Tools: Influence of the K1 Factor

Influence of the K₁ Factor

The evaluation model according to Filß for measurements in open geometry takes into account the attenuation of the emitted radiation in the homogeneous (concrete) matrix with homogeneous activity distribution in 200 L waste containers through a correction factor K₁.

\[ K_1 = \frac{1}{F_0} \cdot \int_{F_0} 1-e^{-\mu \cdot B} dF \]

For 200 L barrels with concrete contents of varying density (ρ ≥ 1 g·cm^-3) and linear attenuation coefficients µ ≥ 0.0578 cm^-1, it is shown that the average value of K₁ for ⁶⁰Co (E ≈ 1250 keV) deviates by a maximum of about 9 % from the value 1, thus having little influence on the calculated activity. For lower energies (e.g., ¹³⁷Cs with 662 keV), the linear attenuation coefficient is greater, meaning the deviation is even smaller. Additionally, it is required in the derivation that the measurement geometry must satisfy S ≥ 2·r.

The application of the evaluation model to (cylindrical) containers of different geometry and other contents, as well as distances S that do not meet the condition S ≥ 2·r, requires a more accurate consideration of the influence of the correction factor K₁.

As an analytical solution of the integral is not possible, a numerical calculation tool is provided for this purpose.

Operation and Display

The calculation tool for K₁ consists of four areas. The upper left area shows a schematic representation of the detector's field of view in lateral view (top) and top view (bottom) with angles φ and θ. The cylindrical container is outlined in yellow, the field of view in front of the container is light blue, and behind the container is blue. Next to it is a two-dimensional representation showing the values for (1 – e^-µ·B(φ,θ) as a function of the angles φ and θ in false-color representation. Below these two images, various parameters can be adjusted with sliders within reasonable limits:

• Distance S of the detector from the container in cm,
• Radius r of the active matrix in cm,
• Linear attenuation coefficient µ of the matrix in cm^-1,
• Degree of filling in % (starting from the bottom of the container)
• Resolution (number of angular increments)

Below this area, the result, the average value for K₁, is displayed.

All inputs immediately affect the graphics and the calculated value for K₁.

Application Examples

• Calculation of K₁ for given values (for example, for input in the calculation tool for measurement uncertainty in open geometry)
• Investigation of the effects on K₁ when varying parameters
• In particular, the deviation from the assumption of a homogeneous matrix due to a reduced filling height can be investigated.
• In combination with the calculation tool for K₂, the deviation from the assumption of a homogeneous activity distribution can be examined (the active matrix is in a smaller (cylindrically symmetric) volume and is surrounded by the same or different material).

Information on Numerical Calculation

The detector is positioned at half the height of the waste container at a distance S. The active matrix (cylindrically symmetric within the container) has a radius r and a height h_fill. The area "seen" by the detector of the active matrix is parameterized by the angles φ and θ. For given angular step sizes, the path lengths B(φ,θ) in the active matrix are calculated, multiplied by the linear attenuation coefficient µ, and then summed. Normalizing over the number of angular positions results in the average value for K₁.

\[ K_1 = \frac{1}{F_0} \cdot \int_{F_0} 1-e^{-\mu \cdot B} dF \approx \frac{1}{N \cdot M} \sum_{i=1}^N \sum_{j=1}^M \left[ 1-e^{-\mu \cdot B(\varphi_i, \vartheta_j)} \right] \]

The accuracy with which the sum value approaches the value of the integral depends on the number of angular steps. The minimum number of steps N and M required for sufficient accuracy was investigated for the parameters S = 72 cm, r = 28 cm, h = 80 cm, h_fill = 100%, µ = 0.0578 cm^-1. For this, the visible area of the detector was divided into N and M equidistant steps with the angles φ_i and θ_j and the average value for K₁ was calculated at this resolution. The reference value K_1,ref was assumed to be the K₁ value for N = M = 5000 (arbitrarily chosen).

For the assumed parameters, an asymptotic approximation to the value 0.7945 results, meaning that sampling with N = N = 100 equidistant steps is sufficient in this case. This needs to be verified for different parameters.

Influence of Various Parameters on K₁

The influence of various parameters on the correction factor K₁ is exemplarily examined for a 200 L barrel. The base parameters in the investigations are assumed to be S = 72 cm, r = 28 cm, h = 80 cm, h = 80 cm, h_fill = 100%, µ = 0.0578 cm^-1.

For other specifications, this investigation can be easily followed with the help of the calculation tool.

Influence of the Filling Height h_fill

A partial filling of the container has a non-linear effect on the calculation of K₁ (see diagram), which leads to partially significant deviations from the linear dependence at filling heights below 50 %.

Calculated K₁ values with the calculation tool (blue line) and under the assumption of a linear dependence on the filling height (black dashed line). The black line shows the percentage deviation of the two values.

Impact of Uncertainties in the Parameters

Typically, the parameters needed to determine K₁ are only known within a certain range of values. The effect on the numerically calculated K₁ value was investigated through a variation analysis.

The calculations were performed for each parameter across its entire (assumed) range of values and were subsequently repeated for each of these values with variations of ± 1 % or ± 10 %. This resulted in a tolerance band for the variation range of K₁ (red or dashed black lines). In the diagrams, the K₁ value for the base parameters is marked with a cross.

Dependency of the K₁ value as a function of the radius r of the active matrix.

Dependency of the K₁ value as a function of the distance S of the detector from the container surface.

Dependency of the K₁ value as a function of the container height.

Dependency of the K₁ value on the height of the active matrix, i.e., the filling level.

Dependency of the K₁ value as a function of the linear attenuation coefficient of the active matrix.

Based on these results, the uncertainties for individual parameters due to inaccurate knowledge of the true values can be estimated as follows:

Comparison of values for K₁ determined by various methods

In practical application, the correction factor K₁ is often calculated using the approximation introduced by Filß or the approach of a radial attenuation running through the center of the barrel. A comparison of these two methods with the results of the calculation tool shows deviations in the range of 7 to 11 % for various linear attenuation coefficients. By more accurately considering the upper and lower regions through the calculation tool, smaller values for the correction factor are calculated.

Values for the correction factor K₁ calculated using various methods for three different linear attenuation coefficients of the active matrix.