46.9. Comparing with the Exponential Attenuation Law

Table of Contents

Understanding the Exponential Attenuation Law in the Context of Your Simulation

In this part of the radiation shielding example you connect your Monte Carlo results to a simple analytical model, the exponential attenuation law. The goal is to see how well your simulated transmission through a shield agrees with the theoretical prediction, and to understand why differences appear.

The Exponential Attenuation Law

When a narrow, monoenergetic gamma beam passes through a homogeneous material, its intensity decreases with depth according to the exponential attenuation law. If $I_0$ is the initial intensity and $I(x)$ is the intensity after the beam has passed through a thickness $x$ of material, the law is

$$
I(x) = I_0 \, e^{-\mu x},
$$

where $\mu$ is the linear attenuation coefficient of the material at the gamma energy you are using.

It is often more convenient to work with transmission $T(x)$, which is the fraction of photons that pass through the shield without being removed from the narrow beam:

$$
T(x) = \frac{I(x)}{I_0} = e^{-\mu x}.
$$

In a Monte Carlo simulation you do not measure a continuous intensity. Instead, you count individual photons.

If $N_0$ is the number of photons emitted toward the shield and $N(x)$ is the number that reach the detector behind the shield, then

$$
T(x) = \frac{N(x)}{N_0},
$$

and if the exponential model holds, $T(x)$ should be close to $e^{-\mu x}$.

Key formula (exponential attenuation law):
$$
I(x) = I_0 \, e^{-\mu x}, \qquad T(x) = \frac{I(x)}{I_0} = e^{-\mu x}.
$$

The coefficient $\mu$ depends on both the material and the photon energy. For a fixed gamma energy, lead has a much larger $\mu$ than aluminum, so the same thickness of lead attenuates the beam more strongly.

Relating Simulation Counts to the Analytical Model

In your shielding project you have already set up a gamma source, a shield with adjustable thickness, and a way to record transmitted photons. To compare with the exponential law, you must express your simulation results in the same language: transmission as a function of thickness.

For a given shield thickness $x$, the simulation gives you a number of transmitted photons, for example in an actor that records which particles enter a detector volume behind the shield. Suppose you run the simulation several times, each time with a different shield thickness, but with the same source configuration and the same number of primary events.

For each thickness $x_i$ you can compute the simulated transmission

$$
T_{\text{sim}}(x_i) = \frac{N_{\text{trans}}(x_i)}{N_0},
$$

where $N_{\text{trans}}(x_i)$ is the number of photons that passed through the shield and reached your scoring region, and $N_0$ is the number of primary photons emitted in the simulation.

If you keep $N_0$ the same for all thicknesses, you can estimate it from a reference run with no shield. In that case

$$
N_0 \approx N_{\text{trans}}(x = 0),
$$

so the transmission at thickness $x$ becomes

$$
T_{\text{sim}}(x) \approx \frac{N_{\text{trans}}(x)}{N_{\text{trans}}(0)}.
$$

This way you do not need to know the absolute activity or flux, only relative counts.

Important practical rule:
Use the simulation without a shield as your reference. For each thickness $x$,
$$
T_{\text{sim}}(x) \approx \frac{\text{transmitted counts at }x}{\text{transmitted counts at }0}.
$$

If you keep the number of primaries constant, this ratio automatically gives you a transmission that is directly comparable to $e^{-\mu x}$.

Getting or Estimating the Attenuation Coefficient

To compare $T_{\text{sim}}(x)$ with $e^{-\mu x}$ you need the value of $\mu$ for your material and gamma energy. There are two natural approaches: using tabulated data or extracting $\mu$ from your simulation results.

Using tabulated values

You can obtain $\mu$ from databases such as the NIST XCOM tables. There you find the mass attenuation coefficient $\mu / \rho$ [cm²/g] for a given material and photon energy. To convert it to the linear attenuation coefficient $\mu$ [1/cm], multiply by the material density $\rho$ [g/cm³]:

$$
\mu = \left( \frac{\mu}{\rho} \right) \rho.
$$

For example, if at 1 MeV you find $\mu / \rho = 0.070\ \text{cm}^2/\text{g}$ and the density of lead is $\rho = 11.34\ \text{g/cm}^3$, then

$$
\mu = 0.070 \times 11.34 \approx 0.79\ \text{cm}^{-1}.
$$

With this $\mu$ you can calculate the theoretical transmission for any thickness $x$:

$$
T_{\text{theory}}(x) = e^{-\mu x}.
$$

Fitting $\mu$ from your Monte Carlo data

You can also use your simulation results to estimate an effective attenuation coefficient. In that case you treat the analytical law as a model for your data and fit $\mu$ to make the model match the statistics from GATE.

Since the law is exponential, it is convenient to work with logarithms. Taking the natural logarithm of both sides of $T(x) = e^{-\mu x}$ gives

$$
\ln T(x) = -\mu x.
$$

If you plot $\ln T_{\text{sim}}(x)$ versus $x$, the points should lie approximately on a straight line with slope $-\,\mu$. A simple linear fit then gives you an estimate of $\mu$:

$$
\ln T_{\text{sim}}(x_i) \approx -\mu_{\text{fit}} x_i.
$$

You can do this fit in Python after exporting your simulated transmissions for several shield thicknesses.

Key idea for extracting $\mu$:
Fit a straight line to $\ln T_{\text{sim}}(x)$ versus $x$.
The slope is $-\mu_{\text{fit}}$.

This approach is especially useful if you are not sure which database value to use, or if your setup deviates from the ideal assumptions of the simple law and you want an effective coefficient that incorporates those effects.

Step-by-Step Comparison Procedure

To make a clear comparison between your simulation and the exponential law, you can proceed through the following conceptual steps. The exact implementation details depend on how you choose to process the output, but the logic is always the same.

First, choose a gamma energy and a shielding material. Also decide on a set of shield thicknesses, for example 0, 1, 2, 3, and 5 cm. For each thickness $x_i$ you run a simulation with the same source configuration and the same number of primary events. You record the number of photons that reach your detector region behind the shield, $N_{\text{trans}}(x_i)$.

Second, use the simulation without a shield as your reference. Let $N_{\text{trans}}(0)$ be the number of transmitted photons in the run with zero thickness. For each other thickness $x_i$ compute the simulated transmission

$$
T_{\text{sim}}(x_i) = \frac{N_{\text{trans}}(x_i)}{N_{\text{trans}}(0)}.
$$

This gives you a set of points $(x_i, T_{\text{sim}}(x_i))$.

Third, obtain a theoretical value for $\mu$. You can either use a tabulated $\mu$ for your energy and material, or fit $\mu_{\text{fit}}$ from your simulated transmissions using the linear relation between $\ln T$ and $x$. Once you choose a value for $\mu$, calculate the theoretical transmission for each thickness

$$
T_{\text{theory}}(x_i) = e^{-\mu x_i}.
$$

Fourth, compare the theoretical and simulated transmissions. It is convenient to summarize this comparison in a small table.

For example, suppose you simulate 1 MeV photons in lead:

Thickness $x$ [cm]	$T_{\text{sim}}(x)$	$T_{\text{theory}}(x)$	Relative difference
0	1.00	1.00	0 %
1	0.46	0.45	about 2 %
2	0.22	0.20	about 10 %
3	0.08	0.09	about 11 %
5	0.02	0.01	about 33 %

The exact numbers depend on how many events you simulate. The relative difference column helps you see where the simulation and the analytical model agree and where they start to diverge.

Fifth, you can visualize the comparison by plotting both $T_{\text{sim}}(x)$ and $T_{\text{theory}}(x)$ as a function of thickness. If you plot them on a linear scale you see the absolute differences. If you plot $\ln T$ versus $x$ you see whether the simulated transmissions follow a straight line.

Understanding Differences between Simulation and Theory

Even with a well configured simulation, you should not expect perfect agreement with the simple exponential law. Several effects can cause differences. Understanding them is part of the learning goal of this example.

The exponential attenuation law describes the attenuation of a narrow, unscattered pencil beam measured in a very small detector directly in line with the beam. It counts only photons that have not undergone any interaction in the shield. In practice, your Monte Carlo geometry is usually less ideal than that.

If your source has a finite size or emits into a cone instead of a true pencil beam, some photons enter the shield at slightly different angles. For oblique incidence the effective path length in the material is longer than the physical thickness, so the attenuation is slightly stronger. This can make $T_{\text{sim}}(x)$ smaller than $T_{\text{theory}}(x)$.

If your detector behind the shield has a finite size, it can detect photons that have been scattered in the shield but still travel roughly toward the detector. The exponential law does not include these scattered contributions, which adds to the difference. For thick shields, a significant fraction of detected photons can be scattered, so the simulation transmission may be larger than the primary-only prediction.

If you use a broad energy spectrum instead of a monoenergetic beam, then each energy has its own $\mu(E)$. The exponential law is strictly valid only for a single energy. In a polyenergetic beam, low-energy photons are attenuated more strongly, and the transmitted beam becomes harder, that is, more weighted toward higher energies. The transmission curve then deviates from a single exponential with constant $\mu$.

Statistical fluctuations also matter. Monte Carlo results are subject to Poisson statistics. If the expected number of transmitted photons is small, especially for thick shields, the relative statistical uncertainty becomes large and the simulated $T_{\text{sim}}(x)$ can fluctuate around the theoretical value. Increasing the number of simulated events reduces these fluctuations and brings the simulation closer to the analytical curve.

Geometric details in the simulation can also create differences. If the shield does not fully cover the beam, or there are small gaps or misalignments, some photons can bypass the intended material and reach the detector, increasing $T_{\text{sim}}(x)$ compared to the ideal analytical case. At the same time, if the shield is larger than the detector and scattered photons are not scored, transmission can be smaller.

Finally, the choice of physics models and production cuts can have a small impact on the exact amount of attenuation, especially if very low-energy secondary photons are either produced or cut off inconsistently with the analytical assumptions. For a simple shielding benchmark you typically choose a well validated electromagnetic physics list and reasonably small production cuts so that the main attenuation processes are accurately modeled.

Interpretation guideline:
Good agreement in the overall exponential trend validates both your shield material and the basic photon physics.
Systematic deviations often reveal geometric effects, scattering into the detector, polyenergetic beams, or insufficient statistics, not fundamental errors.

Using the Comparison to Validate Your Shielding Setup

After you complete the comparison, you can use it as a validation step for your radiation shielding simulation. If the simulated transmission follows the analytical curve within statistical uncertainty for moderate thicknesses, you can be confident that

The shield material and density are correctly defined.
The source is producing the intended energy and direction.
The detector region behind the shield is correctly positioned and scoring the right photons.
The selected physics models describe the main attenuation processes realistically.

If you observe large discrepancies that cannot be explained by the limitations of the analytical model, it is a sign to check your simulation carefully. You can go back to earlier parts of the project and verify the shield geometry, the source definition, and the scoring setup.

In summary, comparing your GATE transmission results with the exponential attenuation law is a powerful way to connect Monte Carlo simulations to simple textbook physics. It helps you develop intuition for how shielding thickness, material, and energy affect transmission, and it serves as a practical consistency check on your simulated geometry and physics configuration.