  Pulse pile-up in pulse-height energy analyzers increases when the incident
rate of pulses increases relative to the inverse of the dead time per pulse of
the detection system. Changes in the observed energy distributions with
incident rate and detector-electronics-formed pulse shape then occur. We focus
on weak high energy tails in X-ray spectra, important for measurements on
partially ionized, warm, pure-hydrogen plasma. A first-principles two-photon
pulse-pile-up model is derived specific to trapezoidal-shaped pulses;
quantitative agreement is found between the measurements and the model
predictions. The modeling is then used to diagnose pulse-pile-up tail artifacts
and mitigate them in relatively low count-rate spectra.
