Letter to the Editor:Comments on “CompositionalAveraging of BackscatterIntensities in Compounds”The electron backscatter coefficient ~h! and the relatedcorrection factor for X-ray intensity ~R! are both stronglydependent on atomic number and although quite good dataare available for pure elements, the derivation of values forcompounds is problematic. This issue is addressed by Don-ovan, Pingitore, and Westphal ~Microscopy and Microanaly-sis, Vol. 9, No. 3, June 2003, pp. 202–215!,inwhichan“electron fraction” averaging method is advocated as animprovement on “traditional” mass fraction averaging, whichis known to be only an approximation. ~The difference,according to Table 3, is only significant, however, for sam-ples containing heavy elements such as Pb and Th.! Newthinking on this topic is welcome, but I believe this proposalshould be treated with caution pending more rigoroustesting.Both h and R are determined by geometrical consider-ations, in which electron stopping power must also be takeninto account. Furthermore, R ~but not h! depends on theenergy distribution of the backscattered electrons ~whichaffects the X-ray intensity they would have contributed hadthey not been backscattered!, and it is necessary to know theform of the distribution for a compound. The theoreticalconsiderations offered in support of the proposed averagingmethod are, therefore, incomplete. Likewise, the experimen-tal h data are of limited value, and the observations onisotopes are not only unsurprising but fail to illuminate thequestion of R averaging for compounds.Direct experimental determination of R is difficult, andadjusting the R averaging method to obtain “correct” analyt-ical results is a dubious procedure, given the uncertaintiesin the other corrections. Therefore the best prospect forimprovement in this area is, in principle, to apply MonteCarlo modeling. However, the model needs to be rigorousin its treatment of scattering and energy loss, where it isusual to make simplifying approximations. It should in anycase be noted that there is no a priori reason why any singleR averaging method should be strictly applicable to allcombinations of different elements.S.J.B. ReedDepartment of Earth SciencesUniversity of CambridgeCambridge CB2 3EQUnited KingdomResponse to Reed’s “Commentson ‘Compositional Averaging ofBackscatter Intensities inCompounds’”We agree with many of Stephen Reed’s comments, particu-larly his suggestion that our proposal should be treated withcaution and subjected to rigorous evaluation. This is gener-ically true of any hypothesis in science.Reed states at the outset that h and R are both stronglydependent on atomic number and we agree with this state-ment. Therefore we advocate the atomic number basedelectron fraction approach ~itself an approximation!, in partbecause it contains one less faulty assumption—that massaffects elastic scattering of electrons in solids.The difficulty with the latter assumption can be dem-onstrated by considering that the maximum energy transferto a recoiling nucleus actually occurs in the case of perfectlyelastic scattering with a scattering angle of 1808. ~For inelas-tic scattering, the effect is smaller.! The kinetic energy of therecoiling nucleus of mass M isdE 5 2Mme2~me1 M !2v2where meis the mass of the electron and v is the initialvelocity of the electron. Since meis much smaller than M,me1 M equals M to a very good approximation, so we canexpress this asdE ' 4meMEwhere E is the initial kinetic energy of the electron and dE isthe energy taken from the electron and carried off by therecoiling nucleus. The largest energy loss is for hydrogenwhere me/M ; 1/2000. So for each purely elastic collisionwith protons in pure hydrogen, the electron loses a maxi-mum 0.2% of its kinetic energy. For the vast majority ofcollisions, with small scattering angles, the energy loss ismuch less than this. For heavier nuclei, such as iron, it issmaller still ~byafactorofA!.Thedifference in this quan-tity, for two isotopes of the same element, is even smallerthan this ~e.g., for56Fe and57Fe the difference in fractionalenergy loss per nuclear collision is, at most, 6 3 1027!.The point is, for electrons slowing in solids, nuclearstopping power—the stopping power due to collisions withnuclei—is completely negligible compared with electronic9~6! 03-063 1/210/06/03 2:11 pmMAM03-063Microsc. Microanal. 9, 1–2, 2003DOI: 10.1017/S1431927603030630MicroscopyANDMicroanalysis© MICROSCOPY SOCIETY OF AMERICA 2003stopping power, that is, the stopping power due to collisionswith electrons. For this reason, we agree with Reed that theisotope measurements are unsurprising. However, severalresearchers in the field have publicly expressed surprise andskepticism at this result, so we felt compelled to make thepoint explicitly in our paper—that these measurementsprovide empirical evidence of what is known from funda-mental principles of physics.We further agree, as stated in the paper, that stoppingpowerisinvolvedinRloss, and that the average of thispower loss can be correctly calculated using mass fractionsbut only because mass-based units are utilized in its calcu-lation. It should be clear, as noted above, that the introduc-tion of mass in the stopping power calculation is arbitraryand that these mass-based units eventually cancel out, leav-ing the electron column density as the determining factor inboth stopping power and backscatter averaging. In any case,our paper focuses on the elastic scattering portion of thebackscatter correction, which, as we have shown, is negligi-bly affected by mass.We concur with Reed that a fast Monte Carlo method,which incorporates detailed physics of electron scattering andenergy loss, would be a useful tool, especially for “on-line”microanalysis, but for the moment we must live with simpli-fying approximations. We did perform many high-precisioncalculations to model backscatter averaging in compoundsusing the NIST MQ software but did not include them in thepaper. These calculations show that, for various compounds~BaTiSi3O9,Sb2S3, PbSiO3,etc.!containing elements withmoderate to large differences in A/Z, mass fraction averaging~Fig. 1a! does not perform as well, even when compared tothe “simple” electron fraction model ~Fig. 1b!, where the onlydifference is the substitution of Z for A in the calculation ofaverage Z. After adjusting the “simple” electron fractionmodel, using Z0.8to compensate for
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