Calculation of mean atomic number

[Donggao Zhao]

Hi, John, a paper by Howell et al. (1998, Scanning, Volume20, Issue1, Pages 35-40) list a few methods for calculating mean atomic number of a compound. What method do you use to calculate MAN for the MAN background correction? Thanks, Donggao

[Probeman]

Hi, John, a paper by Howell et al. (1998, Scanning, Volume20, Issue1, Pages 35-40) list a few methods for calculating mean atomic number of a compound. What method do you use to calculate MAN for the MAN background correction? Thanks, Donggao

Yes, one can find a number of methods to calculate average atomic number in the literature. And not much agreement!  The Standard application has a menu to list a number of these "zbar" calculations as seen here:



But from basic physics we know we can eliminate mass fraction averaging methods from consideration as detailed in these papers:

https://epmalab.uoregon.edu/publ/average_atomic_number_and_electron_backscattering_in_compounds.pdf

https://epmalab.uoregon.edu/publ/Compositional%20Averaging%20of%20Backscatter-%20Reed%20and%20Response%20to%20Reed%20(M&M,%202003).pdf

A discussion on the forum can also be found here:

https://probesoftware.com/smf/index.php?topic=1111.0

From my own work it appears that both backscatter and continuum production can best be modeled by utilizing a Z (electron) Fraction averaging method with an exponent of 0.666 or Z^2/3 which reflects the geometric screening efficiency of the Yukawa Potential model.

I am working on a new paper with several co-authors that should provide additional support for this hypothesis.

For the MAN correction you can use this Z Fraction zbar method as shown here, with any exponent you prefer (with an option to calculate the exponent as energy dependent from Moy et al., 2021):

https://probesoftware.com/smf/index.php?topic=4.msg10036#msg10036

[Donggao Zhao]

Thanks! I realize that this is a rather complicated problem. However, is there a generally accepted simple spreadsheet to calculate the mean atomic number of a compound, such as CaCo3? Looking forward to your new paper!

[Probeman]

I don't have a spreadsheet, but the calculation of a Z Fraction average atomic number is simple.  Here is the code:

Code: [Select]

Sub ConvertCalculateZbarFrac(lchan As Integer, atomfrac() As Single, atomnums() As Integer, exponent As Single, zbar As Single)
' Calculate a Z fraction Zbar based on passed data
'  lchan = number of elements in arrays
'  atomfrac = atomic fractions
'  atomnums = atomic numbers
'  exponent = exponent for fraction calculation
'  zbar = returned zbar based on atomic fractions

ierror = False
On Error GoTo ConvertCalculateZbarFracError

Dim i As Integer
Dim sum As Single

ReDim fracdata(1 To lchan%) As Single

' Calculate sum for fraction
sum! = 0#
For i% = 1 To lchan%
sum! = sum! + atomfrac!(i%) * atomnums%(i%) ^ exponent!
Next i%
If sum! = 0# Then GoTo ConvertCalculateZbarFracBadSum

' Calculate Z fractions (also known as electron fraction)
For i% = 1 To lchan%
fracdata!(i%) = (atomfrac!(i%) * atomnums%(i%) ^ exponent!) / sum!
Next i%

' Calculate Z bar
zbar! = 0
For i% = 1 To lchan%
zbar! = zbar! + fracdata!(i%) * atomnums%(i%)
Next i%

Exit Sub

' Errors
ConvertCalculateZbarFracError:
Screen.MousePointer = vbDefault
MsgBox Error$, vbOKOnly + vbCritical, "ConvertCalculateZbarFrac"
ierror = True
Exit Sub

ConvertCalculateZbarFracBadSum:
Screen.MousePointer = vbDefault
msg$ = "Bad sum in fraction calculation"
MsgBox msg$, vbOKOnly + vbExclamation, "ConvertCalculateZbarFrac"
ierror = True
Exit Sub

End Sub


[Probeman]

Remember, the complete code for CalcZAF and Standard are found here:

https://github.com/openmicroanalysis/calczaf

[Donggao Zhao]

Reed listed MAN for many common minerals, which were calculated using the atomic numbers and the mass fractions (weight %) of the elements in the compound (Reed, 2005, Electron microprobe analysis and scanning electron microscopy in geology. Cambridge university press, p.54-60).

[Probeman]

Yes he did and he was wrong to do so.   

We know, from isotope studies and physics considerations, that mass has a negligible effect in EPMA electron solid interactions. In fact, the use of mass fractions adds a bias to average atomic number calculations equal to the differences in the A/Z ratios of the elements in the compounds.  This is discussed here for backscatter yield but it also applies to continuum average Z calculations because both productions are dominated by Yukawa Potential screening (it's all electrodynamics!):

https://epmalab.uoregon.edu/publ/Compositional%20Averaging%20of%20Backscatter-%20Reed%20and%20Response%20to%20Reed%20(M&M,%202003).pdf

We are currently writing up the physics details of all this in a new paper.

In the meantime Ben Buse also notes the mass bias effects here for backscatter:

https://probesoftware.com/smf/index.php?topic=1111.0

And Ben Wade discusses what he thought was a fluorescence problem but is actually an average Z mass bias problem here for MAN continuum calculations:

https://probesoftware.com/smf/index.php?topic=1221.0

So for example, for a compound such as Mg2SiO4 the A/Z ratios are almost identical so the mass bias doesn't matter:

ELEM:       Si      Mg       O
CONC:    .1996   .3455   .4549
ELEC:    .2000   .3429   .4571
%DIF:    .1895  -.7639   .4972
ATOM:    .1428   .2858   .5714
ELAS:    .2207   .3597   .4196
A/Z :   2.0061  2.0254  2.0000

Zbar (Mass/Electron fraction Zbar % difference) =  .078360

Whereas a compound such as zircon ZrSiO4 gives this:

ELEM:       Zr      Si       O
CONC:    .4976   .1532   .3491
ELEC:    .4651   .1628   .3721
%DIF:  -6.5352  6.2470  6.5733
ATOM:    .1667   .1667   .6667
ELAS:    .5817   .1442   .2741
A/Z :   2.2805  2.0061  2.0000

Zbar (Mass/Electron fraction Zbar % difference) =  3.95775


So the mass bias depends on the specific compound. One of the biggest errors are seen in PbSiO3:

ELEM:       Pb      Si       O
CONC:    .7315   .0991   .1694
ELEC:    .6834   .1166   .1999
%DIF:  -6.5704 17.6732 18.0347
ATOM:    .2001   .2000   .6000
ELAS:    .8119   .0775   .1106
A/Z :   2.5267  2.0061  2.0000

Zbar (Mass/Electron fraction Zbar % difference) =  5.50260
Zbar (Mass fraction) =  62.7264
Zbar (Electron fraction) =  59.2748
Zbar (Elastic fraction) =  68.5453
Zbar (Atomic fraction) =  24.0058

Zbar (Saldick and Allen, for backscatter) =  59.2748
Zbar (Joyet et al.) =  37.7218
Zbar (Everhart) =  78.8973

Zbar (Donovan Z^0.5) =  40.5500
Zbar (Donovan Z^0.667, Yukawa Potential, Z^2/3) =  47.0473
Zbar (Donovan Z^0.70) =  48.3298
Zbar (Donovan Z^0.707, 1/SQRT(2)) =  48.6009
Zbar (Donovan Z^0.80) =  52.1549
Zbar (Donovan Z^0.85) =  54.0144
Zbar (Donovan Z^0.90) =  55.8260
Zbar (Bocker and Hehenkamp for continuum) =  32.3601
Zbar (Duncumb Log(Mass) for continuum) =  46.4013

So your average Z is either 63 (using mass fractions) or 47 (using Z fractions).  That's a big difference!

[Rom]

Greetings!
These data for PbSiO3 have been taken by you from "Output/calculate Alternative Zbars".
My computer gives a bit different numbers, for instance Zbar (Donovan Z^0.667, Yukawa Potential, Z^2/3) =  47.0463  (the same composition, density is 10 gr/cm3) - a little brief question why?

If I open "Standard/List all standard names and average Z", the system gives me Zbar (from mass fraction) =  62.725. Is it only for my knowledge or the system uses these data for calculation if so - where?

You told that in "MAN area of Probe software" the system automatically uses data from the first calculation method of Zbar (not Zbar from mass fraction). But what Zbar data the system uses for ZAF-correction methods?

[Probeman]

These data for PbSiO3 have been taken by you from "Output/calculate Alternative Zbars".
My computer gives a bit different numbers, for instance Zbar (Donovan Z^0.667, Yukawa Potential, Z^2/3) =  47.0463  (the same composition, density is 10 gr/cm3) - a little brief question why?

Density is not important for this calculation. As for a difference in the 5th decimal place of the average Z-bar I would not worry about it.

If I open "Standard/List all standard names and average Z", the system gives me Zbar (from mass fraction) =  62.725. Is it only for my knowledge or the system uses these data for calculation if so - where?

Yeah, it just lists the mass fraction zbar as a default.

You told that in "MAN area of Probe software" the system automatically uses data from the first calculation method of Zbar (not Zbar from mass fraction). But what Zbar data the system uses for ZAF-correction methods?

I did not say that. The MAN calculations uses whatever Zbar option is selected by the user in the Assign MAN Fits dialog.  Mass fraction by default, or Z fraction if selected.

I did say, for the interpolated MAN method, the program automatically utilizes the off-peak background acquisition method for each MAN standard, along with the off peak background fit options selected by the user.

No EPMA matrix corrections utilize the Z fraction calculations at this time.  This is new EPMA physics and the work is continuing. Attached is an abstract describing some of our earlier work on this topic.

[Rom]

"No EPMA matrix corrections utilize the Z fraction calculations at this time."
Could you please explain me more. My knowledge at this point close to zero and I thought 
5 of 10 corrections we can use in PFE use Z-correction (the first correction in ZAF).
Could you recommend me something to return to the right way.
Thank you!   

[Probeman]

"No EPMA matrix corrections utilize the Z fraction calculations at this time."
Could you please explain me more. My knowledge at this point close to zero and I thought 
5 of 10 corrections we can use in PFE use Z-correction (the first correction in ZAF).
Could you recommend me something to return to the right way.

This is recent EPMA physics work, there is not a lot of reading I can suggest except for the discussions on this forum.

If you read the abstract I posted you will see that backscatter is not affected by mass.  This is supported by empirical data and Monte Carlo modeling. 

However, in the analytical physics models in Probe for EPMA (and all other analytical models that I know of),  the Z component of the ZAF correction is combined with the absorption correction in phi-rho-z modeling, and because the absorption correction utilizes mass fractions (e.g., mass thickness because they utilize mass absorption coefficients which are normalized to mass!), they incorrectly conflate these calculations with the backscatter correction which should not be mass based. Electron backscatter (and continuum production for that matter) is solely based on electrodynamics.  Note that the energy loss terms also utilize mass fractions because they utilize mass thickness.

But a properly physics based approach would not utilize mass fractions because mass has essentially no effect on our EPMA physics.  We can see this in more rigorous physics based approaches such as that in Penepma/Penelope Monte Carlo methods, which utilize attenuation coefficients and do not include the implicit mass biases from elements with different A/Z ratios.

You mentioned the example of the difference in Zbar for PbSiO3 when calculated using mass fractions versus Z (Yukawa) fractions. This large difference in Zbar is due to the large difference in the A/Z ratios of Si vs. Pb.  The abstract attached in the post above shows some examples of other compounds with large differences in A/Z.  Consider that for a compound such as PbS, the differences in average Z for mass versus Yukawa Z fraction are ~20%.  Since the backscatter correction contribution to the total matrix correction is approximately 20% in such a material, the relative error we might expect is ~4%, which we can easily observe empirically.

We are currently working on new analytical physics models that will not include these mass biases which should dramatically improve EPMA accuracy in these cases.

[Rom]

".... total matrix correction is approximately 20% in such a material, the relative error we might expect is ~4%, which we can easily observe empirically."

And practically as well: we have several rel.% mistake if we use PbSiO3 standard for Pb and try to measure pure (close to 100%) Pb.

"However, in the analytical physics models in Probe for EPMA (and all other analytical models that I know of),  the Z component of the ZAF correction is combined with the absorption correction in phi-rho-z modeling, and because the absorption correction utilizes mass fractions (e.g., mass thickness because they utilize mass absorption coefficients which are normalized to mass!), they incorrectly conflate these calculations with the backscatter correction which should not be mass based."

From this statement I make a conclusion you have an assumption all popular physics fundamental correction models have a weak point [th]at conflate the mass based corrections with the backscatter (electrodynamic) correction. Am I right?

[Probeman]

".... total matrix correction is approximately 20% in such a material, the relative error we might expect is ~4%, which we can easily observe empirically."

And practically as well: we have several rel.% mistake if we use PbSiO3 standard for Pb and try to measure pure (close to 100%) Pb.

"However, in the analytical physics models in Probe for EPMA (and all other analytical models that I know of),  the Z component of the ZAF correction is combined with the absorption correction in phi-rho-z modeling, and because the absorption correction utilizes mass fractions (e.g., mass thickness because they utilize mass absorption coefficients which are normalized to mass!), they incorrectly conflate these calculations with the backscatter correction which should not be mass based."

From this statement I make a conclusion you have an assumption all popular physics fundamental correction models have a weak point [th]at conflate the mass based corrections with the backscatter (electrodynamic) correction. Am I right?

That is our hypothesis at this time.

You might be interested in this topic also:

https://probesoftware.com/smf/index.php?topic=1111.0

[Probeman]

We added a secondary menu in the Standard application for displaying the standard Zbars calculated using the Yukawa Z fraction (Z^0.667):

[Rom]

It is becoming clear. Thank you!