Silicon drift detectors (SDDs) revolutionized spectroscopy in fields as diverse as geology and dentistry. For a subset of experiments at ultra-fast, x-ray free-electron lasers (FELs), SDDs can make substantial contributions. Often the unknown spectrum is interesting, carrying science data, or the background measurement is useful to identify unexpected signals. Many measurements involve only several discrete photon energies known a priori, allowing single event decomposition of pile-up and spectroscopic photon counting. We designed a pulse function and demonstrated that the signal amplitude and rise time are obtained for each pulse by fitting, thus removing the need for pulse shaping. By avoiding pulse shaping, rise times of tens of nanoseconds resulted in reduced pulse pile-up and allowed decomposition of remaining pulse pile-up at photon separation times down to hundreds of nanoseconds while yielding time-of-arrival information with precision of 10 nanoseconds. Waveform fitting yields simultaneously high energy resolution and high counting rates (2 orders of magnitude higher than current digital pulse processors). We showed that pile-up spectrum fitting is relatively simple and preferable to pile-up spectrum deconvolution. We developed a photon pile-up statistical model for constant intensity sources, extended it to variable intensity sources (typical for FELs) and used it to fit a complex pile-up spectrum. We subsequently developed a Bayesian pile-up decomposition method that allows decomposing pile-up of single events with up to 6 photons from 6 monochromatic lines with 99% accuracy. The usefulness of SDDs will continue into the x-ray FEL era of science. Their successors, the ePixS hybrid pixel detectors, already offer hundreds of pixels, each with similar performance to an SDD, in a compact, robust and affordable package
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