Handbook of Radioactivity Analysis 3rd Edition
In the measurement process, the object to be observed is always affected by an undetermined interaction between the observer and the observed. As a result, the measured magnitudes are always reproduced with a certain inherent uncertainty caused by the instrument. This uncertainty in the measurements makes the use of error theory essential. When we measure radioactive substances, the situation becomes even more complicated, because the radioactivity decay is a random process. In radioactivity, counting two types of fluctuations are basically generated, one related to the activity of the sample, when the half-life of the radionuclide is short, and another caused by the random nature of radioactivity decay, which modifies the disintegration rates with time. Since the measurement of radioactivity involves values with different degrees of reliability and validity, the principles of counting statistics must be applied (Larson, 1969, and Eadie et al. 1971).
In many types of measurement, such as mass, volume, time, length, etc., the measured quantity has a given value and only the measurement conditions introduce statistical variations. The situation is different, however, in radioactivity measurements. The radioactive decay process follows Poison statistics, so a sample’s activity value is not a specific value but a mean value that varies with time. If we measure the emission of a radioactive source, as shown in Table 2.1, repeated measures are not equal. But there is a clear condition: if the measuring equipment does not introduce a perturbation, the emission and measurement rates must follow the same statistical law, so their mean and variance rates are equal from a statistical perspective. Example 2.8 represents a detailed analysis of the set of measurements shown in Table 2.1. We see that they are not affected by the counter. Example 2.9 presents another case in which the counting rates are affected by the counter even though the differences between measurements are smaller.
In this section, we shall study all basic characteristics of both the Poisson and the normal (or Gaussian) distributions, and their relation to radioactivity counting statistics. Although the Poisson distribution involves all processes of radioactivity decay, and therefore the detection of particles and radiation, the normal distribution is applied more often by far and is better known in the majority of cases. Since the application of the Poisson distribution to counting statistics may be tedious and time consuming, and considering that both distributions give identical results, when the total number of counts becomes large, our final objective will be the determination of some characteristic parameters, which will allow one to exchange the Poisson distribution by the normal one.
The Poisson Distribution
The Poisson distribution describes a random process for which the occurrence probability of a certain event is constant and small. This distribution not only concerns radioactivity counting statistics (Helstrom, 1968) or nuclear decay, but is also applied to evaluate, in a more or less approximate way, many other processes Garwood (1936), Grau Malonda (1999). Some examples of daily life that verify Poisson statistics are, for example, the number of phone calls received at a phone switchboard several minutes before noon, the number of annual strikes in a factory, the number of misprints on a book page, the number of times a piece of a machine fails in a given period of time, and the number of fatal traffic accidents each week in a city. Przyborowski andWilenski (1935) present an application of Poisson law and construct rules to minimize the chance of errors in tests and samples.
In radioactivity decay four aspects are fulfilled: all radioactive nuclei have the same decay probability for a given time period, the decay process of one nucleus is not affected by the decay of other nuclei, the total number of nuclei and measurement time intervals are sufficiently large, and the nuclei half-life is long compared with the detection pulse. Therefore, the radioactivity decay is a random process, in which a discontinuous random variable is defined as the number of times a decay event takes place in a continuous period of time t. Additionally, the probability of one decay event occurring in a time increment Dt must be asymptotically proportional to Dt, independently of the time value in the interval Dt and all previous decay events
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