Improving EDS Quantitative Accuracy with the eZAF MACC Database

An Al/Si binary specimen example in the September 2021 issue of EDAX Insight demonstrated that the results fall within approximately 20% relative deviation, provided the Si concentration remains below 30%.

Applying the SCC-based empirical database can improve accuracy in limited concentration ranges for dedicated applications. However, the deviations can be amplified if the sample's composition resides outside of the dedicated concentration range. The SCC factors are utilized to tune the Generation of X-rays for unique elements and an array of excitation energies. The use of SCC is limited to the handling of only non-linear effects. Because of this, a secondary eZAF-dedicated database was introduced. This two-dimensional database focuses on interelement dependencies, such as the Absorption.

Results and Discussion

How is the Absorption modeled with eZAF? In principle, there are two roads to considering the Generation and Absorption: The traditional ZAF and the more than 200 years old method Φ(ρz):

ZAF-approach:

Equations 1 & 2

The Generation equation models electron shell excitation by integrating all electron energies from the primary electron energy (E₀) down to the shell’s critical excitation energy (E_c). This calculation accounts for the energy-dependent cross-section Q(E) and the energy loss of electrons as they penetrate the specimen. It provides a complete depth distribution of X-ray generation, factoring in absorption effects, which vary with depth. Backscatter loss (R) is also included to determine the total X-ray generation by the electrons.

Absorption is handled separately within the ZAF model through an additional equation. This equation integrates the depth distribution of generated X-rays while considering absorption paths and Mass Absorption Coefficients (MACs, µ/ρ). The total absorption is then normalized against the same integral calculated without absorption effects, yielding the relative absorption—the fraction of total generated X-rays. The resulting quotient is equivalent to the Generation term, illustrating the nonlinear nature of absorption processes.

In the classical ZAF approach, a formula for Φ(ρz) is required for analytical integration.

Φ(ρz)-approach:

Equations 3

The newer Φ(ρz)-approach was initially developed from the ability to measure depth distributions, with the curves extended and improved by Monte-Carlo calculations. The requirement or limitation of the curves is no longer required as they are numerically integrated. In theory, the ZAF approach can be solved using numerical integration of the equations. In this way, the depth-distribution model is merely the differentiator, rather than the kind of notation, when comparing the ZAF model and the Φ(ρz)-model.

Figure 1. The blue curve is the eZAF model using depth distribution, which is based on Love/Scott ¹ with modification for tilted specimens ⁶. It was based on the Monte Carlo (MC) method calculations. The gray line is a comparison of a typical measurement or MC-calculated Φ(ρz). Image Credit: Gatan, Inc.

A Φ(ρz) approach has no leading benefit over ZAF. Furthermore, regarding a statement by Heinrich, the ZAF is more appropriately suited to keeping the range of effects separate, which improves each of the best, achieving a better-combined approach and the best possible full correction model.²

The Φ(ρz) equation is based on measured depth distributions that use a specific MAC database. The Φ(ρz) includes a measured database fundamentally in genes. It is not a first principle-based standardless approach.

A number of investigations have indicated that the range of depth distribution models, including the triangular Love/Scott model, can demonstrate comparable results.

The parameters with the heaviest influence are the MACs, or the µ/ρ-values in model equations, which are the atomic data dependent on X-ray energy and material transmission.

The Quant calculations use a single value for each element on the ID list, which is the selected line energy for all element materials.

A mean µ/ρ, calculated through composition, is used. A calculated database is a requirement where all of the element's possible alpha-lines, K-, L-, and M-radiation, are potentially absorbed into any possible element. The needed values are around 3 x 100 x 100.

Figure 2. The used MAC values for an eZAF Quant evaluation with seven elements that are assumed in the specimen. Image Credit: Gatan, Inc.

The concept is to apply a two-dimensional database at the MAC use level in the absorption model to empirically adjust A-correction. This method is required where element combinations have a huge absorption effect in element interdependencies.

Net-counts = X-ray Generation * SCC * Absorption (MACC) * Fluorescence * Efficiency (E) * ECF

In the eZAF model, it is applied where the MACs are utilized in the exponential functions and called Mass Absorption Coefficients Correction (MACC). There is the possibility of adjusting the energy absorptions in the element lines in absorber elements where possible. It influences and adjusts the Quant evaluation only when the dedicated element and absorption constellation are actually in the specimen. This does not influence all other evaluation cases, such as the pure element Si quantitative evaluation, which is unaffected by the MACC for Si in Al, due to the fact that no Al is actually inside the pure Si specimen.

Heinrich’s recommendation has been followed here, with a separate Generation database improvement through SCC and decoupling the Absorption improvement ability using MACC.

Figure 3a shows results obtained by optimizing the eZAF Absorption correction using modified MAC and the MACC database.

The results were improved, with the maximum number of deviations landing at around 8% relative (Figure 3a) over the composition area.

The initial results without using the database had a maximum relative deviation of around 20%, which could only be improved with SCC factors for a limited composition area to around 10% (for “Customized Standardless”). This included the disadvantage outside the adjusted/ dedicated composition area; the results achieved runaways up to 30% relatively.

Figure 3. a) Calculated concentration results by eZAF for a binary Al/Si example specimen (blue Al; red Si) over the Si nominal concentration, all in % units, an adapted MACC database was used for Si-K X-rays in Al. The broad light-red line is the Si net-count raw data curve, arbitrary units, not yet ZAF corrected. b) The same result, but in addition to the MACC, the SCC was adapted for a 50%/50% sample. Image Credit: Gatan, Inc.

The single value MACC offered is a massive improvement over the complete composition range of the pattern binary sample.

As shown in Figure 3b, the SCC can also be adjusted and applied to a 50%/50% spectrum. The curve shifts visibly by a factor; in theory, the curve shape is no longer bent. This SCC adaption does not enhance all result deviations.

The MACC database is not dedicated to changing access for the software operator. This would be an adjustment to the non-linear part of the eZAF model. If the adjustment is not performed correctly, it can have lasting effects on the overall analytical performance in other combinations of elements. A person with skill and experience is needed to adapt the values in the database to ensure step-by-step accuracy improvement going forward.

The central element of the eZAF Quant accuracy questions is found in the non-linear interelement matrix effects and X-ray excitation. These must be considered when using theory/models (and are dependent on assumptions made regarding the other elements and weight fractions).

• The SCC database addresses the X-ray generation.
• Separate from this, the MACC database addresses the non-linear interelement effects, well separated from excitation.

The SCC and MACC databases can adjust the models and processes to achieve improved results. This improves the correction models by empirical results but does not affect reducing larger correction needs.

An additional strategy is to avoid massive corrections. Corrections of smaller raw measured data show a reduction in model and atomic data uncertainty, influencing the results. Figure 4a indicates the corrections applied with an 80%Al/20%Si example with eZAF, where MACC improvement has already been considered.

The A-factor for Si is 0.2577, meaning approximately 74% of all X-rays generated are absorbed in the specimen's material. Figure 4b shows that the correction needs of raw measured data are one order of magnitude less.

Figure 4. a) Evaluation result with eZAF, Si-K absorption correction is within about 74%. b) Evaluation result with PeBaZAF, Si absorption correction is about 7.3%. This means the P/B based PeBaZAF correction requirement is one order of magnitude less. Image Credit: Gatan, Inc.

This is a significant advantage and property of PeBaZAF correction. It is caused by the previously discussed linearized P/B curve raw measurement signal (September 2021 issue of EDAX Insight Figure 1).

There are massive variances in correction needs. This means it is little surprise that net-count-based quantification requires measured databases for good overall standardless accuracy.

It is impossible to apply ZAF only based on fundamental parameter physics. The P/B model-based PeBaZAF simplifies things because the uncertainties in parameters and models affect the results less.⁴ The PeBaZAF demands less correction and is independent of specimen absorption and MACs commonly known to have massive uncertainties (Z ≥ 11).⁵ The reasons for this are found in the characteristic X-ray Generation and, in particular, the Absorption, which provide similar effects compared to bremsstrahlung for the same X-ray energy. An MACC database is not a requirement.

Reduced correction needs are also possible with eZAF for further improvements of Quant and to eliminate the huge absorption correction needed for the example. This can be achieved by measuring standards or using already measured standards libraries.

References and Further Reading

1. Love F, Scott V D (1978) Journal of Physics D 11, 1369
2. Sewell, D.A., Love, G. and Scott, V.D. (1987). Universal correction procedure for electron-probe microanalysis. IV. The tilt factor. Journal of Physics D: Applied Physics, 20(12), pp.1567–1573. https://doi.org/10.1088/0022-3727/20/12/003.
3. Heinrich, K.F.J. (1995). An Evaluation of Quantitative Electron Probe Methods. X-Ray Spectrometry in Electron Beam Instruments, pp.305–367. https://doi.org/10.1007/978-1-4615-1825-9_18.
4. Eggert, F. (2018). The P/B-Method, About 50 Years a Hidden Champion. Microscopy and Microanalysis, (online) 24(S1), pp.734–735. https://doi.org/10.1017/s1431927618004166.
5. Eggert, F. (2019). Complementary Standardless Quantitative Methods with EDS. Microscopy and Microanalysis, 25(S2), pp.560–561. https://doi.org/10.1017/s1431927619003532.

This information has been sourced, reviewed and adapted from materials provided by Gatan, Inc.

For more information on this source, please visit Gatan, Inc.