A statistical model of secondary ion emission and attenuation clarifies disparities in quasi-simultaneous arrival coefficients measured with secondary ion mass spectrometry

Rationale: Secondary ion mass spectrometry data collected using electron multiplier detectors are subject to a correction for the quasi-simultaneous arrival (QSA) effect. Published Poisson statistical models indicate that the QSA coefficients, β, should have an invariant value of 0.5, whereas, with one exception, published experimental determinations vary between 0.6 and 1.0, with a mean value of 0.75. Methods: We developed a more complex model, combining both ion emission and attenuation, that predicts the observed range in measured β and elucidates the mechanism of secondary ion formation. For a given aperture setting, any secondary ion has an equal probability of successful transit to the electron multiplier. Binomial statistics can model pass–fail aperture attenuation but require probability distributions of the quasi-simultaneously emitted (QSE) ion tally, per primary ion, as input. Assuming (a) that each primary ion impact results in 0, 1, 2,… secondary ion emissions, randomly, with an average K_s and (b) that there is finite probability (P2) of a further emission process dependent on K_s, the required QSE probability distributions were generated via a combined Poisson–binomial statistical model. Results: The value of β was output as a function of K_s and P2. For values of P2 > 0 and any value of K_s, β always exceeds 0.5. As P2 → 0, β → 0.5; for values of increasing P2 > 0.5 and decreasing K_s < 0.5, β → >1. Conclusions: Were the emission of one ion not to influence the probability of the formation of a second (i.e. model output for P2 = 0), β should always be 0.5. Yet measurements have never reported this value. Consequently, assuming that published β values are correct, emissions of QSE secondary ions do not occur independently, and it may be inferred that there are linked mechanisms of secondary ion formation as shown here.