Quantitative Trace Analysis by Wavelength-Dispersive EPMA

Stephen J. B. Reed

Department of Earth Sciences, University of Cambridge, Cambridge CB2 3EQ, England

Abstract. `̀ Trace'' elements may be de®ned as ele-

ments whose concentrations are of a similar order to

the detection limit. In WD analysis the detection limit

is a function of the `®gure of merit' P2/B, where P is

the pure-element peak intensity and B the background

intensity. With normal analysis conditions detection

limits of �100 ppm are typical, but substantial im-

provements can be achieved by using higher values

of accelerating voltage and beam current. Long count-

ing times are also advantageous, but should preferably

be divided into relatively short alternating peak and

background measurements to minimise the effect of

instrumental drift. Using separate routines for trace and

major element analysis is desirable owing to their

different requirements. As the statistically de®ned

detection limit is reduced, errors due to background

nonlinearity and interferences (overlaps) from other

elemental peaks become more probable. Spectrum

simulation is useful for optimising background offsets

and choice of crystal to minimise interferences, and

estimating interference corrections when these are

necessary. `Blank' standards containing none of the

trace elements of interest are also useful for quantify-

ing background nonlinearity.

Key words: Trace analysis; WDS; EPMA.

In EPMA the presence of the X-ray continuum limits

the detectability of small peaks as de®ned by

statistical criteria. Other microprobe techniques which

are comparatively free of background have substan-

tially lower detection limits, but EPMA is more

widely available and has better spatial resolution. For

`trace element' EPMA, ED spectrometers are handi-

capped by their relatively poor resolution, which not

only is insuf®cient for relevant lines in the spectrum to

be resolved in some cases, but also results in low

peak-to-background ratios, owing to the greater width

of the band of continuum recorded with the peak. The

limited pulse throughput rate (which applies to the

whole spectrum) is a further drawback. As well as

their better resolution, WD spectrometers have the

additional advantage that they record only one

wavelength at a time and thus are not swamped by

major-element peaks when a high beam current is

used to maximise the intensity of trace-element peaks.

Microcalorimeter spectrometers [1] have resolution

comparable to WDS but are even more severely

limited in throughput rate than conventional EDS and

are thus uncompetitive for trace analysis at present.

The following discussion therefore refers solely to

WD analysis.

The minimum measurable peak size (and hence the

elemental detection limit) is ultimately governed by

counting statistics. By using appropriate conditions,

such as high beam current and long counting times,

peaks representing concentrations of the order of

10 ppm are detectable in favourable cases (higher

®gures apply for L and M lines and samples of high

mean atomic number). However, such peaks typically

correspond to as little as �1% of background, and

careful attention to possible errors in the background

correction is essential. Linear interpolation from

measurements made with the spectrometer offset on

each side of the peak may sometimes be subject to

errors caused by curvature of the continuum and

discontinuities related to absorption edges. Also,

when the presence of a neighbouring peak rules out

background measurement on one side, and only the

background on the other side is used for the

correction, errors can arise if no allowance is made

for slope. In trace analysis it is, in addition,

Mikrochim. Acta 132, 145±151 (2000)

particularly important to beware of inter-elemental

interferences due to overlapping lines, which can

cause serious errors. These may be avoidable by

choosing a spectrometer crystal giving higher resolu-

tion or an alternative X-ray line, but some loss of trace

element detectability is entailed owing to the reduced

intensity. Interferences, if present, must be taken into

account by means of a correction re¯ecting the degree

of overlap and the concentration of the interfering

element. In such cases, trace-element analysis cap-

ability is impaired by increased errors from counting

statistics and uncertainties in the correction.

Counting Statistics

The statistics of counting random events is covered in

standard textbooks and for present purposes it is only

necessary to quote the following conclusions, which

refer to the measurement of a small peak super-

imposed on a relatively large background.

1. The detection limit, de®ned as the minimum

concentration required for the peak to be just

detectable at a speci®ed con®dence level for a

given counting time, is proportional to (P2/B)ÿ0.5,

where P is the number of counts recorded for the

pure-element peak and B the number of back-

ground counts for the specimen.

2. The time required to achieve a speci®ed detection

limit is inversely proportional to the `®gure of

merit', P2/B.

3. The maximum precision in the determination of

the background-corrected peak intensity with a

®xed total counting time is obtained with equal

times for peak and background.

Other Random Fluctuations

Fluctuations from other sources can be detected by

comparing the scatter in a series of repeated

measurements with that predicted from counting

statistics (standard deviation� square root of number

of counts). In practice the difference is usually

insigni®cant. As a precautionary measure, peak and

background measurements in trace analysis may be

subdivided into a series of relatively short counting

times (which is also advantageous in counteracting

drift, as discussed below) and the results analysed

statistically in the course of the measuring routine.

This allows outliers to be rejected; also counting

may be continued until a speci®ed precision is

achieved [2].

Detection Limit

Various de®nitions of detection limit are used, of

which the simplest is `3 standard deviations of

background' (3sdbg), obtained by dividing 3 times

the square root of the background count by the peak

count for the pure element (corrected for absorption in

the sample). The fraction of repeated determinations

of the background count lying within this limit is

99.6% according to classical probability theory.

However, the analytical procedure entails measuring

the number of counts on the peak as well as

background: assuming equal counting times the

fraction of repeated determinations of peak minus

background falling below the detection limit as

de®ned above is 96.6%. The 3sdbg detection limit is

thus the concentration for which the presence of the

element can be af®rmed with 96.6% con®dence.

Adapting to an alternative con®dence level merely

requires the substitution of a different standard

deviation multiplier. Various more elaborate expres-

sions exist, but there is little bene®t in attempting to

quantify the detection limit with high precision. In the

present context its main function is to compare

different analytical conditions.

Detection limits show an upward trend as the

atomic number of the element concerned increases,

owing to the lower intensity of the L� and M� lines

used for heavy elements compared to the K� line used

for lower atomic numbers. The mean atomic number

of the sample is also important, because of its effect

on the continuum intensity. Detection limits approach-

ing 1 ppm in favourable cases can be obtained by

using extreme conditions (high accelerating voltage

and beam current, and long counting times) [3].

However, for such results to be meaningful, close

attention must be paid to possible errors caused, for

example, by nonlinear background, or inter-element

interferences.

Optimum Conditions for Trace Analysis

As stated above, trace element analysis bene®ts equally

from maximisation of peak intensity and peak-to-

background ratio, both of which increase with increas-

ing accelerating voltage (V0), though in cases of high

absorption the peak intensity passes through a

146 S. J. B. Reed

maximum and then decreases owing to the increasing

depth of X-ray production (Fig. 1). However, electron

penetration increases approximately as V5=30 , with a

consequent worsening of spatial resolution (see Fig. 1).

Counting Strategy

Optimum choice of counting times is important for

trace element analysis. Prior knowledge of peak and

background intensities enables detection limits to be

predicted and counting times chosen to achieve the

desired result. For this purpose stored or calculated

intensity data may be used, or alternatively a self-

adaptive procedure can be applied, whereby counting

times are adjusted in the light of relatively rapid

preliminary peak and background measurements [2].

To minimise errors due to beam current drift,

contamination build-up, and specimen damage, it is

desirable to divide the long counting times required

for trace analysis into a series of alternating peak and

background measurements. Where the operating soft-

ware has no provision for this, a similar result can be

obtained by repeating the normal analysis routine

using relatively short counting times.

Specimen Damage

For some types of sample the allowable current is

limited by specimen damage. Assuming this is

determined by the energy deposited in the sample

per unit volume (rather than by the energy of the

individual electrons), a higher current can be used

with high V0 because the energy is deposited in a

larger volume. The case for using a high V0 as stated

above is thus enhanced for this type of sample.

Other preventive measures commonly used, namely

enlarging the beam, scanning the beam in a raster, or

moving the specimen during the analysis, all incur a

sacri®ce of spatial resolution. Beam-induced heating

of poor heat-conductors can be reduced substantially

by coating with a material of high thermal conductiv-

ity [4] (the choice between suitable metals such as

aluminium, copper, silver or gold is in¯uenced by the

need to avoid interference with the X-ray spectrum of

the sample). By comparison with the thin carbon coat

normally used for insulating samples (which has little

protective effect), such relatively thick metallic coat-

ings reduce the X-ray intensities signi®cantly, and this

must be taken into account by using similarly coated

standards or applying a correction factor.

Pulse-Height Analysis

Pulse-height analysis not only minimises the con-

tribution of high-order re¯ections of characteristic

X-ray lines where these coincide with the analytical

(®rst-order) line, but may also reduce background by

excluding the contribution of high-order continuum

X-rays (especially for long wavelengths, where dif-

ferential absorption by the counter window enhances

the relative intensity of the high-order continuum).

Background produced by scattered X-rays and

electrons (especially at low Bragg angles) can also

be reduced by suitable threshold setting. When pulse-

height analysis is used, it is important to avoid

excessive count-rates (e.g.,� 104 counts/s), where

pulse shrinkage and broadening of the pulse-height

distribution may occur. These effects can be mini-

mised by using a reduced counter anode voltage

(compensated by increased ampli®er gain).

Choice of Line and Crystal

For a given element there is sometimes a choice of

analytical line: e.g., for atomic numbers 26 to 35 the

K� and L� lines are both within the range of

spectrometers equipped with the usual crystals (like-

wise the L� and M� lines for Z� 70±92). Further-

more, there is a choice of crystal when the wavelength

concerned falls within the region of overlap between

crystals. For trace analysis the optimum line/crystal

Fig. 1. Calculated relative spatial resolution and detection limit asfunction of accelerating voltage; detection limit shown with andwithout absorption in the sample (assuming arbitrary values ofmass absorption coef®cients for illustrative purposes)

Quantitative Trace Analysis by Wavelength-Dispersive EPMA 147

combination is that which gives the maximum P2/B.

However, the presence of spectral interferences

favours the crystal with smaller interplanar spacing

(and therefore higher Bragg angle for a given

wavelength, and better resolution). Using a narrow

spectrometer slit increases P/B but this is outweighed

by the decrease in P and is therefore of little bene®t,

except occasionally for resolving interferences.

Background Corrections

X-Ray Continuum

Background is attributable almost entirely to the

continuous X-ray spectrum (`continuum') generated

by the bombarding electrons. The continuum intensity

I(�) (in photons per second per unit wavelength

interval) is given by the Kramers expression:

I(�) = const. Z��ÿ10 ÿ �ÿ1��ÿ1, in which �0 is the

Duane-Hunt limit as determined by the incident

electron energy. The intensity rises to a maximum at

2�0 and decreases monotonically for � > 2�0 (the

usual region of interest). The observed background

consists of the continuum integrated over the width,

��, of the spectrometer response function. This is

equal to the peak width for an effectively monochro-

matic X-ray line, and decreases with increasing Bragg

angle, enhancing the effect of the downward trend in

I(�) [5]. Other relevant factors are crystal re¯ection

ef®ciency, window absorption, and counter detection

ef®ciency, but these affect background and peak

intensities equally.

Linear Background Interpolation

The normal method for estimating background is to

measure the intensity with the spectrometer offset on

each side of the peak and interpolate linearly to the

peak position. The offset required depends on the

peak width (which increases at the low end of the

Bragg angle range) and the presence of adjacent lines.

Using large offsets may risk errors due to background

nonlinearity, but it is not necessary to move so far

from the peak as to completely avoid the tails: a

moderate contribution from these is acceptable since

it has no effect on the ratio of the sample and standard

background intensities, provided the same offsets are

used for both and there is no difference in the shape of

the peak pro®le [6].

The presence of neighbouring lines on one side of

the peak sometimes requires background to be

measured only on the other side (assuming this is

interference-free). The factor required to allow for the

slope of the continuum can be derived from a `blank'

standard containing none of the trace elements

concerned. This need not necessarily be closely

similar with respect to major elements, since con-

tinuum shape is independent of composition, but must

not have any absorption edges or interfering lines in

the regions of interest.

Background Curvature

In trace-element analysis background nonlinearity

requires consideration, because the amount which is

acceptable (relative to other errors) is as little as 1%.

Curvature is generally small over the relevant

wavelength interval, but the correction required is

not necessarily negligible. The effect depends on the

size of the offsets used and, in terms of concentration,

is a function of the pure-element peak intensity

compared to the continuum. It is thus greatest for a

sample of high mean atomic number and for a

relatively weak X-ray line (see Fig. 2). Corrections

can be derived from a blank standard satisfying the

criteria described in the previous section. If required,

the local shape of the continuum may be represented

by an equation such as a parabola, the coef®cients of

which can be determined experimentally [7].

Fig. 2. Background error expressed as concentration (ppm) causedby continuum curvature (LiF crystal, accelerating voltage ± 25 kV,sample atomic number ± 26) plotted against sine of Bragg angle.The effect of curvature is greater for L� than for K� lines owing tothe lower intensity of the former

148 S. J. B. Reed

Continuum Modelling

Predicting background intensities by calculating the

continuum intensity offers the attractive possibility of

saving counting time and avoiding interpolation and

interference problems associated with spectrometer

offset measurements. The generated continuum inten-

sity can be derived from Kramers' expression, but the

observed intensity also depends on the spectrometer

ef®ciency. Since it is impracticable to derive this

from ®rst principles, an empirical approach must be

adopted. For example, intensities measured at the

wavelengths of relevant lines using a blank standard

containing none of the trace elements of interest can

be scaled to allow for the atomic number difference.

The uncertainty involved in this scaling should

preferably be minimised by using a standard similar

in mean atomic number to the analysed sample. A

somewhat more complex function than the simple

linear Z-dependence in the Kramers expression is

required (except for small differences in Z): this can be

derived from experimental measurements on samples

of different mean atomic number [2, 8, 9, 10]. Correc-

tions for absorption in the specimen should be applied.

Limits to the accuracy obtainable by this method may

inhibit its use for extreme trace-element analysis.

Background Discontinuities

The assumption that background varies smoothly and

monotonically with wavelength is mostly valid, but

discontinuities due to absorption edges need to be taken

into consideration. For example, the Ar K absorption

edge at 0.3871 nm causes an abrupt change in the

ef®ciency of an argon-®lled proportional counter. Also,

absorption edges of major elements in the sample cause

steps owing to the change in the absorption of the

emerging continuum X-rays. Background offsets should

be chosen so that no such edges lie between the peak

and the positions used for measuring background.

A `hole' in the background for LiF occurs at

0.1271 nm, close to the Au L� line, where about 10%

of the intensity is lost owing to re¯ection by the

`wrong' planes in the crystal [11]. Allowance for the

hole must be made if this line is used, or alternatively

the M� line may be used instead.

Interferences

The high resolution of the WD spectrometer ensures

that characteristic X-ray lines are well resolved in

most cases. However, interfering peaks may cause

either over- or under-estimation of element concen-

trations, depending on whether peak or background is

affected most. It is common practice to adopt a `trial

and error' approach in which normal analytical

conditions are used initially and are modi®ed if the

presence of interferences is indicated, for example, by

asymmetry in the background intensity recorded

above and below the peak, or a negative concentra-

tion. It is preferable, however, to identify possible

interferences in advance, for example, by spectrum

simulation (see below).

Interferences in complex spectra can be minimised

by optimal selection of analytical line, spectrometer

crystal and background offsets, but where unavoidable

their effect needs to be corrected by deducting an

appropriate amount from the measured peak and/or

background intensities. To a fair approximation the

overlap correction can be equated to the apparent

concentration of the overlapped element in a standard

containing the overlapping element, scaled in propor-

tion to the concentration of the overlapping element in

the sample. However, it is desirable to take account of

the matrix-dependence of the relative line intensities

involved by applying `ZAF' factors [12]. Trace-

element analysis is inevitably affected adversely by

interferences, on grounds of both statistical precision

and uncertainty in the corrections.

WDS Simulation

The idea of spectrum simulation originated with the

NIST `Desk-Top Spectrum Analyzer' for ED spectra.

Modelling WD spectrometer characteristics is harder,

hence experimentally recorded spectra are used in the

`Virtual WDS' program [13]. This contains a database

of experimentally recorded spectra, which can be scaled

according to accelerating voltage and beam current, and

combined in proportion to assumed element concentra-

tions (using estimated values for `unknown' samples),

thereby obviating the need for time-consuming `real'

wavelength scans. A case where this approach is

valuable is rare-earth element determination in minerals

[14] (see Fig. 3). Another useful feature is the ability to

predict precision and detection limits.

Trace Element Quanti®cation

The principles of quantitative analysis are the same

for trace and major elements: the concentration of

Quantitative Trace Analysis by Wavelength-Dispersive EPMA 149

each element is derived from the ratio of the

background-corrected peak intensity for the sample

to that obtained from a reference standard (either a

pure element or a compound of known composition

containing the element concerned). There is little

bene®t in using as the standard for a trace element a

material in which the relevant element is present in a

similarly low concentration: indeed, this has the

disadvantage that the standard measurement is then

subject to the inaccuracies associated with measuring

small peaks. Also, homogeneous trace-element stan-

dards are hard to ®nd. It is therefore reasonable to use

the same standards for trace elements as for major

elements, though the beam current must be relatively

low to avoid excessive count-rates.

A secondary consideration is that uncertainty in the

matrix corrections is reduced if the standard resem-

bles the sample in composition. However, tests of

matrix correction procedures for major elements show

that errors are generally less than � 5% (relative),

which is usually smaller than the statistical errors for

trace elements; compositional similarity of standards

is thus less important than for major elements.

Combining Major and Trace Element Analysis

Optimum conditions for trace analysis tend to give

excessive count-rates for major elements, as men-

tioned above. This can be overcome by conducting

major-element analysis separately using `normal'

conditions and then combining the results with

trace-element data obtained with higher accelerating

voltage and beam current. However, electron penetra-

tion increases with the accelerating voltage, hence

trace and major element data will refer to different

volumes within the sample. Also, EPMA software is

commonly not adaptable to this strategy, necessitating

the use of a separate trace analysis program [15]. This

approach is facilitated by the fact that trace elements

need not be included in the iterative correction loop,

since they have a negligible effect on the correction

factors.

Separate routines may be avoidable by selecting

X-ray lines for major elements which are less intense

than the normal choice: for example, K� instead of

K�. (The spectrometer crystal can also be selected

with this in view in some cases). An alternative

possibility is to use an ED spectrometer with a small

aperture (the concentration of high X-ray intensity on

a limited area of the detector does not seem to have

any ill effects).

Errors in Trace Analysis

The precision of trace element analysis is ultimately

limited by counting statistics, but can be worsened by

¯uctuations caused, for example, by instrument

instability (though this is usually small). The most

important potential systematic errors are connected

with background subtraction and inter-element inter-

ferences. The effect of uncorrected continuum curva-

ture is generally small. Larger errors (either positive

or negative) can be caused by interferences. Avoid-

ance by using an alternative analytical line usually

entails poorer precision and detection limit, on

account of the lower peak intensity and peak-to-

background ratio. Even if an interference correction is

applied, there are residual errors due to uncertainties

in the correction, and with large interferences trace

analysis is effectively impossible.

A detected peak may be real, while not truly

re¯ecting the presence of that element in the sample.

For example, scattered electrons may excite X-rays

from surrounding objects, or nearby parts of the

specimen itself, though fortunately the focussing

properties of WD spectrometers help to minimise

such extraneous contributions. Surface contamination

from polishing materials etc. is another potential

cause of misleading results, but can be avoided by

suitable choice of materials and thorough specimen

cleaning. Fluorescence in a neighbouring phase

excited by primary X-rays from the analysed spot

can give rise to a spurious result where there is a high

Fig. 3. Spectra of Eu and other rare-earth elements (LiF crystal)produced by `Virtual WDS' program, illustrating interferencesaffecting both Eu L� peak and potential background offsetmeasurement positions

150 S. J. B. Reed

concentration of the element concerned in the

adjacent phase. An example is �500 ppm Ca observed

in a Ca-free silicate mineral adjacent to calcite (40%

Ca) at a distance of 20mm from the interface [16].

There is no way of preventing this, but corrections can

be estimated. Alternatively, in some cases it may be

possible to extract the grains of interest and analyse

them in isolation.

Conclusions

To exploit the trace element capabilities of EPMA

fully it is necessary to use conditions that maximise

the statistical precision with which a small peak can

be distinguished from background. The high accel-

erating voltage and beam current desirable on these

grounds are, however, unsuitable for major element

analysis on account of excessive count-rates, large

ZAF corrections, poor spatial resolution, and speci-

men damage. Separate routines for major and trace

elements are therefore preferable. The long counting

times necessary for the latter should, if possible, be

divided into alternating peak and background mea-

surements to minimise the effect of instrumental drift.

Ultimate detection limits approaching 1 ppm as

de®ned by statistical criteria are obtainable in the

most favourable cases, but for heavy elements which

require the use of L or M lines (which are less intense

than the K lines used for elements in the range Z�11±

30) and for samples of high mean atomic number (and

consequently high continuum intensity) attainable

limits are considerably higher.

For concentrations around a few ppm, the normal

background correction method entailing linear back-

ground interpolation between offset spectrometer

positions is questionable, the accuracy required in

the background determination being of the order of

�1%. Careful attention must therefore be paid to the

possibility of nonlinearity, which may arise from

curvature of the continuum or absorption edges. A

`blank' standard known to contain none of the trace

elements of interest can be used to quantify these

effects. Serious errors can occur when either peak or

background measurements are affected by interfering

peaks of other elements.

Interference corrections can be applied when

necessary, but in such cases the accuracy of trace

element analysis inevitably suffers. Correction factors

can be determined from measurements on pure

elements (or other appropriate standards). A spectrum

simulation program enables interference effects to be

predicted and often avoided by choosing an alter-

native spectrometer crystal or X-ray line and saves

instrument (and operator) time.

References

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Quantitative Trace Analysis by Wavelength-Dispersive EPMA 151