Hi All, a brain teaser
Here's a plot from penepma of metals, alloys and oxides. Errors approximate symbol size
(https://smf.probesoftware.com/gallery/453_24_07_18_6_41_37.png)
It suggests oxides plot lower than metals or alloys, any ideas?
Win x-ray gives good agreement to penepma
(https://smf.probesoftware.com/gallery/453_24_07_18_6_44_49.png)
Materials simulated
(https://smf.probesoftware.com/gallery/453_24_07_18_6_46_41.png)
I should point out that density is not the problem see screenshot results for diamond and amorphous carbon are the same
Interestingly in the paper 'Mean Atomic Number Quantitative Assessment in Backscattered Electron Imaging'
https://doi.org/10.1017/S1431927612013566
They measure different BSE intensities for minerals and metals but attribute this to variations in BSE from day to day
With subsurface charging in insulators you might expect a higher BSE in insulators (e.g. Ghorbel et al. 2005) - but this is not included in monte carlo simulations, and is the reverse of what is simulated
Quote from: Ben Buse on July 24, 2018, 06:47:01 AM
I should point out that density is not the problem see screenshot results for diamond and amorphous carbon are the same
Ha! This brings back memories...
You are correct Ben, the problem is *not* different densities. Also see some Penepma calculations I did many years ago assuming different densities for gold. See attached plot below.
It is also true that measuring reliable BSE values is problematic. But that is not the problem we are seeing with the Monte Carlo calculations. The problem lies in another area of calculation.
Let me give another brain teaser that answers the first brain teaser: how is average atomic number calculated? For pure elements the answer is obvious, it's simply the atomic number of the element, but what about for compounds? How should average Z be calculated for compounds? It's not obvious until you think through the physics.
Think about the physics of elastic scattering. What exactly is involved? It's true that there is a tiny mass effect which is 0.2% energy loss in a 180 degrees momentum exchange between an electron and a hydrogen nucleus (1/2000 mass ratio). And how often does one get a high angle BSE in hydrogen? Not very often. And for heavier elements it's a *smaller* mass effect.
See attached pdf below. Remember to login to see attachments... Here's another clue: the neutron has almost no effect on elastic scattering (~10-7 or so). It is almost purely an electrostatic effect.
So how should average Z be calculated for compounds?
Edit: fix typo in attached mass kinetic effects pdf (it should be 0.2% since 1/2000 * 4)
err....whats on the X and y axis? Is that part of the brain teaser? ;)
Quote from: jon_wade on July 24, 2018, 02:28:32 PM
err....whats on the X and y axis? Is that part of the brain teaser? ;)
Hi Jon,
Eta (BSE coefficient) on the Y axis, and Z (atomic number) on X.
john
Oooh a brain teaser. Perfect for my train commute!
I'll have a crack at this. My initial thought is that its something to do with the fact that eta for mixed phases is calculated as n[mixed] = sum(n(i)*C(i)) where n= eta, i is an index of the elements and C is concentration in weight percent.*
In my mind, the weight percent weighting doesn't make sense for mixed phases - the likelihood that a primary beam electron encounters a particular element depends on the atomic percent, rather than the weight percent. For example, if an electron interacts with an atom in Si metal, it has a 100% chance of that atom being Si. If it interacts with an atom in SiO2, it has a 1 in 3 chance of interacting with Si and a 2 in 3 of being weakly scattered by O.
*From Scanning Electron Microscopy and X-ray Microanalysis, 4th Edition, Goldstein et al
Quote from: JonF on July 25, 2018, 01:07:54 AM
Oooh a brain teaser. Perfect for my train commute!
I'll have a crack at this. My initial thought is that its something to do with the fact that eta for mixed phases is calculated as n[mixed] = sum(n(i)*C(i)) where n= eta, i is an index of the elements and C is concentration in weight percent.*
In my mind, the weight percent weighting doesn't make sense for mixed phases - the likelihood that a primary beam electron encounters a particular element depends on the atomic percent, rather than the weight percent. For example, if an electron interacts with an atom in Si metal, it has a 100% chance of that atom being Si. If it interacts with an atom in SiO2, it has a 1 in 3 chance of interacting with Si and a 2 in 3 of being weakly scattered by O.
*From Scanning Electron Microscopy and X-ray Microanalysis, 4th Edition, Goldstein et al
Hi Jon,
You're on the right track!
You are correct that weight fraction averaging for BSE is really only a proxy for what we should actually be doing. As you surmise, elastic scattering is not affected by neutrons so atomic weight isn't a very physically accurate model for calculating average Z for BSE.
But should we be using atomic fraction for average Z of the BSE signal? That assumes that each atom contributes equally to the BSE signal and we already know that simply isn't the case.
Think about this: a sample of uranium carbide or UC. Is the BSE signal of UC more like carbon (Z=6) or more like uranium (Z = 92) or halfway between carbon and uranium (Z=49)? The answer of course is very much like uranium (Z=92).
So that is why people started using weight fraction instead of atomic fraction. Because weight fraction averaging was a much more accurate model for average Z of BSE than atomic fraction averaging.
But again, atomic weight is not a physically accurate model, so what would be more physically accurate? Think about it from the perspective of the incident electron... what does the BSE (elastically scattered electron) interact with, inside the atom?
Here's another clue for Ben's brain teaser:
He's trying to plot BSE coefficient vs. average Z (atomic number) for pure elements and compounds. For pure elements average Z is just the atomic number (because 1 atom times the atomic number (Z) equals Z). But for compounds the average Z needs to include the atom fraction of each element in the compound, times what fractional quantity?
E.g., in uranium carbide, we have 1 atom of uranium and 1 atom of carbon, so we have a .5 atoms of each element for this UC compound. Now we want to multiply the atom fraction of each element times what physical parameter? Remember, that the BSE coefficient of UC is *much* more like U than C, so what other physical parameter (other than weight fraction) scales like that? Yes, there is no commonly accepted name for this physical quantity, but it pops up in all sorts of physics calculations...
Think of the analogy to weight fraction. We calculate the weight fraction of a compound how?
OK, so here is the answer to Ben's "brain teaser" for plotting BSE coefficients of pure elements and compounds as a function of Z, as described in the first post above.
We know from physics that mass has almost no effect for BSE production. At most the momentum exchange of an incident electron interacting with a hydrogen nucleus (with a 180 degrees elastic scatter) produces an energy loss (besides a change in angle) of around 1/2000 *4 or 0.2%. For higher atomic number elements the energy loss effect is even smaller.
So since the number of protons in the nucleus dominates the BSE effect we should not be using weight fraction (as it's based on atomic weight) to calculate the average Z for comparing BSE coefficients even though it is traditional, and consider a different scaling parameter based on atomic number.
But first let's consider how we calculate weight fraction for a element i in a compound:
wtfrac(i) = n(i) * A(i)/ SUM(n() * A())
where n(i) is the number of atoms of element (i) in the compound, and A(i) is the atomic weight of element (i) in the compound.
But since we know that atomic weight does not scale exactly with atomic number, by utilizing atomic weight for our average Z calculation for plotting our BSE coefficients, we are introducing an error equal to the degree by which the ratio A/Z varies across the periodic table. And remember, for some increases in atomic number, the atomic weight actually *decreases*! Do you know where they are in the periodic table? It's a good trivia question.
But what if we simply substitute atomic number for atomic weight in the above calculation? In that case we have:
zfrac(i) = n(i) * Z(i)/ SUM(n() * Z())
where n(i) is the number of atoms of element (i) in the compound, and Z(i) is the atomic number of element (i) in the compound.
So let's refer to a figure from a paper we published in 2003 where we plotted Monte Carlo calculations (using NIST"s MQ software) of BSE coefficients versus average Z for a number of pure elements and compounds as Ben did in his original post:
(https://smf.probesoftware.com/gallery/395_31_07_18_9_27_28.png)
Note that here we are using weight fractional averaging to calculate average Z for the compounds just as Ben did and it appears very similar to the two other Monte Carlo softwares. Now let's do the same plot but use the Z based fractional averaging (called here, electron fraction averaging, but protons/electrons- it's still Z based):
(https://smf.probesoftware.com/gallery/395_31_07_18_9_30_37.png)
It's a better fit, but still not perfect. What could be the issue?
Well as we have all observed, as the atomic number increases, the amount of backscattered electrons does not increase linearly. The BSE coefficient curves starts to bend down beginning around zinc (Z = 30 or so). Why is this?
It's because of partial "screening" of the nuclear (proton) charge by the inner electron orbitals of atoms with higher Z (as pointed out to us by Dale Newbury from NIST at the time). Basically it means that incident electrons do not always get to "see" the full electrostatic charge of the nucleus in the higher atomic number elements. Therefore they are elastically scattered a little less than one would think, given their atomic number.
So we need to correct for this nuclear screening by the inner orbital electrons effect, and so we tried a couple of exponents and found the following fit of Z^0.8 works beautifully as seen here:
(https://smf.probesoftware.com/gallery/395_31_07_18_9_38_06.png)
Like beads on a string! :)
Now it's true, that we don't commonly utilize this Z fractional weighting everyday, but it's essentially what the Monte Carlo models do when they calculate the elastic scattering cross sections for compounds.
If you are interested in the details I've attached the PDFs of our original paper, Reed's comments to our paper and our response to his comments. The latter is fun reading.
Finally, if anyone is interested in comparing these different average Z calculations, simply open the Standard app in the CalcZAF distribution and check the menu Output | Calculate Alternative Zbars. Here is the calculation of alternative Zbars for the uranium carbide compound I mentioned in a previous post:
ELEM: U C
CONC: .9520 .0480
ELEC: .9388 .0612
%DIF: -1.3854 27.4551
ATOM: .5000 .5000
ELAS: .9741 .0259
A/Z : 2.5873 2.0018
Zbar (Mass/Electron fraction Zbar % difference) = 1.29078
Zbar (Mass fraction) = 87.8689
Zbar (Electron fraction) = 86.7347
Zbar (Elastic fraction) = 89.7720
Zbar (Atomic fraction) = 49.0000
Zbar (Saldick and Allen) = 86.7347
Zbar (Joyet et al.) = 65.1920
Zbar (Everhart) = 91.7179
Zbar (Donovan Z^0.5 for continuum) = 74.5053
Zbar (Donovan Z^0.80 for backscatter) = 83.2973
Zbar (Donovan Z^0.85 for backscatter) = 84.3084
Zbar (Donovan Z^0.90 for backscatter) = 85.2124
Zbar (Bocker and Hehenkamp for continuum) = 64.3464
Zbar (Duncumb Log(Mass) for continuum) = 80.6927
Edit: fix typo in attached mass kinetic effects pdf (it should be 0.2% since 1/2000 * 4)
It's interesting that nuclear screening has come up with regards to BSE coefficient.
I wonder whether it would be possible to see atomic screening 'artefacts', such as the poor screening by 4f electrons (cause of the 'lanthanide contraction') or the 'd-block' contraction, and if this difference would be discernible with our BSE detectors. Could be an interesting experiment!
Hi John,
Thanks for your responses - I didn't know the answer. To summarise, Basically the offset to compounds is explained by nuclear screening - which you account for using a 0.8 exponent and electron fraction equation. The 0.8 is determined by fitting to monte carlo data.
As you point out atomic fraction is rubish, UO2 MAN = 36 compared to 82 using wt fraction, and 79.5 using electron fraction and 73.448 using electron fraction and 0.8 exponent.
Nice paper I'd never come across it before, I'm just recalculating the results using eq. 20. And will post the new results
I've attached a spreadsheet which calculates the MAN using the different equations
Agreed, using atomic fraction alone is daft.
Regarding screening, I was wondering whether an alternative approach would be to see if the atomic radii (itself a function of nuclear screening) would be an alternative?
Here's the data for the metals and the binary compounds. You can see using electron fraction and 0.8 exponent, gives good agreement to metals' line.
Ben
(https://smf.probesoftware.com/gallery/453_07_08_18_5_48_30.png)
And following Jon's good advice, I've added some axes labels, now we've solved the brain teaser ;)
Hi Ben,
So the Z fraction to the 0.8 exponent are the light symbols? And the dark blue M symbols are the pure elements?
Now just do a bit more work on your legend and you'll have an actual plot! :)
john
Yes I think your right! It certainly wouldn't get published!
light blue: exponent 0.8,
pinky purple - mass % calc.
black 'M': metals
Ben
Quote from: JonF on August 06, 2018, 07:25:12 AM
Agreed, using atomic fraction alone is daft.
Regarding screening, I was wondering whether an alternative approach would be to see if the atomic radii (itself a function of nuclear screening) would be an alternative?
Hi Jon,
It would be worth a try.
The electron fraction Z to the 0.8 power is just a model, as weight fraction Z is just a model. The advantage of the electron fraction (or proton fraction or Z fraction whatever you want to call it) model, is that it makes one less faulty assumption than the weight fraction Z model: which assumes that mass affects elastic scattering of electrons.
By the way, the discerning eye will notice something else from the output of alternative Z-bars in the post above:
Zbar (Mass/Electron fraction Zbar % difference) = 1.29078
Zbar (Mass fraction) = 87.8689
Zbar (Electron fraction) = 86.7347
Zbar (Elastic fraction) = 89.7720
Zbar (Atomic fraction) = 49.0000
Zbar (Saldick and Allen) = 86.7347
Zbar (Joyet et al.) = 65.1920
Zbar (Everhart) = 91.7179
Zbar (Donovan Z^0.5 for continuum) = 74.5053
Zbar (Donovan Z^0.80 for backscatter) = 83.2973
Zbar (Donovan Z^0.85 for backscatter) = 84.3084
Zbar (Donovan Z^0.90 for backscatter) = 85.2124
Zbar (Bocker and Hehenkamp for continuum) = 64.3464
Zbar (Duncumb Log(Mass) for continuum) = 80.6927Does anyone see it?
When we did our original literature search for different ways to calculate average Z, we found that none of them worked that well for calculating BSE trends. The one that was closest was Saldick and Allen, 1954. That was when we started thinking about what would be a better model and then we came up with the electron fraction model.
Then we noticed that our model and the Saldick and Allen model gave the same results, even though the equations looked quite different. Then we realized that their expression and our expression were algebraically equivalent! See our paper here for that discussion:
http://epmalab.uoregon.edu/publ/Compositional%20Averaging%20of%20Backscatter%20Intensities%20in%20Compounds%20(M&M,%202003(.pdf
Of course as we pointed out above, the electron fraction Z to the power of 1.0 model was only marginally better than weight fraction Z model, and that was when we tried some different exponents to account for nuclear screening, and found that electron fraction Z to the 0.8 worked really well.
So why didn't the Saldick and Allen get the attention of people in the EPMA community? Well, here's the title of their paper from 1954:
"The Yield of Oxidation of Ferrous Sulfate in Acid Solution by High-Energy Cathode Rays"
That's why! Maybe next time they would choose a title with broader appeal...
Quote from: JonF on August 06, 2018, 04:30:06 AM
It's interesting that nuclear screening has come up with regards to BSE coefficient.
I wonder whether it would be possible to see atomic screening 'artifacts', such as the poor screening by 4f electrons (cause of the 'lanthanide contraction') or the 'd-block' contraction, and if this difference would be discernible with our BSE detectors. Could be an interesting experiment!
Hi Jon,
I think you might be on to something here!
I ran your idea by a couple of physicists at UofO as to whether one might be able to detect subtle variations in the effective nuclear screening for transition series elements in the electron backscatter signal, and they said that it was "plausible". In fact this might be the solution to the famous (infamous?) "Heinrich kink" dilemma, from his 1968 paper, which we know from physical considerations cannot be due to mass effects. Here is a plot of Heinrich's measurements and my own measurements:
(https://smf.probesoftware.com/gallery/395_08_08_18_3_24_19.png)
So at the time (late 1990s) I knew that his measurements from the 60s were good, but I could not explain them. I was guessing something along the lines of subtle differences in electron backscatter diffraction (channeling) between these elements, but I wasn't very convinced by it.
Since I could reproduce Heinrich's measurements 30 years later using my SX100 at UC Berkeley, I know that his measurements were good, but just couldn't convincingly explain them. But I think you just might have!
I am running some high precision Monte Carlo calculations now and will post these in about a week or so when they are finished. In the meantime please see the attached pdf from an old document I wrote up in the 90s which has more details about this.
This could very well deserve a short paper by us together!
john
'Plausible' from physicists is high praise indeed!
That kind of effect from the raw data is more what I was expecting - I was puzzled as to whether the effect would be noticeable having seen the monte carlo simulations showing up a smooth curve (although you mention this in your paper!)
This may well be opening Pandora's box, but I've been wondering if the shielding of the nucleus in pure elements can be observed, whether the effects of bonding would also be discernible in compounds...
Quote from: JonF on August 10, 2018, 03:54:25 AM
'Plausible' from physicists is high praise indeed!
That kind of effect from the raw data is more what I was expecting - I was puzzled as to whether the effect would be noticeable having seen the monte carlo simulations showing up a smooth curve (although you mention this in your paper!)
This may well be opening Pandora's box, but I've been wondering if the shielding of the nucleus in pure elements can be observed, whether the effects of bonding would also be discernible in compounds...
Maybe? But only very high precision measurements can say.
The Heinrich "kink" measurements that was able to reproduce are in the one or two percent level, so these are difficult measurements. I can't wait to see the Penepma Monte Carlo simulations.
The funny thing was that the 1975 NIST Monte Carlo code seemed to correlate with the A/Z curve as Heinrich tried to demonstrate, but I noticed that they used a "multiscattering" electron model where they did lots of averaging to obtain faster results (computers were sloooow in those days). And guess what? When I asked Bob Myklebust about it, he confirmed that they utilized mass fraction to calculate the effects. Ha!
Of course the 1998 MQ NIST Monte Carlo *single* scattering model did not show any such mass effect.
The question now is: how accurately does Penepma model these nuclear screening effects for elastic scattering of electrons? I really should ask Xavier, but I'll let the calculations finish first.
I'm running 100,000 sec per element so I hope that's enough precision. I did remember to optimize for BSE production. It's only on the 3rd element as seen here:
(https://smf.probesoftware.com/gallery/395_10_08_18_10_17_29.png)
I might plot up what I have on Monday though...
Thank you John for the detailed explanation.
Would you still have the raw data (i.e. backscatter coefficients for different compounds) you used in the paper back in 2003? I wonder what Figure 2c would look like if the tabulated Mott or PENELOPE elastic cross sections were used instead Rutherford.
Philippe
Quote from: Philippe Pinard on August 30, 2018, 09:15:59 AM
Thank you John for the detailed explanation.
Would you still have the raw data (i.e. backscatter coefficients for different compounds) you used in the paper back in 2003? I wonder what Figure 2c would look like if the tabulated Mott or PENELOPE elastic cross sections were used instead Rutherford.
Philippe
Hi Philippe,
I'm sure I have the data somewhere but I would have to search.
However Ben Buse plotted up some Penepma calculated BSEs here:
https://smf.probesoftware.com/index.php?topic=1111.msg7478#msg7478
The plot is a little confusing because he included some "original Z" values, so just ignore the violet colored symbols and you'll see that the remaining Z^0.8 symbols plot nicely on a line.
I should probably re-run all the compounds and pure elements I did back in 2003 again with Penepma just for fun.
john
Many moons ago we published a paper (Donovan, Pingitore and Westphal, 2003) on electron backscattering and the effects on that by atomic mass vs. atomic number.
The bottom line is that mass has essentially zero effect on backscatter loss, while the BSE effect is essentially solely related to the effective nuclear charge of the nucleus, therefore the effect is almost entirely due to electrostatics. This claim is something that physical scientists generally accept after a few minutes of thought, but some, at least at the time, seemed surprised by our results.
After the paper was published, we had a response from Stephen Reed who suggested our "proposal should be treated with caution, pending more rigorous testing". Of course we are all for caution and more rigorous testing, but we still stand by the claim today, and more recently we have repeated these measurements and calculations, and found them to be reproducible both empirically, and theoretically, based new measurements and on the latest Monte Carlo models, as seen above in this topic.
For those interested in the gory details, the full paper and the response by Reed and our response to Reed's response are attached to this post:
https://smf.probesoftware.com/index.php?topic=1111.msg7448#msg7448
There will be more to say on this subject, but please feel free to make your own measurements and calculations and share them with us.
Just to follow up on this BSE "brain teaser" which Ben Buse posted about a while ago, attached below is an abstract accepted for M&M 2019 showing some Penepma BSE simulations for pure elements and compounds and also some rough absorbed current measurements which we hope to do more of before M&M, but in the meantime it's worth a look I think.
john
This is an interesting read:
https://en.wikipedia.org/wiki/Slater%27s_rules
It makes me wonder if using a Z fraction^0.7 weighted average atomic number, to compensate for nuclear screening by inner orbital electrons, could be further improved by considering whether the atoms in our beam incident materials are truly neutral atoms or not.
I mean we're knocking electrons off of them constantly, right? Probably a 10-4 effect (for inner orbitals anyway), but maybe someone would be willing to do some calculations...
Here is a pdf of my presentation given this week at M&M 2019, in case anyone who missed the talk is interested (please login to see attachments).
We've made some small changes to the Standard app in the output from the Output | Calculate Alternative Zbars menu.
Here is the additional output for ThSiO4:
(https://smf.probesoftware.com/gallery/1_12_02_23_9_41_47.png)
And here is a small table I put together showing how these calculations vary for a number of compounds:
(https://smf.probesoftware.com/gallery/1_12_02_23_9_42_02.png)
Somewhat surprising is that the Z^0.7 Z fraction concentrations and zbars can be utilized not only for backscatter productions, but also for modeling continuum productions:
https://smf.probesoftware.com/index.php?topic=4.msg10036#msg10036
We are very pleased that our new paper on "An Improved Average Atomic Number Calculation for Estimating Backscatter and Continuum Production in Compounds" is now published in Microscopy and Microanalysis:
https://doi.org/10.1093/micmic/ozad069
Quote from: Probeman on July 19, 2023, 09:29:04 AM
We are very pleased that our new paper on "An Improved Average Atomic Number Calculation for Estimating Backscatter and Continuum Production in Compounds" is now published in Microscopy and Microanalysis:
https://doi.org/10.1093/micmic/ozad069
The best comment we've received so far on the paper was a colleague who stated: "I've never seen an exclamation mark in a scientific paper before". And if I think about it a bit, I don't think I ever have either.
But due to a profound misunderstanding by some in our scientific community, the authors and the editors felt that the exclamation point was indeed necessary! :)
To follow up, in our recently published paper "An Improved Average Atomic Number Calculation for Estimating Backscatter and Continuum Production in Compounds by J. Donovan, A. Ducharme, J. J. Schwab, A. Moy, Z. Gainsforth, B. Wade, and B. McMorran" (https://doi.org/10.1093/micmic/ozad069), it may be unclear to the reader how Eqs. 6 and 7 are derived from the theoretical calculations presented in the "The Differential Scattering Cross Section of the Yukawa Potential" section.
Here is a simple derivation of these equations.
Consider a compound with density ρ and molecular mass M. Its macroscopic cross section is the sum of the microscopic (differential) cross sections of each element j present in the compound (https://smf.probesoftware.com/gallery/1567_04_08_23_1_41_56.png) times the atomic density Nj, or:
(https://smf.probesoftware.com/gallery/1567_04_08_23_1_42_22.png)
The microscopic differential cross section and atomic density have units of cm2/sr and atoms/cm3, respectively. The atomic density of element j is given by:
(https://smf.probesoftware.com/gallery/1567_04_08_23_1_43_04.png) ,
with cj as the mass fraction of element j and Aj as the atomic mass (in g/mol) of the element j. NA is Avogadro's number (in atoms/mol) and ρ is the density of the material (in g/cm3).
The atomic density can also be expressed using atomic fractions aj instead of the mass fraction:
(https://smf.probesoftware.com/gallery/1567_04_08_23_1_43_54.png) with (https://smf.probesoftware.com/gallery/1567_04_08_23_1_44_18.png)
So,
(https://smf.probesoftware.com/gallery/1567_04_08_23_1_44_35.png)
If we define the mean atomic number as the atomic number of each elemental constituent averaged over the macroscopic cross section above,
(https://smf.probesoftware.com/gallery/1567_04_08_23_1_44_53.png)
At small scattering angle, we have shown that dσ/dΩ is equal to (https://smf.probesoftware.com/gallery/1567_04_08_23_1_45_18.png) (p.7 of the paper), hence:
(https://smf.probesoftware.com/gallery/1567_04_08_23_1_45_31.png)
This expression is the mean atomic number we presented in Eqs. 6 and 7 in our paper "An Improved Average Atomic Number Calculation for Estimating Backscatter and Continuum Production in Compounds" (Microscopy and Microanalysis (2023)).
I share credit for this explanation with Andrew Ducharme who improved the math and provided a clear interpretation of the probability of the expected atomic number.
Quote from: Probeman on July 19, 2023, 09:29:04 AM
We are very pleased that our new paper on "An Improved Average Atomic Number Calculation for Estimating Backscatter and Continuum Production in Compounds" is now published in Microscopy and Microanalysis:
https://doi.org/10.1093/micmic/ozad069
Following up on the above paper Aurelien and I published earlier this year, we've implemented the new Z based backscatter corrections in both CalcZAF and Probe for EPMA. This new Donovan and Moy (DAM) backscatter correction is available from the ZAF/Phi-rho-Z options as seen here:
(https://smf.probesoftware.com/gallery/395_03_01_24_10_05_53.png)
To compare the new Z based DAM backscatter correction we implemented above, with an existing mass based backscatter correction, we replaced the Love-Scott backscatter correction in the Armstrong-Brown-Love-Scott matrix correction option with a modified PAP backscatter model using the new Z based fits we prepared using PENEPMA backscatter modeling of pure elements and compounds.
Here are the error distributions comparing measured k-ratios (from the Pouchou2.dat input file with 826 binary k-ratios) with k-ratios, first calculated using the existing Love-Scott backscatter correction:
(https://smf.probesoftware.com/gallery/395_03_01_24_9_51_50.png)
Note the average error is 1.00845 +/- 0.047. Now using the the new DAM backscatter correction we obtain this error distribution:
(https://smf.probesoftware.com/gallery/395_03_01_24_9_52_11.png)
with an average error of 1.00537 +/- 0.036, which is a small but significant improvement over the traditional mass based backscatter correction.
We can see the effects more easily using a partial k-ratio dataset which contains only matrices with a significant backscatter correction, e.g., greater than 10%, by calculating the same error distributions using the input file PouchouZ10.dat shown here for the traditional mass based Love-Scott backscatter correction:
(https://smf.probesoftware.com/gallery/395_03_01_24_9_52_32.png)
which shows an average error of 1.0149 +/- 0.044. Next we examine the same partial k-ratio dataset using the new Z based DAM backscatter correction:
(https://smf.probesoftware.com/gallery/395_03_01_24_10_42_09.png)
which gives an average error of 1.0055 +/- 0.027 which is a quite significant improvement over the mass based backscatter model!
For those interested in the details, the new code is available on the Open Microanalysis GitHub repository:
https://github.com/openmicroanalysis/calczaf
in the file ZAF.BAS. The details are summarized here for the backscatter coefficient procedure:
' BSC5 / CALCULATION OF Donovan and Moy BACKSCATTER COEFFICIENTS FOR PURE ELEMENTS (modified Pouchou and Pichoir #3)
ElseIf ibsc% = 5 Then
For i% = 1 To zaf.in0%
yy! = zaf.Z%(i%)
h1! = 0.002415529 * yy! + 0.290281095 * (1# - Exp(-0.0196309 * Exp(1.333873234 * Log(yy!))))
hb!(i%) = h1!
Next i%
End If
' BSC5 / SAMPLE CALCULATION OF Donovan and Moy BACKSCATTER (modified Pouchou and Pichoir #3)
ElseIf ibsc% = 5 Then
zbar! = 0#
For i1% = 1 To zaf.in0%
zbar! = zbar! + zaf.zfrac!(i1%) * zaf.Z%(i1%)
Next i1%
For i% = 1 To zaf.in0%
eta!(i%) = 0.002415529 * zbar! + 0.290281095 * (1# - Exp(-0.0196309 * Exp(1.333873234 * Log(zbar!))))
Next i%
End If
And here for the backscatter correction itself:
' STDBKS10 / Donovan and Moy BACKSCATTER CORRECTION FOR pure elements (Modified Pouchou and Pichoir #7)
ElseIf ibks% = 10 Then
For i% = 1 To zaf.in1%
If zaf.il%(i%) <= MAXRAY% - 1 Then
meanw! = 0.595 + hb!(i%) / 3.7 + Exp(4.55 * Log(hb!(i%)))
u0! = zaf.v!(i%)
ju0! = 1 + u0! * (Log(u0!) - 1#)
Alpha! = (2# * meanw! - 1#) / (1# - meanw!)
gu0! = (u0! - 1# - (1# - Exp((Alpha! + 1#) * Log(1# / u0!))) / (1# + Alpha!)) / (2# + Alpha!) / ju0!
zaf.r!(i%, i%) = 1# - hb!(i%) * meanw! * (1# - gu0!)
End If
Next i%
End If
' SMPBKS10 / Donovan and Moy BACKSCATTER CORRECTION FOR SAMPLE (modified Pouchou and Pichoir #7)
ElseIf ibks% = 10 Then
For i% = 1 To zaf.in1%
zaf.bks!(i%) = 1#
If zaf.il%(i%) <= MAXRAY% - 1 Then
If eta!(i%) <= 0# Then GoTo ZAFBksNegativeEta
meanw! = 0.595 + eta!(i%) / 3.7 + Exp(4.55 * Log(eta!(i%)))
u0! = zaf.v!(i%)
ju0! = 1 + u0! * (Log(u0!) - 1#)
Alpha! = (2# * meanw! - 1#) / (1# - meanw!)
If Alpha < 0# Then GoTo ZAFBksNegativeAlpha
gu0! = (u0! - 1# - (1# - Exp((Alpha! + 1#) * Log(1# / u0!))) / (1# + Alpha!)) / (2# + Alpha!) / ju0!
zaf.bks!(i%) = 1# - eta!(i%) * meanw! * (1# - gu0!)
End If
Next i%
End If
We believe this new Z based backscatter correction method is now ready for general use by the microanalysis community, particularly for compounds containing elements with significant differences in A/Z ratios:
(https://smf.probesoftware.com/gallery/395_03_01_24_11_12_51.png)
Quote from: Probeman on January 03, 2024, 11:14:07 AM
We believe this new Z based backscatter correction method is now ready for general use by the microanalysis community, particularly for compounds containing elements with significant differences in A/Z ratios:
(https://smf.probesoftware.com/gallery/395_03_01_24_11_12_51.png)
A quick and easy way to evaluate the A/Z ratios in various compounds is to use the Output | Calculate Alternative Zbars menu in the Standard application:
https://smf.probesoftware.com/index.php?topic=1111.msg11640#msg11640
One of the co-authors (Ben McMorran) of our paper published last year on calculating average Z for backscatter and continuum productions in compounds:
https://academic.oup.com/mam/article-abstract/29/4/1436/7224307
recently found an old (1918!) paper by William Duane (yes the Duane of the Duane-Hunt limit), that we really wish we had been able to cite in our paper. Here is the first page, and please read the last sentence carefully.
(https://smf.probesoftware.com/gallery/395_18_03_24_11_05_01.png)
I've attached the full paper below but I'm not sure how Ben found it because in Google Scholar all I can find is this biographical paper which only mentions this paper:
https://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/duane-william.pdf
Anyway, what is so cool about this paper from 1913 is that we have for decades accepted the mass averaging assumption of average Z for compounds as noted in every modern textbook on microanalysis, when in fact the early founders of our field apparently already knew the assumption of mass averaging to be false (albeit though only tested on pure elements in the Duane paper).
But the really clever aspect to this paper by Duane and Shimizu is that it takes advantage of the fact that going from Co to Ni the atomic number increases, but the atomic weight *decreases*! :o
In fact there are two other places in the periodic table where this swapping of the trend in atomic number and atomic weight occurs. See if you can think of them before looking at a periodic table- makes for a great science trivia question... one involves Mendeleev and the other is important to biology and mutation rates!
So yes, Duane and Shimizu only showed that characteristic x-ray production correlates with atomic number, but we've know since the early 2000s (Donovan et al.) that continuum and backscatter productions also follow atomic number rather than atomic weight, from isotope measurements (actually we also performed characteristic x-ray measurements on isotopes in the early 2000s, but we only (finally) published that characteristic x-ray data in our paper from last year).
So where did we all (historically in the field microanalysis) go wrong and start ignoring this early work by Duane and Shimizu which clearly demonstrated that atomic weight is irrelevant to our field?
Hey, it's electrodynamics, all the way down... :)