To avoid excessive complication, consider a periodic start pulse whose period T_s is longer than the combined TAC/MCA dead time T_d. In this case, no start pulses occur when the TAC/MCA cannot respond. The start pulse normally corresponds to the time at which a process is stimulated. The stop input is used to record the time spectrum of the products emitted from that stimulation. The apparatus must be designed to restrict the intensity of the product events so that statistical sampling of the time distribution is possible via single-ion or single-photon counting.

The MCA sorts the analog output of the TAC into a histogram, whose length is equal to the maximum number of channels offered by the MCA. Thus, each channel spans a time interval, Δt, and the start-to-stop time represented by channel i is

t = i Δt           (4)

where i extends from i = 0 to i = i_max. The maximum channel number i_max is typically in the range of 1000 to 16,000.

To demonstrate the minor effect of the detector and timing discriminator dead time, a single, extending dead time, T_e, will be ascribed to that source. T_e is represented in channel numbers by t_e (rounded to the nearest integer value), where

T_e = t_e Δt           (5)

If a time spectrum is accumulated for a preset number of valid start pulses, n₁, and the number of events recorded in channel i is q_i, then the probability of recording an event in channel i for a single valid start pulse is given² by equation (6).

The right-hand side of equation (6) is composed of three probabilities. The probability of an event impinging on the detector and destined for channel i (before dead time losses) is Q_i / n₁. This event cannot be recorded in channel i if it was preceeded by any stop events since the start pulse. The probability of no stop pulses from channel j = 0 to i –1 is given by the first exponential term in equation (6). If the counting rate at the stop input is absolutely zero for i < 0 (no stop pulses preceeding the start pulse) the last exponential term in equation (6) becomes 1. However, most detectors have some low level of background counting rate caused by thermal excitation. Hence, a background stop pulse occuring in the interval from t = –T_e to t = 0 would prevent the desired stop pulses from being detected in the interval from t = 0 to t = T_e. To account for this effect, the last exponential term in equation (6) is the probability of no stop pulses preceeding i = 0 in the time interval t_e. The step function is defined by

U(t_e – i) = 1 for t_e – i > 0         (7)
= 0 for t_e – i ≤ 0

Equation (6) can be used to correct the acquired spectrum, q_i, for dead time losses in order to generate the corrected time spectrum, Q_i. One starts at channel 0 and presumes all Q_j preceeding channel 0 are zero. As one moves channel by channel to the right in the spectrum the Q_j become available from the Q_i calculated for the previous channels. This calculation is repeated until the maximum channel, i_max, has been treated. The resulting set of Q_i is the time spectrum corrected for dead time losses, with one exception. Because the values of Q_i for i ≤ 0 were unknown and presumed zero, the corrected spectrum will be underestimated for values of i up to several times te. This shortcoming can be easily overcome by adding sufficient cable delay to the stop input to move the spectral features of interest out of the affected region. This allows one to ignore the timing discriminator dead time if it is small compared to the measured time span.

Because the counts q_i are sampled from a preset number of start pulses, n₁, the statistical variance in q_i is given by²

Moreover, the variance in the sum of the counts from any channels from j = h to k is

By using a straight-forward propagation-of-errors computation, while ignoring the timing discriminator dead time, the variance in the Q_i calculated via equation (6) is²

The approximation in the last line of equation (10) is highly accurate, because the second term in the square brackets is negligible compared to 1 for practical applications. An alternative expression of the relationship in equation (10) is

In other words, the relative standard deviation in the calculated counts Q_i is determined by the relative standard deviation in the measured counts q_i.

²D.A. Gedcke, Development notes and private communication, Nov.Dec. 1996.