The active matrix is always surrounded by a casing, in which a further attenuation of the emitted radiation takes place. The resulting reduction of the detector count rate is taken into account by the correction factor K2. Its magnitude depends on the number of casings (e.g. container wall, additional shielding layers inside the container, etc.) and their materials. K2 can be described as the product of the attenuation contributions of the individual layers due to the cylindrical symmetry assumed here.
\[ K_2 = \prod_{l=1}^L K_{2,l} = \prod_{l=1}^L \exp{\mu_i \cdot d_l} \]
For 200 L waste barrels without further shielding (i.e. L=1), the approximate value of d ≈ 1.25 · dw (dw thickness of the container wall) is often used for the thickness d.
Both approaches do not consider that the individual areas of the active layer contribute to the detector count rate with different lengths. The consideration of this effect can be accounted for by a corresponding weighting of the individual spatial directions in the calculation of K2.
\[ K_2 = \frac{1}{N \cdot M} \sum_{i=1}^N \sum_{j=1}^M \left[ \prod_{l=1}^L \exp{ \left( \mu_i \cdot d_l(\varphi_i, \vartheta_j)\right)} \right] \cdot w_{K_2}^{i,j} \]
Operation and Display
The calculation tool for K2calculation tool for K2calculation tool for K2calculation tool for K2, which currently allows the consideration of up to three layers for the casing of the active matrix, consists of three areas. In the left area, the values for the thickness of the layers and their linear attenuation coefficients can be set (it is assumed that the layers of the casings have the same thickness in every direction). Furthermore, information about the distance S of the detector from the surface of the outermost casing and its radius R, as well as the details about the active matrix (linear attenuation coefficient and filling height) is possible. The radius of the active matrix is determined from the input of the radius of the outer casing and the layer thicknesses. In the right area, the inputs made are graphically displayed. The results of the numerical calculation are output in the lower area: the values K2 and K2* without and with consideration of the different count rate contributions in the detector, as well as the approximate solution K2approx..
Application Examples
- Calculation of K2 or K2* for given values (for example, for input in the calculation tool measurement uncertainty in open geometry)
- Comparison of K2 and K2* to investigate the effect of the active matrix on the correction factor
- Comparison of the approximate solution K2approx. with K2 and K2*
- Optimization of shielding with given activity of the active matrix (in connection with the use of the calculation tool measurement uncertainty in open geometry
- Determination of the contribution to the uncertainty of K2 with unknown thickness of the casing(s) and/or their material composition(s).
Comparison of Results
Comparison of the results for K2approx., K2 and K2* depending on various parameters and the impact of uncertainties
The following diagrams show the results of variation calculations with the calculation tool for the case of one layer. Starting from the parameters S = 72 cm, h = 80 cm, R = router = 28 cm, µactive = 0.0578 cm-1 and the values for the shielding d1 = 1 cm and µ1 = 0.42 cm-1, one parameter is varied while keeping all other parameters constant. The results and their deviations from each other are graphically represented for the three calculated quantities K2approx., K2 and K2* . An additional diagram shows the corresponding effects assuming an uncertainty of ±10%.
Variation of Detector Distance S from Container Wall
Variation of Radius r of Active Matrix
Variation of Container Height h
Variation of Wall Thickness d1
Variation of Linear Attenuation Coefficient of Shielding Layer µ1
Variation of Filling Level F
Variation of Resolution
Uncertainties - Summary
The results of the above diagrams are summarized in the following table for the initial parameters and their variations. It shows that weighting for the different radiation directions in the active matrix does not lead to any significant reduction in the uncertainty of the correction factor for a (relatively thin) layer.
