Chapter 14 from BASICS OF NUCLEAR PHYSICS AND OF RADIATION DETECTION AND MEASUREMENT - An open-access textbook for nuclear and radiochemistry students by Jukka Lehto

When measuring the activity ($\text{A}_x$) of a radioactive source the primary result is the total (gross) count rate ($\text{R}_g$) obtained from the measurement system (detector, amplifier and pulse counter)

$$R_g = \frac{X_g}{t}$$

[XIV.I]

where $\text{X}_g$ = number of collected total pulses and $\text{t}$ = measurement time. The unit of count rate is pulses per unit time: counts per second (s^-1, cps) or counts per minute (cpm).

The observed gross count rate ($\text{R}_g$) includes, in addition to pulses resulting from the radioactive source (net pulses $\text{X}_n$), also pulses from background ($\text{X}_{bg}$) originating from various sources other than the actual source, such as cosmic radiation, presence of natural radionuclides or pollution radionuclides and electric noise of the measurement system. These background pulses need to be counted separately in the absence of the radioactive source and the background count rate must be subtracted from the gross count rate to obtain the net count rate ($\text{R}_n$) originating from the radioactive source.

$$R_n = \frac{X_g}{t_g} - \frac{X_{bg}}{t_{bg}}$$

[XIV.II]

Activity of the source ($\text{A}_{x}$) is calculated either by comparing the net count rate of the source ($\text{R}_{x}$) to that obtained by measuring a standard source ($\text{R}_{st}$) with a known activity ($\text{A}_{st}$) in identical conditions as the unknown source

\[ A_x = A_{st} \times \left( \frac{R_x}{R_{st}} \right) \]

[XIV.III]

or if the counting efficiency ($\text{E(%)}$) of the counting system is known dividing the net count rate with the counting efficiency.

$$A_x = R_x \left( \frac{E}{100} \right)$$

[XIV.IV]

What kind of uncertainties are involved here and how they are calculated are discussed below.

In every measurement, including radioactivity measurement, there are two types of errors resulting in an uncertainty in the measurement result (Figure XIV.1):

1. Systematic errors arise from erroneous measurement system and they always function into same direction from the right result. If, for example, the activity of the standard is not what it is supposed to be or the settings of the measurement system, such as amplification of pulses, change during the measurement, this causes error to the observed activity. Even if the parallel results were close to each other, i.e. reproducible and precise, the results would not be accurate since they systematically deviate from the real value to certain direction in case of a systematic error.
2. Random errors arise from the fact that the measurement system or the phenomenon measured, orboth, are intrinsically non-deterministic (stochastic):

• The measurement systems are always non-ideal and do not always give the same response even though the measured quantity would have a constant value. For example, alpha particles for a certain transition of a radionuclide have always the same energy but we never obtain a perfect line peak spectrum, but the peak has a broadness depending on the system's limited preciseness in transforming alpha particle energies to electric pulses.
• Some measured phenomena, such as radioactivity, are intrinsically stochastic. We can know the probability of nuclear transformations, decays, at certain time interval but it is impossible to find out their exact number since they vary in a stochastic manner. An important feature of a stochastic error is that the reproducibility and preciseness increases with the number of measurements.

Below we discuss in more detail the uncertainties arising from the stochastic nature of radioactive decay.

Figure XIV.1. Effect of systematic and random error on observed results. Left side: high precision but low accuracy. Right side: low precision but high accuracy.

The variation of radioactive decay events and other stochastic processes with low and constant probabilities are mathematically described with Poisson distribution probability function (Equation XIV.V). The variation of radioactive decays (or particle/photon flux) is a fundamental physical characteristic of the radionuclide. If we consider a large enough number of radionuclides the number of decay rate varies with time following the equation:

$$P_x = \frac{m^x}{x!} \cdot e^{-m}$$

[XIV.V]

where $\text{P}_x$ is the probability for $\text{x}$ number of events occurring in unit time and $\text{m}$ is the most probable number of events. Since the number of decay events can have only integer values the graphical representation of Poisson distribution is a histogram. Poisson distribution is also not symmetrical but is slightly bended to lower values. To make treatment of results simpler the symmetric normal (Gaussian) distribution is used as an approximation to Poisson distribution. Figure XIV.2 shows the difference of Poisson and normal distributions for the probability to observe decay events in a number of identical time intervals. For a large number (<30) of events (m) Poisson and normal distributions are more or less identical.

Figure XIV.2. Poisson and normal distribution functions for the probability (P) to observe radioactive decay events (m) in a number of identical time intervals (http://nau.edu/cefns/labs/electron-microprobe/glg-510-class-notes/statistics/).

The mathematical formulation of normal distribution is as follows:

$$P_x = \frac{1}{{\sigma \sqrt{2\pi}}} \times e^{-\frac{{(x - m)^2}}{{2\sigma^2}}}$$

[XIV.VI]

where $\text{P}_x$ is the appearance probability of a stochastic event, $\text{m}$ is the real value of the events, $\sigma$ is the standard deviation of events at various time intervals.

To present the variation for a number of decay events/pulses the quantity used is standard deviation, σ. Its mathematical expression is

\[\sigma_x = \sqrt{\left( \frac{\sum (x_i - \bar{x})^2}{n-1} \right)}\]

[XIV.VII]

where $\text{x}_i$ is the number of pulses observed, $\bar{x}$ is their arithmetic mean value and $\text{n}$ is the number of measurements. If for example a radioactive source is measured ten times the number of observed pulses may vary as shown in Table XIV.I. From these results one should calculate both the arithmetic mean and the standard deviation (uncertainty) and present the result as (99.0 ± 8.4) imp s^-1 or 99.0 imp s^-1 ± 8.5 %.

Table XIV.I. Variation of pulses in a radioactivity measurement. STDEV = $\sigma$ = standard deviation.

For the normal distribution it applies that if several measurements of radioactive decays are carried out and the mean value of pulses (or pulse rate or activity) is $\bar{x}$ a single measurement has a 68.3% probability to be observed in the range $\bar{x}\pm\sigma$, 95.5% probability to be observed in the range $\bar{x}\pm2\sigma$ and 99.7% to be observed in the range $\bar{x}\pm3\sigma$ . This is illustrated in Figure XIV.3.

Figure XIV.3. Normal distribution and the probability ranges of standard deviation (https://learn.bu.edu/bbcswebdav/pid-826908-dt-content-rid-2073693_1/courses/13sprgmetcj702_ol/week03/metcj702_W03S01T02_normal.html).

Usually instead of several measurements, only one single measurement is carried out. The uncertainty, i.e. standard deviation ($\sigma$), of a single measurement is calculated as a square root of the number of observed pulses ($\text{X}$)

$$\sigma = \sqrt{X}$$

[XIV.VIII]

For the standard deviation derived in this way applies the same rules as presented above: the measured number of pulses has a 68.3% probability to deviate one $\sigma$ value from the "right" value, 95.5% probability to deviate two $\sigma$ values from the "right" value and 99.7% probability to deviate three $\sigma$ values from the "right" value. Right value refers to mean value what would be obtained if several measurements were done. For example, if we observe 100 pulses, the value of $\sigma$ is $\sqrt{100} = 10$ and thus the measured number of pulses has a 68.3% probability to deviate 10% from the "right" value, 95.5% probability to deviate 20% from the "right" value and 99.5% probability to deviate 30% from the "right" value. If we instead collect 10000 pulses the s gets a value $\sqrt{10000} = 100$ and the measured number of pulses has a 68.3% probability to deviate 1% from the "right" value, 95.5% probability to deviate 2% from the "right" value and 99.5% probability to deviate 3% from the "right" value. Thus increasing the number of observed pulses by a factor of 100 we decreased the uncertainty by a factor of 10. This applies to all measurement: the higher the number of collected pulses the lower is the uncertainty which is illustrated in Table XIV.2.

Table XIV.2. Uncertainties of radioactivity measurements with increasing number of collected pulses.

When presenting the results in radioactivity measurement the results should also include the uncertainty. For example in the following ways: 1030 ± 35 ($\sigma$) Bq or 1030 ± 70 (2$\sigma$) Bq or 1030 ± 105 (3$\sigma$) Bq.

The standard deviation of the gross count rate, using 68.3% probability limits, is as follows: $$\sigma_g = \frac{\sqrt{X}}{t}$$

[XIV.IX]

$$R_g \pm \sigma_g = \frac{X_g \pm \sqrt{X_g}}{t}$$

[XIV.X]

The unit of the uncertainty is the same as that of count rate, s^-1 or imp/s.

When the background is determined with a separate measurement and it is subtracted from the gross count rate it brings further uncertainty to the net count rate. The standard deviations of both the gross count rate and the background count rate are separately calculated using the equation XIV.IX and the standard deviation of the net count rate is calculated with equation XIV.XI which is valid for propagation of any standard deviation of combining two standard deviations from summation or subtraction.

\[\sigma_{Rn} = \sqrt{\sigma_{Rg}^2 + \sigma_{Rbg}^2}\]

[XIV.XI]

\[\sigma_{Rn} = \sqrt{\frac{X_g}{t_g^2} + \frac{X_t}{t_{bg}^2}}\]

[XIV.XII]

The activity ($\text{A}$) is typically calculated by comparing the net count rate of the unknown sample ($\text{R}_n$) to that of the standard ($\text{R}_{st}$). The standard deviation of the activity ($\sigma_{A}$) is then calculated with the equation XIV.XIV which is valid for propagation of any standard deviation of a product or a quotient.

$$\sigma_A = \sqrt{\left( \frac{\sigma_{R_n}}{R_n} \right)^2 + \left( \frac{\sigma_{R_{st}}}{R_{st}} \right)^2}$$

[XIV.XIV]