## Quantification

### Introduction

Concentration is not directly proportional to the corrected intensity ratio of an element’s X-ray line because the ratio is affected by X-ray absorption, secondary fluorescence, electron backscattering, and the electron stopping power of the sample. There are several methods of taking these factors into account and determining concentrations from

X-ray counts:

- Calibration curves, which use an empirically determined relationship between counts and concentration;
- Empirically based matrix corrections (Bence-Albee); and
- ZAF (or PAP), which uses fundamental factors to correct for the effects of atomic number, absorption and fluorescence.

These methods are summarized below. Those wishing more detail should examine one of the references.

### Calibration curves

Calibration curves may be constructed to relate X-ray counts and element concentration. However, these curves require a large number of well-characterized standards with compositions that bracket the unknowns. In wet chemical analysis, such as atomic absorption spectroscopy (AAS), it is relatively easy to make solutions with the appropriate concentrations. In contrast, production of standards for microprobe analysis is far more difficult, requiring apparatus to produce homogeneous glasses. In both cases, however, the data used to construct a curve must be taken at identical operating conditions (take-off angle, accelerating voltage and beam current) as will be used during analysis.

*Calibration curve for carbon in nickel steel. Redrafted from Figure 8.26 in Goldstein et al., 1981, Scanning Electron Microscopy and X-ray Microanalysis.*

For geological applications, calibration curves must be constructed for each mineral group; however, because the unknowns approximate the standards in compositions, matrix corrections are unnecessary. The requirement of a great number of standards is especially difficult to satisfy especially with the accuracy required. In addition, this technique does not allow confident analysis of a truly unknown material.

*Calibration curve of Mg-Kα counts per second vs. weight percent MgO for chemically analyzed biotite, chlorite, staurolite, chloritoid, and garnet. The scatter in the data results from absorption (or fluorescence) due to other oxides other than MgO in the samples. Redrafted after Bence and Albee, 1968.*

### Empirical matrix corrections

Empirical correction schemes are based upon empirical determinations of inter-element effects, which are then applied to unknown materials. These empirical factors must be determined for all operating conditions of interest (take-off angle and *E _{0}*). As with the calibration curves the standards must be suitable for microprobe analysis, but many fewer are required.

In 1963-4, Ziebold and Ogilvie developed corrections for some binary metal alloys using an equation of the form

where, C_{A} = concentration (fraction) of element A in the mixture AB, α_{AB} = factor describing matrix effects, and K_{A} = background-corrected intensity ratio of A counts in mixture AB relative to those in pure element A. This equation can be rearranged as

A plot of C_{A}/K_{A} against C_{A} yields a straight line with slope (1 – α_{AB}); thus, X-ray counts collected on a set of variable mixtures allows one to determine α_{AB}. Although this hyperbolic relationship between C_{A} and K_{A} was demonstrated for several alloy and oxide systems, it was difficult to find appropriate intermediate compositions for many binary systems. Ziebold and Ogilvie showed that α corrections also could be developed for some ternary metal alloys. It should be emphasized that the resulting α factors only apply to a particular accelerating voltage and take-off angle.

*Intensity of Fe-Kα x-rays in Different Binary Mixtures. The intensity of Fe-Kα is affected by the nature of the coexisting element. The presence of Ni causes enhanced Fe-Kα x-ray intensity due to fluorescence by Ni-Kα. Conversely, Cr absorbs Fe-Kα x-rays and reduced their total intensity. The presence of Mn in the mixture has almost no effect. The nature and extent of binary interactions can be determined by measuring x-ray intensity for a range of binary mixtures.*

#### Bence-Albee corrections

The most commonly used empirical method of matrix correction in the 1970s was developed by T. Bence and A. Albee; it is still used. The Bence-Albee correction method utilizes ‘alpha factors’ (α) to describe the effects of atomic number, absorption, and fluorescence. The α factors are established for binary systems as shown in the above figure. For cases where the presence of a second element causes the absorption of X-rays from the element in interest, α > 1. Where fluorescence of the second element by the x-rays from the element of interest occurs α < 1. The ideal case where there are no inter-element effects has α = 1.

*Form of the calibration curves for C vs. K and different values of α. Mixtures exhibiting absorption will have upward concave curves with α factors > 1. Those where fluorescence dominates will have concave downward curves with α factors < 1 (redrafted after Bence and Albee 1968).*

Quantifying the shape of the curve is difficult, but a linear relation exists between (C_{A}/K_{A}) and C_{A} and the value of a can be determined by finding the intercept (where C_{A} goes to zero).

*Plot of Ca/Ka vs. Ca for the curves shown above. a linear relationship results when the ratio of C _{A} to K_{A} is plotted against C_{A}. (redrafted after Bence and Albee 1968).*

For the binary system AB we may write:

There are theoretical grounds for the approximate validity of this correction, but it is unreasonable to expect such a simple equation to be exact in light of the complexity of electron scattering processes. Bence and Albee (1968) applied this correction scheme to silicates and oxides (in which absorption effects dominate and fluorescence effects are generally minor) using oxide rather than element concentrations. For a multicomponent system, the concentration C of element A in the unknown can be calculated as:

where, K^{A}_{unk} = background-corrected intensity ratio of the counts on the unknown relative to the counts on a pure element standard, and

#### Example Bence-Albee correction

Unknown concentrations are determined using an iterative process. Initially, K values are inserted into the equation for β in the place of concentrations to find starting concentrations. These new concentrations are reinserted to find new β values and the iteration is repeated until convergence in achieved. Two or three iterations usually result in values very close to the theoretical composition.

As an example, we will calculate the concentration of an “unknown” synthetic grossularite garnet, Ca_{3}Al_{2}Si_{3}O_{12}. The standards that we will use are:

The appropriate α values from Albee and Ray (1970) are:

Note that most of the α values exceed 1.00, indicating that absorption dominates. Recall that, by definition, the α values for an element in its pure oxide is 1.00.

First, we calculate the βs for the standards, for each of the elements/oxides, i:

Assume that the measured x-ray counts (corrected for background and deadtime) are:

First, we make an initial approximation of the composition of the unknown by calculating initial K factors:

Here the subscript “i” is used to indicate the different elements/oxides. Thus, we make an initial composition calculation:

These K-factors (composition estimates) total to 0.9113 (91.13%). We can improve things greatly by making an approximation of the β values using these K values and the α values above. For this approximation, the K values are used as estimates of the concentrations:

We now use these βs and the K values to make a better approximation of the actual concentrations:

The new total is 1.0024. This can be improved to some degree by further iteration. We can refine the estimates of the β values using the concentrations just calculated:

Note that none of the β values changes by much! Using the refined βs to get new concentrations yields:

Our new total is 1.0017 (100.17%). Continued iteration will improve these numbers a bit, but let’s quit here and compare them with the ideal composition of grossularite:

#### ZAF & PAP corrections

Standards with compositions similar to the unknowns are often used with the Bence-Albee method to minimize the differences in absorption between the standards and the unknowns. (In geological materials, the effect of fluorescence is very small and only absorption is significant.) Use of the Bence-Albee reduction scheme was very common in the 1960-70s, because the calculations required very little computing power and memory, but now that very powerful computers are available (and inexpensive), this is no longer an advantage. Today, the more complicated ZAF or PAP correction schemes are usually employed.

#### Introduction

The ZAF correction scheme is based on first principles and provides data reduction for all operating conditions. If there were no inter-element (matrix) effects, the corrected intensity ratio could be converted to concentration using the formula called Castaing’s 1^{st} Approximation:

where, C^{A}_{unk} = concentration of A in the unknown, C^{A}_{std} = concentration of A in the standard, I^{A}_{unk} = background corrected intensity of A x-rays in the unknown, and I^{A}_{std} = background corrected intensity of A x-rays in the standard. This is equivalent to the case of ideality in Bence-Albee (a = 1). However, as we noted, matrix effects due to absorption and fluorescence of X-rays within the sample and atomic number effects are significant. The matrix corrections to be applied to the estimated initial composition from this approximation can be expressed in the form:

where F^{A}_{unk} and F^{A}_{std} are matrix or ZAF factors for unknown and standard respectively. The derivation of these factors is described below. Since the factors are dependent on composition, which is initially unknown, ZAF works in an iterative fashion similar to that of Bence-Albee. The intensity data are used with Castaing’s 1st Approximation to get initial elemental concentrations that are refined in subsequent iterations until they converge. ZAF is not very good for elements with X-ray energies less than 1 keV because of a lack of knowledge of the factors discussed below. For these elements it is best to use a standard of similar composition to minimize matrix effects. For example, when analyzing F in apatite, use a fluorapatite standard rather than a fluorphlogopite standard.

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#### Atomic number correction (Z)

The atomic number effect controls the amount of incident energy from the electron beam that is actually put into the sample. This effect consists of two components: backscattering and electron-stopping power (or retardation). Both are a function of average Z and, to a lesser degree, the accelerating voltage.

Backscattered electrons leave the sample carrying energy without producing X-rays. The fraction of electrons backscattered from the sample, n_{b}, ranges from about 0.12 for Al to 0.55 for U. At lower *Z*, more electrons stay within the sample to produce X-rays. The backscatter correction factor, *F _{b}*, reflects the X-ray intensity lost due to backscattering and is expressed as a fraction (r) relative to the intensity that would be produced with no backscattering:

This fraction in part depends on the overvoltage, but not to a significant degree. The factor rarely exceeds 1.2 for *Z* less than 30.

The stopping-power correction, *S*, relates the amount of incident energy to the amount absorbed by atoms of specific *Z*. The electron-capture cross-section decreases with increasing Z. Thus, light atoms have a higher ratio of atomic number to atomic weight (*Z/A*) and interact with disproportionately larger numbers of incident electrons. The characteristic intensity per unit concentration of a heavy element is less when combined with a light element than in a sample of the pure element. *F _{s}* for a given element is:

The value for S may be calculated using:

where, *A* = atomic weight, *Z* = atomic number, *J* = mean ionization potential, k = constant, and

where, *E _{0}* = accelerating voltage and

*E*= critical excitation potential. As a simplification, the mean ionization potential may be approximated by 11.5 Z.

_{c}Values for S must be calculated for each element separately because each has a different *E _{c}*. The average value of

*S*for the unknown is calculated in a manner analogous to the mass absorption coefficient, using the mass concentration fractions,

*C*, of the elements:

_{i}The net Z correction factor for both effects is: *F _{b}* x

*F*.

_{s}The effects of backscattering and stopping power are opposing and can be canceling. For example, in the analysis of Fe_{2}O_{3}, with pure iron as a standard, the stopping power of oxygen reduces the intensity of Fe-Kα from the oxide relative to that from the pure standard. However, the lower mean atomic number of Fe_{2}O_{3} (15.2) compared with pure iron (Z = 26) results in a smaller backscattering loss for the oxide when compared with the metal. The net effect is that the intensity of Fe-Kα is 7.2% lower (at 15 keV) than calculated on a basis of the relative concentrations in the two materials.

#### Absorption correction (A)

Since X-rays are generated below the surface of the sample, the emergent radiation suffers absorption prior to detection. The absorption correction is a function of the take-off angle (length of path traversed by the X-rays), the distribution of X-ray generation, the wavelength of the emergent X-ray and the elements present. As the take-off angle increases, the intensity of characteristic radiation decreases due to an increase in path length. Less energetic X-rays are more easily absorbed. Absorption can also be strongly affected by surface irregularities — a good sample polish is thus critical. Most X-rays are generated at relatively shallow depths within the excitation volume and relatively close to the beam axis, because it is in this region that electron energies are greatly attenuated by ionization or electron scattering. Several models have been used to describe the depth distribution of X-ray generation termed φ(ρ,Z), “phi-rho-zee”.

*The forms of the depth and lateral generated intensity functions φ(ρ,Z) and ψ(y). The vertical dimension in this figure is depth beneath the sample surface, expressed as mass thickness, ρz. The horizontal dimension is lateral distance from the axis of the electron beam in arbitrary units (after Williams 1987).*

Approximations are commonly used to represent the emission volume because of its complex shape. The simplest is that of Bishop, which is a puck of uniform thickness. Most improvements in quantification have resulted from improvements of the φ(ρ,z) relationship (for example that of Philibert).

*The Bishop rectangular approximation of φ(ρ,Z), compared to experimental data and Philibert’s analytical approximation. The area of the Bishop rectangle is assumed to approximate to that under the φ(ρ,Z) curve, and the continuous variation in φ(ρ,Z) is replaced by the concept of a ‘mean mass depth’ at half the depth of the Bishop rectangle. This greatly simplifies the calculations required to correct for the effects of variable compositions on the depth generation of analytical X-rays (after Williams 1987).*

Once the shape of the emission volume is describe, absorption can be described by Lambert’s Law. The result is an expression that accounts for the effects of take-off angle, accelerating voltage, composition, etc. One such formulation is the Philibert-Duncumb-Heinrick equation. Using this expression the absorption factor for samples examined by microprobe analysis, F_{a}, is expressed:

where,

and Z= atomic number, A = atomic weight, m = mass absorption coefficient, φ = take-off angle, E_{o} = accelerating voltage, and E_{c} = critical ionization potential. For compounds a mean value of h is used:

where, c_{i} = mass concentration of element i. The σ factor above accounts for the voltage dependence of absorption of primary electrons. Errors in the calculation of F_{a} can be decreased by using a high take-off angle to minimize χ and low overvoltages to maximize σ.

#### Fluoresence corrections (F)

The correction for fluorescence is a function of the elements present, their concentrations, their values of E_{c} and mass absorption coefficients, and the take-off angle, φ. The most important factors are the concentrations of fluorescing elements and elements fluoresced. In general, the correction for fluorescence is the least important factor in the ZAF correction. The fluorescent yield decreases with decreasing Z and is not important for the light elements, which dominate geological samples. The factor, F_{f} can be expressed as:

where, C_{i} = concentration of the flourescing element, and ω_{i} = the fluorescent yield of that element.

The fluorescent yield increases rapidly with increasing atomic number and F_{f} is negligible for K-lines of elements below atomic number 20. In silicates and oxides, absorption dominates and fluorescent enhancement is rarely greater than a few percent.

### PAP model

PAP is a general model for calculating X-ray intensities incontrast to ZAF, which is conceived as a matrix correction proceedure. PAP is a general model for the calculation of X-ray intensities and can be used for a wide range of X-ray energies (100 eV to greater than 10 keV) and accelerating voltages (1-40 keV). The improvement over ZAF results from using better expressions for the Z effects (backscattering factor, retardation of the electrons, and effective ionization cross-section) and a better determination of the absorption effects. The latter is the most significant of these improvements and uses a modified version of the φ(ρ,Z) polynomial used in standard ZAF correction scheme to better fit the experimentally determined X-ray distribution. Although PAP corrections take slightly longer to calculate than ZAF corrections, the improvement is well worth it.

*Depth distribution of Mg-Kα in aluminum at 25 keV. Comparison of theoretical functions with experimental results of Castaing and Henoc (after Pouchou & Pichoir, 1984).*

2008 © Copyright James H. Wittke

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