Response to Reed's “Comments on ‘Compositional Averaging of Backscatter Intensities in Compounds’”

We agree with many of Stephen Reed's comments, particularly his suggestion that our proposal should be treated with caution and subjected to rigorous evaluation. This is generically true of any hypothesis in science.

Reed states at the outset that η and R are both strongly dependent on atomic number and we agree with this statement. Therefore we advocate the atomic number based electron fraction approach (itself an approximation), in part because it contains one less faulty assumption—that mass affects elastic scattering of electrons in solids.

The difficulty with the latter assumption can be demonstrated by considering that the maximum energy transfer to a recoiling nucleus actually occurs in the case of perfectly elastic scattering with a scattering angle of 180°. (For inelastic scattering, the effect is smaller.) The kinetic energy of the recoiling nucleus of mass M is

where m_e is the mass of the electron and v is the initial velocity of the electron. Since m_e is much smaller than M, m_e + M equals M to a very good approximation, so we can express this as

where E is the initial kinetic energy of the electron and dE is the energy taken from the electron and carried off by the recoiling nucleus. The largest energy loss is for hydrogen where m_e /M ∼ 1/2000. So for each purely elastic collision with protons in pure hydrogen, the electron loses a maximum 0.2% of its kinetic energy. For the vast majority of collisions, with small scattering angles, the energy loss is much less than this. For heavier nuclei, such as iron, it is smaller still (by a factor of A). The difference in this quantity, for two isotopes of the same element, is even smaller than this (e.g., for ⁵⁶Fe and ⁵⁷Fe the difference in fractional energy loss per nuclear collision is, at most, 6 × 10⁻⁷).

The point is, for electrons slowing in solids, nuclear stopping power—the stopping power due to collisions with nuclei—is completely negligible compared with electronic stopping power, that is, the stopping power due to collisions with electrons. For this reason, we agree with Reed that the isotope measurements are unsurprising. However, several researchers in the field have publicly expressed surprise and skepticism at this result, so we felt compelled to make the point explicitly in our paper—that these measurements provide empirical evidence of what is known from fundamental principles of physics.

We further agree, as stated in the paper, that stopping power is involved in R loss, and that the average of this power loss can be correctly calculated using mass fractions but only because mass-based units are utilized in its calculation. It should be clear, as noted above, that the introduction of mass in the stopping power calculation is arbitrary and that these mass-based units eventually cancel out, leaving the electron column density as the determining factor in both stopping power and backscatter averaging. In any case, our paper focuses on the elastic scattering portion of the backscatter correction, which, as we have shown, is negligibly affected by mass.

We concur with Reed that a fast Monte Carlo method, which incorporates detailed physics of electron scattering and energy loss, would be a useful tool, especially for “on-line” microanalysis, but for the moment we must live with simplifying approximations. We did perform many high-precision calculations to model backscatter averaging in compounds using the NIST MQ software but did not include them in the paper. These calculations show that, for various compounds (BaTiSi₃O₉, Sb₂S₃, PbSiO₃, etc.) containing elements with moderate to large differences in A/Z, mass fraction averaging (Fig. 1a) does not perform as well, even when compared to the “simple” electron fraction model (Fig. 1b), where the only difference is the substitution of Z for A in the calculation of average Z. After adjusting the “simple” electron fraction model, using Z^0.8 to compensate for nuclear screening by inner orbital electrons in higher atomic number atoms (based on experimental backscatter measurements of the Cu-Ag-Au alloys), our model performs surprisingly well (Fig. 1c).

Backscatter intensity calculations using NIST MQ software (1,000,000 trajectories for each data point at 20 keV) for a number of compounds containing elements with moderate to large differences in A/Z. Average Z calculated using mass averaging (a), “simple” electron fraction averaging (b), and electron fraction adjusted for nuclear screening by inner orbital electrons (c).

We are not making the claim that this model is completely rigorous in its treatment of electron scattering and energy loss. It is yet another semi-empirical approximation that may offer some benefit to the analyst in situations where a large atomic number correction is required and problems have been observed with current methods.

While we are encouraged by these results and others described in our paper, we know that this approximation can be significantly improved. We therefore hope that other researchers, including Stephen Reed, will work to quantitatively evaluate this approximation.