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ANALYTICAL CHEMISTRY, VOL. 51, NO. 8, JULY 1979
(21) Nielson, K . K. Ph.D. Dissertation, Brigham Young University, Provo, Utah, April 1975. (22) Gleit, C. E.; Holland, W. D. Anal. Chem. 1962, 3 4 , 1454-1457. (23) Hollahan, J. R. J . Chem., Educ. 1966, 4 3 , A401-A512. (24) Ramirez-Muiioz, J. "Atomic-Absorption Spectroscopy"; Elsevier Publishing Company: New York. 1968; p 352. (25) Christensen, J. J.; Hearty, P. A,; Izatt, R. M. J . Agric. FoodChem. 1976, 2 4 , 811-815. (26) Denbsky, G. Fresenius Z . Anal. Chem. 1973, 267, 350-355. (27) Conneliy, A. L.; Black, W. W. Nucl. Inshum. Methods 1970, 82, 141-148. (28) Harrison, J. F.; Eldred. R. A. Adv. X-ray Anal. 1973, 17, 560-569. (29) Johansson, T. B.; Van Grieken, R. E.; Nelson, J. W.; Winchester, J. W. Anal. Chem. 1975, 4 7 , 855-860. (30) Bearse, R. C.: Close, D. A.; Malanify, J. J.; Umbarger, C. J. f h y s . Rev. A 1973, 7, 1269-1272, (31) Akselsson, R.; Johansson, T. B. Z . Phys. 1974, 266, 245-255.
(32) Nielson, K. K.; Hill, M. W.; Mangelson, N. F Adv. X-ray Anal. 1976, 79. 511-520. (33) Williamson, C . F.; Boujot, J. P.; Picard, J. Commissariat a i'Energie Atomique---France, Report CEA-R 3042, July 1966. (34) Bennett, C. A.; Franklin, N . L. "Statistical Analysis in Chemistry and the Chemical Industry"; John Wiiey and Sons: New York, 1954; p 358. (35) Camp, D. C.; Cooper, J. A,; Rhodes, J. R. X-ray Spectrosc. 1974, 3 , 47-50. (36) Camp, D. C.; VanLehn. A. L.; Rhodes, J. R.; Pradzynski, A. H. X-ray Spectrosc. 1975, 4 , 123-137. (37) Lindgren, 8. W.; McElrath, G. W. "Introduction to Probability and Statistics"; The Macmillan Co.: New York 1959; p 220.
RECEIVED for review January 4,1979. Accepted March 8,1979. Research was supported in part by N I H Grant AM 15615.
X-ray Measurement of X-ray Fluorescence Sample Mass Hideo
Kubo'
and W. R. Smythe"
Department of Physics and Astrophysics, University of Colorado, Boulder, Colorado 80309
The replacement of gravimetric measurements by the use of incoherently scattered X-rays to measure the mass of a sample being analyzed by X-ray fluorescence is reported. Although the method is not exact, it is shown experimentally that a precision of 2 % or better is possible for many samples of biological and medical interest. The effects of X-ray beam polarization and the possibility that coherent and incoherent scattering may not be distinguishable are also considered.
T h e multielement analysis of very small samples by X-ray fluorescence (XRF) is a technique with a large potential for application in biology and medicine ( I ) . A method is reported here in which X-rays scattered from a small sample which is totally immersed in an X-ray beam are used to determine the mass of t h e sample. This method replaces the tedious weighing of these small (-300 pg) samples and avoids possible problems from sample mass loss during handling after weighing. Related techniques have been reported by Hall ( 2 ) and by Jaklevic et al. (3). Hall used the total (coherently plus incoherently) scattered radiation in obtaining t h e concentration of trace elements observed by XRF in biological sections. Jaklevic et al. used incoherently scattered radiation to measure the linear density of a hair as it was analyzed by XRF. T h e method is based on the fact t h a t the intensity of incoherently scattered radiation is nearly proportional t o the number of electrons in t h e sample, which in turn is approximately proportional t o the mass of t h e sample. T h e actual incoherent scattering cross section per unit mass decreases by about a factor of two from helium to uranium (at 20 keV and 90'). For a multielement sample, t h e relevant cross section is a n average of elemental cross sections weighted according to the elemental composition of the sample. This average cross section depends primarily on t h e major constituents in the sample and is relatively insensitive to variation of t h e trace elements. Thus, within many classes of samples, t h e average cross section does not vary significantly, even in t h e presence of spectactular trace element variations. If the X R F system cannot distinguish between coherent and incoherent scattered radiation, then coherent scattering must 'Present address: Department of Medical Physics, Memorial Sloan-Kettering Cancer Center, New York. N.Y. 10021. 0003-2700/79/0351-1194$01.00/0
also be considered. This can occur if t h e energy resolution of the detector is insufficient t o distinguish between thern, or if t h e exciting radiation is not monoenergetic. However, the method may still be useful in such situations. In place of the incoherent cross section, a weighted average of t h e coherent and incoherent cross sections is relevant. From the preceding discussion, it is clear that the usefulness of this technique for X-ray mass measurement, depends on the behavior of the appropriate average X-ray scattering cross section. This behavior has been investigated both theoretically and experimentally for a variety of samples. In this paper a particular XRF system is analyzed to elucidate the physical principles of the method. An understanding of these principles will facilitate the adaption of t h e method to other XRF systems, which may then be empirically calibrated.
EXPERIMENTAL METHOD The X-ray source was a Diano Corporation EA-75 tungsten anode X-ray tube operated at 55 KV. The bremsstrahlung beam was filtered and collimated to a 6-mm diameter. The sample (typically 300 pg) was mounted as a 3-mm disk on a Formvar foil. The Formvar foil was placed in the X-ray beam at 45' so as to suspend the entire sample in the beam. A 5-mm thick lithium drifted silicon X-ray detector (165-eV resolution a t 5.9 keVi observed the target at 90' to the incident beam. The detector pulses were processed by a Xorthern NS-621 pulse-height analyzer, and the pulse-height spectrum was stored in a Digital Equipment Corporation PDP 8/E computer. The X-ray tube anode current was integrated, and a count of the integrator output pulses, corrected for analyzer dead time, was stored in channel zero of the pulse-height spectrum. A typical spectrum obtained with the system is shown in Figure 1. The X-rays detected between 20.43 and 20.99 keV (channels 377 through 389) are called "scattered counts" and are used as a measure of target mass. Scattered counts are produced by the scattering of photons in the low energy tail of the spectrum of the exciting X-ray beam. This region was chosen because: ill it is free of X-ray lines except for the K@line of technetium, (2) there is sufficient intensity in the primary beam, and (3) the absorption of these X-rays in the target is negligible. There are two contributions to the counts in this region, coherent scattering of photons from the beam between 20.43 and 20.99 kel' and incoherent scattering of photons with initial energies between 21.28 and 21.89 keV. Direct measurement of the energy spectrum of this exciting beam has shown that, there are 1.86 photons in the higher energy interval for each photon in the lower energy interval. The cross section per unit mass of a variety of targets was measured in the following way. The sample material was weighed 0 1979 American
Chemical Society
ANALYTICAL CHEMISTRY, VOL 51, NO. 8, JULY 1979
1195
and d u T / d Q = r02(1- sin' 6' cos' 4)
A
where the outgoing polarization states have been summed, ro = e2/mc2is the classical radius of the electron, and (Y = hu/mc2 is the incident photon energy in units of the electron mass. These expressions were adapted from the Klein-Nishina formula as described by Heitler ( 5 ) and a derivation of the Thomson formula by Jackson (6). For the present case of 90" scattering. they reduce to d u c / d Q = r,'
Figure 1. A fluorescent X-ray spectrum from 306 jLg of dried human liver tissue. The X-ray spectrum extends from 2.9 to 26.2 keV. The sample was illuminated with a filtered 55kV bremsstrahlung beam. The number of counts in the region labeled "scattered counts'' is approximatety proportional to the target mass and may be used to measure it
with a Cahn electronic microbalance and mounted on a Formvar foil with a small amount of dilute polystyrene cement and covered with a second Formvar foil. The sample was run in the XRF system and the scattered counts ( S ) ,the sample mass (M),and the number of Coulombs of integrated anode current (C) were recorded. Several blanks, Formvar foils with cement and Formvar covers but no sample material, were also run and the average number of scattered counts per Coulomb (S'/C? was determined. The experimental differential cross section per unit mass (do/dQ),, was then calculated from the expression: (da/dR), = G [ ( S / C ) ( S ' / C ' ) ] / M ~
where G is a geometric calibration factor. The calibration factor G was determined by use of the calculated cross section (da/dQ)c, of Mylar, (C10HB04)m, and the experimental S / C and mass M of each Mylar sample: G = (du/dQ)cM/[(S/C) - S'/C')] Hence, all of the experimental cross sections quoted here are ratio measurements which are normalized to the value calculated for Mylar.
THEORETICAL CALCULATION OF CROSS SECTIONS T h e theoretical coherent and incoherent cross sections shown in Figure 2 were obtained in the following way. T h e incoherent scattering cross section per unit mass is given by: dg,,, / d R = NOS(x ,Z)(dgc / dR) / A where N o is Avogardo's number, S(x,Z) is the incoherent scattering function, x is ( 1 / X ) sin (0/2), 0 is the scattering angle, X is t h e incident photon wave length in A. Z is the atomic number of the scatterer, daC/dQ is the free electron (Compton) scattering cross section and A is the atomic weight of the scatterer. T h e coherent scattering cross section per unit mass is: du,,h/dQ
= No[F(x,Z)]'( d a T / d Q ) / A
where F ( x , Z ) is the atomic form factor and daT/dR is t h e Thomson scattering cross section. Values of S(x,Z) and F(x,Z) from t h e compilation of Hubbell e t al. ( 4 ) were used. T h e electron scattering cross sections for incident radiation whose plane of polarization makes an angle 4 with the scattering plane are: d u C / d Q = 1/2r02[,1 + oc(l - cos
a 1 - cos 1 + CY(l
-
x
012
cos
e) + a(I - sin2 0 cos? 4)
I
(1
+ a ) 2[1/2a2(1+
CY-'
+ sin2 41
and daT/dQ = r: sin' 15. It is interesting to consider the effect of polarization. For t h e 21.6-keV energy considered here, ' / p 2 ( 1+ a)-' = 0.00085, so that both cross sections vary approximately as sin2 4, indicating that most of the scattering comes from X-rays which are polarized in a direction perpendicular t o t h e scattering plane. This also indicates t h a t the division between Compton and Thomson scattering is practically independent of the incoming polarization state. Therefore, we average over incident polarization and evaluate the electron cross sections: d u c / d Q = 1/r02 (1
+ a)-?[1+ a2(1+ a ) - 1 ]= 3.66 X 10-26c m 2/ st eradiaii
and daT/dQ = 1/2r2= 3.97 X cm*/steradian. Using numerical values of S(x,z) and F(x,Z).the incoherent and coherent cross sections were evaluated for 90" scattering with a final photon energy of 20.7 keV. They are plotted as a function of the atomic number of the scattering atom in Figure 2. In the present measurements, both coherent and incoherent scattering contribute and as there are 1.86 photons incident in the incoherent energy band for each photon in the coherent energy band, we divide the coherent cross section by 1.86and add it to the incoherent cross section to obtain an "effective" cross section, daE/dR, for incoherently scattering 21.6-keV photons into t h e detector. This effective cross section is plotted as curve C in Figure 2, and shows the way in which the mass measurement sensitivity varies with atomic number. With these effective cross sections and the elemental weight fractions cyz, the average effective cross section per unit mass for any sample may be calculated: du/dQ =
C CY, (duE/dQ), z
Table I lists the elemental compositions assumed for dry muscle (71, bone (81, and the U.S.Geological survey standard basalt sample BCR-1 ( 9 ) , together with t h e effective cross sections of those elements. T h e compositions assumed for other samples are: polypropylene ( C3H6),; Formvar (C5Hi02),; Mylar (Cl0H8O4),; and vyns 90% CH,CHCl plus 10% CH2CH02CCH3.
RESULTS AND DISCUSSION T h e calculated and measured average scattering cross sections are compared in Table 11. I t is seen t h a t t h e total range in cross section for t h e samples listed is only 28%, verifying for these samples the original premise t h a t the scattered X-ray intensity is only weakly dependent on t h e composition of the sample. In addition, the agreement between the experimental values and the theoretical values is good, indicating t h a t the theory of the scattering process is adequate. T h e results of examining a statistically significant number of samples of a variety of types are presented in Table 111. T h e average standard deviation within a group is 2.6%. Previous experience in making targets of these types has indicated that weighing contributes about 2 % of this amount.
1196
ANALYTICAL CHEMISTRY, VOL. 51, NO. 8, JULY 1979
Table 11. Comparison of the Measured Cross Section Values with the Calculated Average Effective Values for Various Samples (Cross section units are square centimeters per steradian per gram) do/dn calculated measured
sample
.
i~~
~
Z
.
~
23
30
~~
5C
AC
. 63
~7:
ATOMIC NUMBER
Figure 2. 90' differential scattering cross section per unit mass as a function of atomic number. Curve A is the cross section for incoherent scattering of 21.6keV photons. Excluding hydrogen, the curve is rather fiat. Curve 6 is the cross section for coherent scattering of 20.7-keV photons, which corresponds to the same scattered energy as does curve A. Curve C is the "effective" scattering cross section for the present system which doesn't distinguish between coherent and incoherent scattering. It is obtained by adding 5 4 % of the coherent cross section to the incoherent cross section, because the number of photons incident on the sample in the coherent energy interval (20.4 to 21.0 keV) is 5 4 % of that for the incoherent energy interval (21.3 to 21.9 keV). Thus, curve C gives the relative sensitivity to mass as a function of atomic number
Table I. Elemental Composition of Various Samples, from Which Their Average Effective Scattering Cross Sections Were Calculated (Composition values are in per cent b y weight)
element H C N 0 Na Mg
AI Si P K Ca Ti Mn Fe other
duidi2, cm*/sr/g 0.0219 0.0112 0.0112 0.0111 0.0106 0.0110 0.0109 0.0115 0.0114 0.0138 0.0147 0.0144 0.0151 0.0156
d ry muscle,
bone,
USGS BCR-1,
%
%
%
6.2 51.0 13.1 22.3
2.7 28.0 4.5 25.0
0.4
1.4
0.6 1.4
11.8
25.7
0.0117 0.0119 0.0124 0.01 20 0.0127 0.0119 0.0123 0.0126 0.0103 0.0111 0.0120 0.0131
-
0.0117 0.0121 0.0117 0.0123 0.0127 0.0121 0.0131 0.0123 0.0101 0.0118 0.0121
0.0140
%
calibration T 1.7 -5.6 + 2.5 0.0
+ 1.7
+ 6.5 -2.4 -1.9 + 6.3
+ 0.8
+ 6.9
Table 111. Average Experimental Cross Sections for Various Groups of Samples, Illustrating the Small Variation among Different Samples of Biological Interest and the Small Standard Deviation within a Sample Group number av. d o i d n , std. dev., in group cm2/sr/g %
sample aorta brain heart kidney liver lung muscle spleen bone hair blood plasma oil shale
22 18
17 13 17
17 28 16 17 5 28 21
0.0117 0.0122 0.0119 0.0118 0.0120 0.0118 0.0121 0.0117 0.0117 0.0104
0.0126 0.0139
2.2 2.1 2.5 1.7 2.7 1.9
2.7 2.0 2.5 4.9 2.6 3.5
1.5 45.0 2.4 2.1 7.2 25.2 0.2 1.4 5.0 1.3 0.1
9.4 5.0
Mylar dry muscle bone USGS BCR-1 polypropylene Formvar vyns CaCO, LiF MgO, S TiO,
difference,
0.9
T h u s it is inferred t h a t t h e precision of the X-ray mass measurements is at least as good as t h a t of t h e electronic balance method currently employed. T h e theoretical investigation of the effects of polarization leads to the conclusion that the ratio of coherent to incoherent scattering is unaffected by changes in polarization of t h e incident beam. However, changes in polarization (e.g., rotating t h e X-ray tube 90' about the X-beam axis) might affect the calibration of t h e system. If t h e system is such t h a t t h e sum of both coherent and incoherent scattering is used, then the relative amounts of the two will affect the atomic number dependence of the average cross section (Figure 2). T h e ratio of t h e two might be adjusted (by changing t h e spectral distribution of the exciting radiation, for example) to minimize t h e variation of t h e effective cross section with atomic number. Hydrogen deserves separate discussion as it is a unique case. It has approximately twice the number of electrons per unit
mass and thus about twice the cross section per unit mass of t h e other elements. This could cause difficulty in some circumstances. However, it is worth noting that hydrogen is light and usually occurs as water or as a constituent of some other molecule, which considerably diminishes this effect. For example, the calculated cross section per unit mass of water is the same as that of dry muscle or the rock standard, USGS
BCR-1. ACKNOWLEDGMENT T h e technical assistance of R. Bernthal and C. Crouch is gratefully acknowledged.
LITERATURE CITED (1) A . C. Alfrey, L. L. Nunnelley, H. Rudolph, and W. R. Smythe, Adv. X-Ray Anal., 19, 497 (1976). (2) T. Hall, Adv. X-Ray Anal., 1, 297 (1960). (3) J. M. Jaklevic, W. R. French, T. W. Chrkson, and M. R. Greenwood, Adv. X-Ray Anal., 21, 171 (1978). (4) J. H . Hubbell, W. J. Veigie, E. A. Briggs, R . T. Brown, D. T. Cromer, and R. J. Howerton, J . Phys. Chem. Ref. Data, 4, 471 (1975). (5) W. Heltler, "The Quantum Theory of Radiation", 3rd ed.,Oxford University Press, London, 1954, p 217. (6) J. D. Jackson, "Classical Electrodynamics", 2nd ed.,John Wiley & Sons, New York, 1975, p 660. (7) W. S. Snyder et al., Report of the Task Group on Reference Man", Pergamon Press, New York, 1975. (8) . . H. J. M. Bowen, "Trace Elements in Biochemistry",Academic Press, New York, 1966, p 72. (9) F. J. Flanagan, Geochim. Cosmochim. Acta, 37, 1189 (1973).
RECEIEDfor review January 22,1979. Accepted April 6,1979. T h e financial support of t h e National Institute of Arthritis Metabolism, and Digestive Diseases and of t h e Department of Energy is gratefully acknowledged.
