Page 251 - Radiochemistry and nuclear chemistry

Detection and Measurement Techniques 235

The standard deviation o is given in such cases by
o = d'N (8.18)

In these equations P(N) is the probability of the occurrence of the value N while/~ is the
arithmetic mean of all the measured values and N is the measured value. Figure 8.23 shows
the Poisson and Gaussian distributions for 29 = 20 counts. The standard deviation
("statistical error'), according to the theory of errors, indicates a 68 % probability that the
measured value is within +o of the average "true" value ~. For 100 measured counts the
value 100 5- 10 indicates that there is a 68 % probability that the "true" value will be in the
interval between 90 and 110 counts. If the error limit is listed as 2a the probability that the
"true" count will be between these limits is 95.5 %; for 30 it is 99.7 %. Figure 8.24 shows
the relationship between K, the number of standard deviations, and the probability that the
true figure lies outside the limits expressed by K. For example, at K = 1 (i.e. the error is
+ lo) the figure indicates that the probability of the true" value being outside N + o is 0.32
(or 32 %). This agrees with the observation that the probability is 68 %, i.e. the "true" value
is within the limits of + lo. Figure 8.24 can be used to establish a "rule of thumb" for
rejection of unlikely data. If any measurement differs from the average value by more than
five times the probable error it may be rejected as the probability is less than one in a
thousand that this is a true random error. The probable error is the 50 % probability which
corresponds to 0.67a.
From the relationship of o and N it follows that the greater the number of collected counts
the smaller the uncertainty. For high accuracy it is obviously necessary to obtain a large
number of counts either by using samples of high radioactivity or by using long counting
times.
In order to obtain the value of the radioactivity of the sample, corrections must be made
for background activity. If our measurements give N + o counts for the sample and N O +
o 0 for the background count, the correct value becomes

N~o.= (N-No) +_ (o 2 + ~o2) '~ (8.19)

If the sample was counted for a time At and the background for a time At o, the measured
rate of radioactive decay is

R = N/At- NoIAt 0 + [(cr/At) 2 + (aoIAto)2] V2 (8.20)

It is extremely important in dealing with radioactivity to keep in mind at all times the
statistical nature of the count rate. Every measured count has an uncertainty and the
agreement between two counts can only be assessed in terms of the probability reflected in
terms of o.
The statistical nature of radioactive decay also leads to an uneven distribution of decays
in time which is important when handling dead-time corrections and discussing required
system time resolution. Let us first assume that a decay has occurred at time t = 0. What
is then the differential probability that the next decay will take place within a short time
interval, dt, after a time interval t has passed? Two independent processes must then occur
in series. No decay may take place within the time interval from 0 to t, probability P(0),