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 or 0.05%. 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:

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):

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:

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. The sharp eye will notice that we stated the maximum mass effect for an electron interacting with a hydrogen nucleus is 0.2% but that was a typo, since 1/2000 is actually 0.05%. Doh!

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