## letters to the editor

## Comments on *Treatment of EXAFS data taken in the fluorescence mode in non-linear conditions* by G. Ciatto *et al*. (2004). *J. Synchrotron Rad.* **11**, 278–283

^{a}X-ray Instrumentation Associates, 8450 Central Avenue, Newark, CA 94560, USA^{*}Correspondence e-mail: bill@xia.com

We applaud the recent efforts of Ciatto, d'Acapito, Boscherini & Mobilio [*J. Synchrotron Rad.* (2004), **11**, 278–283] to extend the useful operating range of data collection as presented in their recent article *Treatment of EXAFS data taken in the fluorescence mode in non-linear conditions*. Methods that work accurately at ever higher input counting rates not only produce more science in less time but also allow synchrotron sources to be used more effectively. In support of these goals, we would like to suggest that, not only can the accuracy of their method II be further increased, but that it should also have a much wider range of applicability than the authors indicate.

In method II, the measured number of counts *m*_{F}(*E*) in the fluorescence window at each measured energy *E* in the spectrum is corrected to obtain the `true' number of counts *n*_{F}(*E*) that would have been measured in the absence of pile-up according to

where *m*_{T}(*E*) is the total number of counts placed into the complete spectrum during the measurement at *E* and *n*_{T}(*E*) is the `true' number of photons received by the detector during the measurement period. We assert that this correction, optimally applied, is not only always valid but will also give superior results, independent of whether the fluorescence signal is a large or a small fraction of the total input count rate. Our assertion is based on the observation that method II, as represented by the authors through equation (1), is a standard form of spectrometer dead-time correction and can in principal achieve perfectly corrected results provided that dead times can be accurately estimated. Our hope, in writing this *Letter*, is to encourage further experimental testing of this assertion since analyses are particularly sensitive to dead-time losses and thus to the accuracy of their correction.

In the first place, we observe, from the paper's Table 1, that method II gives extremely accurate interatomic distances at all input count rates. The ^{−1}. We suggest that the major source of this error is that the counts from the fast channel in the DXP electronics are themselves suffering from dead-time losses at these input counting rates so that the true input is not being accurately estimated. That is, it is important to remember that the fast channel also suffers dead-time losses according to the paralyzable dead-time formula

where *n*_{Tt}(*E*) is the true number of events input to the fast channel and τ_{f} is the fast-channel's Thus, if the fast-channel's is 0.8 µs, as reported, then, for example, the measured input count rate *n*_{T} of 132 kcount s^{−1} corresponds to a true *n*_{Tt} of about 148.7 kcount s^{−1}, a difference of 12.6%. The effect of this correction can be roughly estimated by scaling the reported 3.47 by 12.6%, which gives 3.91, much closer to the true value. Similar corrections apply at the other input count rates and bring all coordination numbers much closer to the true value. This correction, of course, is only being performed with an estimated fast-channel and, further, is only an approximation to the more rigorous approach of making the fast-channel dead-time correction at each energy *E* and then repeating the data analysis. This estimate strongly suggests, however, that the combination of more accurately measuring τ_{f} and including fast-channel dead-time corrections in the analysis would further enhance the accuracy of the derived coordination numbers, especially at the highest input counting rates where the error is largest.

In the second place, we contend that method II does not work in the case presented by the authors because the majority of input counts occur in the fluorescence window and that therefore the correction signal contains the *E.* Rather, it works to the extent to which the spectrometer's dead-time terms are accurately known and its dead-time losses being accurately corrected for as the input varies with *E.* Thus, if it works well at high concentrations where both the input and required dead-time correction are varying rapidly with *E*, it should work at least as well, and quite possibly even better, in the dilute limit where the input is nearly constant with *E.* In this case the dead-time correction terms will be essentially independent of *E*, as may be appreciated *via* the following simple thought experiment.

Consider the case where we are just below the ^{−1}). At this point (initially neglecting fast-channel dead-time effects) we have *m*_{T} output counts, *n*_{T} input counts and no fluorescence counts (an approximation for the sake of argument) and *m*_{T}/*n*_{T} = 1/e from differentiating the dead-time formula. Assuming the sample is dilute, if we now measure slightly above the the fluorescence will turn on, *n*_{T} will become *n*_{T} + δ, where the edge jump δ << *n*_{T}, while *m*_{T} will remain unchanged owing to our operating point. But how many fluorescent counts will we see in the spectrum? Not zero, because the pile-up is not an energy-dependent phenomenon and affects fluorescent and elastic photons equally. Therefore, if below the edge the fraction of counts placed into the total spectrum was *m*_{T}/*n*_{T}, then above the edge it will be essentially the same, since δ << *n*_{T}. Hence, approximately δ*m*_{T}/*n*_{T} counts will be placed into the fluorescence window and (*n*_{T} − δ)*m*_{T}/*n*_{T} counts into the elastic scatter peak. From this, we see immediately that we can accurately recover δ by multiplying the fluorescent counts by *n*_{T}/*m*_{T} = e and that this correction depends upon correctly estimating the input rather than the magnitude of δ. To refine this correction we should further correct for the fast-channel as discussed earlier. From this result we learn that (i) the method II correction factor *n*_{T}/*m*_{T} works in the limits of both large and small percentage fluorescence signal, and (ii) not surprisingly the amount of signal in the correction factor scales with the fraction of the total counts that are fluorescence counts simply because those fluorescent counts are contributing to the total input and thus to the spectrometer's dead time.

In conclusion, we would be delighted to see the authors correct their present results for fast-channel

and then repeat their careful measurements using both dilute and intermediate samples so that we can truly see just how far non-linear data collection can be pushed for a broad range of experimental conditions.