KAHIBARO
46.7. Calculating Transmission
Table of Contents
Understanding Transmission
Transmission in radiation shielding describes how many photons pass through a shield compared to how many were incident on it. In your GATE simulation this is directly related to how many simulated photons are detected after the shield compared to how many you started with or how many reach a reference plane without any shield.
You will typically compute transmission as a function of shield thickness and material, using the particles recorded by an actor placed behind the shield.
Defining Transmission from Simulation Data
In this project you already have two key ingredients: a gamma source and an actor that records transmitted photons after the shield. Once the simulation is run, transmission is obtained by counting how many photons were transmitted and normalizing this count.
If you simulate a fixed number of primary particles $N_0$ and record $N_t$ photons that pass through the shield and reach your scoring plane or detector, the simplest definition is
\[
T = \frac{N_t}{N_0}.
\]
Here $T$ is the fraction of photons transmitted through the shield. If you then change the shield thickness and repeat the simulation, you can compute a new $T$ for each thickness and study how transmission decreases.
If your configuration uses a reference run without the shield, then you will have a count $N_{\text{ref}}$ recorded at the same scoring plane with identical simulation settings but no shield present. In that case, even if $N_0$ is not explicitly known, you can still define
\[
T = \frac{N_t}{N_{\text{ref}}}.
\]
This ratio measures how strongly the shield reduces the transmitted fluence or number of photons relative to the unshielded case.
Transmission is usually defined as
\[
\boxed{T = \frac{\text{number of transmitted photons}}{\text{number of incident photons}}}
\]
or, using a reference run,
\[
\boxed{T = \frac{\text{counts with shield}}{\text{counts without shield}}}.
\]
When you analyze GATE outputs, be clear which definition you are using. Mixing both definitions within the same analysis will lead to confusing results.
Using GATE Output to Count Photons
To compute $N_t$ and, if needed, $N_{\text{ref}}$, you will typically use an actor that records particles crossing a plane behind the shield. This is often implemented as a phase space actor or a fluence actor attached to a thin scoring volume.
After the simulation, the output file of this actor contains one entry per transmitted particle. Counting entries provides $N_t$. If you run a separate simulation without the shield using exactly the same scoring setup, counting entries in that file gives $N_{\text{ref}}$.
If you use multiple energy bins, you can also calculate energy dependent transmission. In that case, you restrict counts to a particular energy range. For a given energy bin $i$, with $N_{t,i}$ transmitted counts and $N_{\text{ref},i}$ reference counts, the energy dependent transmission is
\[
T_i = \frac{N_{t,i}}{N_{\text{ref},i}}.
\]
You can then study how the shield behaves differently for various photon energies.
Normalization and Acquisition Time
Many shielding simulations are run with a fixed number of primaries, in which case you do not need to worry about acquisition time, since $N_0$ is known from the simulation configuration. However, if you use a time based simulation where the source activity and acquisition duration are specified, the total number of primaries may vary between runs.
In that case, directly comparing $N_t$ between runs is misleading. You must either compute $N_0$ from the source configuration or use a shielded and an unshielded run with the same acquisition time and activity, and compute the ratio $N_t / N_{\text{ref}}$.
The idea is always the same: transmission must compare transmitted photons to the photons that could have been transmitted under the same conditions but without the shield.
Statistical Uncertainty on Transmission
Because GATE uses Monte Carlo methods, the counts $N_t$ and $N_{\text{ref}}$ are affected by statistical fluctuations. As a result, the computed transmission $T$ has an uncertainty.
If $T$ is defined as $T = N_t / N_0$ and $N_0$ is a fixed known number (for example the number of primary events), then you can treat $N_t$ as a Poisson variable with variance $\sigma_{N_t}^2 = N_t$. The relative uncertainty on $T$ is then
\[
\frac{\sigma_T}{T} = \frac{\sqrt{N_t}}{N_t} = \frac{1}{\sqrt{N_t}}.
\]
So the absolute uncertainty is
\[
\sigma_T = \frac{\sqrt{N_t}}{N_0}.
\]
If you instead use a reference run and compute $T = N_t / N_{\text{ref}}$, both numerator and denominator fluctuate. In that case, if $N_t$ and $N_{\text{ref}}$ are large enough, you can use standard error propagation and obtain
\[
\left(\frac{\sigma_T}{T}\right)^2 = \frac{1}{N_t} + \frac{1}{N_{\text{ref}}}.
\]
So the uncertainty is
\[
\sigma_T = T \sqrt{\frac{1}{N_t} + \frac{1}{N_{\text{ref}}}}.
\]
For large counts, treat $N$ as Poisson with variance $N$ and use
\[
\boxed{\frac{\sigma_T}{T} = \frac{1}{\sqrt{N_t}} \quad \text{or} \quad \boxed{\left(\frac{\sigma_T}{T}\right)^2 = \frac{1}{N_t} + \frac{1}{N_{\text{ref}}}}}
\]
to estimate the uncertainty in transmission.
These formulas help you decide how many events you need to reach a desired precision. For example, to obtain a relative uncertainty of 1 percent on $T$ when using $T = N_t / N_0$, you need roughly $N_t \approx 10^4$ transmitted photons.
Building Transmission Curves from Multiple Runs
To study shielding performance, you will likely perform several simulations with different shield thicknesses. For each thickness, you compute a transmission value $T(d)$ and its statistical uncertainty. Here $d$ is the material thickness. The result is a set of pairs $\{d_k, T_k\}$ for $k = 1, 2, \ldots$.
You can then plot $T$ versus $d$. On a linear scale, the curve shows how transmission decreases with thickness. On a semi logarithmic scale, plotting $\ln T$ versus $d$ makes the curve approximately linear for a monoenergetic beam, which is useful when you compare your simulation with the exponential attenuation law later in the project.
If you work with several materials, you can repeat the same procedure for each material. For example, you might compute $T_{\text{lead}}(d)$, $T_{\text{aluminum}}(d)$, and $T_{\text{concrete}}(d)$ and then compare the curves to see which material provides greater attenuation for the same thickness.
Practical Considerations When Counting Transmitted Photons
When you extract $N_t$ from your GATE output, it is important to be consistent about what counts as transmitted. In shielding problems you usually care about primary photons that pass through the shield and still carry significant energy. However, the actor may record:
All photons, including scattered photons.
Photons with very low energy, far below the initial energy.
Particles other than photons, for example electrons.
Depending on the goal of your study, you might want to apply simple filters when counting:
Select only photons within a specific energy window, for instance above some minimum energy threshold.
Ignore secondary particles and count only photons.
Ensure that you count each crossing event appropriately, usually one entry per particle that reaches your scoring plane.
If you are interested in total transmitted photon fluence you will likely count all photons hitting the plane, regardless of history. If your later comparison with the exponential attenuation law assumes uncollided primary photons, then energy and geometry filters become more important and are handled in the later chapter dedicated to that comparison.
Summary
In the radiation shielding example, calculating transmission is a matter of turning GATE output into a simple ratio. You count transmitted photons behind the shield, relate that to the number of incident photons or to a reference run without a shield, and then estimate the statistical uncertainty. By repeating this for different shield thicknesses and materials you build up a set of transmission values that you will later compare with the analytical exponential attenuation law.
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