# Analysis of Secondary Particles as a Complement to Muon Scattering Measurements

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Muon Scattering Tomography and Its Use Cases

^{−2}min

^{−1}at sea level [10]. They have, unlike electrons, photons and neutrons, a high material penetration power, as they have a nearly 200 times higher mass than an electron and mostly interact with atoms through Coulomb scattering. This makes muons induced by cosmic rays a perfect natural source for tomography systems [11]. Muon Scattering Tomography takes advantage of the deflection of the muon trajectory due to the Coulomb scattering, with the deflection angle being proportional to the atomic number Z and the density $\rho $ of the material. By measuring the scattering angles for many muons, the properties of an examined object can be reconstructed, allowing to differentiate between materials with distinct atomic numbers and densities [12,13]. Another technique of utilizing cosmic ray muons for imaging of large-scale structures is the so-called Muon Radiography. For this method, the differential muon absorption is measured, which allows for the discrimination between materials, as the absorption is proportional to the atomic number and density of the examined materials [14,15].

#### 1.2. Complementary Information from Secondary Particles

^{+}e

^{−}pair or Compton scattering.

#### 1.3. Proposed Work

## 2. Materials and Methods

#### 2.1. Simulation Setup

^{2}square area in the $xy-\mathrm{plane}$ located at $z=7.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. The shipping container is modeled by a hollow box with 2 mm thick stainless-steel walls centered at the origin. Its dimensions follow the size of a 20 ft ISO container ($2.44\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 6.06\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 2.59\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$). Four detector planes surround the container, an upper and lower detector plane in the $xy-\mathrm{plane}$ at $z=\pm 1.75\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ with an area of $3.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 7.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, and two detector planes on the sides in the $yz-\mathrm{plane}$ at $x=\pm 1.75\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ with an area of $7.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 3.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. The detectors are modeled as 1 nm thick vacuum and show perfect acceptance and reconstruction efficiency to all particles passing through them with perfect spatial and energy resolution. If a particle is detected in one of the four planes, the following information is assigned to the so-called hit:

- Particle type: electron, photon, neutron
- Hit location: ${x}_{h},\phantom{\rule{3.33333pt}{0ex}}{y}_{h},\phantom{\rule{3.33333pt}{0ex}}{z}_{h}$
- Particle momentum: ${p}_{x}^{h},\phantom{\rule{3.33333pt}{0ex}}{p}_{y}^{h},\phantom{\rule{3.33333pt}{0ex}}{p}_{z}^{h}$
- Particle kinetic energy: ${E}_{kin.}^{h}$

#### 2.2. Analysis of Secondary Particle Kinematics

#### 2.2.1. Kinematic Dependence on Density

^{3}located at $y=-1.75\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ and a thin plate ($1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$) positioned at $y=1.75\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$. Two blocks with different thickness made out of the same material are studied in order to investigate the effect of self-shielding. If secondary particles are produced within an object, they can also interact with the surrounding matter until leaving the object or getting stopped within. This will affect the number of detectable secondary particles and therefore needs to be taken into account.

#### 2.2.2. Kinematic Dependence on Atomic Number

- Magnesium ($\rho =1.74\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3},Z=12$) and Cesium ($\rho =1.87\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3},Z=55$)
- Chromium ($\rho =7.18\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3},Z=24$) and Ytterbium ($\rho =6.73\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3},Z=70$)
- Molybdenum ($\rho =10.22\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3},Z=42$) and Lead ($\rho =11.35\phantom{\rule{3.33333pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3},Z=82$)

^{3}cubes are analyzed independently, with the lower atomic number element positioned at $y=-1.75\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ and the higher atomic number element positioned at $y=1.75\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.

#### 2.3. 3D Map Reconstruction

#### 2.3.1. Particle Detection

#### 2.3.2. Particle Back-Tracing

#### 2.3.3. 3D Map Generation

## 3. Results

#### 3.1. Test Scenarios

**Separate water and lead block**Each cube has a size of 1 m^{3}. Both blocks are positioned at $x=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ and $z=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, with the water cube at $y=1.75\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ and the lead cube at $y=-1.75\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$.**Lead block in water basin**A 1 m^{3}lead cube is located at the center of a water basin ($2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 4.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\times 2.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$) positioned at the origin.**Multiple lead blocks**

^{3}to allow for a sufficient spatial resolution within reasonable computation time and low level of background noise. The parameter sets are chosen manually for each scenario to create a clear visual discrimination between the different materials and objects while maintaining the shape and size of the items. If a measurement out of $M1$–$M9$ is by itself not capable of discriminating between the target objects and background noise, it is defined as negligible and therefore not utilized.

#### 3.2. Separate Water and Lead Block

#### 3.3. Lead Block in Water Basin

#### 3.4. Multiple Lead Blocks

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

UV | Ultraviolet |

3D | Three-dimensional |

CERN | Conseil Européen pour la Recherche Nucléaire |

ISO | International Organization for Standardization |

2D | Two-dimensional |

AI | Artificial intelligence |

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**Figure 1.**A 3D visualization of the simplified geometry. The detector planes are shown in blue, the container model in red and some generic objects representing the container content in green. In addition, an example of a trajectory of an air shower particle (gray line) with secondary particle production (yellow lines) is depicted.

**Figure 2.**The distribution of the kinetic energy ${E}_{kin.}$ as a function of the $y-\mathrm{position}$ of the reconstructed photons (

**left column**) and neutrons (

**right column**) in the upper detector plane. The materials analyzed are Cesium (

**upper row**), Tin (

**middle row**) and Palladium (

**bottom row**). The hits from the empty container are subtracted.

**Figure 3.**The distribution of the kinetic energy ${E}_{kin.}$ as a function of the $y-\mathrm{position}$ of the reconstructed photons (

**left column**) and neutrons (

**right column**) in the sidewise detector planes. The materials analyzed are Cesium (

**upper row**), Tin (

**middle row**) and Palladium (

**bottom row**). The hits from the empty container are subtracted.

**Figure 4.**The distribution of the kinetic energy ${E}_{kin.}$ as a function of the $y-\mathrm{position}$ of the reconstructed photons (

**left column**) and neutrons (

**right column**) in the lower detector plane. The materials analyzed are Cesium (

**upper row**), Tin (

**middle row**) and Palladium (

**bottom row**). The hits from the empty container are subtracted.

**Figure 5.**The distribution of the kinetic energy ${E}_{kin.}$ as a function of the $y-\mathrm{position}$ of the reconstructed electrons in the lower detector plane. The materials analyzed are Cesium (

**top left**), Tin (

**top right**) and Palladium (

**bottom**). The hits from the empty container are subtracted.

**Figure 6.**The distribution of the kinetic energy ${E}_{kin.}$ as a function of the $y-\mathrm{position}$ of the reconstructed photons (

**left column**) and neutrons (

**right column**) in the upper detector plane. The materials analyzed are Magnesium and Cesium (

**upper row**), Chromium and Ytterbium (

**middle row**), as well as Molybdenum and Lead (

**bottom row**). The hits from the empty container are subtracted.

**Figure 7.**The distribution of the kinetic energy ${E}_{kin.}$ as a function of the $y-\mathrm{position}$ of the reconstructed photons (

**left column**) and neutrons (

**right column**) in the sidewise detector planes. The materials analyzed are Magnesium and Cesium (

**upper row**), Chromium and Ytterbium (

**middle row**), as well as Molybdenum and Lead (

**bottom row**). The hits from the empty container are subtracted.

**Figure 8.**The distribution of the kinetic energy ${E}_{kin.}$ as a function of the $y-\mathrm{position}$ of the reconstructed photons (

**left column**) and neutrons (

**right column**) in the lower detector plane. The materials analyzed are Magnesium and Cesium (

**upper row**), Chromium and Ytterbium (

**middle row**), as well as Molybdenum and Lead (

**bottom row**). The hits from the empty container are subtracted.

**Figure 9.**The distribution of the kinetic energy ${E}_{kin.}$ as a function of the $y-\mathrm{position}$ of the reconstructed electrons in the lower detector plane. The materials analyzed are Magnesium and Cesium (

**upper row**), Chromium and Ytterbium (

**middle row**), as well as Molybdenum and Lead (

**bottom row**). The hits from the empty container are subtracted.

**Figure 10.**A 2D visualization of the particle back-tracing process and the score assignment. A hit is shown as a solid dot and its projected trajectory with a dashed line. The total, non-zero voxel scores ${s}_{tot}$ are shown in the corners of each voxel and are also represented by the color.

**Figure 13.**The 3D voxel map of the lead block in the water basin (

**left**) and its 2D projection (

**right**) reconstructed with the parameter set optimized for lead block in water basin as shown in Table 5. The color of the voxel point represents the combined voxel score ${s}_{comb}$.

**Figure 14.**The 3D voxel map of the lead block in the water basin (

**left**) and its 2D projection (

**right**) reconstructed with the parameter set optimized for water as shown in Table 4. The color of the voxel point represents the combined voxel score ${s}_{comb}$.

Photons | Neutrons | Electrons | |
---|---|---|---|

Upper detector | $M1$ | $M2$ | – |

Sidewise detectors | $M3$ | $M4$ | – |

Lower detector—production | $M5$ | $M6$ | – |

Lower detector—absorption | $M7$ | $M8$ | $M9$ |

$\mathit{x}-\mathbf{Position}$ | $\mathit{y}-\mathbf{Position}$ | $\mathit{z}-\mathbf{Position}$ | |
---|---|---|---|

Block 1 | $-0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $-1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $-0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ |

Block 2 | $-0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $-1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ |

Block 3 | $-0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $-0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ |

Block 4 | $0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $-1.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ | $-0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ |

Photons | Neutrons | Electrons | |
---|---|---|---|

Upper detector | 20% | 10% | – |

Sidewise detectors | 20% | 10% | – |

Lower detector—production | – | 10% | – |

Lower detector—absorption | 40% | 40% | 40% |

Photons | Neutrons | Electrons | |
---|---|---|---|

Upper detector | 15% | – | – |

Sidewise detectors | 15% | – | – |

Lower detector—production | 15% | – | – |

Lower detector—absorption | 30% | 30% | 30% |

**Table 5.**The parameter set with the noise thresholds ${t}_{min}$ optimized for a lead block in water basin.

Photons | Neutrons | Electrons | |
---|---|---|---|

Upper detector | 40% | 30% | – |

Sidewise detectors | 40% | 30% | – |

Lower detector—production | – | – | – |

Lower detector—absorption | 60% | 60% | 60% |

**Table 6.**The parameter set with the noise thresholds ${t}_{min}$ optimized for multiple lead blocks.

Photons | Neutrons | Electrons | |
---|---|---|---|

Upper detector | 30% | 20% | – |

Sidewise detectors | 30% | 20% | – |

Lower detector—production | – | 20% | – |

Lower detector—absorption | 50% | 50% | 50% |

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**MDPI and ACS Style**

Pérez Prada, M.; Barnes, S.; Stephan, M.
Analysis of Secondary Particles as a Complement to Muon Scattering Measurements. *Instruments* **2022**, *6*, 66.
https://doi.org/10.3390/instruments6040066

**AMA Style**

Pérez Prada M, Barnes S, Stephan M.
Analysis of Secondary Particles as a Complement to Muon Scattering Measurements. *Instruments*. 2022; 6(4):66.
https://doi.org/10.3390/instruments6040066

**Chicago/Turabian Style**

Pérez Prada, Maximilian, Sarah Barnes, and Maurice Stephan.
2022. "Analysis of Secondary Particles as a Complement to Muon Scattering Measurements" *Instruments* 6, no. 4: 66.
https://doi.org/10.3390/instruments6040066