Dead time, or resolving time, of a counting system is defined as the minimum time that can elapse between the arrival of two successive particles at the detector and the recording of two distinct pulses. The components of dead time consist of the time it takes for the formation of the pulse in the detector itself and for the processing of the detector signal through the preamplifier-amplifi- er-discriminator-scaler (or preamplifier-amplifier-MCA). With modern electron- ics, the longest component of dead time is that of the detector, and for this
74 MEASUREMENT AND DETECTION OF RADIATION
reason, the term "dead time" means the dead time of the detector. The dead-time component of the preamplifier-amplifier-discriminator-scaler can be ignored with any type of detector.
Because of counter dead time, the possibility exists that some particles will not be recorded since the counter will not produce pulses for them. Pulses will not be produced because the counter will be "occupied" with the formation of the signal generated by particles arriving earlier. The counting loss of particles is particularly important in the case of high counting rates. Obviously, the ob- served counting rate should be corrected for the loss of counts due to counter dead time. The rest of this section presents the method for correction as well as a method for the measurement of the dead time.
Suppose 7 is the dead time of the system and g the observed counting rate.
The fraction of time during which the system is insensitive is g7. If n is the true counting rate, the number of counts lost is n(g7). Therefore
and
Equation 2.108 corrects the observed gross counting rate g for the loss of counts due to the dead time of the counter.
Example 2.26 Suppose 7 = 200 p s and g = 30,000 counts/min. What frac- tion of counts is lost because of dead time? What is the true counting rate?
Answer The true counting rate is
Therefore, dead time is responsible for loss of 555 - 500 - 55
555 555 - 10% of the counts
Notice that the product g~ = 0.10, i.e., the product of the dead time and the gross counting rate, is a good indicator of the fraction of counts lost because of dead time.
The dead time is measured with the "two-source" method as follows. Let n,,n,, n,, be the true gross counting rates from the first source only, from the second source only, and from both sources, respectively, and let n, be the true background rate. Let the corresponding observed counting rates be g,, g,, g,,, b.
The following equation holds:
True net True net True net
(counting rate) ,+ (counting rate) ,= (counting rate or
n, + n, = n,, + n, Using Eq. 2.108,
It will be assumed now that b r 4 1, in which case,
(If b~ is not much less than 1, the instruments should be thoroughly checked for possible malfunction before proceeding with the measurement.)
The dead time T can be determined from Eq. 2.109 after g,, g,, g,,, and b are measured. This is achieved by counting radioactive source 1, then sources 1 and 2 together, then only source 2, and finally the background after removing both sources. Equation 2.109 can be rearranged to give:
Equation 2.110 is a second-degree algebraic equation that can be solved for T. It was derived without any approximations.
If the background is negligible, Eq. 2.110 takes the form
g l g 2 g l 2 ~ ~ - 2g1g27 + gl + g2 - g12 = 0 (2.111) Solving for T,
When dead-time correction is necessary, the net counting rate, called "true net counting rate," is given by
It is assumed that the true background rate has been determined earlier with the standard error ub. The standard error of r, a,, is calculated from Eq. 2.113 using Eq. 2.84. If the only sources of error are the gross count G and the
76 MEASUREMENT AND DETECTION OF RADIATION
background, the standard error of r is
If there is an error due to dead-time determination, a third term consisting of that error will appear under the radical of Eq. 2.114.
PROBLEMS
2.1 What is the probability when throwing a die three times of getting a four in any of the throws?
2.2 What is the probability when drawing one card from each of three decks of cards that all three cards will be diamonds?
2.3 A box contains 2000 computer cards. If five faulty cards are expected to be found in the box, what is the probability of findiag two faulty cards in a sample of 250?
2.4 Calculate the average and the standard deviation of the probability density function f(x) =
l/(b - a) when a I x I b. (This pdf is used for the calculation to round off errors.)
2.5 The energy distribution of thermal (slow) neutrons in a light-wave reactor follows very closely the Maxwell-Boltzmam distribution:
N ( E ) dE = ~ & e - ~ / ~ ~ d~
where N(E) dE = number of neutrons with kinetic energy between E and E + dE
k = Boltzmann constant = 1.380662 X 1 0 - 2 3 ~ / 0 K T = temperature, K
A = constant Show that
(a) The mode of this distribution is E = i k T . (b) The mean is E = + k ~ .
2.6 If the average for a large number of counting measurements is 15, what is the probability that a single measurement will produce the result 20?
2.7 For the binomial distribution, prove N
(a) pi:), = 1 (b) 3 = p N (c) u 2 = m(l - p )
n = O
2.8 For the Poisson distribution, prove
2.9 For the normal distribution, show
rn
a P = 1 (b) i = m (c) the variance is o 2
2.10 If n,, n,, . . . , n, are mutually uncorrelated random variables with a common variance u 2 , show that
N - 1 ( n i - Ti) = - u
N
2.11 Show that in a series of N measurements, the result R that minimizes the quantity
is R = E, where 7i is given by Eq. 2.31.
2.12 Prove Eq. 2.62 using tables of the error function.
2.13 As part of a quality control experiment, the lengths of 10 nuclear fuel rods have been measured with the following results in meters:
What is the average length? What is the standard deviation of this series of measurements?
2.14 At a uranium pellet fabrication plant the average pellet density is 17 X lo3 kg/m3 with a standard deviation equal to lo3 kg/m3. What is the probability that a given pellet has a density less than 14 x lo3 kg/m3?
2.15 A radioactive sample was counted once and gave 500 counts in 1 min. The corresponding number for the background is 480 counts. Is the sample radioactive or not? What should one report based on this measurement alone?
2.16 A radioactive sample gave 750 counts in 5 min. When the sample was removed, the scaler recorded 1000 counts in 10 min. What is the net counting rate and its standard percent error?
2.17 Calculate the average net counting rate and its standard error from the data given below:
2.18 A counting experiment has to be performed in 5 min. The approximate gross and background counting rates are 200 counts/min and 50 counts/min, respectively.
(a) Determine the optimum gross and background counting times.
(b) Based on the times obtained in (a), what is the standard percent error of the net counting rate?
2.19 The strength of a radioactive source was measured with a 2 percent standard error by taking a gross count for time t min and a background for time 2t min. Calculate the time t if it is given that the background is 300 counts/min and the gross count 45,000 counts/min.
2.20 The strength of radioactive source is to be measured with a counter that has a background of 120 f 8 counts/min. The approximate gross counting rate is 360 counts/min. How long should one count if the net counting rate is to be measured with an error of 2 percent?
2.21 The buckling B 2 of a cylindrical reactor is given by
where R = reactor radius H = reactor height
If the radius changes by 2 percent and the height by 8 percent, by what percent will B 2 change? Take R = 1 m, H = 2 m.
78 MEASUREMENT AND DETECTION OF RADIATION
2.22 Using Chauvenet's criterion, should any of the scaler readings listed below be rejected?
115 121 103 151
121 105 75 103
105 107 100 108
113 110 101 97
110 109 103 101
2.23 Using the data of Prob. 2.13, what is the value of accepted length x , if the confidence limit is 99.4 percent?
2.24 Prove that for radioactivity measurements the value of MDA is given by the equation MDA = k 2 + 2CDL, if k, = kS = k. Hint: when n = MDA, the variance u 2 = MDA + a:.
2.25 A sample was counted for 5 min and gave 2250 counts; the background, also recorded for 5 min, gave 2050 counts. Is this sample radioactive? Assume confidence limits of both 95% and 90%.
2.26 Determine the dead time of a counter based on the following data obtained with the two-source method:
g l = 14,000 counts/min g 1 2 = 26,000 counts/min g2 = 15,000 counts/min b = 50 counts/min
2.27 If the dead time of a counter is 100 ps, what is the observed counting rate if the loss of counts due to dead time is equal to 5 percent?
2.28 Calculate the true net activity and its standard percent error for a sample that gave 70,000 counts in 2 min. The dead time of the counter is 200 ps. The background is known to be 100 f 1 counts/min.
2.29 Calculate the true net activity and its standard error based on the following data:
G = 100,000 counts obtained in 10 min B = 10,000 counts obtained in 100 min The dead time of the counter is 150 ps.
BIBLIOGRAPHY
Arley, N., and Buck, K. R., Introduction to the Theory of Probability and Statistics, Wiley, New York, 1950.
Beers, Y., Introduction to the Theory of Error, Addison-Wesley, Reading, Mass., 1957.
Bevington, P. R., Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, 1969.
Chemical Rubber Company, Handbook of Chemistry and Physics, Cleveland, Ohio.
Chemical Rubber Company, Standard Mathematical Tables, Cleveland, Ohio.
Eadie, W. T., Dryard, D., James, F. E., Roos, M., and Sadoulet, B., Statistical Methods in Experimental Physics, North-Holland, Amsterdam, 1971.
Jaech, J. L., "Statistical Methods in Nuclear Material Control," TID-26298, U.S. Atomic Energy Commission, 1973.
Johnson, N. L., and Leone, F. C., Statistics and Enperimental Design, Wiley, New York, 1964, Chaps.
1, 3, 4, and 5.
Smith, D. L., "Probability, Statistics, and Data Uncertainties in Nuclear Science and Technology,"
OECD/NEA, Vol. 4, American Nuclear Society, La Grange Park, Illinois, 1991.
REFERENCES
1. Currie, L. A., Anal. Chem. 40(3):586 (1968).
2. Roberson, P. L., and Carlson, R. D., Health Phys. 62:2 (1992).
3. Flanigan, J. A., Rad. Prot. Mgl. Nov.-Dec.:37 (1993).