
0 1.71 0
0 2.10 1.20 0 -0.46 0 45 kV-By equations
FeKa NiKa CrKa
0 1.711
2.099 1.226 0
0
I
I
Table V. lnterelement Coefficientsa for the Ternary System Fe-Ni-Cr
0 1.52 0
1.92 1.01 0
FeKa NiKa CrKa
0 1.532 0
1.895 1.023 0
-0.47 0 -0.27
0 0 -0.419
20 kV-From
FeKa NiKa CrKa
Ni
t
-0.465 0 -0.234
0.5 0’ 0
graphs
0 0 -0.47
20 kV-By
/
-0.50 0 -0.31
I
0.4
C
1
I
0.6
0.8
I 1.0
Figure 7. Binary curves for calcium and silicon in CaCOs-Si02 The data were taken from the paper by Lachance and Traill (37). Arrows denote a specimen where large experimental error seems likely
equations
0 0 -0.449
I
0.2
I
-0.498 0 -0.265
a T h e coefficients are for the effect of one element upon the X-ray line of the element being determined; e.g., the effect of nickel on the determination of iron (when determined from graphs for the 45-kV data) is -0.47.
hyperbolic model. Their equation yielded an average relative error of 0.24% for nickel in the Fe-Ni binaries, compared to our method which gave 0.47%. For iron, there was no difference in the average relative error, with a result of 0.60% for both. In comparison of the two methods, however, one should
Table V I . Results (in Mass Fraction) for Fe-Ni-Cr Alloys by Two Computation Proceduresa Fe
Ni
Cr
Specimen
Chem.
RH
BB
Chem.
RH
BE
5054 5202 5364 3987 1188
0.7250 0.6303 0.4721 0.3431 0.0660
0.7229 0.6277 0.4750 0.3378 0.0655
0.6779 0.5890 0.4710 0.1710 0.0639
0.0015 0.1480 0.2357 0.6552 0.7265
0.0015 0.1507 0.2409 0.651 2 0.7236
0.001 6 0.1590 0.2460 0.8270 0.7520
Chem.
0.2577 0.2130 0.2784 0.0000 0.1540
RH
0.2522 0.2090 0.2675 0.0000 0.1421
BE
0.2740 0.2420 0.2700 0.0001 0.1400
a The values given in columns labeled “Chem.” were determined by wet chemical analysis. RH is the new procedure described in this paper (Rasberry and Heinrich), while BE stands for the hyperbolic approximation procedure (Beattie and Brissey)
Table V I I . Results (in Mass Fraction) for Si02 in Calcium Carbonate-Silica, by Two Computation Proceduresa (Data from Ref. 37) Given
RH
BE
Improvement by R H
(0.923) 0.769 0.61 5 0.538 0.462 0.385 0.231 0.077
0.889 0.772 0.614 0.549 0.468 0.387 0.229 0.076
0.874 0.760 0.605 0.542 0.465 0.388 0.237 0.084
0.01 5 0.006 0.009 -0.007 -0.003 0.001 0.004 0.006
a RH IS the new procedure described in this paper, while BE stands for the hyperbolic approximation procedure.
cence effect) were essentially unchanged while there was an appreciable decrease in the A-coefficients (absorption effect), as expected. In Table VI, analytical results are given for the five FeNi-Cr alloy unknowns as calculated by our model (RH) and by the hyperbolic model of Beattie and Brissey (BB). The coefficients used in the calculation by our method were those listed in Table V as “45 kV-By equations.” Overall, our results show better agreement with the chemical values than do those by the hyperbolic model, and in some instances, notably in the absence of chromium, the improvement is striking. The equation of Claisse and Quintin (36), with crossproduct terms omitted, gave better accuracy than the 88
consider that Claisse’s model requires, a t a minimum, twice as many standards as does ours. Further, the quadratic terms in Claisse’s equation cannot be accurately determined unless the compositions of the standards are properly spaced. Consequently, the application of that equation would have been difficult or impossible, had fewer suitable standards been available to us. Calcium Carbonate-Silica System. We also tested our method on a nonmetallic system. Among information in the literature, the CaCOS-SiOz system was selected because the CaKa line causes significant fluorescence of the SiKa line. The data (Figure 7) covering a range of mass fraction from 0.077 to 0.923 were obtained from a paper by Lachance and Traill (31). As shown in Table VII, a slightly better overall agreement is obtained with our method than with the hyperbolic method. For calcium, in the absence of secondary fluorescence, the two methods are identical. However, it is of interest to observe that the experimental data on the calcium show no systematic deviation from the hyperbolic equation. The data for both elements in one specimen (marked with an arrow in the figure and given in parenthesis in the table), deviate from the pattern of the remainder of the specimens and appear t o be inaccurate. Summary. For the two systems mentioned above, 23 concentrations were determined by our method with a resulting average relative error of 2%. The relative error was 1% or less for 13 determinations and 4% or less for all but 2 of the 23 values.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 1, JANUARY 1974
APPENDIX
h = l -
The general equation for a hyperbola with center a t ( h , k )and asymptotes parallel to C and R is
(C - h ) ( R - k )
=
z
(AI)
where h, h, and z are constants. The analytical curve (Equation 1) passes through the two points (0,O) and (1,l). For a hyperbola passing through these points, ( A l ) yields: 2 =
h(l
- h)
(A2)
and
k
(A3)
Substituting into Equation A1 we get:
R=
C 1 1 - -(1 - C ) k
(A4)
which reduces to Equation 1in the text, with a = - l / k Received for review April 6, 1973. Accepted August 3, 1973. Mention of specific items of equipment in this paper does not imply the endorsement of the National Bureau of Standards for products of any manufacturer.
Hydrogen-Bond Studies of the Near Infrared Combination Bands of the Butylamines J. E. Sinsheimer and Anne M. Keuhnelianl Coiiege of Pharmacy University of Michigan. Ann Arbor. Mlch 48704
Deuteration, concentration, and temperature studies are employed to establish the 1.99-Fm band of primary butylamines to be free N-H absorption and the 2.02-pm band to be hydrogen-bonded N-H absorption. The effect of amine structure on absorptivity of these bands is noted and correlation of Taft + g * values to the 2.02-pm absorbance observed. The absorbance at 2.02 p m is also correlated to solvent-solute hydrogen bonding. The interrelationships of concentration, amine structure, and solvent-solute hydrogen bonding as to their effects on the 1.99- and 2.02-pm combination bands of primary butylamines are discussed.
The most intense absorption for primary amines in the near infrared (NIR) occurs in the regions of 2.0 and 1.5 km ( I ) . The former is observed only with primary amines and results from the combination of N-H bending and stretching modes. The latter occurs with both primary and secondary amines and is the first overtone of the N-H stretching vibration. The combination bands of primary aromatic amines have been studied by Whetsel et al. (2-4) and those for aliphatic amines by Lohman and Norteman (5). Lohman and Norteman investigated both the overtone and the combination bands for the quantitative analysis of mixtures of primary and secondary aliphatic amines. In the course of this work, they noted two combination bands for primary amines. Two different solvent-associated states were postulated. A 2.023-pm band was described only as essentially chemical in nature and a second, 1.998-km band as the usual solvent-solute interaction. However, study of their results in regard to change in solvents and temperature strongly suggests the possibility Present address, Purdue University at Fort Wayne, Fort Wayne, Ind. 46805 ( 1 ) R. F. Goddu, Advan. Ana/. Chem. Instrum., 1, 347 (1960). ( 2 ) K. B Whetsel. W E Roberson. and M . W . Krell. Ana/. Chem . 30. 1594 ( 1 9 5 8 ) . (31 / b i d . . 30. 1598 (19581 (41 /bid 32. 1281 (19601 ( 5 ) F H Lohman and W E. Norternan, J r , A n a / Chem.. 35. 707 (1963)
of a free (1.998-pm) band and a hydrogen-bonded (Hbonded) (2.023-pm) band which is not limited to solventsolute interaction. It is the purpose of this paper to establish these facts and to explore the utility of these bands for H-bonding studies.
EXPERIMENTAL Apparatus. All spectra were obtained on a Beckman DK2-A Ratio Recording Spectrophotometer calibrated for wavelength accuracy against 1,2,4-trichlorobenzene. Unless otherwise noted, a constant temperature cell thermostated at 25 "C was used. The following instrument parameters were employed: period, 0.2; speed, 10 cm/min a t 0.025 pm/cm; sensitivity, 50. The 0.16% v/v amine solutions were recorded at 90-100'70 transmittance (Tj, all other solutions at 0-1 absorbance. Cyclohexylamine spectra were resolved by use of a Du Pont 310 curve resolver adjusted t o standard Gaussian curves Reagents. The amines were Eastman white label grade chemicals and were used without further purification. n-Butylamine, because of its hygroscopic nature, was distilled and stored over KOH. Absence of 1.98-pm HzO absorption was used to monitor removal of HzO from this amine. All solvents were new bottles of reagent grade chemicals and were used without further purification. Preservative alcohol present in reagent CHC13 was removed by passage through a column of alumina and its removal monitored by absence of absorption at 1.98 pm, This CHC13 was stored in a brown bottle and used within 5 days. p-Dioxane was dried by storing over sodium metal for 48 hr; removal of HzO was monitored by absence of absorption at 1.98 pm. Procedures. Liquid Spectra. Amine spectra were recorded as pure liquids in a 2-mm cell with a single silica plate in the reference beam. Vapor Phase Spectra. A few drops of liquid amine were added to a 10-cm cell thermostated a t 70 "C and the cell atmosphere was saturated with vapor before recording spectra. Deuteration of n-Butylamine. A 10-ml aliquot of a 10% solution of n-butylamine in ether was shaken with 4.0 ml DzO for 4 hr. The solution was then dried over 10 grams of XaZSO4 and subse. quently over sodium, filtered, and the ether portion evaporated. A neat liquid spectrum of the resulting deuterated n-butylamine was obtained. Temperature Dependence. Solutions (5'70 v/vj of the four butylamines were prepared in hexane, benzene, and CHC13. Spectra were obtained a t 10, 30, 45, and, in the case of benzene, a t 60 " c in a thermostated cell. Under these conditions, tert-butylamine in hexane did not yield a clear solution.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 1, JANUARY 1974
89
