T_D = T_P + T_W

where T_W is the width of the pulse above the noise level, and T_P is the time from the start of the pulse until the point at which the subsequent ADC detects peak amplitude and closes its linear gate  (Fig. 22). Note that the period T_P receives double weighting because a second pulse that arrives during this period also causes the first pulse to be rejected due to pile-up. The dead time is an extending dead time, and the unpiled-up output rate r_o for the amplifier is related to the input counting rate r_i from the detector by the throughput equation

r_o = r_i exp[–r_i (T_P + T_W)]

Figure 24 illustrates this equation for amplifier shaping time constants ranging from 0.5 to 10 µs. The amplifier output counting rate reaches its maximum when r_i = 1/T_D. It is clear from Fig. 24 that higher counting rates require shorter shaping time constants.

When the ADC is part of the spectroscopy system, the dead times of the amplifier and the ADC are in series. The combination of the amplifier extending dead time followed by ADC non-extending dead time T_M yields a throughput described by

r_i
r_o =                                                                                                    
exp[r_i(T_W + T_P)] + r_i [T_M – (T_W – T_P)] U [ T_M – (T_W – T_P)]

where U [T_M – (T_W – T_P)] is a unit step function that changes value from 0 to 1 when T_M is greater than (T_W – T_P).

Figure 24. Plot of the Unpiled-Up Amplifier Output Rate as a Function
of Input Rate for Six Values of Shaping Time Constants.