970
ANALYTICAL CHEMISTRY
arc' iiottxl with coiicentratioii cshanges at a fixed frequeiicy. In Figures 8 and 9 there is a reversal of the direction of thc conducta n c ~change with excess reagent' aft'er the end point, is passed.
Further investigations in progress should lead to an explanation of thew fact,s. The dat,a of Anrlcrsnn, Bettis, and Reviiisnii ( 1 ) shtrn srirnc. Firnilar d i ~ ~ ~ i ~ i i io 1f ~c~irve ~ i 1 ~forin 1 ~ o n frrqucnry. iCKVOWLEDGMEYT
The authors exprebs their thanks to Joseph Collett, \rho coilstructed the cell and aided a i t h some preliminary experimental work on this equipment. They also appreciate the permisqiori o t the G ~ n ~ rRadio al Po. to us? Figure 1
LITERATURE CITED
Kermit. Hettis. E. S.. and Rerimon. D.. .Isa~. CHRM.. 22, 743 11950). (2) Hlnedel. 11. .J.. arid lfelmstadt. H. V., Ihid..22, 734 (1950). 1 8 ) General Radio Co., ('ambridge, Mass.. "Operating Itlstrrictioiis for Type 821-.\ Twin-T ImpPriatice-lfeasuring Cilcnit. Form 11 i .Indersoii.
568-C'."
(4).leiisen, E'. W., and Parrack, .I.I,.. I s u . Est. P H E V .\SAL. ., ED., 18, 595 (1946). (5) dinclair, D. B., Proc. I f & . Radio. E n y r u . . 28, 310 (194Uj. !6) West. P. IT., Burkhalter, T. R., a u d fit,oussard. I,., h s i i . . CHEM., 22, 460 (1950). K w r . l v r . o Sovpiiibcr 13, IHT,O
Polychromatic X-Ray Beams N o n1 inea r Re la t ions h ip between Composition and Absorbance H . A . LIEHtIAFSKY AND 1'. I). ZEMANY Keseurch Lnborutory, Geriernl Electric Co., Srhenectudy, Y. Y.
For monochromatic x-ra? beams, the m a b s absorption coefficient of a mixture (or solution) stands i n additiie relationship to the coefficients of the elements. free or comhitied, contained therein. The existence of this relationship facilitates the u s e of' x-ray absorption methods in analytical cheniistr?. Ordinarily the additi, it? exists also for polychromatic beams, but marked deviations therefrom occur in the zirconium-hafnium system. The prese n t iniestigation proves t h a t such deiiations are traceable to absorption edges. For selected pairs of elements, the occiirrence of an absorption edge can cause an inxer4t)n i n the relationship o f their ah-
HE: improved mranh not+ :tvailxble for n i e a ~ ~ r i n \-r:i) g intermiti- have operated t o iricrmw the uwfulnrh* of \-r;t\ absorption method. in :+naIjti('a1 chrmihtrj ( 3 ) .ilthough it i i desirable in principle to use monorhrornatic x-rays in applyilly i nhich the concomitant redurthese methocis, there are ~ a w tor tion in beam intensit? r a n n o t (or nerd not) he tolcratcrl, and pol! cBhroiiiatic beams are cotiwilut~iitIy fmployed I t is well knonn that \-r:i> :thorption is ail atomics propert), :tnd that the mas$ at)>oiptioncweficient (9)of a (tompound or ot a mixture can be coniputtd I i o i n t host, i d thfs rli~mi~tits, piovidilcl tht. t)cs:ini is monochroni;ct w . For example, collsitlei, :i s:t~llplccoiitaiiiiiig ouly o ~ or ~ both c 01 compound,* A H : I I I ~CD, n~ and n being the proportioiw t)y weight of A and C in these compounds. The mass absorption cvwfficient of a sample, S , containing the proportion hy weight .c o f compound :1R is then given hy t tit, binary
Superscripts ident,ify mass :~baorpt,ioncoefficients. Equation 1 contains only t,\\-ovariables, z and p?,, and t,he mass :ibsorption coefficient. of the mixture is an additive function of the coefficients for the four constituent elements. -1s a conwquence, the relationship het\vtwl x-ray ahorbanr-c~: r n t l w n ~ ~)(i~itioii will alpo 1)e linwr For s;rniples t q u a l in mass.
sorptioii coefficients for Innnochromatic beamsthat is, the element haring the greater coefficient a t ware lengths below that of the edge may have the qmaller coefficient a t wave lengths above it. If the inversion occurs within the wave-length range of a pol) chromatic beam, the deyiations in question ma) result. In the two q s t e m s studied, these deviations take the form of absorbance maxima as composition is \aried, hut other forms are possible. While the occurrence of these deviations restricts the usefulness of pol? chromatic heams in chemical anal) sis, i t H i l l often be possible to cam) out the analysis by x-ra? methods under other experimental cnnditions.
This paper is concerned with the deviations from linearity that occur wit,h samples of equal mass when the incident h a m is polyrhroma t,ic,.
('tin
%IRCONIA-HAFNI4 MIXTURES
Such deviations were first, encountered (6) when the x-ray absorptions of five known zirconia-hafnia mixtures, kindly lent by H. €I. Willard, LTniversity of Michigan, were investigated on a 1ahor:ttoi-y photometer (Q), in which x-ray intensity was measured t)y ineans of a detector consisting e~sent,iallyof a phosphor and a niultiplier phototube. The work on these mixtures was done by the direct method ( 4 ) ,in which the intensity of the x-ray heam is d j u s t e d to some standard initial value by adjuiiting the x-ray tube voltage and current until the desired output current is obtained with a standard thickness of aluminum in the beam, the voltage :icros:s the detector and the amplifier setting being fixed. The \\.(sighed sample contained in the cell was then carefully placed in the heam to ensure proper alignment,, and an average value of output current was obtained, usually from ten readings of the nmnieter taken 10 seconds apart,. Owing to the difficulties of working with powdered samples of this size (about 10 m g ) , very painstaking manipulation JTas required. Cnder the simplest conditions, the expected relationships among x-ray absorbance, output currents, and composition (as - characterized by mass absorption coefficients) in the foregoing rsperiinents may he written
V O L U M E 2 3 , NO. 7, J U L Y 1 9 5 1 log [11/19] = log [i,/ip] = m
( p
971
2 - p$)/2.303
a
(2)
where the I’s represent x-ray intensities, and the i’s output currents for samples 1 and 2, which differ in composition, but are identical in mass m and cross-sectional area a , and have the mass absorption coefficients shown. (The superscripts SI and 8 2
SODIUM NITRATE-POTASSIUM NITRATE SOLUTIONS CONTAINING 2 GRAMS TOTAL SALT AND 20 GRAMS OF WATER.
40MG. OF ZIRCONIA-HAFNIA MIXTURES,
mean sample 1 and sample 2.) If these coefficients obey Equation l, a straight line should evidently result when the output currents are plotted on a logarithmic scale against composition. The curve in Figure 1 shows the experimental results corrected to a sample weight of precisely 40 mg. -1molybdenum target was used in these experiments. To show that this curvature was not usually obtained, similar measurements were made under comparable experimental conditions on aqueous solutions of sodium and potassium nitrates. Within the experimental error these results lie on a straight line, and this proves the applicability of Equation 2. ( I t is proof also that the detection and amplification systems are linear-i.e., that I and z are proportional over the range of experimental conditions.) The sodium-potassium nitrate case is typical of most system- so far investigated. THE DEVIATION EXPLAINED
The probable cause of the curvature in Figure 1 is clear from data in Figure 2 , which lead t o these qualitative conclusions. Any effect involving the L absorption edge of hafnium i p negligible, owing to the low intensity of the x-ray beam tritnsmitted t,hrough the sample in this wave-length region. At wave lengths below its K absorption edge, the mass absorption coefficient of zirconium exceeds that of hafnium. At wave 1engt.hs immediakly above this edge, the reverse is true. This inversion of the mass absorption coefficient relationship could produce a minimum in a plot like Figure 2-i.e., a deviation from linearity in the relationship between absorbance and coniposition. Because the K , line of molybdenum lies on the long wavelength side of the absorption edge involved, the minimum was probably enhanced. -ittempts to account quantitatively for the curvature in Figure 1 were only moderately successful. The following calculation was carried through for each composition investigated. The percentage of total radiant energy, I x a in the emission spectrum Figure 1. Data for Two Systems was estimated for each interval of 0.05 A. From the correspondOne shows deviation from linearity (see text) caused by inversion in ing calculated absorption coefficient of the sample, the transrelationship of two m a s s ahsorption coefficients mitted intensity, Ix, for that interval was calculated and plotted against A. Graphical integration then gave a quantity taken M A S S ABSORPTION COEFFICIENTS as proportional to the intensity of the OF ZIRCONIUM A N D H A F N I U M transmitted x-ray beam. When these quanH f (L) ABSORPTION EDGE-A tities were plotted against composition in X-Zr the manner of Figure 1, a curve roughly 0 - H f -6 resembling that in the figure was obtained. F 200 Further confirmation was provided by the results of similar calculations on a simplified system consisting of two hypothetical elements, A and C (cf. Equation l ) , with absorption coefficients related to each other somewhat as are those of zirconium and WAVE L E N G T H (A) hafnium. 10
20 40 60 80 PERCENT POTASSIUM NITRATE OR HAFNIA
0
I00
d i
X-RAY E M I S S I O N S P E C T R U M F R O M MOLYBDENUM -TARGET TUBE
K W
z
W
-I
s 0 I-
b
5 0 W
6 a
20 IO
02
0.4
0.6
08
I .O
WAVE L E N G T H
Figure 2.
(4
1.2
1.4
1.6
Auxiliary Data Relating to Deviation from Linearity
These data have qualitative significance only. The L ahsorption edges of hafnium are n o t shown i n detail. Mass absorption coefficients were calculated by use of t h e Xa law from data given for neighboring elements i n (3). The intensity distribution for the molybdenum target was calculated from the results of Treloar (5)
COPPER-SILVER NITRATE SOLUTIONS
Finally, measurements were made to show the absorption of x-rays from a copper target by aqueous solutions of copper and silver nitrates. This combination waa selected because the pertinent data (,9) for it resemble those of Figure 2, one difference being the greater spread between the lowwave-length absorption edge (the K edge of d v e r a t 0.48 A . ) and the characteristic lines (for copper, near 1.5 A . ) in the incident beam. Each of the nitrate solutions was made up to contain 10 g r a m of dissolved metal per liter. The measurements were
912
ANALYTICAL CHEMISTRY
done hj- the comparative method ( 6 ) with aluminum as the standard. Typical results are given in Figure 3, in which a maximum corresponds t o a minimum in Figure 1 owing to the difference in ordinates. It is clear immediately that deviations from linearity do occur, for a maximum was obtained under some conditions with a sample of each of the volumes listed. The magnitude of the deviations is, of course, reduced by the appreciable absorption of x-rays by the nonmetallic elements in thesamples.
alumiiium foil therein (4). The resulting values are given in Figure 3 alongside most experimental points. Effective wave lengths obtained as just described often serve for the estimation of absorbance (Equation 2) when polychromatic beams are used and Equation 1 is valid-e.g., in the case of the nitrate solutions of Figure 1. When the deviations under discussion occur, however, Figure 3 shows that absorbances may pass through maxima while effective wave lengths vary in one direction. Circumstances such as this tend to reduce the usefulness of the concept of effective wave length. For a given volume of sample Figure 3 shows that the tendency to produce a maximum is greater as the voltage on the x-ray tube is increased. This effect is no doubt due to the concomitant change in the energy distribution in the beam incident on the sample, but the authors have not attempted to prove this quantitatively, partly because all the requisite data are not available. I n the case of curves I11 and V, however, such proof is scarcely required. For curve 111, the short wave-length limit (0.43 A. at 28.5 kv.) is almost coincident with the absorption edge (that of silver near 0.48 A.) responsible for the maximum. Accordingly, the beam will contain very little energy in the neighborhood of the absorption edge, and no appreciable deviation from linearity traceable to this absorption edge is to be expected (contrast with Figure 1). There is no such close coincidence for curve T’. CONCLUSIOri
VEASURED EFFECTIVE WAVE L E N G T H G I V E N AT E A C H POINT
1
j
20 80
0 100
Figure 3.
I11
v
I
I
60 40
80
60
20
100 %Aq NO, SOLUTION 0 %Cu(NO,),SOLUTION
Deviation from Linearity for Copper and Silver Yitrate Volume of Solution,
MI.
Curve
I I1 IV
I
40
50
.
25 10 10 25
Solutions of
Peak Voltage, Kv. 34.5 34.5 34.5 42.0
It appears to be established that the presence of absorption edges can lead to deviations from the linear mass absorption coefficient relatiowhip when a polychromatic x-ray beam is incident upon a sample containing more than one element. It is certain that this article does not describe all the systems in which such deviations are possible, and it is probable that these may take forms different from those shown in Figures 1 and 3. Although the occurrence of these deviations restricts the usefulness of polychromatic b e a m for purposes of chemical analysis, this restriction should not prove generally serious, provided the investigator is forewarned. When such deviations occur, it will often be possible to carry out the analysis by x-ray methods under other experimental conditions.
28.5
Under the simplest conditions, the ordinates (mils of aluminum equivalent to 1 ml. of solution) should be identical for all solutions of copper nitrate alone. That they are not is probably due t o another type of deviation commonly encountered with polychromatic beams-namely, the well-known variation of p m with wave length for monochromatic beams. As a consequence of this variation, the mass absorption coefficient of a sample for a polychromatic beam decreases with the thickness of the sample because the effective wave length decreases continuously as the beam progresses through the sample. This deviation is more pronounced for aluminum (owing to its lower mass absorption coefficient) than for copper, and the equivalent thickness of aluminum per milliliter of copper nitrate consequently increases with the thickness of the copper nitrate sample. The same argument applies t o solutions of silver nitrate (in this connection, see 1 , Table XIII, and 4 , Figure 3). When polychromatic x-ray beams are used in absorptiometry, it is ordinarily useful to consider their “effective r a v e lengths,” these being the wave lengths of the corresponding monochromatic beams toward which the sample exhibits the same absorption coefficient. The true effective wave lengths of the beams operating on the solutions of Figure 3 are unknown. In default of these, effective wave lengths of the beams emerging from the solutions were measured by determining the absorbances of thin
ACKNOWLEDGMENT
The authors wish to thank Kurt Berman for doing the calculations on hypothetical elements A and C, and Grace Poellmitz for carrying out the measurements on the copper-silver nitrate solutions. LITERATURE CITED
(1) Calingaert, G., Lamb, F. W.,Miller, H. L., and Soakes, G. E., *4N.4L. CHEM., 22, 1238 (1960). (2) Compton, A. H., and Allison, S. K., “X-Rays in Theory and Experiment,” Xew York, D. Van Nostrand Co., 1935. (3) Liebhafsky, H. A., ANAL.CHEM.,21, 17 (1949). (4) Liebhafsky, H. A., Smith, H. M., Tanis, H. E., and Winslow, E. H., Ibid., 19, 861 (1947). (5) Treloar, L. R. G., Phil. M a g . , 6, 1008 (1928). (6) Zemany, P. D., Winslow, E. H., Poellmitz, G. S., and Liebhafsky, H. A., -4XAL. CHEM.,21, 493 (1949). RECEIVED December 20, 1960.
CORRECTION. I n the article on “Design and Operational Characteristics of Cartesian hfanostats” [AXAL. CHEM.,23, 157 (1951)] the model referred to as the Ritzer manostat should have been the Holzschuh-Long manostat. ROGERGILMONT
