CAMAC Instruments -Counting Statistics with Finite Dead Time

Counting Statistics with Finite Dead Time

If the amplifier and multichannel analyzer had zero dead time, the statistical variance in the counts recorded in any channel of memory would be σ_q²= q, where q is the number of counts recorded in the channel during a counting time t. However, the dead times in the amplifier and the MCA not only supress the recorded counts according to equation (2), but they alter the variance as well. Several authors have calculated the effect of the dead time on the variance for systems incorporating a single dead time of either the extending or non-extending type.* Although the equation for cascaded dead times is not readily available, the single dead time equations indicate that the variance for the recorded counts can be expected to be less than q. Furthermore, this deviation from σ_q² = q is highly sensitive to the percent dead time losses.

One way to correct for the dead time losses is to measure the counts, q, recorded in the "real" time t, and use equation 2 to calculate the counts, Q, that would have been observed with zero dead time. (The "real" time is the time measured by a clock that does not turn off during dead time intervals.) Under those circumstances, the statistical variance in the corrected counts calculated via equation (2) will be larger than σ_Q² = Q, and the magnification will escalate with increasing percent dead time.* In other words, dead time losses degrade the accuracy of the calculated detector counting rate.

A more practical alternative is to use a live time clock to correct for the dead time losses. An "ideal" live time clock* is a clock that a) is turned off for the entire time that the spectrometer is unable to record an event arriving at the detector, and b) records one event for each dead time interval. A live time clock is applicable only to random events uniformly distributed in time (constant counting rate). If the events at the detector obey Poisson statistics, then it can be shown that the variance in the number of events, m, recorded in the live time, t_L, is*

σ_m² = m (3)

The counts at the detector before dead time losses can be calculated as

(4)

or the counting rate at the detector can be computed as

r_i = M/t = m/t_L (5)

It follows rather simply that the percent standard deviation in r_i, r_o, M or m is given by

(6)

Table 1 summarizes the number of counts required to reach a desired level of precision in measuring the counting rate, r_i.

Table 1. Statistical Precision with an Ideal Live Time Clock.

Number of Counts in Live Time t_L	Percent Standard Deviation
1	100%
100	10%
10,000	1%
1,000,000	0.1%

Equation (3) can also be extended to the sum of the counts over any number of channels in the MCA memory, i.e.,

(7)

where N is the sum of the counts m_i in channels j through k. * shows how this variance applies to the subtraction of background under peaks.

*Ron Jenkins, R.W. Gould, and Dale Gedcke, Quantitative X-Ray Spectrometry, (New York and Basel: Marcel Dekker, Inc.,)1981, pp. 209-287, First Edition.