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UofO- Geology 619. Electron Beam MicroAnalysis- Theory and Application Electron Probe MicroAnalysis - (EPMA). Quantitative Analysis: Intensities to Concentrations. Modified from Fournelle, 2006.
Electron Beam MicroAnalysis- Theory and ApplicationElectron Probe MicroAnalysis -(EPMA)
Intensities to Concentrations
Modified from Fournelle, 2006
Recall Castaing’s approach to quantitative analysis, where specimen intensities are ratioed to standard intensities:
where K is the “K ratio” for element i, I is the X-ray intensity of the phase and subscript i is the element.
With the counts acquired on BOTH unknowns and standards on the same instrument, under the same operating conditions, we assume that many physical parameters of the machine that would be needed in a rigorous physical model cancel each other out (same in numerator and denominator).
Castaing’s “first approximation” follows this approach. The composition C of element i of the unknown is the K ratio times the composition of the standard. In the simple case where the standard is the pure element, then, the fraction K is roughly equal to the fraction of the element in the unknown.
However, it was immediately obvious to Castaing that the raw data had to be corrected in order to achieve the full potential of this new approach to quantitative microanalysis.
The next two slides give a graphic demonstration of the need for development of a correction procedure.
This plot of Fe Ka X-ray intensity data demonstrates why we must correct for matrix effects. Here 3 Fe alloys show distinct variations. Consider the 3 alloys at 40% Fe. X-ray intensity of the Fe-Ni alloy is ~5% higher than for the Fe-Mn, and the Fe-Cr is ~5% lower than the Fe-Mn. Thus, we cannot use the raw X-ray intensity to determine the compositions of the Fe-Ni and Fe-Cr alloys.
(Note the hyperbolic functionality of the upper and lower curves)
Note that the Fe-Mn alloys plot along a 1:1 line, and so is a good reference.
In his 1951 Ph.D. thesis, Castaing laid out the two approaches that could be used to apply matrix corrections to the data:
In addition to absorption (A) and fluorescence (F), there are two other matrix corrections based upon the atomic number (Z) of the material: one dealing with electron backscattering, the other with electron penetration (or stopping). These deal with corrections to the generation of X-rays.C is composition as wt% element (or elemental fraction).
We will now go through all these corrections in some detail, starting with the Z correction, which has two parts, the stopping power correction, and the backscatter correction.
Note that all these corrections require close attention to exactly what feature’s value is being input: the target (matrix), or the X-ray in question.
Incident electrons lose energy in inelastic interactions with the inner shell electrons of the target. The “stopping power” (energy lost by HV electrons per unit mass penetrated) is not constant but drops with increasing Z. A higher number of X-rays will be produced in higher Z targets. Thus, if the mean Z of the unknown is higher than that of the standard, a downward correction in the composition must be applied. The stopping power correction factor is “S”, and can be approximated by:
Reed, 1996, Fig. 8.6, p. 135
Stopping power of pure elements for 20 keV electrons
where J=11.5+ Z and Emean= (E0+Ec)/2
(J is the mean ionization energy; J, Z and A are of the target, Emean is of the X-ray)
As we discussed earlier, the fraction of high energy incident elections that are backscattered (h) increases with atomic number. There then will be relatively less incident electrons penetrating into higher Z specimens, resulting in a smaller number of X-rays. Thus, if the mean Z of the unknown is higher than that of the standard, a upward correction in the composition must be applied. The backscatter correction factor is “R”.
R can be approximated by
where W = Ec/E0 (the inverse of overvoltage), and Z is of the target, and W is of the X-ray
The total atomic number correction is formed by multiplication of the R and S of the unknown and standard thusly:
Z = Rstd/Runk * Sunk/Sstd
Overall the backscatter and the stopping power corrections tend to cancel each other out.But if there is a (small) correction, it is usually in the direction of the backscatter correction.
The intensity I of X-rays that pass through a substance are subject to attenuation of their initial intensity I0 by the material over the distance they travel within the material. The attenuation follows an exponential decay with a characteristic linear attenuation length 1/m, where m is the (linear) absorption coefficient. Beers Law can also be expressed in terms of mass, using density terms:
I = I0 exp -(m/r)(r Z)
where (m/r) is the mass absorption coefficient (cm2/g), r is the material density (g/cm3), and Z is the distance (cm)
Als-Nielsen and McMorrow, 2001, Fig 1.10, p. 19
Mass absorption coefficients (MACs) have been tabulated* for many X-rays through many substances (though some are extrapolations). They exist as a matrix of numbers: absorption of a particular X-ray line (emitter, e.g. Ga ka) by a absorber or target (e.g. As) will have one value (51.5). Note that the absorption of As Ka by Ga is a totally different phenomenon with a distinct MAC (221.4) .
Emitter = X-ray (here, Ka)
Absorber = matrix material
*See following discussion
Goldstein et al, 1992, p. 750.
X-rays produced within the material will be propagated in all directions, and will suffer attenuation in the process. Note that the path length of travel of the X-ray to the spectrometer is z cosecy, where y (psi) is the takeoff angle (cosec = 1/sin).
Castaing’s approach was to integrate the Beer’s Law equation over the depth at the given y, producing the absorption correction factor f(c) where c is defined as m cosec y where m is the MAC.
The absorption (A) correction is then defined as
A= f(c)std / f(c)sample
Reed, 1993,, p. 219
To be able to correct for this absorption of the measured X-rays, we need to know how the production of X-rays varies with depth (Z) in the material.
The distribution of X-rays as a function of depth is known as the f(rz) [phi-rho-z] function, where a “mass depth” parameter is used instead of simple z (bottom right).
The f(rz) function is defined as the intensity generated in a thin layer at some depth z, relative to that generated in an isolated layer of the same thickness.
Reed, 1993, p. 219
where c = m cosec y , s is a measure of electron absorption and depends on effective electron energy, where
The Philibert approximation breaks down, however, at the near surface, creating errors when dealing with low energy light elements, and we need to go to more complicated and accurate forms of the f(rz) function.
Reed, 1993, p. 219
(combining the absorption correction with stopping power and backscatter loss)
To be able to correct properly for absorption -- particularly for light elements, the exact shape of the f(rz) [phi-rho-z] curve must be known. Each X-ray has its own curve. There are 3 main parameters that affect the shape of the curve:
Reed, 1993, p. 220
The f(rz) [phi-rho-z] curves are usually determined by the “tracer method”, where successive layers are deposited by vacuum evaporation. The tracer layer B is deposited atop substrate A, with successive layers of A deposited on top.
Characteristic X-rays from the tracer element are measured (“emitted”) and then a generation curve is calculated by correcting each step for absorption and fluorescence effects
What constitutes a model?
Is the following case, the x-ray intensity as a function of depth can be modeled using the geometry of a rectangle!
And it works!
The X-rays produced within a specimen have the potential for producing a second generation of X-rays: this is secondary fluorescence, generally shortened to fluorescence. This occurs when the characteristic X-ray has an energy greater than the absorption edge energy of another element present in the specimen.
As we saw earlier, Ni Ka (7.48 keV) is able to fluoresce Fe Ka (Ec 7.11 keV). This effect is maximized when there is a small amount of the fluoresced element present, e.g. Fe in a Ni-Fe alloy.
Reed gives an example where the Fe intensity is 142% of what it “should be”.
Also, the continuum above an absorption edge also causes fluorescence, although this is generally weak.
Reed, 1996, Fig. 8.10, p. 139
Secondary fluorescence is an important issue that must be appreciated. Generated X-rays are not scattered nearly as much as incident electrons, and thus the generated X-rays can travel relatively long distances (50 um in Fig 3.49) within the specimen and produce a second generation of X-rays. If the specimen (and standards) are relatively large (=homogeneous), this is not a problem. However, if minor or trace elements are being analyzed in small grains (Phase 1 in Fig 16.10) and the host phase (2) has high abundance, an error may be made in the EPMA analysis.
Goldstein et al. p. 142; Reed 1993, p. 258
Secondary fluorescence is a potential source of analytical error across linear boundaries, either horizontal (e.g. thin films) or vertical (e.g., diffusion couples).
In the example here of a vertical interface between untreated Cu and Co, there is NO diffusion. However, the resulting EPMA profiles clearly imply there is diffusion. There is NO diffusion – there is only secondary fluorescence across the boundary. Cu Ka X-rays can excite Co, to the extent that there is apparently 1 wt% Co about 15 um away from the boundary within the Cu. But Co Ka cannot excite Cu, so only the continuum X-rays can create secondary fluorescence, which is less – but certainly distinguishable, an apparent 0.5 wt% Cu at 10 um from the boundary in the Co.
Reed 1993, p. 259-260
A recent article (below left) reports an innovative approach to correcting the secondary fluorescence (SF) in diffusion couples and from adjacent phases. This utilizes a complex Monte Carlo program called PENELOPE (Penetration and Energy Loss of Positrons and Electrons) that permits complicated geometric models of electron and X-ray behavior in materials. SF can be simulated in a model that represents the actual specimen (e.g. Fig 1 below), and then subtracted from the observed data (right figure).
and then the K-ratios are input into an automated matrix correction program.
To run, the correction calculations must assume an initial composition for the unknown -- because the magnitude of each factor is proportional to the abundance of the element times its correction in a pure end member. The assumed composition is a normalized (to 100%) value of the K-ratio. Based upon the first iteration with this assumed composition, the result gives a more truer composition, which then is the input for the second iteration. The process is iterated until convergence, usually 3-5 times.
* Probe for EPMA does the interference correction within the matrix correction, a far better approach compared to the normal (antiquainted) procedure of correcting the data after the matrix correction is completed.
One currently widely used matrix correction program is CITZAF, developed by John Armstrong (then CIT, now NIST) and implemented in our Probe for EPMA software. There are several options, which we elucidate here, but that generally we do not modify them from the default values. Probably the only parameter you would ever modify would be mass absorption coefficients (there are different ones for the light elements).
In the early decades of probing when computer power was negligible, the alpha correction technique was widely used, as it required less number crunching and relied mainly on empirical data and less on complex physical models and physics. Today, however, there may be a rekindled interest in this approach, as it may “work better” in many cases.
In 1963-4, Ziebold and Ogilvie* developed a corrections for some binary metal alloys, with an equation in the form
where a12 is the a-factor for element 1 in the binary with element 2, K is the K-ratio, and composition (fractional) is C. This equation can be rearranged in the form
If experimental data exist for binary alloys, then a plot of C1/K1 versus C1 is a straight line with a slope of (1- a12), leading to determination of a12. Such a hyperbolic relationship between C1 and K1 was shown to be correct for several alloy and oxide systems, but it was difficult to find appropriate intermediate compositions for many binary systems.
*Quantitative Analysis with the Electron Microanalyzer, Analytical Chemistry, Vol 35, May 1963, p. 621-627; An Empirical Method for Electron Microanalysis, Analytical Chemistry, Vol 36, Feb. 1964, p. 322-327.
Ziebold and Ogilvie showed that a corrections could be developed for some ternary metal alloys, with an equation in the form
where a123 is the a-factor for element 1 in the ternary with elements 2 and 3, and is defined as
This equation can be rearranged
Similar relationships can be written for elements 2 and 3, and used to calculate a-factors for the 3 binary systems of the ternary.These a-factors were limited to a particular E0 and takeoff angle.
Bence and Albee* in 1968 showed that this approach could be extended to silicates and other minerals, i.e. a system of n components, where for the nth component a b-factor could be found
where an1 is the a-factor for the n1 binary.
These factors were determined for a limited set of conditions, i.e. 15 and 20 keV, and take off angles of 52.5° and 38.5°.
The 1968 Bence and Albee paper is one of the most highly cited papers in the geological literature (over ~20,000 citations).
* Empirical correction factors for the electron microanalysis of silicates and oxides, J. Geology, Vol. 76, p. 382-403; also see Albee and Ray, Correction Factors for Electron Probe Microanalysis of Silicates, Oxides, Carbonates, Phosphates, and Sulfates, Analytical Chemistry, Vol 42, Oct 1970, p. 1408-1414.
One can also use a multi-standard calibration curve method.
Generally this is only for certain unique situations such as trace carbon in steel.
In 1988, John Armstrong* reviewed the Bence-Albee (a-factor) correction scheme for EPMA of oxide and silicate minerals. He evaluated the old factors, and revised some, using a -factors calculated from newer ZAF and f(rz)algorithms, and showed “that with some modifications the a -factor corrections can be as accurate as any other correction procedure currently available and much easier and quicker to process.”
*Bence-Albee after 20 years: review of the accuracy of a-factor correction procedures for oxide and silicate minerals, in Microbeam Analysis-1988, p. 469-76.
Armstrong also reviewed† ZAF and f(rz) corrections and suggested that some of these correction algorithms “produce poorer results in the analysis of silicate and oxide minerals than some of the earlier corrections”. He specifically was referring to various corrections that were optimized for metal alloys
† Quantitative analysis of silicate and oxide materials: comparison of Monte Carlo, ZAF and f(rz) procedures, in Microbeam Analysis-1988, p. 239+
The matrix corrections assume that all elements present (and interacting with the X-rays) will be included. There are situations, however, where either an element cannot be measured, or not easily, and thus the analyst must make explicit in the quantitative setup the presence of unanalyzed element/s -- and how they are to be input into the correction.
Typically oxygen (in silicates) is calculated “by stoichometry”. Elements can also be defined in set amounts, or relative proportions, or “by difference” – although this later method is somewhat dangerous as it assumes that there are no other elements present.
* Tingle, T.N., Neuhoff, P., Ostgergren, P., Jones, R.E. and Donovan, J.J. (1996) The effect of “missing” unanalyzed oxygen on quantitative electron probe microanalysis of hydrous silicate and oxide minerals. GSA Abstracts, 28, 212.
One complication for oxygen is variable valence states of elements such as Fe. Robust software will allow you to enter case by case different valence states.
In some cases, if oxygen is not included, there can be errors in the matrix corrections of some elements, as the presence of O, OH, and H2O can affect the actually measured elements, as there may be significant absorption of those x-rays by the oxygen present*.
Which has LOTS of oxygen which typically is “unanalyzed” and therefore not involved in the matrix correction
Solution: Iterate a fixed amount of H2O (16 atoms of H = 1.76 wt% H plus stoichometric O) per formula to achieve good results.
As shown in the bottom analysis where the H2O is missing, there is up to 3% relative error for cations.
The ZAF corrections require accurate and precise knowledge about many physical parameters, such as
As David Joy points out in his 2001 article “Constants for Microanalysis”, there are problems in our knowledge of many parameters:
At the Eugene EPMA workshop in September 2003, John Armstrong reviewed the state of EPMA matrix corrections
We have discussed various ways to correct the raw data, the goal being to come up with the most accurate and precise analytical procedures to give us the most trustworthy data.
We have just mentioned that everything is not as rosy as one would hope.
So, can we trust the numbers we get out of the probe? In many/most cases, given care, yes. But we cannot blindly look at the electron probe and computer as a black box!
Stay tuned for an upcoming installment, where we discuss standards, accuracy and precision in EPMA.