Penepma/Penfluor/Fanal derived Alpha Factors For Matrix Corrections

[John Donovan]

Latest version of the matrix.mdb file contains a large number of new binaries.

197K k-ratios for 1562 binary pairs.

[Probeman]

Here is the error histogram plotted for the full Pouchou2.dat dataset using specific compositions calculated using Penfluor/Fanal (Penepma for primary intensities and Fanal for analytical fluorescence):



Quite good really, but lets look closer at a few of the outliers by plotting k-ratio error as a function of concentration as seen here (the data labels are the line numbers in the first column in the Pouchou2.dat file):



As one can see, many of these are at very high overvoltages where the absorption correction is very large, except for the Au La measurement at 13 keV which is a low overvoltage measurement.

For comparison with the analytical phi-rho-z calculations here is the same plot using the default phi-rho-z in CalcZAF:



The Pouchou2.dat file is included with your CalcZAF.msi installation, but I attach it below for convenience.

[Philippe Pinard]

For comparison, results using the full-blown PENEPMA.



The points in red are those highlighted by Probeman in the PENFLUOR/FANAL graph. Note that the limits of the y-axis are different.

Edit by John: This is excellent!  Would it be possible to re-plot the data with the same axes as the others?  Which version of Penepma was utilized?  Do you see a difference between say Penepma 2008 and 2012?

[Philippe Pinard]

Same scaling.



The results are from PENEPMA/PENELOPE 2011. The simulations were run until an relative uncertainty of 1% was obtained for the x-ray line of interest. The detector was centered as the specified take-off angle and had an opening of +- 9 deg.

[John Donovan]

Hi Philippe,
I am surprised that there is so much difference between the Penfluor/Fanal and Penepema results given that the main difference between them is the treatment of fluorescence (Fanal does the fluorescence analytically).

I've got a back burner project to start calculating k-ratios for alpha factors from the full Penepma, but now it's going to get more attention...

Ok, here's another thing I realize after some thinking: if I compare the two graphs directly:



and



it appears to me that the high concentrations errors are more similar and that the Penfluor errors increase as the concentrations are lower. Since the default in Penfluor is to only calculate 3600 sec for each voltage this could mean that the Penfluor k-ratios used to calculate the alpha-factors are of insufficient precision.

I am going to do some comparisons with the full blown Penepma and check this idea...

No, it wasn't a precision issue after all. Instead I found a silly mistake with the formatting of the takeoff angle in the Fanal input file!  Re-extracting the data using Fanal provides significantly improved error distributions and many of the remaining outliers seem to be highly fluorescent systems- see updated graphs above.

Edit by John: it could be the low overvoltage situations (Au La) may be precision related- I will check that as well.

[Probeman]

A new Matrix.mdb database has been released for CalcZAF and Probe for EPMA which includes some light element binaries (many more coming soon) and newly recalculated k-ratios with double precision pure element standards for improved calculations.

To see what binaries are currently available please see the following link:

http://probesoftware.com/download/Calculated%20Matrix%20Binaries.txt

I would be very pleased if you have any experimental data comparing these new fast Monte-Carlo alpha factors with traditional analytical calculations.

[Probeman]

OK, I've recalculated the Pouchou binaries using Penfluor/Fanal (without B Ka just as Philippe did above) and here is what I get now:



As you can see the outliers are now all very low overvoltage situations. This is because I used the default 3600 seconds per beam energy to obtain the intensities and at very low overvoltages the precision is insufficient under these circumstances. But this is very promising compared to the analytical codes in CalcZAF.

The advantage of the Penfluor code is that once you have a composition, you can extract multiple elements, x-rays at different takeoff angles and beam energies in seconds.  Because these intensities will be used to calculate k-ratios for alpha factors for matrix corrections for arbitrary compositions within the binary one cannot specify the precision in advance, as one can with Philippe's Penepma runs shown above.

Next I will compare to a 7200 sec per beam energy calculation to see how that improves the precision for these low overvoltage situations. Meanwhile the Monte-Carlo modeling continues...

[John Donovan]

The latest version of CalcZAF (and Probe for EPMA) now contain an updated matrix.mdb k-ratio database for over 2000 binaries at beam energies between 1 and 50 keV for a total of over 229,000 k-ratios.  That's about half the periodic table calculated at normal precision using the Penepma (Penfluor/Fanal) code.

Most of the new additions are for REEs and actinide elements.

For a list of the binaries already calculated see this link:

http://probesoftware.com/download/Calculated%20Matrix%20Binaries.txt

Oh, and there are a significant number of new light element binaries for carbon, nitrogen, etc.  Please remember that while these light element matrix corrections are well modeled, they do not include chemical bonding effects which affect peak shape and position.  For these chemical bonding effects, area peak factor (APF) corrections are still required as discussed here:

http://probesoftware.com/smf/index.php?topic=248.msg1186#msg1186

[Probeman]

To compare the difference between the default precision Penfluor/Fanal calculation with higher precision calculations see the graphs below of the Ag, Cu, Au alloy measurements in the Pouchou2.dat (without the Cu La measurements).  That gives us 232 measurements to compare, and the nice thing about this system is that there are all sorts of physics here, high absorption and large fluorescences.

First here is the original CalcZAF (Armstrong/Reed) calculation without beta fluorescence for reference:



This can be compared to the new CalcZAF fluorescence code with beta lines fluorescence seen here:

http://probesoftware.com/smf/index.php?topic=490.msg2710#msg2710

Now here are the same measurements, but calculated using Penfluor/Fanal derived k-ratio alpha factors using the default precision (3600 sec per beam energy). All k-ratios for 11 compositions in each binary all precalculated in advance:



The high side outlier is a low overvoltage situation.

Now here are the same data, but this time using "high precision" Penfluor k-ratios (7200 sec per beam energy):



Similar, but note that the variance is improved considerably over the default precision calculations (much smaller).  And the low overvoltage outlier is no longer visible!

In fact these 7200 sec k-ratios fitted to alpha factors compares quite well to the full Penfluor/Fanal calculations on the specific Ag, Cu, Au conditions and compositions in the Pouchou2 database as seen here:

AverageA .995590
StdDevA .026136
MinimumA .881889
MaximumA 1.12403


The work continues...

[Probeman]

I'm looking to further test the Penepma derived k-ratio alpha factor matrix corrections for extreme fluorescence situations.

The Pouchou binary k-ratio database was apparently selected to avoid large fluorescence situations and mainly test the Z and A terms in the matrix corrections at that time.

But if anyone has a high quality set of measurements on stainless steel or high alloy steel standard materials and they would be willing to share it, I would be very interested in obtaining such a database for testing these Monte-Carlo characteristic (and continuum) fluorescence matrix corrections. The elements of interest would be Fe, Ni, Cr, Co, W, Nb, etc, etc...

[Probeman]

I have recently improved the handling of Penepma Monte-Carlo derived k-ratios for alpha factor matrix corrections to show whether the binary is question has already been calculated and loaded or not (if not, the calculation falls through to the current phi-rho-z method to analytically calculate the missing k-ratios).

This is can seen in the following output from CalcZAF. Normally, the binary has been loaded from the Penepma matrix.mdb database (or not,) in both emitter-matrix directions, for example as seen here:

 NON-LINEAR Alpha Factors, Takeoff= 40, KeV= 20

P=1, C=.9900, K=.9883, Alpha=1.1735
P=2, C=.9500, K=.9416, Alpha=1.1793
P=3, C=.9000, K=.8753, Alpha=1.2822
P=4, C=.8000, K=.7586, Alpha=1.2729
P=5, C=.6000, K=.5420, Alpha=1.2675
P=6, C=.5000, K=.4421, Alpha=1.2620
P=7, C=.4000, K=.3467, Alpha=1.2563
P=8, C=.2000, K=.1668, Alpha=1.2485
P=9, C=.1000, K=.0818, Alpha=1.2475
P=10, C=.0500, K=.0405, Alpha=1.2461
P=11, C=.0100, K=.0080, Alpha=1.2505
Xray  Matrix   Alpha1  Alpha2  Alpha3  Alpha4 %AvgDev   *from Penepma 2012 Calculations
pr la in p    4.6916  3.1333  2.6693 -3.4302 1.49763

P=1, C=.0100, K=.0048, Alpha=2.1103
P=2, C=.0500, K=.0245, Alpha=2.0975
P=3, C=.9000, K=.8059, Alpha=2.1672
P=4, C=.8000, K=.6481, Alpha=2.1724
P=5, C=.6000, K=.4089, Alpha=2.1688
P=6, C=.5000, K=.3160, Alpha=2.1645
P=7, C=.4000, K=.2360, Alpha=2.1576
P=8, C=.2000, K=.1053, Alpha=2.1252
P=9, C=.1000, K=.0501, Alpha=2.1059
P=10, C=.0500, K=.0245, Alpha=2.0952
P=11, C=.0100, K=.0048, Alpha=2.0868
Xray  Matrix   Alpha1  Alpha2  Alpha3  Alpha4 %AvgDev   *from Penepma 2012 Calculations
p ka in pr    2.6245   .6932   .2787  -.5319 .295767


In this example one element binary (emitting element and matrix element) are found in the matrix.mdb database, but the other binary is not:

NON-LINEAR Alpha Factors, Takeoff= 40, KeV= 15

P=1, C=.9900, K=.9889, Alpha=1.1146
P=2, C=.9500, K=.9440, Alpha=1.1280
P=3, C=.9000, K=.8877, Alpha=1.1389
P=4, C=.8000, K=.7765, Alpha=1.1511
P=5, C=.6000, K=.5636, Alpha=1.1615
P=6, C=.5000, K=.4621, Alpha=1.1641
P=7, C=.4000, K=.3638, Alpha=1.1660
P=8, C=.2000, K=.1762, Alpha=1.1686
P=9, C=.1000, K=.0868, Alpha=1.1696
P=10, C=.0500, K=.0430, Alpha=1.1700
P=11, C=.0100, K=.0086, Alpha=1.1703
Xray  Matrix   Alpha1  Alpha2  Alpha3  Alpha4 %AvgDev
Rb la in c    1.8420   .6163   .4769  -.6695 .159128

P=1, C=.0100, K=.0009, Alpha=11.3800
P=2, C=.0500, K=.0046, Alpha=11.4613
P=3, C=.9000, K=.5033, Alpha=8.8820
P=4, C=.8000, K=.3013, Alpha=9.2778
P=5, C=.6000, K=.1342, Alpha=9.6748
P=6, C=.5000, K=.0932, Alpha=9.7260
P=7, C=.4000, K=.0646, Alpha=9.6565
P=8, C=.2000, K=.0259, Alpha=9.3892
P=9, C=.1000, K=.0120, Alpha=9.1567
P=10, C=.0500, K=.0058, Alpha=9.0326
P=11, C=.0100, K=.0011, Alpha=8.9032
Xray  Matrix   Alpha1  Alpha2  Alpha3  Alpha4 %AvgDev   *from Penepma 2012 Calculations
c ka in Rb   69.4274 51.9633 47.0268-59.1578 8.38180

[John Donovan]

This is merely a sanity check comparing the fit error from using polynomial (3 coefficient) and non-linear (4 coefficient) alpha factors for the Pouchou Au, Cu, Ag dataset with the actual calculated k-ratios from Penepma (Penfluor/Fanal).

As you can see, both fit methods look excellent, the non-linear fit is slightly closer to 1.000 (no error) and has a slightly worse deviation, than the polynomial fit alpha factors.

The reason why the comparison is not exactly always equal to one, is due to limitations of precision in the Monte Carlo calculations, so there will be always be some small amount of variance in the fit (see image attachments below). But, here is where it gets interesting: because the fit is essentially regressing the "noise" it should provide a *more accurate* calculation than just using a single Monte Carlo calculation!

Interesting that the alpha factor magnitudes are similar but the slopes are different!

[Probeman]

I been making progress on utilizing Penepma Monte-Carlo derived k-ratios for fast matrix corrections in CalcZAF (and Probe for EPMA and CalcImage), and can show some new results (with newly extracted k-ratios with self-consistent precision levels) starting with the Au, Cu, Ag binary systems in the Pouchou k-ratio database.

First Au, Cu and Ag binaries calculated at 3600 per keV (x10) times 11 binary compositions (from 1 to 99%) or 110 hours per binary system:



Note that while the preliminary Monte Carlo calculations take a long time, the actual matrix correction runs in seconds, quite comparable to the traditional analytical expressions we all use every day. Next here are the same Au, Cu and Ag binaries but calculated to a significantly higher precision at 14400 sec/keV (x10) times 11 binary compositions or 440 hours per binary system:



Note that the higher precision calculation gives both a better error average (closer to 1.000) and has a smaller variance. This is mostly due to several very low overvoltage situations in the Pouchou database.  Note: as we attempt even lower overvoltage measurements on FEG instruments to reduce our interaction volumes these, very low overvoltage calculations become much more critical.

But the good news is that already we are already getting better accuracy with the above fast Monte-Carlo k-ratios compared to the traditional analytical expressions. For comparison check the same binary compositions calculated using the JTA/Reed phi-rho-z, first without beta line fluorescence:



and now with the beta fluorescence:



The average error calculation with beta fluorescence is slightly higher than without beta lines (as one would expect), but the variance is improved (smaller) by including beta lines. I will have more on this, as I am still adjusting the relative line weights in the Reed fluorescence correction for more exotic fluorescence situations...

[Probeman]

I've updated the matrix.mdb file for many new binaries. In fact most (though not all) of the binary systems in the Pouchou database have been calculated. For example, I have not yet extracted k-ratios for 75 degree takeoff angles yet!

But enough binaries have been calculated that it's worth comparing to traditional analytical physics expression accuracy. Both calculations below were done using the Pouchou database *without* B Ka and Cu La (there are bigger analytical issues with these emission lines!).

First here is the JTA phi-rho-z with beta fluorescence:



A decent average error of 1.01 and a variance of around 4.5%.  Now let's try the Penepma derived fast Monte-Carlo k-ratios fitted using polynomial alpha factors.



As you can see, an even better average error and the variance has decreased to around 3.4%!  I believe the era of fast Monte-Carlo has arrived...

[Mike Jercinovic]

Pretty impressive!