Quantitative PIXE microanalysis of thick specimens

Section III Analytical methodology

NIIIMB

Nuclear Instruments and Methods in Physics Research B77 (1993) 95-109 North-Holland

Quantitative J.L. Campbell,

Beam Interactions with Materials&Atoms

PIXE microanalysis of thick specimens D. Higuchi,

Guelph - Waterloo Program for

J.A. Maxwell and W.J. Teesdale Graduate Work in Physics, University of Guekh, Guelph,

Ontario NlG 2W1, Canada

Methods of standardization in quantitative micro-PIXE analysis are reviewed and various issues that bear on analytical accuracy are explored; pertinent recent work on Si(Li) X-ray detector response is included and some geochemical examples are drawn upon. Extension of the GUPIX software to deal with multilayer targets, including secondary fluorescence within and between layers, is reported, analytical examples include alloy foils and multilayer solar cell structures.

1. Introduction In an elegant review of standardization methods in electron probe microanalysis, Newbury [ 11worked from the general functional relationship between the intensity I(Z) of a particular X-ray of an elemental constituent Z of the specimen and its concentration C,:

I(Z)

=f(C,,

ME, IE);

(1)

here ME represents matrix effects, i.e. the influence of constituents other than Z upon I(Z); IE represents instrumental effects such as detection efficiency. Our intent here is emulate for pPIXE the development presented by Newbury for EPMA. Examples of both major element and trace element analysis will provide illustrations and will identify some residual problems in PIXE quantitation. Extraction of a desired concentration C, via eq. (1) from a measured X-ray intensity I(Z) is predicated upon the reliable extraction of I(Z) from a Si(Li) X-ray spectrum that may be rendered very complex by a multiplicity of overlapping peaks. This necessitates attention to both intrinsic and extrinsic effects that cause Si(Li) line shapes to deviate from the Gaussian ideal. Although various references are cited, this article is not intended as a review paper covering the entire literature, but rather as a distillation of recent experience.

2. Standardization 2.1. General principles

Newbury [l] discussed four means of solving eq. (1). At the empirical extreme, working curves relating I(Z) to C, may be constructed if one can obtain or synthesize a suite of standards which share identical ME with 0168-583X/93/$06.00

the specimens, and contain the particular elements that are of interest in accurately known concentrations. These standards are analyzed under precisely the same conditions as the specimen and so the IE need not be quantitatively known. Specimen concentrations are read off the curves by interpolating at the observed X-ray intensity values. This is practical in a few cases; bone specimens, for example, can be mimicked by hydroxy-apatite [2] with appropriate trace elements incorporated. But for a proton microprobe facility such as ours, dealing with a wide range of geochemical and materials science specimen types, the working curve method is far from feasible. And it would be quite impractical for some of the more complex applications, e.g. multilayer specimens, with which p,-PIXE is now (dealing. Some flexibility is gained by working with the measrelative to ured ratio SZ,R of “sensitivities” I(Z)/C, that measured for a reference constituent (I(R)/C,), which is usually a high-concentration element. The SZ,R values have to be determined by measurements on known multielement standards and the sensitivities obviously are subject to matrix effects. This approach offers the possibility of some degree of cancellation of matrix effects between the elements Z and R, but the cancellation is only partial. A major shift away from these empirical approaches is possible if the physics of the beam-specimen interaction is well enough known (and an accurate data base exists) that ME may be calculated. In EPMA the so-called ZAP approach provides the ratio of ME between a specimen and some standard, whose similarity to the specimen need only be general rather than precise. In PIXE, which actually better satisfies the stated condition, ME may be calculated either in absolute terms for the specimen or as a ratio between specimen and standard. Now that standards need not mimic the specimen precisely, flexibility is much in-

0 1993 - Elsevier Science Publishers B.V. All rights reserved

III. ANALYTICAL METHODOLOGY

J.L. Campbellet al. / PIXE microanalysisof thick specimens

96

creased; a wide range of specimen types may be handled with a very small suite of well-chosen standards. The greatest flexibility is achieved by reverting to a “first principles” approach that abandons standards entirely. This too is possible in p,-PIXE, given the simple physics and the excellent data base. But it demands absolute measurement of proton beam flux, which is less straightforward than superficial examination might suggest. At an early stage in thick target PIXE’s development, Clayton 131 demonstrated lo15% systematic errors in such analyses based on first principles. The discrepancies were a function of specimen type, and Clayton corrected them by analyzing reference materials of the appropriate type. By so doing, he effectively demonstrated the efficacy of the third of Newbury’s approaches, as outlined in the previous paragraph. 2.2. The matrix correction approach The PIXE version of eq. (1) has appeared frequently in the literature; it results from an integration of the X-ray intensities recorded from each point on the linear path, within the specimen, of the decelerating proton:

X

(2) In this equation, for the K shell case: Z(Z) is the intensity in a particular K X-ray line, usually but not necessarily the dominant K, or Kal; Np is the number of protons of energy E, incident at angle a! to the normal upon the specimen, and uz(E) the ionization cross-section for the K shell; wz is the corresponding fluorescence yield and b, the fraction of K X-rays in the selected line; N, is Avogadro’s number; A, is the atomic mass of element Z; S(E) is the proton stopping power; 6ro is the X-ray take-off angle; flci is the absolute efficiency of detection, which involves both Si(Li) detector solid angle and intrinsic efficiency of the silicon crystal; tZ is the attenuation of the X-rays in any absorber (filter) placed between specimen and detector; p/p is the attenuation coefficient within the specimen matrix for the X-ray line of interest. The L and M-shell equivalents are essentially the same, albeit with the additional complication of subshells; they thus involve the Coster-Kronig probabilities that transfer the proton-induced vacancies among subshells of a major shell prior to X-ray emission. The practical unit of integrated beam charge being

the microcoulomb (l.&), we define lr(Z, Cz,,) as the computed or theoretical X-ray intensity per PC per steradian and per unit concentration. The Cz,, are the concentrations of the major elements which comprise the specimen matrix and thus determine ME. Then the measured intensity for Q ~J,Cof charge is

There are several important points to note. First, the IE, viz. the terms 0, &. and tz, factor out; they will therefore cancel in the comparison of specimen (SP) and standard (ST) for X-rays of a particular element z: IS’(Z) ___=IS’(Z)

c;p I,“‘(Z,

Cz,,)

c;= I,“(&

Cz,,)

.

(4)

In addition, measurement of Q needs to be correct only in relative terms between specimen and standard. The ME are represented by the term I&Z, Cz,,), which involves both the compound stopping power and the compound attenuation coefficient, which are concentration-weighted sums’ of these quantities over the major constituents. If the C,,, subset is known, e.g. from prior EPMA analysis, then the I,(Z, C,,) are immediately calculable, and eq. (4) offers a one-step solution for the trace element concentration subset CZ,t. The further advantage of this ratio approach for each Z-value of interest is the additional cancellation of fluorescence yields and relative X-ray line intensities, and the partial cancellation of ionization crosssections within the integrals. The degree of partial cancellation of stopping power and attenuation coefficients will increase with the similarity of the specimen and the standard. We have discussed the accuracy of the data base in some detail elsewhere [4] and with one exception will not repeat that here. We simply note that the data base is sufficiently well known that a generic, as opposed to a specific, similarity between specimen and standard will suffice. The exception pertains to elements of very low atomic number, which are of increasing interest to PIXE analysts, and will be dealt with in section 3.4. The situation is more complex, but entirely tractable, when the major element concentrations Cz, are unknown, a situation that we encounter with ‘increasing frequency in our geochemical work. The approach here is to begin by estimating a set of values for the Cz,,, compute the corresponding I,(Z, C,,,), fit the measured X-ray spectrum by least squares to obtain I(Z), and via eq. (4) deduce a new set of C,,,. This process is iterated until a self-consistent set of Cz,, is obtained. Computing overheads have long ceased to be a deterrent to this inclusion of one iterative process (spectrum fit) within the iterations of another (concentration determination).

J.L. Campb~~ et at. / PIXE ~icroa~a~~isofthick specimens

Eq. (4) presents a choice as regards the standard. We indicated above that, given an accurate data base from which to calculate ME, only a general as opposed to a close resemblance was necessary between specimen and standard. In this spirit we use synthetic sulfide minerals, containing known quantities of one or two trace elements, to standardize measurements upon a wide variety of oxide, silicate and other minerals. However if the concentration of one major element in the specimen is known (e.g. from EPMA) then the specimen itself may serve as standard; an example is the analysis by Rogers et al. [5] of the NIST standard 610 (trace metals in glass), effected relative to the major constituent, calcium. This confers the further advantage of cancelling any residual errors in proton stopping powers, which are now identical in specimen and standard. The iterative solution discussed above for major elements has been incorporated in the GUPIX software package (GUPIX92 is the current version) for fitting spectra and providing concentrations 161.A useful concept, due originally to Lagarde et al. [7], is that the matrix may include an “invisible element”, i.e. one whose X-rays are too low in energy to appear in the spectrum. The condition is then imposed that all concentrations sum to 100% and the invisible element concentration is iterated with the others in the computation of concentrations. GUPIX92 allows there to be more than one invisible element, with the necessary proviso that their ~ncentrations remain in a specified ratio; this would permit for example the presence of SiO, as a defined species. EPMA practice suggests a useful variant for the specific case of the wide range of silicate minerals where the elements are all present in oxide form. This is to deduce the oxygen contribution for each oxide by applying the stoichiometric ratio to each cation concentration; one then can compare the total of all element concentrations to 100% as a further check of the analysis. Table 1 Results of PIXE analyses of Fe/Ni Standard

NIST SRM 1159 (thick target)

lnvar FOBS a (10 pm thickness)

The characteristic X-rays of major elements may induce significant secondary fluorescence both of trace elements and other major elements. We have incorporated a full treatment of secondary fluorescence in GUPIX92. Its accuracy has been examined elsewhere [S] and we have further comments later in this regard.

3. Major element anaiysis: practical aspects Micro-PIXE has to date been concerned much more frequently with trace elements than with major elements, the latter frequently being predetermined by EPMA in a complementary use of the two methods. However, geochemists (for example) who have to purchase time outside their laboratory for both techniques may well be accommodated in full by micro-PIXE, to their obvious economic benefit. The advantages and disadvantages of the standardization approach embodied in eq. (4) i.e. of an approach based on computation of ME and cancellation of IE are now explored. The examples used are mainly from our own materials science and geochemistry experience. 3.1. Attenuation and secondary fluorescence uncertainties

Pure single-element standards are convenient and attractive in that they preferentially transmit their own characteristic X-ray lines, thus minimizing the impact of X-ray attenuation coefficient errors on the analysis. In our work on silver and platinum-group elements as traces in sulfide minerals, it is sometimes necessary to examine major element variations. An example is iron and nickel in pentlandite, where there is interest in a smuggested [91 linear correlation between the trace Pd content and the Ni/Fe ratio. The major and trace analyses can be effected in a single measurement with

alloys. Cases A use theoretical p/p Element

Si Cr Mn Fe Ni Ti Cr Mn Fe Ni

Reference

0.5 0.069 0.3 51.0 48.2 -

64.0 36.0

91

for Fe; B uses measured I( /p Concentration [%I

10

PIXE (A)

PIXE (B)

0.24 0.07 0.29 51.7 47.3 0.1 0.12 0.34 63.2 34.2

0.24 0.07 0.29 51.2 48.4 0.1 0.12 0.34 63.4 35.3

0.01 0.003 0.01 0.1 0.1 0.005 0.004 0.007 0.1 0.1

a Mn + Si + C < 1%. hence Fe and Ni slightly below nominal values. III. ANALYTICAL METHODOLOGY

98

J.L. Campbell et al. / PIXE microanalysb of thick specimens

3 MeV protons, using a 0.35 mm Al absorber to preferentially suppress the dominant Ni and Fe peaks. Due to the absorber, no Si X-rays are detected and so Si is treated as an “invisible element”, the total Csi + C, + C, being 100%. The standards are pure iron and nickel targets. This appears straightforward, but we felt that the procedure warranted testing. An appropriate way to do this is by analysis of the NIST nickel-iron alloy standard reference material (SRM 1159>, since it has 48% of nickel and 51% of iron. We find consistently (see table 1) that we underestimate the nickel concentration and overestimate that of iron. This is unlikely to arise from inhomogeneity in the standard, since the microbeam is rastered over an area 0.4 x 0.4 mm. We have observed the effect before [8], in conventional broad-beam thick-target PIXE with a different detector. And (see section 5 below) we observe it also in thin Invar alloy foils. This discrepancy is not attributable to either ionization cross-sections or stopping powers since we are dealing with elements which are separated by only two units of atomic number; any discrepancies would be the same for both, and would tend to cancel. Suspicion therefore falls on the attenuation coefficients p/p, and specifically on the largest of these, that for the nickel K, line in iron, since this quantity strongly influences both the matrix attenuation for NiK, and the secondary fluorescence of iron. While the p/p theoretical data base used here [lo] is felt to be accurate generally to 2-3%, its authors warn of anomalies at X-ray ener-

gies above and within 1 keV of an absorption edge. A detailed experimental study of this region for iron [II] suggests a p/p value for NiK, that is 6% above theory. Fig. 1 presents a comparison of the measured and theoretical p/p values in the vicinity of the iron K absorption edge. When the g/p values from ref. [ll] are substituted, together with the corresponding photoelectric cross-sections, in the GUPIX data base [6], agreement between our data and the recommended concentrations improves significantly. We do not have similarly extensive experimental p/p values for attenuation in nickel, and so the correction of the data base is necessarily incomplete. The point that we wish to demonstrate is simply PIXE’s susceptibility to error in attenuation coefficients at X-ray energies near absorption edges. This kind of error is likely to be encountered for other element pairs spaced by one or two Z-values. GUPIX has therefore been modified to provide a user-friendly means by which a user may substitute preferred experimental p/p values for those in the data base. In addition, these observations argue against the use of semiempirical formulae for attenuation coefficients, since these cannot accommodate the oscillatory behaviour (see fig. 1) near an absorption edge. 3.2. Light major element amlysis at low proton energy If the matrix is comprised mainly of low-Z elements, then the presence of any absorber between specimen and detector is precluded and the intense X-rays of the

Energy (keV) Fig. I. Difference A between measured and XCOM data base values of X-ray attenuation coefficient in iron at energies near the K absorption edge.

99

J.L. Campbell et al. / PIXE microanalysis of thick specimens

major elements will dominate the throughput of the electronic system to an extent that very few trace element X-rays will be recorded. Simultaneous determination of C,,, and Cz,t is therefore not possible. Further problems are the deleterious effects of scattered protons on the Si(Li) detector response, together with long-term damage to the crystal [12]. We therefore derive 0.75 MeV protons from a 1.5 MeV Hz beam, which is transported under the same beam optical conditions as the 3 MeV H+ beam used in subsequent trace element analysis. At 0.75 MeV the scattered protons do not penetrate the beryllium window of our detector. The resulting high counting rates necessitate increasing the specimen-detector distance and inserting a 0.3 mm collimator. GUPIX92 offers the facility of storing the major element concentration files that are obtained in a suite of 0.75 MeV measurements; these are recalled by GUPIX92 in order to generate matrix corrections when it is processing subsequent trace element data taken in the same sequence at 3 MeV proton energy. Since pure single-element standards are not available for all light elements, recourse may be necessary to compound standards. As an example, table 2 shows results for analyses of the two minerals sanidine (spectrum in fig. 2) and bustamite where the mineral albite was employed as a standard for the elements sodium, aluminum and silicon; single-element standards were used for higher-Z elements. The oxygen concentration was found by assigning the normal valence to each cation species and deducing the oxygen corresponding

Table 2 u-PIXE analyses of the minerals bustamite and sanidine an albite a standard; R = recommended, M = measured Element

Concentration

[%I

Bustamite R 0 Na Mg AI Si K Ca Mn Fe Zn Ba Total

38.39 0.04 0.13 0.02 22.46 0.05 13.56 18.83 6.32 0.20

Sanidine M 38.5 kO.4 0.11+0.1 0.14kO.03 0 22.3 +O.l 0 13.8 +0.3 19.6 +0.9 6.1 +0.9 0 0

100

using

100.6

R

M

46.28 2.23 9.93 30.23 10.05 0.14 _

45.4kO.3 2.2 * 0.15 0 9.710.12 29.7 i 0.2 10.3 + 0.2 0 0 0 0 1.0+0.2

0.98 99.85

98.3

a AIbite concentrations are: 48.76% 0; 8.60% Na; 10.34% Al; 32.03% Si; 0.18% K 0.09% Ca.

-~~,“‘;““~““;,“1 1

ENERGY

Fig. 2. PIXE

spectrum

of sanidine protons.

(KeV)

recorded

using 0.75 MeV

to the content of sodium, aluminum, etc. Not only is there good agreement between recommended and Imeasured concentrations for the various elements, including oxygen, but the total element content in each case is close to lOO%, which provides a check on consistency. The advent of ultrathin window (UTW) detectors permits recording of the oxygen lines and averts the need for an invisible element in the computation. The two prices paid for this complete coverage of the X-ray spectrum would be (i) a smaller detector (since only small-diameter windows are feasible) and (ii) entry of scattered protons. The first is of no matter for major elements since count rates remain adequate, but the decreased efficiency would be a detriment in trace analysis of the same specimen. Musket [12] has demonstrated removal of the scattered protons in conventional PIXE by interposing an arrangement of compact permanent magnets (see fig. 3) between specimen and detector. It remains to be determined if this solution is compatible with a microbeam. If it proved to be compatible, one could conduct both major and trace element analysis with 3 MeV protons, which would be much more convenient than the mode described here, which involves switching between two energies. III. ANALYTICAL

METHODOLOGY

100

J.L. Campbell et al. / PIXE microanalysis of thick specimens

3.3. IE for light elements

For elements of low atomic number the cancellation of intrinsic detector efficiency implicit in eq. (4) is very important. Various factors cause significant uncertainties in the intrinsic efficiency. The beryllium window thickness is unlikely to be known precisely; a 10% error at 8 pm thickness causes 8% error in the recorded intensity of sodium K X-rays. Any ice layer [13] will cause significant and time-dependent attenuation which is difficult to estimate, necessitating regular checks via for example the intensity ratio of copper L and K X-rays, and subsequent deicing, if necessary. Another and a more complex issue, is that of the widely misunderstood frontal silicon layer in which charge collection is incomplete. The thickness tICC of this layer is conventionally determined by regarding the ICC layer as an inert silicon absorber of photons, and measuring the ratio of intensities in the Gaussian (A,) and the low-energy tailing portion (A,) of a peak. For K X-rays of elements like iron and copper a significant portion of the overall tailing is due to the intrinsic Lorentzian X-ray line shape [14], and if this fact is ignored an overestimate of the ICC layer thickness will result. This necessitates using photons from a monochromator, as done by Krumrey et al. [15], who showed that, rather than being well-defined, the tICC value deduced in this manner was a linearly increasing function of photon energy. Subsequent to verifying this observation, we have developed [16] an extended model for the ICC layer. Every X-ray interaction in the layer is taken to result in a degraded event due to imperfect collection of the electron-hole pairs liberated; further degraded events arise when photoelectrons from interactions in the subsequent “perfect”

/Out

(to Si(Li)

)

I

I

2

4

I

I

I

I

8

10

12

(keV)

EN&

Fig. 4. Comparison of measured tail-to-Gaussian intensity ratios with the model of eq. (5) (upper curve), and a model based on electron escape only (lower curve).

region penetrate back into the ICC layer. In this model the tail-to-peak ratio is

4 -= AG

[l -

exp(-k%C)I

+J

exp(--CLtICd-J

’

where J=lj:_+“~O.5(1-~)Pe~(-~r)dr,

(6)

and R is the photoelectron range. This expression differs from the conventional one in the appearance of the term J; it provides an excellent fit to the photon energy dependence of AT/A, (fig. 4) and hence a unique t,,, value. If the ICC thickness value were determined by the conventional approach (J absent from eq. (5)) with monoenergetic 5.9 keV K, X-rays of manganese, then WI the rIcc value obtained would be a factor 1.7 times larger than the true t,,,; use of the erroneous value would then cause an error of some 6.4% in efficiency for the sulfur K X-ray line. If a radionuclide 55Fe source were used to provide the Mn K X-rays, the error would be even larger, due to the enhancement of the low energy tailing by the Lorentzian contribution. In sum, the rapidly changing efficiency below 5 keV, subject to large errors, is best cancelled from the analysis by using standards as per eq. (4). Where single-element standards do not exist, e.g. Na, S, P, there are convenient mineral standards containing these elements in the form of simple compounds. 3.4. Data base for light elements

Distance

(inches)

Fig. 3. Compact magnetic deflection unit for removal of scattered beam particles from the axis of a UTW Si(Li) detector [El.

The GUPIX data base employs the most sophisticated ECPSSR K ionization cross-sections available to date [17], together with Bambynek’s 1984 compilation

J.L. Campbell et al. / PIXE microanalysis of thick specimens

101

3.5. Detector line shapes for light elements

.

.

t R

1.06

.

. .

. 1.00 .

.

Fig. 5. Ratio R of reference X-ray production cross-sections [19] to those predicted by ECPSSR theory [17] and the K fluorescence yields of Bambynek [18].

of K fluorescence yields as reported by Hubbell [18]. Paul’s useful set of reference cross-sections [19] is based on critical assessment of all measured data, interpreted via Krause’s fluorescence yield compilation [20]. In fig. 5 we compare the X-ray production crosssections (~~0~ from GUPIX with the Paul/Krause values i.e. the best measured values. Below 2 = 20 there is a widening discrepancy which merits investigation. In the PIXE analysis context, comparison of specimen and standard for each element using eq. (5) will ensure significant (although not complete) cancellation of errors inherent in uK and wK. GUPIX does not deal with elements lighter than sodium. Recent very careful K X-ray production measurements for 6 I Z I 12 by Yu et al. 1211using 0.75-4.5 MeV protons suggest that the ECPSSR cross-sections somewhat underestimate measured values, the discrepancy widening as Z decreases; carbon, however, departs from this well-defined trend, an observation that may suggest problems with the fluorescence yields [20] used. These yields are of order 0.001 and directly measured values are presumably subject to large uncertainty. Willemsen and Kuiper [22] have reported measurements of the K X-ray yields of N, 0, and F; they compared these to predictions of the BEA theory using the same fluorescence yields, and found large discrepancies. When K-shell fluorescence yields deduced from experimental measurements of the complementary Auger yields were used, the agreement between measured and predicted X-ray intensities was much better, although there is still uncertainty due to possible effects of chemical bonding upon the wK values. All of this emphasises that it is better to use standards as opposed to relying on the data base, with the proviso that there could even be differences in fluorescence yield between specimen and standard.

Since low-energy tailing is maximum at X-ray energies just above the silicon K absorption edge energy of 1.74 keV and then drops by a factor of 10 for the K X-rays of Si, Al, etc., spectrum fitting has to contend with rather variable tailing contributions in light element work. Unless it is accounted for in systematic fashion in a spectrum-fitting code, this tailing can cause error in the intensities of weak peaks that are superimposed on the tails of intense peaks of slightly higher energy. A further problem noted by many authors is a time dependence of the tailing intensity and shape; we have observed this phenomenon and we have noted that in our case it correlated with the existence of an ice layer (observed via loss of efficiency for low energy X-rays). When the ice layer was removed using the heating device that is integral to Link Analytical Si(Li) detectors, the tailing became minimum and constant. This suggests that the variable tailing is connected to electric field distortion caused by the ice layer. In addition, preferential absorption of the background continuum in the silicon ICC layer produces a “notch” in the backgr?und that can detract from accuracy of Si concentrations in specimens; this phenomenon is well-known in EPMA. Musket [13] has drawn attention to a similar phenomenon in oxygen determination caused by differential absorption of the background continuum in the oxygen of the ice layer; X-rays of energy just above the K edge (0.532 keV) are strongly absorbed, while those having energies just below the edge are preferentially transmitted. The resulting “bump” in the background may be interpreted as an excess of oxygen in the specimen if the specimen is fitted by least-squares programs which assume monotonic background. This is only observed for windowless or ultrathin window detectors. While these detectors are advantageous in providing oxygen data, there are some risks as regards the accuracy of that data if the detector is not adequately deiced. The alternative solution for light major elements in minerals is then to retain Be-window detectors and to adopt the conventional EPMA procedure of complementing aach observed element (Na, Mg, Al, . . . ) with oxygen as determined by its oxide stoichiometry; this ensures proper matrix corrections. 3.6. Specimens of finite thickness: use of backscattered protons The above discussion is predicated upon specimens that are sufficiently thick to stop the proton beam or at least to reduce its energy to a level at which X-ray production is negligible; this is typically 50 pm in our geochernical work. However, if the beam penetrates

III. ANALYTICAL, METHODOLOGY

J.L. Campbell et al. / PIXE microanalysis of thick specimens

102

_” 4

ENERGY Fig. 6. PIXE spectrum

of a Cd,Hg,_,Te

film deposited

jc

400

303

200

0 ”

150

12

(keV) on an indium-tin-oxide

coated

jca

j”

glass substrate.

BS

:

,” 250 c 3

10

6

6

I

:_

.,

...

,..,

..‘,

‘.

.‘.

..

‘..

50

,_.

I

0 1500.3

L750.0

2000.0

2250.0

energy/

energy/ Fig. 7. Backscatter

keV

and PIXE spectra

..’

2500.0

keV

2750.0

r-

of a fly-ash particle

[23].

3ooo.a

J.L. Campbell et al. / PIXE microanalysis

the specimen, then either the thickness or the emergent energy is needed in order to compute ME. This situation is best avoided, but use of a substrate that emits an intense characteristic X-ray can provide a solution. As an example, we have had to measure the composition of multielement films deposited on glass substrates and having unknown thickness. Fortunately the high barium content of the glass results in the presence of barium L X-rays in the spectrum, as shown in fig. 6. The barium concentration is first determined by analysis of the glass alone. Then a multilayer version of GUPIX92 is used to process the spectrum from the compound specimen. Different estimates of the film thickness are made until one is obtained that reproduces the correct barium concentration in the substrate. The film element concentrations from that particular fit are then accepted. We develop this theme further in section 5. An alternative means of thickness measurement, which simultaneously provides major element concentration ratios is the spectroscopy of backscattered protons as demonstrated by Jaksic et al. [23]. While the scattering is not pure Rutherfordian, there is a sufficient data base of inelastic scattering cross-section to enable the creation of a successful BS spectrum simulation program; this then provides quantitative measures of thickness and of light element concentrations. The example we offer is that of a fly-ash particle shown in fig. 7.

4. Trace element analysis: practical aspects We turn now to the more frequent application of p-PIXE i.e. the determination of trace element concentrations in a known matrix. 4.1. Trace element standards and homogeneity The standardization approach embodied in eq. (4) now demands a standard matrix of some general similarity to the specimen, and containing each trace element of interest. These are various standard reference materials in solid form which contain a suitably large number of trace elements that one is able to interpolate with very little uncertainty for those missing. A degree of error will be incurred in fitting the complex PIXE spectrum with many trace element contributions. The analysis of Rogers et al. 151 of the standard reference material NIST 610, a glass containing various rare earths, is a good example. Some authors will prefer to adopt standards in which there are fewer elements, and in which the concentrations are at the minor as opposed to the trace level, and to seek to supplement these with a reliable means of interpolation in Z.

103

qf thick specimens

YI ,n loo

““I

10

““““‘I

15

20

15

20

“I

”

”

”

25

30

25

30

0

1

-10

10

ENERGY Fig. 8. Details of the fit to PIXE spectrum of pelletized CCU-1 reference material. The residuals are in units of one standard deviation.

While some SRMs may be used directly (e.g. the NIST glass and the NlST Fe/Ni alloy mentioned earlier), others exist in a form that is not immediately amenable to PIXE use. An example is the CANMET reference copper concentrate CCU-1, which is a powder mixture comprising 82% chalcopyrite, 9% pyrite, 9% sphalerite and a trace of pyrrhotite. Element concentrations are accurately known via an interlaboratory analysis [24], in which atomic absorption of digested samples was the main method. We took PIXE spectra from this material mixed with CFll cellulose (15%) and pressed under 25000 PSI to form pellets. To compensate for expected inhomogeneity at the micron level, the 10 ym wide beam was rastered over a 400 x 400 km area. The resulting PIXE spectrum, shown in fig. 8, shows the kind of problem that arises in such standards. The fit, as judged by the residuals, is bad in the pileup region of copper and zinc; as the microbeam scans, it encounters “clumps” of zinc (i.e. sphalerite), causing a momentarily high count rate with excessive pileup. The “pileup element” method [25] used to fit the spectrum assumes constant count rate conditions; it cannot cope with varying conditions. In a sequence of analyses of ten spots by this method, the error given by the fitting program for major element concentrations corresponded closely to the expected standard III. ANALYTICAL METHODOLOGY

J.L. Campbell et al. / PIXE microanalysis of thick specimens

104 Table 3 Variation in major element CCIJ-1 standard Element

concentrations

Concentration [%I

Standard deviation [%I Internal

30.80 24.71 3.22

Fe cu Zn

in pelletized

External 2.4 2.4 10.1

0.36 0.21 0.18

Table 4 X-ray intensity variation among ten replicate PIXE analyses of synthetic pyrrhotites (Fess) containing trace elements at 0.1% concentration Beam spot diameter [wnl Standard A: Fe,S (Cu, Pd) 5x5 cTh a, 50x50 uh

deviation from simple counting statistics. But the standard deviations for ten replicate measurements (see table 3) were considerably larger, and clearly suggested a much greater heterogeneity for zinc than for iron and copper. Recognizing the need to homogenize geological standard materials on the scale of the volume sampled by the beam, Ryan et al. [26] fused US Geological Survey rock standards to form glasses. In two cases (BCR-1 and AGV-l), EPMA analysis demonstrated homogeneity in major and minor elements, but some trace elements measured by p-PIXE at eight spots showed spreads in concentration more than twice the statistically expected range. The summed spectrum gave concentrations that agreed to 1% on average with recommended values, lending credence to the observa-

Standard B: Fe,S (Se, P:) 5X5 "h “e 50X.50 uh fie

Standard deviation a [%I Fe

Cu

1.82 0.29 0.43 0.28

2.25 1.84 1.68 1.82

Se

Pd 2.29 2.10 3.67 2.08

10.67 1.27 6.68 1.22

1.38 0.27 0.69 0.27

1.79 2.48 2.63 2.44

a (TV= observed standard deviation, a, = expected standard deviation ( = square root of counts).

tion of heterogeneity. The GSP-1 material was markedly heterogeneous due to the distribution of minute grains of zircon and quartz. For these reasons we have preferred to conduct our geochemical work with synthetic sulfide minerals prepared by CANMET [27]. But even these simple systems can be heterogeneous. Table 4 shows the varia-

(b) 105

ti g lo4

iI p: lo5 V E

IO0 t.

100 r 14

16

16

20

14

16

16

20

14

16

16

20

14

16

16

20

ENERGY ENERGY Fig. 9. Details of fits to a zirconium PIXE spectrum: (a) no scattering step is included in the line shape, and yttrium is included in the element list; (b) a scatter step is included.

105

J.L. Campbellet al. / PIXE microanalysisof thickspecimens

tion in X-ray intensity from ten separate beam spots on two synthetic pyrrhotites (Fe,S), one containing 0.1% each of Cu and Pd, and the other 0.1% each of Se and Pd. The observed standard deviation crh is compared with the value qe expected from conventional square root containing error. In each case the iron is inhomogeneous at the 5 p,rn scale and this persists even with a 50 pm spot. The trace elements Cu and Pd are homogeneous at both scales, but the Se is markedly inhomogeneous. This demonstrates the need to check even simple standards for inhomogeneity. The multielement standards prepared by Ryan et al. [26] have the attraction of providing standardization across a wide range of atomic numbers; it is straightforward to interpret the X-ray yields rS7.@) for absent elements, facilitating use of eq. (4). On the other hand standards such as our synthetic sulfides, with rather high concentrations of just one or two elements, provide spectra very quickly and fitting these should incur very little error; but it is then necessary to have a reliable interpolation scheme, an aspect we take up in section 4.4 below. 4.2. Instrumental effects Most trace element work involves higher-Z trace elements in a lower-2 matrix, although the archaeometry area provides notable exceptions where clever use has been made of compound critical filtering to improve the peg-to-background ratio of low energy lines in the presence of intense higher energy lines [28]. One is typically working with X-rays in the energy range 5-35 keV. The problem of intrinsic detector efficiency becomes much more tractable than in the low energy region (see section 3.3). The intrinsic efficiency is essentially constant up to 15 keV and its falloff beyond that is determined mainly by the silicon crystal thickness, which may be accurately determined with convenient radionuclide standard sources; depending on the effort devoted to this task, & can be known with some 1% standard deviation. Two further instrumental effects merit attention. Both relate to the detector line shape. At the low X-ray energies discussed in section 3.5, the departure of line shape from the hypothetical Gaussian form is a quasi-exponential low-energy tailing that arises from loss of charge near the detector’s front surface. The intrinsic effect decreases with increasing X-ray energy and becomes negligible at 15 keV. At around 15 keV a different form of tailing sets in, this being extrinsic in origin; it arises from Compton scattering of X-rays in the specimen and on nearby structures in the specimen chamber. In our experience these tails are quite well represented by truncated flat shelves, terminating at an energy equal to the energy loss incurred in 180” scattering, provided the detector is collimated and thus does

105

IO’

10*

10Z

10’

LO@L.

7

20

’ 25

,

I

30

ENERGY (kev)

Fig. 10. Fits to the PIXE spectrum of tin: (a) with no Lorentzian contribution; (b) with a Lorentzian contribution (r = 11.2 eV).

not “see” chamber structures responsible for multiple scattering. In fig. 9 we illustrate the need to describe this Compton tail in fitting a zirconium metal spectrum. If the tail is omitted and GUPIX is permitted to include yttrium in its element list, it generates an yttrium concentration around 450 ppm, which is spurious. This has necessitated careful attention to tail description in our work on colour-banded zircons, where yttrium may play a role in zone coloration. The second issue is that of the intrinsic Lorentzian shape of an X-ray line. In a special version of GUPIX we have introduced the option of convoluting the Gaussian plus tailing line shape with this Lorentzian distribution to generate the Voigt function that truly represents the line in question. It has been customary in Si(Li) spectroscopy to neglect this effect entirely on the grounds that ~rentzian widths are much smaller III. ANALYTICAL METHODOLOGY

106

J.L. Campbellet al. / PIXE microanalysisof thick specimens

than typical detector resolution. The fits to a tin K X-ray spectrum in fig. 10 show that this neglect is not entirely justified even though the Lorentzian width of 11.2 eV is significantly less than the full width at half maximum value of 300 eV. Fitting with Gaussian line shapes causes a 2% underestimate of the I& line intensity, as compared to using Voigtians. For zirconium K,,, where the Lorentzian width is smaller, the effect is 1%. These Lorentzian effects probably do not justify in routine analysis the computation time needed to properly represents the line shapes. But they should be borne in mind as a source of small discrepancies. 4.3. The H due

.---/ /

,-._

method

We and others have found the following method

rapid and convenient in performing large numbers of analyses for trace elements in known or predetermined matrices. Rather than standardizing with many elements and cancelling IE as in eq. (4) we determine the properties of any Si(Li) detector very carefully, so that the instrumental effects are accurately known, and then we rely on a standard containing only one or two trace elements. Returning to eq. (3) we first recognize that if secondary electron suppression is not perfect then Q may merely be proportional to collected charge rather than equal to it; alternatively Q may be measured indirectly e.g. by sampling beam particles scattered off a rotating vane: in each case some correction factor fQ should multiply the measured quantity Q so that Qf, is the actual charge in FC. We combine the unknown fp and the solid angle fl into an instrumental constant H that effectively characterizes the geometry and charge calibration of the PIXE system. Thus I(Z)

i

=~Q&,W,(-G

Cz,,).

(7)

A measurement of I(Z) for a standard containing concentration C, of a single element then provides a value for H, and this in principle standardizes the instrument for all elements. It is interesting to note that this is precisely the same as the approach taken by Fei and Van Espen [29] in their recent work on quantitative analysis by X-ray fluorescence. Fig. 11 shows H-values obtained in this way using pure single-element standards. Protons of 0.75 MeV energy were used for K X-rays of nickel and titanium and L X-rays of niobium, silver and tin, and 3 MeV protons were used for K X-rays of nickel and higher-Z elements. Particular attention was paid to determining the efficiency and line shape of the Link Si(Li) detector used [16]. The result indicates that the data base and the IE determination are reliable enough that a single element is indeed capable of providing the parameter H. A similar demonstration of the constancy

Fig. 11. H values measured for a well-characterized Si(Li) detector. The inverted triangles represent L X-ray data from Nb, Ag, Sn using 0.75 MeV protons. The filled circles represent K X-ray data taken with proton energies of 0.75 MeV (Ti, Ni) and 3 MeV (Ni and higher-Z elements).

of the instrumental constant in the X-ray fluorescence analysis case can be found in ref. 1291. There is in fig. 11 an indication of a slight decrease in H for the K X-rays of elements of Z > 40. This is pursued in fig. 12, where K X-ray data were taken for a wide range of single-element standards using 3 MeV protons and a different detector, whose 5 mm nominal thickness should ensure constant efficiency over a wide range of X-ray energies. These results demonstrate the hazards of inadequate attention to IE. In fitting the K X-ray spectra with the GUPIX code, Compton tails were neglected for the elements in the atomic number region 40 to 52, as were the effects of the finite Lorentzian linewidths. The data in the lower portion of the figure have had these effects incorporated, and only now does H become a constant, as it should be. Obviously the simplicity of the H value method can result in error if exploited naively. We observed that, at 3 MeV, L, X-rays give a slightly but significantly different H value from that of K, X-rays, which presumably reflects errors in the ionization cross-section data base. We handle this in GUPIX by the empirical manoeuvre of defining separate parameters H, and H,. Finally in fig. 12 we observe that the falloff in the H, value at low 2 arises because we have not yet characterized this particular detector’s line shape at low photon energies. We handle this in GUPIX by allowing both H, and H, to be functions of X-ray energy; they are measured for a few elements, stored in files, and GUPIX interpolates for other elements as needed. The above has shorn us that for K X-rays, H is indeed a well-defined quantity for a particular set of targets, namely pure metal standards. With a much more limited set of synthetic sulfide minerals contain-

107

J.L. Campbell et al. / PIXE microanalysis of thick specimens

Table 5 H values measured a using metallic iron and iron sulfide

ing different trace elements we have also demonstrated an acceptably well-defined H [27]. Our argument that standards need not closely mimic specimens requires us to demonstrate that the same H value is obtained for very different materials. Were we to find different H values for metals and nonconducting targets, this would suggest inadequate suppression of secondary electrons, and our arguments would fail on grounds of imperfect charge measurement. To explore this we measured H for metallic iron and for two carboncoated pyrrhotite (Fe,S) specimens using 3 MeV protons, with a 125 ym mylar X-ray absorber. As shown in table 5, the H values were very consistent. This observation emphasises the university of the method. In sum then, H, should be a constant for a wellcharacterized system. Data base errors however demand separate H, and HL values. If the detector characterization is somewhat superficial, IE errors can be compensated for by permitting HK and H, to be X-ray energy dependent. Taken to its extreme of course, this argument would return us to the multielement calibration implicit in eq. (4); but in practice only

Hs

t 8%

.

z 1.4T+ x

.

l.*

. . .

10

Energy

z :

," 1.4 x

I

1.0

e.*.., l '

.

. .

0. .

.

.-

25

(keV)

. '.

..

. .

l*

.

. 1

~~~~‘~~~~I~~~,/~‘~‘I~~‘~I~~ 0

5

10

15

20

0.0404+ 0.0008

5. Multilayer specimens

.

20

15

Iron sulfide B

0.0406+ 0.0002 0.0403* + 0.0004 - 0.0007 0.0395k 0.0008

a few standard elements suffice to define the slight variation in H that results from an incomplete knowledge of instrumental effects.

J l *

0.0410+ 0.0005

Hi%

a Each value is the mean of five measurements, with the error representing the range of values.

1.6 3 0

Iron Iron sulfide A

25

Fig. 12. H values for K, X-rays, measured using 3 MeV protons. The upper data result from spectrum fits which neglected both low-energy line shape distortion by Compton scattering and the intrinsic Lorentiian linewidth. These effects were included in computing the lower data set.

With the standardization of homogeneous specimens now rather well developed, there has recently been increasing emphasis on more complex specimen types and especially on those comprising several distinct layers each of which individually is homogeneous. As a result k-PIXE has begun to supplement and extend Rutherford backscattering of He ions, previously regarded as the ion beam technique of choice for multilayer specimens. Rickards and Zironi [30] presented a detailed analytic treatment of X-ray generation in a target comprising up to three layers each composed of one or ,more defined elements. Given the layer thicknesses, which are crucial in determining both the slowing down of the proton and the attenuation of X-rays, they could compute the intensity ratios of K, X-rays from elements in different layers. This necessitated computing secondary fluorescence of an element in one layer by X-rays emanating from a different layer, which was effected by an extension of the classical secondary fluorescence formalism of Reuter et al. 1311.An application that illustrated p-PIXE’s potential was determination of the thickness of a titanium film (0.02-2 rJ,m) deposited on a steel substrate. Fig. 13 demonstrates that at appropriate proton energies of a few hundred keV, a comparison of measured and predicted K, X-ray intensity ratios provides the titanium thickness with uncertainty of order 10%. The sensitivity of this method derives from the strong influence of the film thickness upon the substrate X-ray intensity. Demortier [32] has also demonstrated this technique, his examples including films of gold on copper, zinc on iron, and nickel on copper. In the Zn/Fe case, for example a layer thickness range of 2-35 pm corresponds to more than a hundredfold variation in the intensity ratio of Fe K, to Zn K, when 3 MeV protons are used, providing a sensitive thickIII. ANALYTICAL METHODOLOGY

J.L. Campbell et al. / PIXE microanalysb of thick specimens

108

keV

,011 ”

I

0.1

01

I

I

+

1.0

t,

(mg /cm2) Fig. 13. Calculated ratio Ti/Fe of K, X-ray intensities for a Ti layer of thickness t1 on a steel substrate [30]. Proton energies between 400 and 700 keV were used. The circles and crosses are measured points for two samples (the dashed lines represent a different choice of ionization cross-sections in the data base).

ness measurement; RBS using the same protons can only measure up to 15 Km. The next degree of complexity is the elemental analysis of a buried layer. One example is the treatment of fluid inclusions in materials of Ryan et al. [33]. Here the major and trace element concentrations of the host matrix are determined by conventional PIXE. The depth and thickness of a buried layer i.e. a fluid inclusion are then determined by optical methods. When the beam is directed onto the inclusion, one has an unknown layer at reasonably well-defined position within a well-characterized host. Ryan et al. [33] used ratio from aqueous inclusions the chlorine K,/K, containing 30% NaCl as a check on the optically determined depth. This method provides an accurate trace element analysis of the buried layer. The variety of multilayer problems precludes writing a computer code that will deal with every case. But if the code user is willing to do some repetitive work, he can exploit p,-PIXE’s multilayer potential. In extending GUPIX92 to cope with multiple layers it was first necessary to modify our secondary fluorescence formalism from the thick target case to the case of a single target of “intermediate thickness”. Our published treatment [8] built upon the classical treatment of Reuter et al. [31], avoided the hitherto used approximations regarding jump ratios at absorption edges, and employed a modern data base. It gave good

agreement with recommended concentrations when used to analyze standard reference materials, and slight differences in predictions relative to other treatments supported our use of exact photoelectric cross-sections as opposed to jump ratios. The modification used for a single “intermediate thickness” target was tested on 64% Fe/36% Ni foils of 10 km thickness. As table 1 shows, the results deviated slightly from expectation until measured p/p values for NiKa in iron were substituted into the GUPIX data base; then, as in the thick target case, rather better agreement resulted. The program was then extended to deal with secondary fluorescence among multiple layers. The resulting code incorporated in GUPIX92 then provides (a> computation of X-ray yields from a multilayer structure; by comparison of measured intensity ratios with these, layer thicknesses may be deduced; (b) element concentration determination in one or more layers which are distinct in their elements; this case involves a least-squares fit of the spectrum in which the multilayer ME are the determinants of the intensity ratios of lines for any given trace element. However, as suggested above, intelligent use of the code can render tractable problems that are too complex to treat in a single run of the code. As an example, we are currently analyzing nonstoichiometric films of cadmium-mercury-telluride, thickness unknown, deposited on a glass substrate, with an intervening layer of nonstoichiometric indium-tinoxide whose thickness is unknown [34]. We take kPIXE spectra of the L X-ray region using three targets viz. glass, glass + ITO, glass + IT0 + CMT, a typical spectrum of the complete structure, which is a solar cell, is in fig. 6; the corresponding H values are measured using single-element standards. As indicated earlier, the barium content of the glass provides a means of determining the successive IT0 and CMT thicknesses. The ITO/glass spectrum is fitted using different IT0 thicknesses until the correct Ba concentration is produced; this provides the IT0 concentrations. Then the process is repeated for the CMT/ ITO/glass spectrum to determine the CMT thickness and concentrations. These various examples suggest that we can expect to see an increasing number of specimens. of complex structure being analyzed quantitative by CL-PIXE,which will also provide a check on homogeneity.

Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada. We thank Dr. Louis Cabri for the CANMET synthetic sulfide standards and Dr. Gerald Czamanske for the CCU-1 standards.

J.L. Campbell et al. / PIXE microanalysis of thick specimens

References [I] D.E. Newbury, J. Trace and Microprobe Tech. 32 (1986) 103. [2] M. Hyvonen-Dabek, J. Raisanen and J.T. Dabek, J. Radioanal. Chem. 63 (1981) 163. [3] E. Clayton, Nucl. Instr. and Meth. 191 (19811567. [4] J.L. Campbell, W.J. Teesdale and J.-X. Wang, Nucl. Instr. and Meth. B50 (1990) 189. [5] P.S.Z. Rogers, C.J. Duffy and T.M. Benjamin, Nucl. Instr. and Meth. B22 (1987) 133. [6] J.A. Maxwell, J.L. Campbell and W.J. Teesdale, Nucl. Instr. and Meth. B43 (1988) 218. [7] G. Lagarde, J. Cailleret and C. Heitz, Nucl. Instr. and Meth. 205 (1983) 545. [8] J.L. Campbell, J.-X. Wang, J.A. Maxwell and W.J. Teesdale, Nucl. Instr. and Meth. B43 (1989) 539. [9] G.K. Czamanske, V.E. Kunilov, L.J. Cabri, M.L. Zientek, L.C. Calk and A.P.Likhachev, Can. Mineralogist 30 (1992) 249. [lo] M.J. Berger and J.H. Hubbell, National Bureau of Standards Rep. 87-3597 (NIST, Gaithersburg, 1987). [ll] N.K. Del Grande and A.J. Oliver, Bull. Am. Phys. Sot. 16 (1971) 545. [12] R.G. Musket, Nucl. Instr. and Meth. B15 (1986) 735. [13] R.G. Musket, National Bureau of Standards Spec. Pub. 604 (NIST, Gaithersburg, 1981). [14] J.L. Campbell and J.-X. Wang, X-Ray Spectrom. 21 (1992) 223. [U] M. Krumrey, E. Tegeler and G. Ulm, Rev. Sci. Instr. 60 (1989) 2287. [16] J.L. Campbell and J.-X. Wang, X-Ray Spectrom. 20 (1991) 191. [17] M.H. Chen and B. Crasemann, At. Data Nucl. Data Tables 33 (1985) 217; M.H. Chen, private communication. [18] J.H. Hubbell, National Institute of Standards and Technology Rep. 89-4144 (NIST, Gaithersburg, 1989).

109

[19] H. Paul and J. Sacher, At. Data and Nucl. Data Tables 42 (1989) 105. [20] M.O. Krause, J. Phys. Chem. Ref. Data 8 (1979) 307. [21] Y.C. Yu, M.R. McNeir, D.L. Weathers, J.L. Duggan, F.D. McDaniel and G. Lapicki, Phys. Rev. A9 (1991) 5702. [22] M.F.C. Willemsen and A.E.T. Kuiper, Nucl. Instr. and Meth. B61 (1991) 213. [23] M. Jaksic, G. Grime, J. Henderson and F. Watt, Nucl. Instr. and Meth. B54 (199i) 491. [24] G.H. Faye, W.S. Bowman and R. Sutarno, CANMET Rep. 79-16, Copper concentrate CCU-1, a certified reference material (Dept. of Energy, Mines and Resources, Ottawa, Canada). [25] G.I. Johansson, X-Ray Spectrom. 11 (1982) 194. [26] C.G. Ryan, D.R. Cousens, S.H. Sie, W.L. Griffin, G.F. Suter and E. Clayton, Nucl. Instr. and Meth. B47 (1990) 55. [27] L.J. Cabri, J.L. Campbell, J.H.G. Laflamme, R.G. Leigh, J.A. Maxwell and J.D. Scott, Can. Mineralogist 23 (1985) 133. [28] C.P. Swann and S.J. Fleming, Nucl. Instr. and Meth. B49 (1990) 65. [29] F. Hei and P. Van Espen, Anal. Chem. 63 (1991) 2237. [30] J. Rickards and E.P. Zironi, Nucl. Instr. and Meth. B29 (1987) 527. 1311W. Reuter, A. Lurio, F. Cardone and J. Ziegler, J. Appl. Phys. 46 (1975) 3194. [32] G. Demortier, S. Mathot and B. Van Gystaeyen, Nucl. Instr. and Meth. B49 (1990) 46. [33] C.G. Ryan, D.R. Cousens, C.A. Heinrich, W.L. Griffin, S.H. Sie and T.P. Mernagh, Nucl. Instr. and Meth. B54 (1991) 292. [34] J.L. Campbell, J.A. Maxwell and W.J. Teesdale, to be published.

III. ANALYTICAL METHODOLOGY