Dead Time Effects in Counting and Timing

Dead Time Effects in Counting or Timing

The dead time loss experienced when using fast timing discriminators for either counting or timing measurements is typically controlled by the counting rate and two dominant cascaded dead times, T_e and T_ne. T_e is the extending dead time caused by the width of the analog pulse at the noise discriminator threshold. It is an extending dead time because a second analog pulse occurring during a preceeding pulse extends the dead time by one pulse width from the second pulse’s arrival time, and the second pulse will not be counted. T_e is normally determined by the detector response and any pulse shaping added by an amplifier interposed between the detector and the discriminator. T_ne is the longest non-extending dead time following the noise discriminator. Non-extending dead time implies that a pulse arriving during the dead time created by a previous pulse will not be recorded and will not extend the dead time. The minimum value for Tne is the dead time set by the output driver for the discriminator. If the discriminator drives another device which also contributes a non-extending dead time, T_ne is assigned the value of the larger of the two non-extending dead times. This latter assignment is an adequately accurate representation of Tne for most practical cases.^{1, 2}

For the general case in time spectrometry where R(t) (the instantaneous counting rate of photons or ions at the detector) varies with time, the instantaneous counting rate after dead time losses, r(t), is given by³

Cascaded Dead Times, Variable Counting Rate:

(2a)

where time t₁ is distinguished from t only for the purpose of integration over the time interval, and U(T_ne–T_e) is a unit step function defined by

U(T_ne–T_e) = 0 for T_ne ≤ T_e
= 1 for T_ne > T_e (2b)

In this special case of variable counting rate, R(t) and r(t) can be interpreted as the probability per unit time of observing an event at the input to the detector and at the output of the cascaded dead times, respectively, at the time t. In a practical measurement, a process is stimulated at time t = 0, and R(t) represents the probability of events from the process arriving at the detector as a function of time. Because of the cascaded dead times, the probability of recording events as a function of time is given by r(t). To build up a statistically significant time spectrum, one must repeat the stimulation n times (where n is a large number) while summing the resulting time spectra. Generally, this is accomplished in a digital histogramming memory, which has finite time bin widths, Δt. Consequently, the number of counts recorded in a bin at time t is predicted to be

q(t) = n r(t) Δt (3)

The time spectrum recorded in histogram form is described by q(t). The practical application of equations (2) and (3) to time digitizers is explained in the introduction to Counters, Ratemeters, and Multichannel Scalers.

For the simplifying case where R(t) is constant over time, R(t) = R and r(t) = r, leading to

Cascaded Dead Times, Constant Counting Rate:

r = R exp [–RT_e] [1–U(T_ne–T_e) r (T_ne–T_e)] (4a)

(4b)

If the extending dead time is larger than the non-extending dead time, then the non-extending dead time is irrelevant and equations (2) and (4) become:

Extending Dead Time, Variable Counting Rate:

(5)

Extending Dead Time, Constant Counting Rate:

r = R exp[–RT_e] (6)

Equations (5) and (6) are the equations for a single, extending dead time.

If the extending dead time is negligible compared to the non-extending dead time, then equations (2) and (4) simplify to the relations for a single non-extending dead time:

Non-Extending Dead Time, Variable Counting Rate:

(7)

Non-Extending Dead Time, Constant Counting Rate:

r = R [1–rT_ne] (8a)

(8b)

The above equations allow one to estimate the dead time losses when the counting rate is known (or predicted) and the dead times T_e and T_ne have been measured (either by an oscilloscope or by graphing r versus R). These equations can also be used to correct or the dead time losses if the losses are not excessive. If the dead time losses are less than 15%, the extending, non-extending, and cascaded dead time equations all yield values of r(t)/R(t) that agree within 1%, provided

T = T_e + U(T_ne–T_e) (T_ne–T_e) (9)

is substituted for the single dead time in the extending and non-extending equations. This permits considerable simplification of the computation in exchange for a tolerable limit on the dead time loss.

¹Jörg W. Müller, Nucl. Instr. Meth. 112, (1973), 47–57; Figure 3.
²D.R. Beaman, et al., J. of Physics E: Sci. Instr. (1972), 5, 767–776.
³D.A. Gedcke, Development notes and private communication, Nov. 1996.