14.7.1 Statistical Error of Radioactivity Measurement

The activity of radioactive substances is usually measured indirectly. This means that the number of the particles or photons counted in the detector is proportional to the radioactivity, the number of decompositions in a unit of time (see Section 4.1.2). As mentioned in Section 4.1.1, radioactive decay is a statistical pro­cess, so repeated measurements give a statistical distribution around a mean value. On the basis of the statistical laws, the statistical error of the measurements can be determined accurately.

The statistical laws postulate that as the measured counts (N) increase, the abso­lute error (AN) also increases. The relative error (ДN/N), however, decreases. When N tends to be infinity, the relative error tends to be zero.

AN

lim 0 14.9)

N! N N

The real value of N is determined by the time, which can be chosen for the mea­surements. Thus, the relative error can be decreased under a certain value at most.

Let’s measure the intensity of a radioactive sample k times under the same conditions. The counts are Ni, N2,.. .,Nj,…, Nk, and their mean value is N. In principle, the distribution of the values (PN) can be given by Poisson’s discrete probability distribution:

P — <№) є-®

For sufficiently large values, the probability distribution obtained for the measured values and Poisson’s probability distribution function is the same (Figure 14.12). For smaller values, the regular process of radioactive decay was disturbed by an external factor (e. g., the instability of the measuring tool, the aging of the detector, the change in the position of the sample). On the basis of Poisson’s distribution, the standard deviation expected for N counted impulses (s. d.) is:

s. d. — VN e Xt

At Xt« 1 (i. e., when the activity of the radioactive sample remains the same during the measuring time):

s. d. — PN <14.12)

As seen in Eq. (14.8), the standard deviation can be calculated for one measure­ment when the counts are high enough.

In the case of the many counted impulses (N > 100), the Poisson distribution becomes identical to the Gauss distribution, which is simpler and accurate enough:

The value and the distribution of the differences between the individual mea­sured values and the mean value

Д = Nj — N (14.14)

usually fulfill the Gauss distribution (Figure 14.12).

The degree of accuracy is the degree of closeness of the measurements to the actual value. When 50% of the measurements are within this given value, it is called “probable deviation.” The mean error or the standard deviation is the error obtained for 68.27% of the measurements.

If the Gauss probability distribution function is integrated from N — VN to N + VN, 0.6827 is obtained. If the Gauss probability distribution is valid, 68.27% of the measurements fall into the interval N ± VN. Accordingly, the standard devi­ation is ± VN, the square root of the mean value. This value is equal to the value obtained in Poisson’s probability distribution.

When the Gauss distribution function is integrated from N — c VN to N + cVN, a certain portion of the measurement falls into the interval N ± c VN. Table 14.2 shows these probabilities.

In most cases, the standard deviation is used. This value can be calculated as follows:

As discussed previously, 68.27% of the measurements fall into the interval

n ± vN.

The standard deviation of the mean value (SD) is less than the standard deviation of the individual measurements (sd) because the mean value is obtained from k independent measurements. The time of the measurements is t, and the number of the counted impulses is N. The standard deviation of the mean value is expressed as:

The standard deviation of the mean value per time (in other words, the standard deviation of the activity or intensity) is:

SD = y/NTk t t

As seen in Eq. (14.17), the standard deviation can be decreased by increasing the measuring time and the number of the measurements.