I recently implemented a new regression method for alpha factor matrix corrections as described here:
http://probesoftware.com/smf/index.php?topic=40.msg2762#msg2762This new regression method improves the accuracy for situations in which very large fluorescence or absorption effects are present. This method can also be abused in situations in which some noise is present, for example low precision (low overvoltage or low concentration) Monte-Carlo derived k-ratios.
But first a short history of these regression efforts to model electron-solid interactions for binary pairs of elements over time as seen in this graphic:
The original method was proposed by Bob Ogilvie and others at MIT and applied to geological samples at Cal Tech (Bence and Albee) in the early 70's was to allow empirically measured k-ratios in binary compositions of known standards to be utilized for matrix corrections for complex materials. The Cal Tech implementation was originally based on assuming that the matrix correction did not vary as a function of the binary compositional range to facilitate the use of slide rules for calculations and therefore assumed the matrix correction at a 50:50 constant composition applied to all compositions for that binary.
This is not as crazy an assumption as one might think, for there are some binaries in which the matrix corrections are relatively constant over the binary range, for example as seen here:
Furthermore, for silicate matrices, which was the target main effort to improve matrix correct accuracy at Cal Tech, the matrix is mostly oxygen and therefore the matrix corrections were relatively more constant than in other systems.
However, it was soon realized by Mark Rivers (in the late 70's), then at UC Berkeley, that by rearranging the traditional alpha expression to solve for alpha, one could perform almost any regression equation to the alpha factor calculations. He applied a linear assumption which is far better than the constant assumption as seen here for the same dataset:
This was an enormous improvement in accuracy especially in non-geological matrices such as sulfides and other minerals and alloys.
But for instances where significant absorption or fluorescence effects are present John Armstrong in the late 80's (then at Cal Tech), noted that a polynomial fit would handle even these situations quite well as seen here for the famous Fe Ni binary:
Further details on the Rivers linear and Armstrong polynomial alpha factor methods can be found here:
http://epmalab.uoregon.edu/bence.htmHowever, the keen eye will will notice that the polynomial fit isn't perfect and with the addition of a four coefficient non-linear regression even these extreme situations are dealt with properly as seen here for the same system:
Note that the average deviation for this non-linear expression is significantly improved, compared to the polynomial basis. For the record, the polynomial regression basis for the Mark Rivers alpha expression, (C/K - C)/(1 - C), is seen here:
alpha = coeff1 + conc * coeff2 + conc ^ 2 * coeff3
The corresponding beta expression (for summing binary alpha factors for complex matrices) for the polynomial fit is seen here:
beta = (coeff1 + conc * coeff2 + conc ^ 2 * coeff3) * conc
The regression basis for this new non-linear four coefficient fit is seen in the alpha expression here:
alpha = coeff1 + conc * coeff2 + conc ^ 2 * coeff3 + Exp(conc) * coeff4
and the corresponding beta expression for the non-linear fit is seen here:
beta = (coeff1 + conc * coeff2 + conc ^ 2 * coeff3 + Exp(conc) * coeff4) * conc
Remember: all this fitting takes place in hyperbolic space due to the nature of the alpha concentration - kratio relationship as first described by Ogilvie!
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