Radiation Detection and Measurement

Chapter 1: Radon detection

Figure #1: Decay chain of radon-222, until lead-210.

From this brief depiction of the radon-222 chain, it can be easily seen why this radionuclide is considered the second major cause of lung cancer. In order to monitor the levels of radon in the environment, the following quantities are introduced:

• The radon concentration [Bq/m³];
• The radon exposure, equal to the concentration integrated in time [Bqh/m³];
• The potential alpha energy (PAE), as the total energy emitted as alpha particles by a nucleus of Rn222 during its decay chain (usually, until Pb210) [MeV];
• The potential alpha energy concentration (PAEC), sum of the PAE of all the nuclei present in a unit of volume [MeV/l or J/m³]. The relation between PAE and PAEC is: A_iPAEC = C_iPAE_i Σ λ_i ιι

Note that, as far as the PEAC is concerned, the beta emitters are the most important, since their decay time is longer than the one of the alpha emitters (i.e. λ_α < λ_β).

• The working level (WL) is the value of PEAC associated with a concentration of 3700 Bq/m³ in equilibrium: WL = 2.08 × 10^-5 J m^-3;
• The working level month (WLM), as the exposure resulting after a month of work (170 h) at 1 WL.

Typically, radon daughters are not in equilibrium with radon itself, because they can attach to particulate, walls, floor, etc. Under these conditions, the PEAC is lower than the one expected under equilibrium. For this reason, we define also the following quantities:

• The equilibrium equivalent radon concentration (EEC), as the radon concentration in equilibrium with its daughters whose PAEC is equal to a given mixture of radon daughters not in equilibrium;
• The equilibrium factor (EF), which is the ratio between the EEC and the actual radon concentration in a given environment. A commonly accepted equilibrium factor indoor is EF = 0.4.

All the measurements available are referred to this equilibrium factor; however, if the EF differs, the measured concentration has to be divided by the ratio between the actual EF and 0.4.

In normal conditions, radon-222 is emanated from soils or rocks that are rich in uranium. It can also be found in water and it can easily migrate from the soil/water to the air, thus decaying into its progeny. Its measurement is relatively cheap, and the monitoring of its concentration is done through passive tracks detectors. CR-39 is an example of them. This material is a plastic polymer, used in the production of glass: an ionizing particle which penetrates this material transfers energy to the electrons, breaking the polymeric chains. The tracks can be made visible with a chemical treatment, called etching. The opening of the track is then about 5-20 µm, depending on the type and energy of the hadrons.

Figure #2: On the left, a CR-39 casing, with dimensions; on the right, an example of tracks, left by a perpendicular radiation (right) and an inclined one (left).

The number of tracks is then read by an automatic reader; their density, ρ, is linked to the exposure, through a calibration factor, C_f:

Exposure = C_f ρ

A typical value of C_f is 0.4 cm^-2 m³ kBq^-1 h^-1. The radon concentration is then easily calculated, knowing the duration of the measurement. In Italy, the law sets limits for the concentration of radon at:

• 300 Bq/m³ in existing residential buildings;
• 200 Bq/m³ in new residential buildings (built after 2025);
• 300 Bq/m³ in workplaces.

Chapter 2: Statistical analysis and data handling

2.1 Characterization of data

In the study of radiation, we take under consideration a series of random measurements, (x₁, x₂, ..., x_n), that we can consider independent. They are typically acquired through a radiation counter, such as a Geiger-Muller counter. These measurements can be treated and elaborated in different ways, the simplest one of which is to calculate their experimental mean, that is:

x̄_e = (1/n) Σx_i

A convenient representation of the data is given by the so-called frequency distribution function, F(x):

F(x) = (number of values x_i within a bin Δx) / (number of measurements, n)

It is easy to see how F(x) is normalized, that is:

ΣF(x) = 1

Through the use of F(x), it is possible to calculate the experimental mean:

x̄_e = Σx_iF(x_i)

Another step to follow in the data handling is to evaluate the spread of the data around the mean value. This can be effectively done through the so-called sample variance.

Let us consider the residual of any data point as:

s_i = x_i – x̄_e

where x is the mean calculated on an infinite number of measurements (also considered as the true value or the expectation value). Since s_i and σ_i can be either positive or negative, their sum cannot be useful, being zero. It is better to consider the square of these quantities and sum them up. Their mean value is called variance:

s̄² = (1/n) Σs_i²

Since x is unknown — it is impossible to practically calculate it — the variance σ² cannot be evaluated. Its best estimate is thus s̄², divided by n-1, in order to account for the dependence of x̄_e in the experimental data set:

s̄² = (1/(n-1)) Σ(x_i – x̄_e)²

This quantity can also be written as:

s̄² = Σ(x_i – x̄_e)²F(x_i)

The variance, say Var(x), has the following properties:

• Var(x+cost) = Var(x);
• Var(x+y) = Var(x) + Var(y);
• Var(cost⋅x) = cost²⋅Var(x);
• Var(x⋅y) = y²⋅Var(x) + x²⋅Var(y)

with x and y two independent sets of experimental data.

Because x is a variable quantity, through different measurements, it has its own variance, Var(x̄_e), calculated as follows:

Var(x̄_e) = (1/n(n-1)) Σ(x_i – x̄_e)²

Finally, the square root of the variance is called standard deviation:

s = √( (1/(n-1)) Σ(x_i – x̄_e)² )

This quantity has to be introduced in order to have a spread that is homogeneous with the data in terms of unit of measure.

2.2 Statistical models

The frequency distribution function F(x) is a distribution assessed "a posteriori", that is after the measurements. Anyway, under certain conditions, we can predict the distribution function, P(x), that will describe the results of many repetitions of a given measurement.

A way to build a statistical model is to consider a binary process (e.g., the decay of a nucleus: it decays or it doesn't), in which we consider only two possible outcomes: success or failure.

If we consider, for example, the rolling of a dice and the rolling of a certain number as a success, we know that there is a probability p for it to happen. The probability of failure (a different number is rolled) is thus 1-p. After n trials, we will have x success and n-x failures, with total probability:

P(x) = (n!/x!(n-x)!) p^x(1-p)^n-x

called Binomial distribution. The mean and the variance of this distribution are:

• x̄ = np
• σ² = np(1-p)

This model is the most general and is widely applicable to all constant-p processes. Unfortunately, it is computationally cumbersome in radioactive decay due to the high number of n.

For very low probabilities, p << 1, the Binomial distribution can be simplified in the so-called Poisson distribution:

P(x) = (e^-np (np)^x) / x!

In terms of decay measurement, it means that the decay time of the observed nucleus is bigger than the observation time. Then the number of radioactive nuclei remains essentially constant during the observation, and the probability of recording a count from a given nucleus in the sample is small.

The third important distribution is the Gaussian distribution, which is a further simplification if the average number of success is relatively large (say greater than 30):

P(x) = (1/√(2πσ²)) e^{-(x-x̄)2/2σ2}

The main difference between the Poisson distribution and the Gaussian distribution is that the first one is discrete, while the second one is continuous. For the Gaussian distribution especially, we can rely on the well-known confidence intervals, that is intervals of known probability, in which a measurement can fall. They are:

• P(x̄ ± σ) = 68%
• P(x̄ ± 1.64σ) = 90%
• P(x̄ ± 1.94σ) = 95%
• P(x̄ ± 2σ) = 95.4%

We shall notice that all the above models are identical for processes with a small individual success probability p but with a large enough number of trials, so that the expected mean number of successes is large.

2.3 Χ² test

This test is used to compare an experimental distribution to a theoretical one. Chi-squared is simply another parameter of the experimental data distribution and is defined as:

χ² = Σ((x_i - x̄_e)² / x̄_e)

It is closely related to the sample variance, and the two are related by:

χ² = (N-1)s̄² / σ²

Now, if the experimental data are closely modeled by a Poisson distribution, s̄² ≈ σ² and σ = √x̄_e. Furthermore, we have chosen x̄_e to be equal to Σx_i/n. Therefore, the degree to which χ² differs from N-1 is a corresponding measure of the departure of the data from the predictions of the Poisson distribution.

Since chi-squared is a random variable, it is associated with a distribution. For each curve, p gives the probability that a random sample of N numbers from a true Poisson distribution would have a larger value χ²/N-1 than that of the ordinate. This ratio is called reduced chi-squared.

Figure #3: A plot of chi-squared distribution.

The reduced chi-squared can show also if an experimental distribution is too good to be true. In fact, we would expect a value between 0.6 and 1.5, thus not too close to zero and not too higher than one.

2.4 T-Student

This time we want to compare the sample mean, x̄, with the expectation value, x (the exact value). The t-Student test uses the variable t, defined as follows, to do that:

t = (x̄_e - x̄) / (s/ √N)

Since s is an approximation, t will be a random value, and thus, it will have its own distribution, depending on the number of degrees of freedom, N-1. If the size of the sample is infinite, this distribution is a Gaussian distribution, centered in zero.

This test is used when the comparison of two averages of two separate populations of measurements is done. However, these variables have to be independent from each other, normally distributed, and with the same variance. Under these conditions, the parameter t is given by:

t = (x̄₁ - x̄₂) / √((s₁²(N₁-1) + s₂²(N₂-1)) / (N₁+N₂-2)(1/N₁ + 1/N₂))

2.5 Poisson distribution in a time domain

For our purposes, the treatment of data from radiations, it is important to ask the following question: if a Poisson event happens at time t₀, what is the probability P(t) to obtain another Poisson event at time t₁+dt?

This probability is equal to the product of the probability of no event in the interval t = t₁-t₀ — say P(0) — and the probability of an event in the interval dt. Let us now call r the number of events per unit of time — this is called count rate in radiation detection; we obtain:

P(t) dt = P(0) × r dt

Since the event is a Poisson event, we have:

P(0) = e^-rt

We obtain by substitution:

P(t) dt = r e^-rt dt

In radiation measurements, this distribution is typically used in order to evaluate the average decay time:

∞∫ tP(t) dt = 1/r

2.6 Uncertainty assessment

We define the uncertainty as the parameter associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measured. In radiation measurement, we want to evaluate the probability distribution of the effective radiation present, Y, measuring a radiation X and taking into account the background signal X₁, as well as the different correction factors, X₂, X₃, ..., X_N. The often-used model is the following:

Y = (X - X₁) / (X₂ X₃ ... X_N) = W(X - X₁)

in which W is called the calibration factor.

To assess the uncertainty associated with Y — called combined uncertainty, u_C(Y) — we have to assess the uncertainties related to every single input variable and the associated probability distribution, as well as the correlation of these variables:

There are two types of evaluation methods of standard uncertainty, suggested by the ISO/IEC GUIDE 98-3:2008:

• Type A evaluation, founded on the frequency distribution of the data; this kind of assessment can be used just if one is reasonably sure that the random variable has a Gaussian distribution;
• Type B evaluation, based on a priori distribution, usually taken from previous measurements, experience, reference data, etc.

2.7 Error propagation

Often data are processed through multiplication, addition, or other functional manipulation to arrive at a derived number of more immediate interest. The values that are produced by these processing steps will be distributed in a way that is dependent on both the original distribution and the types of operations carried out.

If x, y, z, ... are directly measured counts or related variables for which we know the standard deviations σ_x, σ_y, σ_z, ..., then the standard deviation for any quantity u derived from these counts can be calculated from:

σ_u² = (&partial;u/&partial;x)²σ_x² + (&partial;u/&partial;y)²σ_y² + (&partial;u/&partial;z)²σ_z² + ...

where u=u(x, y, z, ...). This equation is generally known as the error propagation formula and is applicable to almost all situations in nuclear measurements. The variables x, y, z, ..., however, must be chosen so that they are truly independent in order to avoid the effects of correlation.

If the variables are correlated in some way, the formula mentioned above includes other mixed terms, called terms of covariance. For the simple case of two input variables we have:

σ_u² = (&partial;u/&partial;x)²σ_x² + (&partial;u/&partial;y)²σ_y² + 2 σ_x,y(&partial;u/&partial;x)(&partial;u/&partial;y)

in which σ_x,y is called covariance, and it is calculated as:

σ_x,y = (1/(N-1)) Σ (x_i – x̄) (y_i – ȳ)

Usually, the term of covariance is calculated in a different form, introducing the correlation coefficient r(x,y):

r(x,y) = σ_x,y / (σ_x σ_y)

If r(x,y)=1 then there is a positive (direct) correlation between the variables; if r(x,y)=0 they are uncorrelated; if r(x,y)=-1 the correlation is negative (indirect).

The probability distribution of u is then given by the so-called Central Limit Theorem, which states that, under certain conditions, the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The Central Limit Theorem is significant because it shows the very important role played by the variances of the probability distributions of the input quantities. It implies that a distribution converges towards the normal distribution as the number of input quantities contributing to the variance of u increases and that the convergence will be more rapid the closer the values of the terms of the error propagation formula are to each other.

2.8 Limits of detectability

There are some circumstances under which it is convenient to estimate the smallest signal that can be detected reliably in order to set a detection limit for the counting system. In the simplest case, a counting system is set up to detect the radiation of interest and the total number of counts recorded for equal periods of time as different samples are put in place. Let N_T, be the number of counts recorded with an unknown sample and N_B be the number of recorded counts when a blank sample is substituted to determine the background level. The net counts resulting are calculated as:

N_S = N_T - N_B

To make a decision whether the sample contains activity, N_S is then compared with a critical level L_C that will be determined in the analysis that follows. A simple protocol is followed: if N_S is less than L_C, it is concluded that the sample does not contain activity, whereas if N_S exceeds L_C, it is assumed that some real activity is present.

With the statistical fluctuations that are inevitable in any counting measurement, there will be many instances of positive N_S that will be observed even for samples with no activity. One would therefore like to choose a value of L_C that is high enough to minimize the likelihood of such false positives while keeping it low to reduce the possibility of missing real activity when some is actually present (false negatives).

If we assume that the measured counts are normally distributed, then also N_S will have a Gaussian distribution.