“Particle Mass Effect”

Fe k-ratio (bulk K411)

Monte Carlo Simulation Fe in K411 Spheres, E0 = 20 keV

1.2

1.0

0.8

0.6

0.4

0.2

0.0

. Table 23.3  Composition of K411 glass and analysis of flat, bulk target at E0 = 20 keV (Standards: Mg, Si, CaF2, Fe; oxygen by stoichiometry)

NIST SRM 470 certificate values

. Fig. 23.30  Monte Carlo calculation of the emission of Fe K-L2,3 from K411 spheres of various diameters at E0 = 20 keV. The intensity is normalized by the Fe K-L2,3 intensity calculated for a flat bulk target of K411

for an incident beam energy of 20 keV. For this plot, the Fe K-L2,3 intensity emitted from the particle has been divided by the intensity calculated for a flat bulk K411 target to determine the k-ratio. Note that as the particle diameter increases from zero, the Fe K-L2,3 k-ratio increases and asymptotically approaches unity (i.e., equivalent to the flat bulk target) for a diameter of approximately 10 μm. Thus, for the case K411 at E0 = 20 keV, bulk behavior for Fe K-L2,3 (6.400 keV) is

observed for spherical particles with diameters of 10 μm and greater.

“Particle Absorption Effect”

The surface curvature of a particle can reduce the absorption path to the detector compared to the interaction volume in a flat bulk target, as shown schematically in .  Fig. 23.31. Since X-ray absorption depends exponentially on the absorption path length, surface curvature can significantly modify the measured intensity, creating the “particle absorption effect.” The magnitude of the particle absorption effect depends strongly on the energy of the characteristic photons involved. For the example shown in .  Fig. 23.32, the Fe K-L2,3

. Fig. 23.31  Schematic illustration of the difference in the X-ray absorption path length in a particle and in a flat bulk target of the same material

23.6 · Particle Analysis

. Fig. 23.32  Emission of Fe

K-L2,3, O K-L2 and Mg K-L2,3 (normalized to O K-L2 and Mg K-L2,3 from bulk K411); note excursion

in the O K-L2,3 and Mg K-L2,3 above unity (i.e., exceeding the

emission from a flat, bulk target)

k-ratio (K411 bulk)

Monte Carlo calculations of K411 Spheres (E0 = 20 keV)

1.6

1.8

1.2

1.0

Diameter (micrometers)

radiation at 6.40 keV is sufficiently energetic that it undergoes relatively little absorption in the bulk case so that the modification of the absorption path length by particle surface curvature produces a negligible effect. However, the lower energy characteristic photons such as O K-L2 (0.525 keV) and Mg K-L2,3 (1.254 keV) in K411 suffer significant absorption in a flat bulk target, so that when these photons are generated in a spherical particle, the reduced absorption path in the direction toward the EDS leads to an increase in X-ray emission compared to a flat bulk target.

.  Figure 23.32 shows that as the particle diameter increases

from zero, the k-ratios for O K-L2 and Mg K-L2,3 initially increase similarly to FeKα as a result of the particle mass

effect. However, for larger particles the reduced absorption path of the curved particle surface causes the emitted O K-L2 k-ratio to actually exceed unity (i.e., higher emission than bulk behavior) for a particle diameter of 1.6 μm, reaching a maximum of 1.35 relative to bulk at a particle diameter

of 2.8 μm. For Mg K-L2,3, the emission exceeds unity for a particle diameter of 2 μm and reaches a maximum of 1.17 at

a diameter of 3.0 μm. For particle diameters beyond the

intensity maxima, the O K-L2 and Mg K-L2,3 k-ratios gradually decrease with increasing particle diameter, asymptoti-

cally approaching the equivalent of bulk behavior at 25 μm

diameter for O K-L2, and 18 μm diameter for Mg K-L2,3. Thus, for spherical particles of the K-411 composition mea-

sured with E0 = 20 keV, effectively bulk behavior is observed for all characteristic X-ray energies for particles with diameters greater than 25 μm for a beam position at the top center of the particle (detector take-off angle 40°). As demonstrated in .  Fig. 23.23, deviations in the beam placement either toward the EDS or away have significant effects due to the

modification of the absorption path. Particle geometry and its impact on X-ray absorption must be considered when selecting beam locations on a particle for analysis. It is critical that the analyst always be aware of the position of the EDS detector relative to the X-ray source, as demonstrated in

.  Fig. 23.24, to minimize the effects of particle geometry.

23.6.5\ Uncertainty in Quantitative Analysis

of Particles

Quantitative analysis of particles is performed by following the same k-ratio/matrix correction protocol used for flat, bulk specimens. However, it must be recognized that particle geometry modification of the interaction of beam electrons and the subsequent propagation of X-rays introduce factors which violate the fundamental assumption of the bulk quantification method, namely that the only reason the X-ray intensities measured in the target being analyzed are different from the standards is that the composition(s) is different. Thus, with the impact of the geometric factors, the analytical accuracy of the conventional k-ratio/matrix correction protocol is inevitably compromised. The critical question to consider is the degree to which the uncertainty budget is increased by the systematic error contribution of the particle effects.

The Analytical Total Reveals the Impact of Particle Effects

The analytical total is the sum of the calculated concentrations, including oxygen by stoichiometry if calculated.

.  Table 23.3 shows the results of the analysis of K411 glass in the form of a flat, bulk target. The beam energy was

E0 = 20 keV, and the standards used pure elements Mg, Si, and Fe, while SRM 470 K412 glass was used as the Ca standard. The analytical total was 0.9789 (the sum of the SRM certificate values is 0.9886), and the relative errors were all well within the ±5 % error envelope, with the largest error at −2.2 % relative for Ca.

.  Table 23.4 gives the results of applying the k-ratio/ matrix correction factor protocol to the analysis of a 1.3-μm-diameter spherical particle of K411 glass, with the EDS spectrum collected with the beam placed at the center of the particle image. The intensity of each elemental constituent has been measured relative to the same suite of standards: pure elements Mg, Si, and Fe, and CaF2 as the Ca standard. Oxygen has been calculated by the method of assumed stoichiometry. The analytical total is 0.3480, and the relative errors for the raw calculated concentrations are large and negative, for example, −70 % relative for Mg and −56 % relative for Fe. These large negative relative errors (a negative relative error indicates that the calculated concentration underestimates the true concentration) for both the low pho-

ton energy peaks (e.g., Mg K-L2,3 and Si K-L2,3) and the high photon energy peaks (e.g., Ca Kα and Fe Kα) are a result of

23 the particle mass effect reducing all X-ray intensities compared to bulk behavior. Clearly, these raw concentrations

have such large systematic errors as to offer no realistic meaning. To compensate for the mass effect and thus place the concentrations on a meaningful basis, internal normalization can be applied:

Note that normalization is only useful if all constituents present in the analyzed volume are included in the total, including any such as oxygen that are calculated by assumed stoichiometry rather than measured directly. After normalization, the relative errors are reduced in magnitude, as given in .  Table 23.4, but the values for Mg (−14 %) and Fe (+26 %) remain well outside the bulk analysis error histogram.

.  Table 23.5 presents similar measurements and calculations for a 6.1-μm-diameter K411 particle for which the analytical total is 1.091. This particle diameter is sufficiently large so that the X-ray production for the higher energy photons,

Ca K-L2,3 and Fe K-L2,3, has nearly reached equivalence to the flat, bulk target, resulting in relative errors of −5 % or less.

For this particle size, the lower energy photon peaks, Mg

K-L2,3 and Si K-L2,3, are still strongly influenced by the particle absorption effect, causing relative errors that are large and

positive, since more of these low energy photons escape than