Evaluation Methods - Models
Evaluation procedures and their mathematical models are directly linked to the measuring methods. Not every evaluation procedure is suitable for the analysis of the measured data from every measuring method. Accordingly, the chosen measuring method usually specifies the evaluation procedure to be used.
In this section, we will become acquainted with the essential mathematical models (i.e., formulas) applicable for analyzing measurement data in practice. We will discuss their areas of application and, most importantly, their validity, without showcasing their mathematical derivations. These can be traced through the corresponding links in the section "Experts."
The starting point is the various measuring methods:
as well as the (almost) universal correlation underlying the model descriptions of various evaluation procedures between activity A and measured count rate Z for a characteristic line of a nuclide
\[ A = T \cdot Z \]
The transfer function T describes the respective mathematical model and depends on energy and model-specific parameters.
Measurement in open geometry
Evaluation procedure according to Filß
The mathematical model for evaluating a measurement in open geometry (with and without rotation) according to Filß is described by the equation
\[ A = M \left[ \frac{1}{\epsilon} \cdot \frac{1}{\eta} \cdot \left( \frac{\mu }{\rho} \right) \cdot \frac{1}{F_0} \cdot \frac{K_2}{K_1}\right] \cdot Z \]
and is based on the following assumptions:
The content must
- completely fill the container,
- be homogeneous (applies to material and activity distribution), and
- the density and/or material composition in the container must be known.
If these assumptions are (mostly) fulfilled, then an activity determination from the measured count rate Z can be achieved using the transfer function T, our model description.
\[ T = \left[ \frac{1}{\epsilon} \cdot \frac{1}{\eta} \cdot \left( \frac{\mu }{\rho} \right) \cdot \frac{1}{F_0} \cdot \frac{K_2}{K_1}\right] \]
Note:
The additional quantity M occurring in the equation to calculate activity A corresponds to the mass of the container's contents, known as the net mass. The derivation of the model description originally arose to calculate the specific activity a from the count rate Z. By multiplying the specific activity a by the net mass M, the corresponding activity A is obtained.
Significance of the individual parameters in the transfer function T:
The transfer function T establishes the relationship between the measured count rate Z and the sought activity A. Its partially energy- and nuclide-dependent parameters must be selected appropriately for each characteristic line to be evaluated.
Emission probability η
A radionuclide can emit gamma radiation with one or more discrete energies. The probability of emission at a specific energy is given by the emission probability. The values are tabulated and partially publicly accessible (e.g., Lara, IAEA, NDS, KAERI, JAEA). It is noteworthy that the evaluation software of measurement systems does not always take into account the most current transition probabilities or correct known erroneous values.
For the identified characteristic line, the corresponding emission probability is chosen from one of the external databases or that integrated within the evaluation software. The (mostly) also tabulated associated uncertainties are used to determine the uncertainty of the calculated activity (Note: these uncertainties are usually so small that they do not affect the uncertainty of the determined activity).
When using the values, attention must be paid to the correct unit! The values are mostly given as a decimal number, sometimes also in percentage (where a value of 1.0 corresponds to a value of 100%). If a value is greater than 1, it is mostly a percentage indication!
Effectiveness ε of the detector for a point source on the container surface
The evaluation model according to Filß is based on an effectiveness calibration with a point source located on the container wall facing the detector at the same height as the detector at a specified distance S. The distance S corresponds to the distance the container to be measured will subsequently occupy.
The energy-dependent effectiveness of the detector describes in this case the probability with which gamma radiation emitted by the point source hits the detector and is detected in the photopeak.
Thus, the effectiveness depends on the detector system used, the energy, and the geometric measurement arrangement. If there is a change, a new effectiveness calibration must be performed under the new boundary conditions (e.g., a different detector system). If only the distance changes, then the effectiveness values can usually be adjusted according to
\[ \epsilon_{n} = \epsilon_{a} \cdot \frac{S_a^2}{ S_n^2} \]
If the calibration measurement at the distance Sa yielded the effectiveness εa, then εn is the “corrected” effectiveness at the distance Sn.
In the calibration measurements, effectiveness values can only be determined for some few energy values, which should cover the entire relevant energy range. However, in evaluations, effectiveness values for deviating energies are often needed. In these cases, an efficiency curve is determined from the discrete efficiency values, from which the sought efficiency values can be read or calculated (with the help of fitted equations).
Note:
For the distance S, it should be: \(S > 2 \cdot r \) with container radius r.
Mass attenuation coefficient (μ/ρ) of the matrix
The material properties of the homogeneous contents of the container, in which the activities are uniformly (i.e., homogeneously) distributed – these are the assumptions for the model – are described by the energy-dependent mass attenuation coefficient (unit g·cm-2). This can be retrieved for elements and material compositions from databases (e.g., NIST tables 3 and 4).
This assumes prior knowledge of the material composition of the contents of the container. If this is not exactly known, then this "lack of knowledge" can be transferred into the uncertainty of the activity calculation: from the tables, one determines a minimum and maximum value for the mass attenuation coefficient that corresponds with the (to be documented) "personal" assumptions of the "lack of knowledge," computes the average used for further calculations, and takes the deviations of the average from the two extreme values as uncertainty (this is an empirical approach to quantify the "lack of knowledge").
Cross-sectional area F0 of the container projected onto a sphere
The derivation of the evaluation model assumes a point detector. The cross-sectional area of the container (unit cm2) is mapped onto a sphere around the point detector. The radius of the sphere corresponds to the distance S from the detector to the container (Note: corresponds to the position of the calibration source during the effectiveness calibration).
If the cross-sectional area of the container is \( F_{\infty} = 2 \cdot r \cdot h \), with the container radius r and the container height h, then the projected cross-sectional area \( F_0 \) is calculated as
\[ F_0 = F_{\infty} \cdot \frac{S^2}{(S + r)^2} \]
Correction factor K1 for the attenuation in the matrix
The gamma radiation emitted by the activity distribution within the container must first traverse a distance within the container before reaching the detector. It is attenuated according to the material properties of the so-called active matrix (= homogeneous material distribution in which a radionuclide is homogeneously distributed). Depending on the location of the emission of gamma radiation, differing amounts of gamma radiation reach the detector due to varying path lengths in the active matrix. This attenuation is taken into account by the correction factor K1, which is determined by averaging the field of view of the container surface in the detector and calculated with
\[ K_1 = 1 - \exp(-\mu \cdot B) \]
where B is the diameter of the container (\( B = 2 \cdot r \)). The energy-dependent linear attenuation coefficient μ is expressed through the mass attenuation coefficient and the density ρ of the active matrix as
\[ \mu = \left( \frac{\mu}{\rho} \cdot \rho \right) \]
Using the mass attenuation coefficient and the density of the active matrix is often simpler than determining the corresponding linear attenuation coefficient, as only a few databases are known for the latter.
In many practical applications, the value of K1 lies in the range of 1.0 or slightly below. To determine the correction factor K1 based on all parameters, a corresponding tool is available.
Correction factor K2 for the attenuation in the container wall and internal shields
Similarly to the attenuation of gamma radiation in the active matrix, the attenuation while passing through the container wall must also be accounted for. This is achieved through the correction factor K2. If only one container wall is present (i.e., the active matrix is only surrounded by the container wall with a wall thickness of w), then K2 can be approximately calculated with
\[ K_2 = \exp(-\mu_w \cdot w) \]
where \(\mu_w \) is the energy-dependent linear attenuation coefficient of the wall material. This can alternatively be calculated through the mass attenuation coefficient and the density ρw of the wall material:
\[ \mu_w = \left( \frac{\mu_w}{\rho_w} \right) \cdot \rho_w \]
The use of the collimator, which is aligned with the rotational axis of the measurement system, restricts the area "seen" by the detector to a small section. Therefore, the actual thickness of the wall material can be used as a good approximation for the wall thickness (Note: in contrast to the measurement in open geometry, where an effective wall thickness is assumed).
In some cases, the container holding the active matrix is placed in additional containers and/or surrounded by an inactive shielding layer. In this case, the factor K2 expands to
\[ K_2 = \exp \left( - \sum_{i=1}^N \left( \mu_{w_i} \cdot w_i \right) \right) \]
for the N layers of shielding with the linear attenuation coefficients and thicknesses \(\mu_{w_i}\) and \( w_i \). It should be noted that the thicknesses \( w_i\) must be adjusted according to their effective thicknesses. Here, the tool for determining the correction factor K2 based on all parameters is also useful. It calculates the K2 factors for various specifications of wall thicknesses (actual wall thickness, effective wall thickness).
Measurement in collimated geometry
Evaluation procedure according to Filß
The mathematical model for evaluating a measurement in collimated geometry (i.e., by scanning the measuring object in the spiral scan or multi-disk scan mode) according to Filß is described by the equation
\[ A = M \cdot a = M \cdot H^{'} \cdot \left( \frac {\mu}{\rho} \right) \cdot \frac{K_2 \cdot K_3}{K_1} \cdot \frac{1}{\eta} \cdot Z \]
and is based on the following assumptions:
- The waste container is a cylindrical vessel.
- The contents of the active matrix are homogeneous, i.e., it holds for
− the density: ρ = constant (unit: g·cm-3).
− the specific activity: a = constant (unit: Bq·g-1).
− the activity concentration: CA = constant (unit: Bq·cm-3). - The material of the waste product is known.
- If the radionuclide-bearing material is in a cylindrical inner container, then it is symmetrically positioned with respect to the axis of rotation in the waste container.
- The attenuation of gamma rays while passing through the container wall and additional internal shields with thicknesses wi and linear attenuation coefficients \(\mu_{w_i}\) is described by a correction factor (average shielding factor) K2, which typically assumes values close to 1.
- The energy- and material-dependent calibration factor \(H^{‘}\) is determined in calibration measurements with appropriate point, area, or volume sources.
If these assumptions are (mostly) met, then an activity determination from the measured count rate Z can be made using the transfer function T, our model description.
\[ T = \left[H^{'} \cdot \left( \frac{\mu }{\rho} \right) \cdot \frac{K_2 \cdot K_3}{K_1}\right] \]
Significance of the individual parameters in the transfer function T:
The transfer function T establishes the relationship between the measured count rate Z and the sought activity A. Its partially energy- and nuclide-dependent parameters must be selected appropriately for each characteristic line to be evaluated.
Energy-dependent calibration factor \(H^{'}\)
In the literature and also in the original work of Filß, the calibration factor H is defined by
\[ H = \frac{\eta_0 \cdot A_0}{Z_0 \cdot F_0} \cdot \left( \frac{\mu}{\rho} \right) \]
The data of the used calibration source determine the parameters
- \(\eta_0\) (emission probability of the considered characteristic line),
- \(A_0\) (activity) and
- \(Z_0\) (measured count rate of the characteristic line).
The applied collimator specifies the size of the parameter \(F_0\).
All these parameters are independent of the properties of the measurement object. The mass attenuation coefficient (μ/ρ) is, however, linked to the matrix properties. A significant simplification in practical application arises when a calibration quantity \( H^{'} \) independent of the mass attenuation coefficient is introduced
\[ H^{'} = \frac{\eta_0 \cdot A_0}{Z_0 \cdot F_0} \]
which is linked to the original calibration factor H through the relationship
\[ H = H^{'} \cdot \left( \frac{\mu}{\rho} \right) \]
\(H^{'} \) can be determined for different collimators and energies in calibration measurements, the values stored in tables, and retrieved in evaluations.
Note:
\(H^{'} \) values for energies not determined by calibration measurements can typically be determined by linear interpolation between two adjacent energies when a sufficient number of measured energies is available:
\[H^{'}(E) \approx \frac{H^{'}(E_1) - H^{'}(E_2)}{E_1 - E_2} \]
Mass attenuation coefficient (μ/ρ) of the matrix
The material properties of the homogeneous contents of the container, in which the activities are evenly (i.e., homogeneously) distributed – these are the assumptions for the model – are described by the energy-dependent mass attenuation coefficient (unit g·cm-2). This can be retrieved for elements and material compositions from databases (e.g., NIST tables 3 and 4).
This assumes prior knowledge of the material composition of the contents of the container. If this is not exactly known, then this "lack of knowledge" can be transferred into the uncertainty of the activity calculation: from the tables, one determines a minimum and a maximum value for the mass attenuation coefficient that corresponds with the (to be documented) "personal" assumptions of the "lack of knowledge," computes the average used for further calculations, and takes the deviations of the average from the two extreme values as uncertainty (this is an empirical approach to quantify the "lack of knowledge").
Correction factor K1 for the attenuation in the matrix
The gamma radiation emitted by the activity distribution within the container must first traverse a distance within the container before reaching the detector. It is attenuated according to the material properties of the so-called active matrix (= homogeneous material distribution in which a radionuclide is homogeneously distributed). Depending on the location of the emission of gamma radiation, differing amounts of gamma radiation reach the detector due to varying path lengths in the active matrix. This attenuation is taken into account by the correction factor K1, which is determined by averaging the field of view of the container surface in the detector and can be computed with
\[ K_1 = 1 - \exp(-\mu \cdot B) \]
where B is the diameter of the container (\( B = 2 \cdot r \)). The energy-dependent linear attenuation coefficient μ can be expressed through the mass attenuation coefficient and the density ρ of the active matrix as
\[ \mu = \left( \frac{\mu}{\rho} \right) \cdot \rho \]
Using the mass attenuation coefficient and the density of the active matrix is often simpler than determining the corresponding linear attenuation coefficient, as only a few databases are known for the latter.
In many practical application cases, the value of K1 lies in the range of 1.0 or slightly below. To determine the correction factor K1 based on all parameters, a corresponding tool is available.
Correction factor K2 for the attenuation in the wall of the container and inner shields
Similarly to the attenuation of gamma radiation in the active matrix, the attenuation while passing through the wall of the container must also be accounted for. This is achieved through the correction factor K2. If only one wall is present (i.e., the active matrix is only surrounded by the wall with a wall thickness of w), then K2 can be approximately calculated with
\[ K_2 = \exp(-\mu_w \cdot w) \]
where \(\mu_w \) is the energy-dependent linear attenuation coefficient of the wall material. This can alternatively be calculated through the mass attenuation coefficient and the density ρw of the wall material:
\[ \mu_w = \left( \frac{\mu_w}{\rho_w} \right) \cdot \rho_w \]
The use of the collimator, which is aligned with the rotational axis of the measurement system, restricts the area "seen" by the detector to a small section. Therefore, the actual thickness of the wall material can be used as a good approximation for the wall thickness (Note: in contrast to the measurement in open geometry, where an effective wall thickness is assumed).
In some cases, the container that holds the active matrix is positioned in additional containers and/or is surrounded by an inactive shielding layer. In this case, the factor K2 expands to
\[ K_2 = \exp \left( - \sum_{i=1}^N \left( \mu_{w_i} \cdot w_i \right) \right) \]
for the N layers of shielding with the linear attenuation coefficients and thicknesses \(\mu_{w_i}\) and \( w_i \). It should be noted that the thicknesses \( w_i\) must be adjusted according to their effective thicknesses. Here, the tool for determining the correction factor K2 based on all parameters is also useful. It calculates the K2 factors for various specifications of wall thicknesses (actual wall thickness, effective wall thickness).
Correction factor K3 for the fraction of measuring time in the total measuring time, while the active matrix is in the detector's field of view
In segmented gamma scan measurements, the collimated detector scans the waste container, i.e., it is possible that the active matrix is not in the field of view of the collimated detector during the entire measurement. This property is taken into account by the correction factor K3.
\[ K_3 = \frac{T}{T^*} \]
T is the total measurement time of the segmented gamma scan measurement, T* the time during which the collimated detector sees the active matrix.
Note:
This definition of the correction factor K3 only applies when the total measurement time T and the segment measurement time T* can be determined (e.g., from information on segment spectra and spatial distributions). Often, alternatively, the ratio of the total scan height h to the height h* of the active matrix can be used:
\[ K_3 = \frac{h}{h^*} \]
The height h* can, for example, be determined from the evaluation of the generated energy-specific spatial distributions.
