2798
Anal. Chem. 1084, 56,2798-2801
Atomic Number Correction in Electron-Probe X-ray Microanalysis of Curved Samples and Particles Andrzej A. Markowicz' and Ren6 E. Van Grieken* Department of Chemistry, University of Antwerp (U.I.A.),B-2610 Antwerp- Wilrijk, Belgium
New expressions are presented for the backscatter coetflclent R , the stopplng-power factor S, and hence for the atomic number correctlon 2,In electron-probe X-ray mlcroanalysls of curved samples and spherlcal particles. The results obtalned via the new expressions are compared wlfh those of the conventlonally used formulas for flat bulk Specimens. It appeared that the latter expresslons can adequately be used for partlcles consisting of elements wlth slmllar atomlc number If normallzatlon to 100% Is carrled out In an tteratlve ZAFtorrectlon procedure. I n other cases, the new equations for the Z-correction can advantageously be applled. The S-factor appears to be the most crttlcal component and requlres exact evaluatlon of the partlcle slze.
v, the electron-backscatter factor
One of the more challenging aspects of electron-probe X-ray microanalysis (EPXMA) is that of individual particle characterization. However, the quantitative analysis of the particles requires rather sophisticated correction procedures; these have been recently reviewed by Small (I). A theoretical correction method, based on geometric modeling of the particle shape, as proposed by Armstrong and Buseck (2), seems to be the only one existing procedure which can be used efficiently for this purpose (3, 4). Another method, the peakto-background method (5), is also promising for quantitative analysis of particles but still requires improvements (6). Only little effort has been made hitherto toward a rigorous treatment of the atomic number correction. In both quantitative methods for particles, the atomic number correction factors (2) are calculated according to the well-known correction procedure for thick polished specimens. However, there is not sufficient theoretical evidence to support this approach fully. The aim of the present work is to show the influence of the curvature on the 2-correction of curved bulk samples and particles as well as to evaluate more rigorously the whole 2-correction for unpolished particles. For reasons of symmetry and relative simplicity the calculations have been limited to spherical particles, but the conclusions drawn for this case can be extended to other particle shapes as well.
THEORY For flat specimens the atomic number correction 2 can be treated by considering the electron-backscatter factor R and the stopping power fador S separately. The approach by Love et al. (7)gives a more rigorous 2-correction procedure than those proposed either by Duncumb and Reed (8)or by Philibert and Tixier (9). Electron-Backscatter Factor. In the 2-correction for unpolished particles, their curvature (or tilting) should also be taken into account because it influences both the electron-backscatter coefficient and the electron-backscatterfactor R. For tilted flat single-element bulk samples the backscatter On leave from the Institute of Physics and Nuclear Techniques, Academy of Mining and Metallurgy, Cracow, Poland. 0003-2700/84/0356-27B8$0 1.50/0
coefficient qflat can be described approximately by (10) qflat =
+ cos 0)P
1/(1
(1)
where 0 = the tilting angle and p = 9/2i1/2, for pure element i with an atomic number Zi. Using this equation and treating the surface of a curved bulk sample as having variable tilting angles, one can approximate the backscatter coefficient for spherical particles, qsphere, by Taphere
=
sin 20 d0 (1 + cos 0)P
=I2
(2)
This expression is only valid if the particle thickness exceeds the backscatter electron generation range and when the whole sample area is irradiated homogeneously, by a defocused beam or in the raster scan mode. After eq 2 was integrated, the following new equation was obtained
The results of the calculations using equation 3 will be shown below. An additional modification of the electron-backscatter coefficient should be considered for samples thinner than the backscatter electron generation range. According to experimental results and Monte Carlo calculations (11, 12), the backscatter-electron coefficient for thin films, qthi,,, reaches its bulk sample value (eq 1 with 0 = 0) for a film thickness of only 30% of the electron penetration range de, while it is approximately a linear function of the film thickness t below 0.3de Or qthin = (q/0.3de)t,for t < 0.3de. In analogy, for small particles, the electron-backscatter coefficient qptc could be modified as qptc =
-D
0.3de
qptc = q for
for
D < 0.3de
D > 0.3de
(4b)
where D = the effective thickness of the particle, approximated by the diameter for the spherical particles, and q = the electron-backscatter coefficient for the bulk specimen. One may wonder whether it is justified to combine the electron backscatter coefficient modifications for curved samples (eq 3) and thin samples (eq 4),i.e., substituting q in eq 4 by ?sphere, to obtain an adequate value for small spherical particles. As will be seen below, the modification given by eq 4 is important only for very small particles resulting in very small values of qPk. Irrespective of whether q,phere (eq 3 ) or vflat (eq 1 with 0 = 0) is used, the electron-backscatter correction will become negligible in this case. A subsequent problem in the analysis of particles by EPXMA arises in the calculation of the electron-backscatter factor R when the particle thickness is smaller than the range for the generation of characteristic X-rays of a given element. In such a case one can modify, in the following way, the formula proposed by Love et al. (7)for the R factor in thick polished specimens. The number of ionizations, I , produced in a particle by an incident electron can be calculated after 0 1984 American Chemlcal Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1984
modification of the expression of Green and Cosslett (13)
I = K1
" In U d U =
For particles of diameters greater than the range for the generation of characteristic X-rays of a given element and for bulk samples eq 9 reduces to
U,
K,(~UolU n d U - S "1 1 n
U d U ) (5)
where K1 = a constant which depends upon the atomic number
U = E/Ec
Uo = Eo/E,
Ui = E , / E c
E = the energy of the electron, E, = the critical ionization potential for the X-ray line of interest, Eo = the energy of the incident electrons, and, E , = the mean energy of electrons leaving the particles (E, > E, for the assumed particles). El depends on E,, the composition, shape, and size of the particles. After integrating and invoking a commonly used approximation ( I d ) , one obtains
I = K[(u, - 1 p 7 - (u,- 1)1.671
(6)
where K = 0.365K1. In analogy to the approach used by Love et al. (7) for bulk samples, the number of ionizations that would have been produced by the backscattered electrons, IbkSc, if they had remained in the particle, can be calculated by (7)
.Iu, 0 '
Ibksc =
2799
dq
=(U
- l)1.67 dU =
For high values of Uo and U,, eq 7 can be approximated by ( 7 )
I~~~~= Kqpk(u- 1 ~ - Kq 7 PtC(6'1)1.67
(8)
6 = o.5u0(1 + qptc) 6'=o.5u1(1 + qPk)
(84
which is exactly the expression proposed by Love et al. (7). For a multicomponent specimen the backscatter coefficient(Tptcor ??sphere) is given by (12)
= Cwiqi (12) where wi and vi = the weight fraction and the backscatter coefficient of the ith element, respectively. Stopping-PowerFactor. To complete the atomic number correction procedure in the analysis of particles by EPXMA, the stopping-power factor S should be reevaluated. By modifying the limits of the integral in the expression proposed by Love et al. (7),one can obtain 7
where Q = the ionization cross section and -dE/d(pz) = the stopping power. After integration and utilization of a simplication which has been justified by Love et al. (7), the following new expression is obtained for the stopping-power factor Sptcfor particles
where f(U) = [ U l n U + (1 - U)] X
where
(8b)
For low U, values, and especially for U , < 2/(1 + qptc), Le., for a narrow integration range, the factor (U - l)1.67 in the last term of eq 7 does not vary greatly in the integration range and can be approximated by its midpoint value, namely [0.5(U1 + 1) - l]1.67. Eventually, this is equivalent to the requirement, in the case of low VI values, of substituting 0' in eq 8 by
6'= 0.5(U1 + 1)
(84
Finally, the electron-backscatter factor for a particle, Rpk, can thus be calculated according to the new expression
Ji = the mean ionization potential of the ith element in a particle. For thick particles of diameters greater than the range for the generation of characteristic X-rays of a given element and for bulk specimens, eq 14 simplifies to
which is again the corresponding expression proposed by Love et al. (7).
RESULTS AND DISCUSSION As is seen from eq 3 the backscatter coefficient for curved bulk samples, vgphere, is a function of atomic number of the element. Table I presents the calculated results for the backscatter coefficients obtained for curved and flat bulk Relevant qpk values can be calculated via eq 4. specimens of several pure elements, according to eq 3 and 1 The mean energy of electrons leaving a spherical particle, (for 8 = 0), respectively. As can be seen from the last column E,, can be calculated from Whelan's equation (15) of Table I, the relative differences of the backscatter coefficients obtained for bulk flat and spherical samples are quite large, especially for low-2 elements. This fact raises the question as to which extent the electron-backscatter factors where 2 and A = mean atomic and atomic mass numbers of R calculated according to classical eq 11 are affected for curved the specimen, respectively, p = density of the specimen, g ~ m - ~ , specimens. Table I1 shows the results of the calculations using D = the diameter of the spherical particle, km, and Eo and eq 11 and the values of ??sphere and vflatfrom Table I. It is seen E, = the energies of the electrons, keV. It follows that eq 9 that, although the differences between vSphm and llflat are large, can be used only for spherical particles of diameter D < the differences between the corresponding Rsphere and Rflat O.O33(A/Zp) values are not so critical.
2800
ANALYTICAL CHEMISTRY, VOL. 56, NO. 14, DECEMBER 1884
Table I. Comparison of the Backscatter Coefficients 9#,hers (eq 3) and vnat (eq 1 for B = 0) for Several Single-Element Bulk Specimens
0
x
Sulfur Iron Molybdenum
A, 70 =
atomic no. 8 12 16 20 26 42
ltflat
0.232 0.296 0.346 0.404 0.429 0.511
0.110 0.165 0.210 0.248 0.294 0.382
((vflat - %phere)/ %phere) loo
-53 -44 -39 -39 -31 -25
x
0'
5
10
Normalized diameter
Table 11. Comparison of the Calculated Backscatter Factors, Rephereand Rnpt,Obtained for Single-Element Bulk Specimens Using vsphem and Values, Respectively ( E , = 20 keV) A. % =
8 12 16 20 26 42
0.900 0.865 0.842 0.815 0.835 0.709
0.960 0.939 0.922 0.915 0.922 0.819
6.7 8.6 9.5 12.3 10.0 15.5
In order to show the influence of the particle size and the curvature on the electron-backscatter factor RPt, given by the proposed new eq 9, calculations have been carried out for particles of which the thickness is less than the characteristic X-ray generation range d for a given element. Table I11 presents the results of these RPt, factors obtained by using eq 9 for several single-element particles. For particles larger than 0.3 times the electron penetration range de (i.e., larger than 1.5 pm for S and 0.4 pm for Fe and Mo), the electron-backscatter factor R,, appears to vary little with the particle diameter D (maximum about 5% in the case of Fe). For such particles, R can thus be calculated by using the expression derived for thick specimens, as has been assumed by Armstrong and Buseck ( 2 ) . However, there are significant discrepancies between the results obtained for flat particles of a certain size when using classical formulas for t and those obtained when taking their curvature into account (eq 4b and 3). This implies that for such relatively large spherical particles the proposed modification of R (eq 9) influences the results mostly by the curvature effect, reflected in the choice of 7 (eq 4b and 3). For very small particles, the differences between bulk and particle electron-backscatter factors become quite substantial, particularly for high-2 elements. Finally, calculations of the stopping-power factor S,,! (eq 14) have been carried out for the same single-element particles;
Figure 1. Theoretical relationship between the normalized stoppingpower factor (Le., the ratlo of the stopping-power factor SPlcfor particles, eq 14, to the S-value for bulk samples, eq 15) and the normalized diameter (Le., the ratio of the particle diameter D to the X-ray generation range d ) for several single-element spherical particles.
the results are shown graphically in Figure 1. As can be seen in Figure 1,the stopping-power factor is a strong function of the particle diameter, especially for very small particles and/or a t high overvoltages. This implies that the knowledge of the size of the particles can be very critical in applying an accurate atomic number correction, especially for particles consisting of elements with considerably different atomic numbers leading to different X-ray generation ranges and different normalized diameters.
CONCLUSIONS New expressions have been presented for the backscatter coefficient Q, the electron-backscatter factor R, and the stopping-power factor S for electron microprobe analysis of spherical particles. The limitations of applying bulk specimen atomic number corrections for unpolished particles can thus be evaluated. I t appeared that, in comparison to the results of the conventionally used corresponding formulas for flat bulk specimens, (1)the backscatter coefficients qsphere, which take into account the sample curvature (eq 3), can deviate by as much as 50%, especially for l o w 4 elements, (2) nevertheless, the electron-backscatter factors Rsphere, calculated for curved bulk specimens via osphere values, are only lower by ca. 7% for low-2 elements to ca. 16% for high-2 elements, (3) the exact R,, values differ by 1-4% only, except for particles much smaller than the X-ray generation range (D< 0.3de),and (4) the exact SPkvalues can be up to 10 times higher, in the case of submicrometer particles, making S by far the most critical factor in the atomic number correction for particles. The determination of the size the of particles can present serious problems in the calculation of the stopping-power factor, especially for particles composed of elements of considerably different atomic numbers, as was seen from Figure 1. However, because of the similarity of the effects for similar elements, the influence of the particle size on the factor S as
Table 111. Comparison of the Values of the Electron-Backscatter Factors R,,,(eq 9) vs. the Diameter of the Single-Element Particles, when ?sphere and vnat Backscatter Coefficients Are Used ( E , = 20 keV) molybdenum (d" = 1.1 pm)
iron (d" = 1.2 pm)
sulfur (d" = 4.9 pm)
D , wn
(Rpdsphere
(Rpdflat
0.2 0.5 1.0 2.0 3.0 4.0 4.5 5.0b
0.986 0.960 0.905 0.831 0.833 0.835 0.837 0.842
0.991 0.977 0.948 0.916 0.917 0.918 0.920 0.922
pm
0.1 0.3 0.5 0.7 0.9 1.0 1.1 1.3'
(&)sphere
0.976 0.889 0.801 0.807 0.817 0.836 0.837 0.838
(Rptc)flat
D , hrn
(Rptc)sphere
(Rpte)flat
0.985 0.938 0.895 0.900 0.909 0.922 0.922 0.922
0.1 0.3 0.5 0.8 1.0 1.2*
0.947 0.759 0.694 0.697 0.701 0.709
0.963 0.845 0.807 0.809 0.813 0.819
' d = characteristic generation range for S-K, Fe-K, and Mo-L X-rays. 'Calculations have been carried out by using eq 11.
Anal. Chem. 1984, 56, 2801-2805
well as the influence of the curvature on the electron-backscatter factor R can be eliminated or a t least minimized for particles consisting of elements with similar atomic numbers, by using normalization of the analysis results to 100% in an iterative ZAF-correction procedure. Hence, in normal practice, the use of the conventional 2-correction formulas for flat bulk specimens will often be justified. In a case when not all matrix elements present are determined, i.e., when normalization of the results to 100% cannot be applied (e.g., in the analysis of particles consisting of carbon or organic compounds), the expressions presented in this paper can be used for an adequte calculation of the atomic number correction. The fact that the influence of the particle size can be accounted for in a versatile way by the proposed expression can also enable one to use particles of any diameter as standards in EPXMA of different particles. In practice, it might be difficult to determine the size of real particles accurately. This problem, however, is common to all existing correction procedures for quantitative analysis of particles, e.g., the absorption correction in the Armstrong-Buseck procedure (2). Moreover, using eq 9 and 14 will at the very least allow the estimation of the errors in the 2-correction due to uncertainties in the particle sizing for any analytical case. Although only spherical particles have been considered in this paper, some conclusions can also be extended to other geometrical shapes. Registry No. Sulfur, 7704-34-9;iron, 7439-89-6;molybdenum, 7439-98-1.
2801
LITERATURE CITED Small, J. A. Scanning Electron Microsc. 1981, 1 , 447-461. Armstrong, J. T.; Buseck, P. R. Anal. Chem. 1975, 4 7 , 2178-2192. Wernisch, J.; Kettner, E. Mikrochim. Acta 1982, 1 , 73-63. Van Dyck, P.; Storms H.; Van Grieken R. J . Phys., Colloq. (Orsay, F r . ) 1984, C2, 781-784. Small, J. A.; Heinrich, K. F. J.; Newbury, D. E.; Myklebust, R. L. Scanning Nectron Microsc. 1979, 2 , 807-816. Markowicz, A. A.; Van Grieken, R. E. Anal. Chem. 1984, 56, 241R250R. Love, G.; Cox, M. G.; Scott, V. D. J . Phys. D . 1978, 1 1 , 7-21. Duncumb, P.; Reed, S. J. B. In "Quantitative Electron Probe Microanalysis"; Heinrich, K. F. J., Ed.; National Bureau of Standards: Washington, DC, 1966; NBS Spec. Pub/. ( U S . ) 1988, No. 298, 133-154. Phlllbert, J.; Tixier, R. In "Quantitative Electron Probe Microanalysis"; Heinrich, K. F. J. Ed.; National Bureau of Standards: Washington, DC, 1966; NBS Spec. Pub/. ( U . S . ) 1988, No. 298, 13-33. Arnal, F.; Verdler P.;Vlncensini, P. D. C.R . Acad. Sci. (Paris) 1989, 268, 1526. Reed, S. J. B. "Electron Microprobe Analysis"; Cambridge University Press: New York, 1975; Chapter 13. Goldstein, J. I.; Newbury, D. E.; Echlin, P.; Joy D. C.; Fiori, Ch.; Lifshin, E. "Scanning Electron Microscopy and X-Ray Microanalysis"; Plenum Press: New York and London, 1981. Green, M.; Cosslett, V. E. J . Phys. D . 1988, 1 , 425-436. Dyson, N. A. "X-Rays in Atomic and Nuclear Physics"; Longman Group Limited: London, 1973. Armstrong, J. T., Ph.D. Dissertation, Arizona State University, Tempe, 1978.
RECEIVED for review April 6,1984. Accepted July 24, 1984. This work was partially financed by the Belgian Ministry of Science Policy under Grant 80-85/ 10. Grateful acknowledgment is made to the Belgian National Foundation for Scientific Research for providing a sabbatical leave grant to A.A.M.
Thermal Lens Absorption Measurements by Flow Injection into Supercritical Fluid Solvents R. A. Leach' and J.
M.Harris*
Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
Supercrltlcal fluld solvents are shown to possess outstanding properties for thermal lens absorption measurements. By use of CO, near Its critical point, a greater than 150 tlmes increase In sensltlvlty relative to CCI,, combined wlth the low background absorption of CO,, has allowed detectlon limits of A,,, = 2 X lo-' to be obtained In a 1 cm path length cell wlth 50-mW laser power. Experimental results obtained agree well with theoretical predictions for both the thermal lens enhancement and tlme constant In CO,. Flow injectlon of samples into the supercritical fluid stream allows convenient and rapid transfer from ambient laboratory conditions to the high pressure, controlled temperature environment for detection.
Supercritical fluids or dense gases are a class of solvents which are generating interest as both chromatographic mobile phases (1-4) and extraction solvents (5). This interest is primarily due to the physical properties of supercritical fluids, which include liquidlike densities, along with diffusivities and LPresent address: E. I. du P o n t de Nemours a n d Co., Pioneering Euksearch L a b , Experimental Station, Wilmington, DE 19898.
viscosities which fall somewhere between those for normal gases and liquids. These properties provide increased chromatographic efficiency, compared to normal liquids, while eliminating the high molecular weight restrictions of conventional gas chromatography. Analyte detection has presented technical challenges in supercritical fluid methodology development primarily due to problems associated with the high pressures of the fluid. The successful direct interfacing of flame ionization (6) and mass spectrometry (7, 8) to pressurized supercritical fluids indicates that gas chromatographic detectors may be equally suitable for supercritical fluid chromatography. Optical spectroscopic detection also appears promising since most mobile phases employed are molecules with simple electronic structure which are generally UV-visible transparent (4)while having relatively few vibrational modes leading to simple infrared spectra with numerous windows (9). Optical absorption measurements in supercritical fluids by thermooptical methods were recently described ( I O ) , where the heat deposited in a sample due to absorption and nonradiative decay of excited species acts to modify the refractive index of the sample. The magnitude of the refractive index disturbance due to absorption depends on the optical power absorbed by the sample, the nonradiative quantum yield of the absorber, and the thermooptical properties of the solvent.
0003-2700/64/0356-2801$01.50/00 1984 American Chemical Society
