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Anal. Chem. 1985, 57,2885-2889 ( I O ) Prange, A,; Knochel, A.; Michaelis, W. Anal. Chim. Acta 1985, 172,
79. (11) (12) (13) (14)
Kaiser, H.;Specker, H. Z . Anal. Chem. 1956, 149, 46. Kaiser, H. Z . Anal. Chem., 1965, 2 0 9 , 1. Currie, L. A. Anal. Chem. 1988, 4 0 , 568. American Chemical Society Committee on Environmental Improvement Anal. Chem. 1980, 52, 2242.
(15) American Chemical Society Committee on Environmental Improvement Anal. Chem. 1983, 55, 2210. (16) Prange, A. Ph.D. Thesis, University of Hamburg, 1983. (17) May, T. W.; Kane, D. A. Anal. Chim. Acta 1984, 161, 387.
RECEIVED for review April 22, 1985. Accepted July 24, 1985.
Bremsstrahlung Background in Electron Probe X-ray Microanalysis of Thin Films Andrzej A. Markowicz,‘ Hedwig M. Storms, and Ren6 E. Van Grieken* Department of Chemistry, University of Antwerp (U.I.A.),B-2610 Antwerp- Wilrijk, Belgium
New equations are presented to describe the contlnuum Bremsstrahlung intensity In EPXMA spectra of thin films. The assumed model takes into account that electrons, havlng passed through the film, retain a fraction of the energy that cannot be converted Into Bremsstrahlung, unlike the case of bulk samples for which the Bremsstrahlung Is descrlbed by the well-known Kramers’ law. Experimental verlflcatlon of the proposed equatlons wlth single-element thin films showed a satisfactory agreement, contrary to Kramers’ law, partlcularly for low and medium Bremsstrahlung energies. The llmitations and possible Improvements of the proposed equations are discussed.
The X-ray continuum generated by the inelastic interaction of an electron is invariably problematic in electron probe X-ray microanalysis-EPXMA (1). First of all, fitting a function through the background portion of an EPXMA spectrum is not easy; several experimental fitting routines and mathematical approaches such as the Simplex method have been proposed in this context lately. Secondly, for bulk multielement specimens, the theoretical prediction of the continuum Bremsstrahlung radiation is not trivial; indeed it has been known for several years that the commonly used Kramers’ formula can lead to significant errors. In this context some improvement is offered by several modified versions of Kramers’ formula, recently developed for multielement bulk specimen. A third problem is connected with the description of the X-ray continuum generated in specimens of thickness smaller than the continuum X-ray generation range. This problem arises in the analysis of both thin films and particles by EPXMA. Some authors ( 2 ) have reported that even a modified Kramers’ equation, which incorporates a secondorder term, leads to results that are too high for particles smaller than 10 Irm. The lack of an adequate method for the theoretical prediction of the continuum distribution is a major obstacle for the effective application of the peak-to-background method in quantitative EPXMA of particles. An additional problem in EPXMA of small particles is the background contribution from the substrate; it can be approximated via a recently developed empirical procedure (3). In the present paper we propose a theoretical model for the shape of the X-ray continuum radiation generated in multielement thin films or particles. The new simplified ex-
pression for the continuum radiation, combined with known formulas for the self-absorption of X-ray and electron backscattering, has been verified experimentally for single-element thin films. In the derivation of the new equation, some approximations had to be made due to the lack of knowledge concerning the shape of the interaction volume, the distribution of the electrons within the interaction volume, and the anisotropy of the continuum radiation for different energies of the continuum photons and for different film thickness. Some further developments of the presented approach are suggested.
THEORY The number of continuum photons, dIco,with energy E,, within a photon energy range AE,, which are generated by one electron of energy E , from an infinitely thin composite film of thickness d(pz), can be calculated by using the expression (4)
where A , and J, are the atomic mass and mean ionization potential of element i , respectively, p is the density of the specimen, and n is the number of the elements in the specimen. Of course, this expression does not take into account the X-ray absorption and the electron backscattering effects. Also the anisotropy of the continuum emission is not considered. The energy loss of the electrons per unit of distance traveled in the specimen, dE/d(pz), is given by the Bethe equation (5)
The limitations of the application of the Bethe equation have been discussed elsewhere previously (6). The limits for the sum in eq 2 and in the subsequent expressions are the same as in eq 1 and will not be repeated in the equations below. The calculation of the number of continuum photons generated in a specimen of finite thickness t will be different when the considered photon energy E,, is either smaller or greater than the mean remaining energy E l of the electrons after passing a distance. This mean energy E , can be calculated from Whelan’s equation (7)
‘On leave f r o m the I n s t i t u t e for Physics a n d Nuclear Techniques, Academy of M i n i n g a n d M e t a l l u r g y , Cracow, Poland. 0003-2700/85/0357-2885$01.50/00 1985 American Chemical Society
-
(3)
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 14, DECEMBER 1985
0
where
I
E01.69- -0.033A cos 6
Flgure 1. Representation of the geometry used in the calculation of the Bremsstrahlung from a film with thickness t (see text).
sin /3 = t / d y
where 2 and A are the mean atomic and mass numbers of the specimen, respectively, t is the film thickness, pm, and Eo is the energy of the incident electrons, keV. The part of the continuum Bremsstrahlung radiation of energy greater than E , can be calculated adequately by using the procedure developed previously for bulk multielement samples (6). On the contrary, the part of the continuum Bremsstrahlung radiation of the energy smaller than E , should be calculated in quite a different way. In this case only a part of the electron energy (from Eo to E,) can be converted, inter alia, into Bremsstrahlung radiation. This leads to a reduction of the number of continuum photons of energy E, < El compared to that generated in the bulk specimen within the same energy region. Above the energy E , the Bremsstrahlung spectra generated in finite and bulk specimen of the same composition should be identical. This qualitative theoretical conclusion wil be confirmed below by experimental results. But the problem arises how to calculate the number of continuum photons of energy E, < El due to the fact that some electrons obliquely leaving the presumed film of thickness t have an energy below E,, as is seen from Figure 1, and only those electrons leaving perpendicularly can be described by the energy El. If the distribution of the electrons within the interaction volume is spherically isotropic, and the hemisphere in Figure 1presents approximately the generation range for X-ray continuum of a certain energy E,, the leaving electrons have mean energies within the range E, to E, (directions OA and OB, respectively). In that case one can assume that the part of the electrons within the solid angle Qbulk whose projection is represented by p lose their energy down to E, and can be treated in the same way as the electrons in bulk specimen. The assumption about the shape of the electron interaction volume can be valid only for high density p and relatively low energies of incident electrons Eo (8,9). The solid angle, Ob&, in which the electrons lose their energy from Eo to E, can be calculated by Qbulk
= 2 r sin p
(4)
Combining eq 1-3 one can calculate I,,, the total number of continuum photons of energy E, within an energy range AE,, generated from a thin composite film, when the absorption of X-rays and the electron backscatter effects are not considered. For El < E, < Eo,the formula for Bremsstrahlung from bulk samples (6) can be used
I,, = const-lEoK a, E , E,
dE
with
K=
For 0
< E , < E,, this equation should be modified into
0.5917
where d, is the generation range for the X-ray contipuum of energy E, expressed by
The first term in eq 5b corresponds to the number of continuum photons of energy E, generated within the solid angle Qbulk, while the second one gives the mean number of continuum photons generated within the solid angle ( 2 -~Qbdk). For single element thin films, eq 5a and 5b simplify to
I,, = const-Z(Eo m” E, for E,
- E,)
< E, < Eo and
+ I 219@8 ( --)
I,, = const-Z E,
sin @(E,- E,)
d;[ Eo - E01.69-
90-6
0
(1 - sin p) x
ZP
t
0.033A cos 0
0.5917
1)
Equation 5b is not valid when the interaction volume cannot be approximated by a hemisphere, as is the case for specimens of low atomic number and relatively high energy of incident electrons. In this case the electron interaction volume has a distinctive pear shape with the length substantially greater than the width. Then it would be better to assume that the first term in eq 5b reduces to zero and the number of continuum photons of energy E,(O < E, < E,) can be expressed by
For an assumed spherically isotropic distribution, eq 5a and 5b are rigourously valid. However, because the assumed interaction volume shape cannot be realistic, eq 7 can certainly not describe adequately the continuum generation in the extreme cases of a very thin film and of a film whose thickness approaches the X-ray continuum generation range. To eliminate the coarse approximations made in the derivation of the equations and to make the theoretical approach more general, one should be able to introduce: (1)the accurate shape of the electron interaction volume, (2) the real distribution of the electrons within the interaction volume, and (3) the angular distribution of the emitted continuum photons for different energies and for different film thicknesses. Hence it is obvious that the development of general expressions for continuum radiation requires further extensive and detailed research on electron beam-specimen interaction.
EXPERIMENTAL SECTION Equations 6a, 6b, and 7 have been verified for thin single element films of Fe (0.5 pm and 0.75 pm) and A1 (0.75 pm). The X-ray spectra were collected with a JEOL JXA-733 electron probe X-ray microanalyzer equipped with the TN-2000 energy-dispersive
ANALYTICAL CHEMISTRY, VOL. 57, NO. 14, DECEMBER 1985
Flgure 2. Measured EPXMA spectra for bulk and thin Fe specimens. (The arrows a and b indicate the energy keV for 0.75 pm and 0.5 pm Fe foil, respectlvely.)
2887
E , equal to 12.8 keV and 15.5
a -
ft
Channel number
1
I
I
4
Figure 3. Measured EPXMA spectra for bulk and thin AI specimens. (The arrow indlcates the energy E , = 5.8 keV.)
X-ray detection system of Tracor Northern. The foils were excited with an electron beam current and high voltage of 2 nA and 20 keV for Fe and 10 nA and 10 keV (or 20 keV) for Al, respectively.
RESULTS AND DISCUSSION In order to compare the theoretical results predicted by the proposed equations with experimental data, one should take self-absorption of X-rays and electron backscattering effects into account. The self-absorption correction factor ftffor thin films of thickness t can be calculated by using the classical formula (8)
Eo = 10 keV
. 0.75 pm AI film
I t I
OO
5
10
E,[keVI
Figure 4. Calculated self-absorption correction factors for (b) AI specimens.
where 4(pz) is the X-ray depth distribution function, p is the mass absorption coefficient of the specimen for X-rays of energy E,, and $ is the takeoff angle for detected X-rays. In the calculation of the self-absorption correction factor for continuum X-rays it was assumed that the depth distribution function for continuum X-rays is the same as that for characteristic radiation of the same energy (10). The ~ ( P zfunction ) given by Brown and Packwood (11)was used. Equation 8 will also be used for bulk specimen with substitution of d, instead oft. The electron backscatter correction factor for continuum radiation has been calculated by using the expression proposed by Statham for bulk specimens (6,12),The required electron backscatter correction factor for characteristic radiation has been calculated by the expression of Love et al. (13)for bulk specimens and by a recently developed equation for particles (14).
Figures 2 and 3 show the X-ray spectra obtained for Fe and A1 foils, respectively. The experimental points for the spectra of thin film and bulk overlap completely above the energy E, predicted by eq 3. It can also be seen that in the lower energy
(a) Fe and
part of the spectra (E, < E,) the Bremsstrahlung intensity of thin films is considerably lower than for the bulk specimen. Hence, the general shape of the Bremsstrahlung spectra for thin films, as shown in Figures 2 and 3, confirms fully the theoretical qualitative predictions presented previously. For the quantitative comparison between the experimental data and the proposed formulas, the theoretical results obtained from eq 6b and 7 have been corrected for self-absorption and electron backscattering effects. Figure 4 shows the self-absorption correction factor vs. the energy of continuum photons for bulk Fe and A1 specimen, and for thin films of the same elements. As can be seen substantial differences exist in the case of Fe for low energies as well as just above the FeK absorption edge. The electron backscatter correctionfactore are almost the same for the considered bulk specimens and thin films for both characteristic and continuum radiation; the relative differences between bulk and thin film electron backscatter correction factors are within 0.5-1.5% for A1 and 1-370 for Fe. Figures 5-8 present a comparison of the theoretical results corrected for self-absorption and electron backscattering effects with the experimental data. To eliminate differences
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 14, DECEMBER 1985 to= 20 keV 0.5 pm Fe film
t
.tf
x
1
Eo =20keV 0 . 7 5 ~ AI 1 ~film x -theory I c Is -exp.
-theory t Is-exp.
i I
10
0'91
0.8 'I
"vO 5 10 15 E,[keVI Flgure 5. Comparison of experlmental Bremsstrahlung results (0)and theoretical predictions by the proposed formulas (X) and by Kramers' law (horizontal solld line).
*
I
Eo = 20 keV 0.75 pm Fe film x -theory I ? Is-exp
if
0 6' 0
1
I
1
5
10
15
w
E,[keVI
Flgure 6. Comparison of experimental Bremsstrahlung results (0)and theoretical predictions by the proposed formulas (X) and by Kramers' law (horizontal solid line). Ec = I 0 keV 075 pm AI film - theory 1 c I s -exp
0
0
8
1
5
-
10
E,ikeVI
Flgure 7. Comparison of experimental Bremsstrahlung results (0)and theoretical predictions by the proposed formulas (X) and by Kramers' law (horlzontal solid line).
in detection efficiency, the data have been normalized to the bulk specimen results and the figures show the dependence of both the theoretical and experimental ratio of the number of continuum photons obtained for thin film and bulk specimen Icotf/Ic>lkas a function of the energy of continuum X-rays. Moreover, to reduce the errors due to counting statistics this ratio has been calculated for an energy interval AE, = 0.5 keV. The theoretical Zctf values for Figures 5-7 have been obtained by using eq 6a and 6b and those for Figure 8 using eq 6a and 7 . The theoretical number of continuum photons, generated in bulk single element specimen, Ic>lk, has been calculated by Kramers' formula
m!J
IcobUlk = const---Z(Eo E,
- E,)
The relevant self-absorption correction factors for bulk specimens have been calculated according to eq 8 and the electron backscatter correction factors using the formula proposed by Love et al. (13). The first conclusion that can be drawn from Figures 5-8 is that the experimentally obtained data for thin films differ drastically from those predicted by Kramers' formula; indeed,
01 I
0
I
5
10
1
+
15 E, [keVI
Flgure 8. Comparison of experimental Bremsstrahlung results (0)and theoretical predictions by the proposed formulas (X) and by Kramers' law (horizontal solid line).
the experimental ratios I,tf/Ic>lkare not close to unity, as would be the case when Kramers' law were valid. Comparison of Figures 5 and 6 shows that the experimental ratios differ more from unity with decreasing film thickness. Inspection of Figures 6 and 8 shows larger deviations for the matrix of lower atomic number, while comparison of Figures 7 and 8 indicates a larger discrepancy with increasing incident electron beam energy. All these observations have one single base: the deviations from Kramers' law increase as the film thickness becomes smaller relative to the continuum X-ray generation depth. Only the high energy part of the spectrum can be approximated by Kramers' law, namely, for E, > El, Le., above the mean energy of the electrons after passing a path length equivalent to the film thickness. Yet, the low- and mediumenergy range of continuum Bremsstrahlung radiation is of greatest importance of EPXMA. The new expression, although derived using coarse approximations, gives much better results than Kramers' formula. This is evident from the match of theoretical and experimental data points in Figures 5-8. Contrary to the case of Kramers' equation, the general shape of the Bremsstrahlung spectrum as a function of energy E, is approximated reasonably well by the proposed new equations. Moreover, relative discrepancies between experiment and theory are markedly lower for the new formulas than for Kramers' equation; e.g., it can be seen from Figure 8 that, for a 0.75-km A1 film and Eo = 20 keV, and for the Bremsstrahlung energy E, around 5 keV, Bremsstrahlung intensities predicted by Kramers' law are too high by a factor of 3, while the new equations yield accurate results within a few percent. More generally, it may be noticed that below, e.g., 10 keV Bremsstrahlung energy, the relative discrepancies are never higher than ca. 10% in the four cases considered (Figures 5-8). The agreement is particularly satisfactory for the low and medium energy range. However, the Ico"f/Ic,bu'k results predicted by the new formulas are still considerably higher than the experimental ones for higher energies of continuum photons. One of the possible reasons of these discrepancies, apart from the coarse approximations that can result in some errors, is differences in anisotropy of the continuum radiation generated in thin films and bulk specimen; these are especially important for continuum X-rays of high energy (15). As mentioned above the angular distribution of the generated continuum radiation should be taken into account to calculate exactly the number of photons within the solid angle subtended by the Si(Li) detector. An approach similar to that presented in the paper can be applied for the prediction of continuum Bremsstrahlung radiation generated in particles although one should be aware of additional difficulties due to spurious background. This paper is a first step toward the development of the generally
Anal. Chern. 1985, 57, 2889-2891
valid formula for the theoretical description of the continuum Bremsstrahlung radiation generated in a specimen of finite size, which presents one of the most serious problems in today's EPXMA.
LITERATURE CITED Llfshin, E. Proc., Annu. Conf.-Microbeam
Anal. SOC. 1964, 19,
215-220. Small, J. A.; Heinrich, K. F. J.; Newbury, D. E.; Myklebust, R. L. Scannlng Electron Mlcrosc. 1979, II 807-816. Aden, G. D.; Buseck, P. R. Proc., Annu. Conf.-Microbeam Anal. I
2889
num Press: New York and London, 1981. (9) Brown, J. D. "Electron Beam Interaction with Solids"; Proceedings of the 1st Pfefferkorn Conference; SEM, Inc.: AMF O'Hare, Chicago, IL, 1984;pp 137-144. (10) Statham, P. J. X-Ray Spectrom. 1976, 5 , 154-1138, (11) Brown, J. D.; Packwood, R. H. X-Ray Spectrom. 1982, 11, 187-193. (12) Statham, P. J. Proc ., Anno. Conf.-Microbeam Anal. Soc. 1979, 14,
247-253. (13) Love, G.;Cox, M. G.; Scott, V. D. J . Phys. D 1978, 1 1 , 7-21. (14) Markowicz, A. A,; Van Grleken, R. E. Anal. Chem. 1984, 5 6 , 2798-2801. (15) Dyson, N. A. "X-Rays In Atomic and Nuclear Physics"; Longman Group Ltd., London, 1973.
SOC.1983, 18, 195-201. Fiori, C. E.; Swyt, C. R.; Ellis, J. R. R o c . , Annu. Conf.-Mlcrobeam Anal. SOC. 1982, 17, 57-71. Heinrich, K. F. J. "Electron Beam X d a y Mlcroanalysis"; Van Nostrand Relnhold: New York, 1981. Markowlcz, A. A.; Van Grleken, R. E. Anal. Chem. 1984, 56,
2049-2051. Armstrong, J. T.; Ph.D. Dlssertatlon, Arizona State University, 1978. Goldsteln, J. I.; Newbury, D. E.; Echlln, P.; Joy, D. C.; Fiorl, Ch.; Llfshin, E. "Scanning Electron Microscopy and X-Ray Microanalysis"; Pie-
RECEIVED for review December 18,1984. Resubmitted August 8, 1985. Accepted August 8, 1985. we wish to acknowledge partial financial support from the Belgian National Foundation for Scientific Research which provided a sabbatical and from the Ministry for leave grant to Science Policy, under Contract 80-85/ 10.
Determination of Magnesium in Alumina Ceramics by Atomic Absorption Spectrometry after Separation by Cation Exchange Chromatography Tjaart N. van der Walt*' and Franz W. E. Strelow
National Chemical Research Laboratory, CSIR, P.O. Box 395,Pretoria 0001, Republic of South Africa
A method Is presented for the determlnatlon of traces of magneslum In alumlna ceramlcs. After dissolution In an orthophosphoric acid-sulfuric acid mixture the magnesium Is separated from the large excess of aluminum by catlon exchange chromatography, using a 4 % cross-ilnked resin and 0.50 M oxallc acid as eluting agent. Magnesium Is flnally determlned by atomlc absorption spectrometry uslng an acetylene-nltrous oxide flame. By use of Suprapur reagents and beakers made of Teflon, contamlnatlon can be reduced to ca. 2 pg wlth a varlatlon between multlple blank runs of ca. 0.4 pg. About 3 ppm of magnesium In 1-g samples can be determlned wlth approximately the same varlatlon whlie larger amounts of magnesium (200-300 ppm in the alumlna ceramics) show a variation of only k l ppm.
Addition of small amounts of magnesium oxide to alumina below the solubility limit at sintering temperature enhances the density of the alumina ceramic and the normal grain growth rates ( I ) . Above the solubility limit magnesium oxide has a negative influence on densification and acts as a grain growth inhibitor. Accurate information on the concentration of traces of magnesium oxide in alumina ceramics is therefore important and relates to their physical properties. Atomic absorption spectrometry is probably the most attractive and simple method for the determination of small amounts or traces of magnesium. Unfortunately, aluminum causes a serious signal depression with the acetylene-air flame, even when present in low concentrations (2, 3). No interference occurs with up to 50 ppm aluminum present in sohPresent address: Isotope P r o d u c t i o n Centre, B u i l d i n g 3000, AEC, P r i v a t e B a g X 256, P r e t o r i a 0001, Republic o f S o u t h Africa.
tion when the acetylene-nitrous oxide flame is used (3). In order to determine traces of magnesium (from a few to several hundred parts per million) in alumina ceramics, very much higher concentrations of aluminum can be expected to be present in solution, up to ca. 20 000 ppm. In addition there may be even larger amounts of other elements, probably alkali metals, from fusion mixtures, should a fusion step be required. Such an aluminum and solids content will result in interference in the flame. In order to obtain reliable results, especially at low magnesium concentrations, it was therefore decided to combine a recently developed cation exchange separation of magnesium from aluminum (3) with atomic absorption spectrometry, using the acetylene-nitrous oxide flame. There remained two problems to be solved. Firstly, alumina sintered at high temperatures (ca. 1650 "C) is quite difficult to dissolve. A suitable dissolution procedure was therefore developed. Secondly, magnesium contamination introduced during the dissolution step and subsequently in the analysis had to be kept at a minimum, which was still practical for normal laboratory use and also reproducible. Methods for the dissolution of alumina previously employed in this laboratory included fusions with either potassium pyrosulfate, sodium peroxide, or sodium carbonate. Procedures using fusion with sodium hydroxide ( 4 , 5), mixtures of soda and borax (6, 7) or potassium hydrogen sulfate (8), or simply dissolution in hydrofluoric acid (9), phosphoric acid (IO), or in mixtures of acids in a PTFE bomb ( 5 , I I ) have also been described. Mendlina et al. (12) dissolved fusion alumina in a mixture of orthophosphoric acid and sulfuric acid or hydrochloric acid. Although it was felt that the alkaline fusions named above were not very suitable because of their large magnesium blank values, further investigation revealed that they were also not very effecive in dissolving magnesium-doped alumina sintered under pressure at high temperatures. The
0003-2700/85/0357-2889$01.50/00 1985 American Chemical Society
