KAHIBARO
24.9. Comparing Simulation Results
Table of Contents
Looking at Results from Different Shielding Configurations
In a radiation shielding study you do not just want plots from a single configuration. The real goal is to compare how different materials and thicknesses perform under the same irradiation conditions, and to understand what those differences mean physically. This chapter focuses on how to compare results from your shielding simulations in a clear and quantitative way once each configuration has already been simulated and you have transmission or attenuation values available.
Normalizing Results Across Simulations
Before comparing shielding configurations you must ensure that the results from different simulations can be put on the same scale. In a Monte Carlo simulation the most natural normalization is the number of primary particles fired at the source.
If you denote by $N_\text{inc}$ the number of incident primary photons simulated for a given run, and by $N_\text{det}$ the number of particles transmitted through the shield and recorded in your scoring region, the basic transmission is
$$
T = \frac{N_\text{det}}{N_\text{inc}}.
$$
For a meaningful comparison between different shields, always compare dimensionless quantities, such as:
- Transmission: $T = N_\text{det} / N_\text{inc}$
- Attenuation factor: $A = 1 / T$
- Attenuation coefficient: $\mu$ extracted from $T = e^{-\mu x}$
Never compare raw counts from simulations with different $N_\text{inc}$ or different scoring definitions.
If you used the same number of primary particles and identical source and detector definitions in all runs, then you can safely compare $N_\text{det}$ directly across runs because the normalization is implicit. However, computing and storing $T$ explicitly in your analysis code avoids confusion and helps later when you overlay results from several runs on the same plot.
When your analysis includes energy information, for example transmitted energy spectra, you should still normalize those spectra in a consistent way. A simple choice is to divide each histogram bin content by $N_\text{inc}$ so that the vertical axis represents a probability per event or per primary rather than a raw count.
Comparing Different Shielding Materials
If you have simulated several materials at the same thickness, for example lead, aluminum, and concrete slabs of a fixed thickness, you can compare their performance in a straightforward way. The essential question is: for the same incident radiation, which material reduces the flux more efficiently?
A simple comparison table might look like this:
| Material | Thickness $x$ (cm) | Transmission $T$ | Attenuation $1 - T$ |
|---|---|---|---|
| Lead | 2.0 | 0.05 | 0.95 |
| Aluminum | 2.0 | 0.40 | 0.60 |
| Concrete | 2.0 | 0.30 | 0.70 |
From this type of summary you can already see which material gives the lowest transmission for the same thickness. If your simulation also records energy deposition in a detector behind the shield, you can create a similar table using deposited dose or average deposited energy per incident photon instead of transmission.
It is often useful to connect your simulation results with the exponential attenuation model that is frequently used in shielding calculations. For a monoenergetic beam and a uniform slab, the transmission is approximated by
$$
T(x) = e^{-\mu x},
$$
where $x$ is the shield thickness and $\mu$ is the linear attenuation coefficient of the material for the given photon energy.
From a single simulation at thickness $x$ you can extract an effective attenuation coefficient by rearranging this formula:
$$
\mu_\text{eff} = -\frac{1}{x}\,\ln T(x).
$$
If you do this for each material, you can present a comparison like:
| Material | Thickness $x$ (cm) | Transmission $T$ | $\mu_\text{eff}$ (cm$^{-1}$) |
|---|---|---|---|
| Lead | 2.0 | 0.05 | $\mu_\text{Pb}$ |
| Aluminum | 2.0 | 0.40 | $\mu_\text{Al}$ |
| Concrete | 2.0 | 0.30 | $\mu_\text{conc}$ |
Even though your Geant4 simulation uses detailed physics and multiple interaction mechanisms, this effective coefficient helps relate the results to simple analytical formulas and to reference shielding data.
When comparing different materials, pay attention to the shape of the transmitted energy spectra as well as overall transmission. High‑Z materials such as lead tend to produce more high‑energy scattered photons and characteristic X rays, while lower‑Z materials often produce broader Compton‑scattered spectra. Geant4 can show these differences directly in the spectra recorded in your scoring region.
Comparing Different Shield Thicknesses
When you simulate several thicknesses of the same material, you can study how the transmission changes with thickness and check how well it follows the expected exponential trend. For example, you might simulate thicknesses $x = 0, 1, 2, 5, 10$ cm and compute transmission for each.
A convenient way to present the results is a plot of $T(x)$ versus $x$ for a given material. If the incident beam is approximately monoenergetic and narrow, you will often see a roughly exponential decrease of $T$ on a linear scale. On a semi‑logarithmic plot where you show $\ln T$ versus $x$, an ideal exponential attenuation would appear as a straight line. You can compare your Geant4 data points with such a line extracted from a fit or from reference values of $\mu$.
If you have simulated several materials at multiple thicknesses, you can overlay curves of $T(x)$ for each material. The result might look conceptually like this: at $x = 0$ cm all curves start at $T = 1$, and as $x$ increases lead drops the fastest, then concrete, then aluminum. You can use the same approach for other quantities such as transmitted dose.
To compare thickness performance in a more compact way, you can present half‑value layer (HVL) and tenth‑value layer (TVL) extracted from your simulation. The HVL is defined as the thickness for which the transmission is $T = 0.5$, and the TVL for $T = 0.1$. From your discrete simulation points you can estimate these values, for example by interpolating between thicknesses where the transmission crosses 0.5 or 0.1. Then you can compare which material achieves the same transmission with a smaller thickness, which is often a practical design question.
Including Statistical Uncertainties
Because Geant4 uses Monte Carlo sampling, every quantity you compute from the simulation has a statistical uncertainty that depends on the number of events. When comparing different shielding configurations, you should include these uncertainties to know whether an observed difference is statistically meaningful.
For a simple count $N_\text{det}$ with $N_\text{inc}$ primaries, if each primary has a probability $T$ of being transmitted, then the distribution of $N_\text{det}$ is approximately binomial. For large $N_\text{inc}$ and small $T$, it is also well approximated by a Poisson distribution. In either case, the variance on $N_\text{det}$ is roughly
$$
\sigma_{N_\text{det}}^2 \approx N_\text{det},
$$
so the standard deviation is $\sigma_{N_\text{det}} \approx \sqrt{N_\text{det}}$.
The transmission $T = N_\text{det} / N_\text{inc}$ then has an uncertainty
$$
\sigma_T = \frac{\sigma_{N_\text{det}}}{N_\text{inc}} \approx \frac{\sqrt{N_\text{det}}}{N_\text{inc}}.
$$
If you use an attenuation coefficient extracted from $T$, the uncertainty propagates through the logarithm. With
$$
\mu_\text{eff} = -\frac{1}{x}\,\ln T,
$$
the standard error on $\mu_\text{eff}$ is
$$
\sigma_{\mu_\text{eff}} = \frac{1}{x}\,\frac{\sigma_T}{T}.
$$
Whenever you present transmission or attenuation curves from Monte Carlo simulations, include error bars based on the statistical uncertainties. Do not over‑interpret small differences between configurations when the error bars overlap significantly.
In practice, you can compute these uncertainties during the analysis stage using the accumulated counts from your runs, or you can use the error handling built into the Geant4 analysis system for histograms and profiles.
When you compare two shield configurations, for example two materials at the same thickness, and their transmissions are $T_1 \pm \sigma_{T_1}$ and $T_2 \pm \sigma_{T_2}$, you can estimate the significance of their difference as
$$
\Delta T = T_1 - T_2,
$$
with an uncertainty
$$
\sigma_{\Delta T} = \sqrt{\sigma_{T_1}^2 + \sigma_{T_2}^2}.
$$
If $|\Delta T|$ is smaller than or comparable to $\sigma_{\Delta T}$, then the two results are statistically compatible at the level of your simulation statistics.
Visual Presentation of Shielding Performance
Once you have normalized results and uncertainties for different materials and thicknesses, the last step is to present them in a way that is easy to interpret. This is especially important when the shielding study is intended to guide design decisions.
Typical visualizations include transmission versus thickness plots with different materials shown as separate curves, dose behind the shield as a function of thickness, and bar charts of $1 - T$ for a fixed thickness comparing materials. When energy information is available, comparative spectra of transmitted photons for different shields can reveal how the shield changes the radiation quality. For example, you might see that one material suppresses the primary peak effectively but produces a noticeable secondary peak at lower energies from scattered photons.
In all such comparisons, clearly label the incident beam conditions, such as photon energy and beam geometry, so that the viewer understands that all curves correspond to identical source definitions and scoring conditions. You can then point out patterns, such as which material achieves a certain transmission at the smallest thickness, or how much thicker a lighter material must be to match the attenuation of a denser one.
By combining normalized quantities, effective attenuation parameters, statistical uncertainties, and clear visualizations, you can extract meaningful conclusions from your Geant4 shielding simulations, and you can compare different design options with confidence based on the same underlying Monte Carlo model.
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