Derivation of the Transfer Function according to Filß in Open Geometry

Preliminary Remarks

The following detailed derivation largely follows the publication by Peter Filß.

A 200 l barrel is assumed as the container. However, the derivation can be applied to any other cylindrical objects for which the related assumptions hold.

The fundamental idea of the derivation is to divide the container into small volume elements ΔV (so-called voxels). For each of these non-overlapping voxels that occupy the entire volume of the container, its contribution to the count rate in the detector is determined and all contributions are summed. A key point is the transition from a finite to an infinitesimal small voxel volume.

For the derivation of the transfer function, the detector is initially assumed to be point-like. A calibration source, also assumed to be point-like, is mounted on the outer surface of the container at half height. The detector is located at the same height at a distance S from the calibration source.

A sphere with radius S can now be placed around the point-like detector (red circles in the following figures). This radius corresponds to the distance of the calibration source from the detector and is used to determine the detector efficiency. It is utilized that a point-like detector can be regarded as a quasi-spherical detector. This means that the detection effectiveness for detecting all photons of energy E that hit the detector from any point on the surface of the sphere is the same.

Note:
Typically, detector crystals are mostly cylindrical in shape, deviating from the assumption of a point-like detector. Consequently, the detector effectiveness is not constant on the surface of the sphere.

For any point in the container, the connecting line between this point and the detector passes through the area F₀ on the surface of the sphere. The dimensions of F₀ are determined by the distance S as well as the height h and width (2∙r) of the container.

The area F₀ is marked by blue lines in the figures. The area F₀ is now covered by non-overlapping area elements ΔF. The total of all lines from the detector passing through the area element ΔF intersects a partial volume in the container, which is represented in the figures as gray marked areas. Photons emitted from radionuclides located in this partial volume towards the detector must consequently pass through this area ΔF.

Note:
Scattering effects do not need to be considered in this derivation, as they are generally associated with an energy loss of the photons. For the activity determination of a radionuclide, only the corresponding information from the photopeaks is utilized, which contains contributions from photons that have not experienced energy loss on their way to the detector.

The (point-like) detector and the area element ΔF define a cone, whose cross-sectional area increases quadratically with the distance from the detector. The opening angle of this cone is determined by the cross-sectional area ΔF.

The application of the distance law (1/r² law) results in the relationship between the size of the area element ΔF at a distance S from the detector and the area element ΔF₁ at any distance S₁

\[ \frac{\Delta F}{S^2}=\frac{\Delta F_1}{S^2_1} \]

The distance S₁ of the area element ΔF₁ from the detector is extended by the distances z_w and x compared to the distance S of the area element ΔF. z_w is the distance between the center of the area element ΔF lying on the surface of the sphere and the container surface. The size x describes the depth at which the area element ΔF₁ is located within the container, i.e., how far this area is from the container surface. All distances refer to the connecting line of the detector to point B, which describes the intersection of the connecting line with the container wall on the side opposite the detector.

The size of an area element ∆F_x at any position x in the container along the connecting line between the detector and point B is thus determined as:

\[ \Delta F_x = \Delta F \cdot \left(\frac{S + z_w + x}{S} \right)^2 \]

Each of these area elements ∆F_x along the connecting line in the container contributes to the count rate Z measured by the detector. For a homogeneous activity concentration C_A in the container (unit Bq∙cm^-3), which is one of the assumptions of the derivation, the contribution of an area element ∆F_x to the count rate ∆Z_x depends on the following quantities:

• the emission probability η of the considered gamma line of energy E,
• the efficiency ε(E) of the detector for the detection of a photon of energy E, and
• the distance of the area element ∆F_x from the detector (see previous equation).

This results in the contribution of a volume element ∆V(x) = ∆F_x · ∆x to the count rate ∆Z_x as follows:

\[ \Delta Z_x = \epsilon \cdot \eta \cdot C_A \cdot \left( \frac{S}{S + z_w + x} \right) ^2 \cdot \Delta F_x \cdot \Delta x \]

So far, the effect of the attenuation of photons on the way from the area element ∆F_x to the detector and the resulting reduced contribution of ∆Z_x to the count rate Z has not yet been considered.

For the distance within the container, i.e., in the matrix (e.g., the waste product), attenuation can be accounted for by the factor exp(-μ⋅x). μ is the material- and energy-dependent linear attenuation coefficient for the matrix material. Its values can be retrieved from databases when the material composition is known.

Further attenuation occurs when passing through the container wall. This can also be accounted for by the factor exp(-μ_w⋅w^*). μ_w here is the linear attenuation coefficient of the wall material, and w^* the average wall thickness. Since radiation usually does not pass perpendicularly but at an angle through the container wall, this is considered by an average wall thickness that is approximately 1.25 times the actual wall thickness w for 200 L barrels.

For historical reasons, this contribution to attenuation is described using the average shielding factor K₂ in the following notation:

\[ K_2 = \exp(\mu_w \cdot w^*) = \frac{1}{\exp(-\mu \cdot w^*)} \]

In summary, the contribution of the volume element ∆V(x) = ∆F_x⋅∆x to the count rate ∆Z_x:

\[ \Delta Z_x = \epsilon \cdot \eta \cdot C_A \cdot \frac{1}{K_2} \cdot \exp(-\mu \cdot x) \cdot \left( \frac{S}{S + z_w + x} \right) ^2 \cdot \Delta F_x \cdot \Delta x \]

The total count rate ∆Z can be calculated by summing all values of ∆Z_x along the path from one container wall to the opposite one. x takes all values between 0 and the distance to the respective point B.

For infinitesimal thin volume elements ∆V(x), the summation can be replaced by integration over all values of x:

\[ \Delta Z_x = \epsilon \cdot \eta \cdot C_A \cdot \frac{1}{K_2} \cdot \int_0 ^B \exp(-\mu \cdot x) \cdot \left( \frac{S}{S + z_w + x} \right) ^2 \cdot \Delta F_x \cdot dx \]

Considering the relationship between ∆F_x and ∆F results in:

\[ \Delta Z_x = \epsilon \cdot \eta \cdot C_A \cdot \frac{1}{K_2} \cdot \int_0 ^B \exp(-\mu \cdot x) \cdot \left( \frac{S \cdot (S + z_w + x)}{S \cdot (S + z_w + x)} \right) ^2 \cdot \Delta F \cdot dx \]

And after cancelling and moving the area element ∆F in front of the integral:

\[ \Delta Z_x = \epsilon \cdot \eta \cdot C_A \cdot \frac{1}{K_2} \cdot \Delta F \cdot \int_0 ^B \exp(-\mu \cdot x) \cdot dx \]

After analytically solving the integral, it results in:

\[ \Delta Z_x = \epsilon \cdot \eta \cdot C_A \cdot \frac{1}{K_2} \cdot \Delta F \cdot \frac{1}{\mu} \cdot \left[1- \exp(-\mu \cdot B) \right] \]

When the activity concentration C_A is expressed by the specific activity a and the density ρ

\[ C_A = a\cdot \rho \]

the contribution ∆Z from the entire volume behind ∆F to the count rate Z becomes:

\[ \Delta Z_x = \epsilon \cdot \eta \cdot a \cdot \frac{1}{K_2} \cdot \Delta F \cdot \frac{\rho}{\mu} \cdot \left[1- \exp(-\mu \cdot B) \right] \]

The total count rate Z is then obtained by summing the contributions of all area elements ∆F on F₀. Since the transition to infinitesimal area elements is again carried out here, the summation can also be replaced by integration over the entire area F₀. This describes the area of the projection of the container onto the surface of the sphere with radius S around the (point-like) detector:

\[ Z = \int_{F_0} \Delta Z \]

or

\[ \Delta Z_x = \epsilon \cdot \eta \cdot a \cdot \frac{1}{K_2} \cdot \frac{\rho}{\mu} \cdot \int_{F_0} \left[1- \exp(-\mu \cdot B) \right] dF \]

The integration cannot be performed analytically, meaning there is no formulaic description of the result of the integration. Depending on the distance of the barrel from the detector (i.e., the size S) and the value of [1 - exp(-μ ⋅ B)], the integration can either be carried out using approximations or computed numerically.