An Update on Electron Backscatter and Mass Effects

In 2003 we (Donovan, Pingtore and Westphal) published a paper on the lack of a significant mass effect for electron backscatter intensities (from both theoritical considerations and experimental measurements) for the purposes of improving the calculation of electron loss in quantitative analysis in EPMA, where the calculation is traditionally based on the mass fraction weighted sum of the atomic numbers in a compound. The original paper and a comment from Reed along with a response by us are linked below:

• Donovan, et. al., "Compositional Averaging of Backscatter Intensities in Compounds" (M&M, 2003)
• Donovan, "Compositional Averaging of Backscatter Intensities in Compounds"- Comment by Reed and Response to Reed (M&M, 2003)

Although not included in our original paper, in the response to Reed we noted that Monte-Carlo backscatter calculations using NIST's MQ software showed agreement with our proposed simple electron fraction formulation using a Z^0.8 term to account for nuclear screening effects for high atomic number elements.

Subsequently one of us (Donovan) followed up these calculations using the much more rigorous Penepma (Penelope) Monte-Carlo software and the results are linked below:

• Penepma BSE calculations with 1,000,000 electron trajectories for pure elements and various compounds
• Penepma BSE calculations for electron fraction Z^0.7 only for pure elements and various compound

It is interesting to note that whereas the NIST MQ software gave a best fit using an electron fraction term of Z^0.8, the Penepma software gives a best fit using Z^0.7, though the differences are very small. For comparison, the experimental data in our paper (Fig. 6 on page 213) showed the best fit to the Au-Cu-Ag experimental data was Z^0.8 and the best fit to the moderate atomic number materials was around Z^0.7, which is consistent with both sets of Monte-carlo calculations.

An additional note is with regard the so called "Heinrich Kink", where in a paper published in 1968, Kurt Heinrich claimed a mass effect on backscatter intensities as a function of atomic number. Unfortunately this hypothesis was not only in conflict with the well known fact that electrostatic effects dominate gravitational (mass) effects by many orders of magnitude at the atomic scale, but also with the basic physics of momentum exchange. That is to say, the largest mass effect possible is with an electron interacting with a hydrogen atom where in the case of elastic scattering with 180 degrees of momentum exchange, an effect of 4/2000 (0.2%) is possible as described in the response to Reed's comments. However, this tiny momentum effect is even smaller in electron interactions with heavier atoms.

In fact Heinrich's complete dataset (taken from a table in his 1968 paper and plotted here) shows that this correlation between backscatter yield and A/Z does not hold true for elements outside the selected elements he showed graphically in the same paper.

Subsequently, in our 2003 paper we empirically showed this claimed mass effect on backscatter yields to be false, based on isotope measurements of pure elements where the only difference was atomic mass. Careful measurements of these materials with the same atomic number, but different masses, clearly showed no effect on electron backscatter, continuum or characteristic x-ray productions at the precision levels attained in multiple measurements.

However, the experimentally measured variation of backscatter intensity (as originally shown by Heinrich in 1968) is valid and can be reproduced with careful absorbed current measurements using a voltage biased sample as shown in the following link:

• Comparison of experimental BSE (absorbed current) measurements with Monte-carlo calculations using 10^7 trajectories

These measured BSE variations may have a basis in electron channeling effects or perhaps different surface oxidation rates due to alternating electron shell filling in this area of the periodic table, as the measured effects are very small. However, what is clear from both the isotope measuremens and physical theory is that these variations are not related to atomic weight (shown as a scaled A/Z curve in the above graphs).

Subsequent high precision modeling using Penepma (Penelope) with even 100,000,000 electron trajectories shows yet again that these BSE variations are not correlated with mass effects as claimed by Heinrich:

• Penepma BSE calculations with 10^8 electron trajectories

In summary, the use of mass fractions to calculate average atomic numbers or backscatter yields in compounds is convenient , but not physically based. A more accurate formulation is obtained by using a fractional averaging method based on atomic number, with a correction for nuclear screening effects for higher atomic number elements as we have shown. An even more accurate calculation could utilize tables of empirical elastic scattering cross sections and a fractional averaging method based on atomic numbers and the number of atoms in a compound as we have proposed.

We are pleased to accept further suggestions from commentators.

John Donovan, 03/09/2011
donovan@uoregon.edu