The ZAF Model for Correction of Matrix Effects Upon Measured X‑ray Intensities
The prime method used for data reduction by the JEOL 8200 EPMA to account for the effects described above is the ZAF Model. This model assumes an initially linear relation between generated X-ray intensity and concentration of a given element (Castaing’s approximation, 1951). Any deviation from the initial, linear relation is ‘corrected’ by a series of multiplicative factors that account for the effects of atomic number (Z – stopping power, back-scattering factor and X-ray production power), absorption (A) and fluorescence (F), each of which are calculated. Mathematically, the ‘net’, matrix-effect-corrected X-ray intensities start with a comparison of raw, measured and background-corrected X-ray intensities for an element in an unknown sample, Iunk , against that of a standard, Istd, with a known concentration of a given element, ‘X’. The ratio of these two intensities is called the ‘K-ratio’, and is commonly denoted as ‘K’. The ratio of the concentration of element ‘X’ in the unknown relative to that in the standard is proportional to the K-ratio; by using simple algebra, we can solve for the concentration of element ‘X’ in the unknown sample, or so it would seem. Thus,
where the K-ratio is equal to 
. The key issue to understand here, is that the K-ratio is only proportional to the ratio of respective concentrations in the unknown and standard. In order to achieve ‘equality’ (more accurately described as ‘best fit’) between the 2 sides of the equation, we need to mathematically integrate the matrix effect corrections, by multiplying the K-ratio by the Z, A and F corrective factors, as follows:
In essence, the ZAF corrections are only calculated and iteratively determined quantities – they are iterated over several cycles until they reach self-consistency. The Z-correction accounts for the fact that heavier elements produce and absorb more X-rays than lighter elements. From a physical point of view, this makes sense, especially in the case of binary compounds such as PbS, where there is a large disparity in Z between the constituent atoms. Even though the stoichiometry of PbS is 1:1 atomically, the Pb will produce more X-rays than S will in a PbS crystal, simply because the Pb is much larger than and has a more complex electronic configuration than S. Therefore, the uncorrected K-ratio would tell us, erroneously, that there is more Pb than S, on an atomic basis, and that the K-ratio needs to be corrected.
Both the absorption correction factor, A, and the fluorescence correction factor, F, are dependent on Z, and are highly influenced by crystalline structure and orientation of the structure in the electron beam. The absorption correction, A, is required to account for absorption of primary, characteristic X-rays that we want to measure on their way out of a sample – without the absorption correction, the intensities of certain X-rays will be apparently reduced, giving us anomalously low concentration results. The fluorescence correction, F, accounts for the secondary fluorescence of target atoms by primary X-rays (i.e. X-rays produced from collisions of electrons from our source with target atoms) or high-energy, secondary electrons (electrons ionized from target atoms). The fluorescence effects cause our measured X-ray intensities to be higher than expected, and result in falsely high concentrations and totals. In most cases, absorption effects outweigh fluorescence effects, owing to the concept of mass-absorption coefficients. In most substances, the absorption correction commonly dominates the fluorescence correction, especially in the case of heavy trace elements such as REEs, whose mass absorption coefficients (MACs) tend to be very large compared with those of lighter elements that may make up the bulk of the matrix in which the REEs are present (e.g., La, Ce in apatites). MACs will be discussed in more detail, later in this guide.
The iteratively corrective ZAF data reduction model is used to transform raw data into results that are as accurate and as precise as possible. Much of the remainder of this guide will discuss accuracy and precision of results, and the factors that affect those two parameters, as well as the basic counting statistics that apply to quantitative analyses. Ultimately, primary standards must analyze within analytical error in order to gauge the accuracy and precision of the analytical method and suitability of the correction procedure. The question then becomes, “What is an acceptable analytical error?”. In order to answer this question, we need to recognize sources of error and find ways to minimize them, or at least account for them mathematically.