Preamble

Application of Momentum Coupled Muon Scattering Tomography in Estimating Special Nuclear Materials Zi-Yang Chen, Jia-Ning Liu, Da-Ming Liu, Yu-Lei Li, Hui Gong, Yi Wang, Dong Han, and Bao-hong Guo 1 Department of Engineering Physics, Tsinghua University, Beijing 100084, China China Institute of Nuclear Industry Strategy, CINIS, No. 43, FuCheng Road, 100048, Beijing, China Muon scattering tomography has emerged as a prominent research direction in recent years, showing great potential in various applications such as cargo inspection and nuclear material detection. An increasing number of studies have highlighted the critical importance of incorporating muon momentum information, which can be obtained by Time of Flight (ToF) system as an alternative, to enhance image quality and rapid inspection capability in muon scattering tomography. MRPC (Multi-gap Resistive Plate Chamber) detectors are charac- terized by excellent time resolution and are widely used in ToF systems. Our resent research shows the time resolution of very narrow gaps MRPC reaching 16 ps and the spatial resolution of sealed MRPC achieving 0.5 mm. Therefore, MRPC holds great potential in muon scattering tomography. In this work, we employ Geant4 toolkit to simulate a realistic and detailed muon scattering tomography system consisting of MRPC-ToF and MRPC trajectory detection modules to estimate the internal structure of the classical Steve Fetter hypothetical model. The results indicate that momentum information significantly contributes to improving the quality of muon scattering tomography. To further improve imaging performance, a specialized neural network Unet is applied to the reconstructed images to extract nuclear material regions, thereby enhancing the effectiveness and resolution of muon scattering tomography.

Keywords

Muon Scattering Tomography, MRPC, Time of Flight, Geant4, U-Net

INTRODUCTION

Muon scattering tomography (MST) has been rapidly de- veloping in the nuclear technology field[ ]. By exploit- ing the naturally generated cosmic-ray muons and the fact that muon scattering angles are sensitive to the atomic num- ber of the material traversed, MST enables internal structure detection or imaging of objects under investigation without artificial radioactive sources and radiation damage harm. At present, MST can be applied in border security[ ], e.g., sus- picious woods inspection in cargo containers[ can also be applied in the management of nuclear fuel and waste[ ], as well as reactor imaging[ ] and indus- trial sector[ ]. Depending on different applications and re- quirements, muon scattering imaging technology can be di- vided into two main development directions: the first focuses on rapid detection capability[ ], where muon informa- tion collected within an relatively short exposure time is an- alyzed to quickly determine the presence or absence of tar- get materials, which could not provide direct MST image; the second focuses on fine imaging capability[ ], aimed at producing high-quality images of the internal structure of ob- jects.

Regardless of whether for rapid detection or detailed imag- ing, growing evidence shows that muon momentum infor- mation is critical for enhancing the performance of muon scattering tomography systems[ Worldwide, vari- ous approaches have been explored for estimating muon mo- mentum within MST systems. Some studies introduce ex- tra scattering materials into the system, analyzing the degree of muon scattering in these media to infer momentum[ Yi Wang, Tsinghua University, Beijing 100084, others exploit the energy sensitivity of Cherenkov detectors, employing multilayer Cherenkov detectors to classify muon momentum[ ]. In our previous work[ ], we proposed in- corporating a Time of Flight (ToF) system into the MST sys- tem to measure and calculate muon momenta, focusing on improving the system’s rapid detection capability for special nuclear materials.

In the field of MST, multiple detector technologies have been employed for muon trajectory measurement, includ- ing scintillation detectors[ ], drift tubes[ ], RPCs[ ] and MRPCs. Among them, MRPCs (Multi-gap Resistive Plate Chambers)[ ] are outstanding gas detectors, offering high efficiency for charged particle detection, excellent position resolution and orthogonal readout capability, making them an excellent choice for muon detection. Moreover, MRPCs are renowned for their ultrahigh time resolution[ ] and have been widely used in high-energy physics experiments[ Based on our earlier work, we improved the system design and developed a Multi-functional Cosmic Ray Imaging Sys- tem (MCRIS) composed entirely of MRPC detectors, with efficacious capability of momentum measurement.

The main content of this paper is the detailed construc- tion of a simulated MCRIS system using Geant4 toolkit[ followed by an in-depth investigation of its imaging perfor- mance for a specific nuclear device: the hypothetical nuclear device model proposed by Professor Steve Fetter, an expert in nuclear arms control and nonproliferation. Our study demon- strates the importance and necessity of incorporating momen- tum information in MST, and further employs a U-Net algo- rithm to optimize the imaging results, extracting the structural features of interest while reducing muon exposure time.

METHODS

Principle of Muon Scattering Tomography Cosmic-ray muons, with an average energy of about 4 GeV,

are generally regarded as minimum ionizing particles. Af- 65

ter passing through matter, muons undergo multiple Coulomb scattering (MCS). The distribution of the multiple scattering angles is strongly correlated with the atomic number Z of the material along muon path. This relationship is expressed in Equation (

σ θ = 13 . 6 MeV βcp

[1 + 0 038 ln( Based on our previous work[ ], we introduced the con- cepts of the nominal muon scattering angle and the nom-

inal scattering density per unit path length λ nom . The nominal 74

muon scattering angle is defined by the following expression:

θ nom = θ i p i √ L i (2) 76

In Equation ( represent respectively the scatter- ing angle, momentum and traversed path length of a certain

muon. The significance of θ nom and λ nom is that λ nom accounts 79

for the effects of the incident muon momentum and muon path, thereby ensuring that the nominal scattering density per

unit path length, which is directly related to the material’s 82

scattering length , depends solely on the distribution of the nominal muon scattering angles in Equation ( ), where is a constant:

i =1 θ nom ∝ 1 L rad (3) 86

λ nom = 1 Np 2 0

In other words, the distribution of of muons directly re- flects , or atomic number Z of objects under investigation Simulation in Geant4 In Geant4 simulation, two types of MRPC detector sys- tems are modeled: a high position resolution MRPC (named Hp_MRPC) system and a high time resolution MRPC (named Ht_MRPC) system. The Hp_MRPC system consists of two modules, each composed of three MRPCs for muon track reconstruction, while the Ht_MRPC system contains two Ht_MRPCs in total. Both detector types were constructed in Geant4 following the actual experimental configuration.

Specifically, each Ht_MRPC comprises four chambers with 32 gas gaps, where each gap has a thickness of 0.128 mm and the glass plates are 0.4 mm thick. Each Hp_MRPC consists of a single chamber with five gas gaps of 0.25 mm and glass plates of 0.7 mm. For both detector types, the PCB thickness is 1 mm; the Mylar layer is 0.5 mm; and the honeycomb sup- port panel is 10 mm, all consistent with the experimentally measured specifications.

To better match experimental performance, 2 MRPC types are calibrated using the measured spatial and temporal resolu- tions described in the literature. For the Hp_MRPC, a spatial resolution of 0.5 mm (close to 0.404 mm in [ ]) is applied.

The muon hit position is determined by averaging the ioniza- 111

tion electrons’ positions across all gas gaps, with additional

Gaussian smearing ( σ = 0.5 mm, µ = 0) independently ap- 113

plied to the x and x coordinates (we suppose z coordinate is along vertical direction in reality). For the Ht_MRPC, a time resolution of 20 ps (close to 16.7 ps in [ ]) is assumed.

The muon arrival time is obtained by averaging the ioniza- 117

tion times of the primary charges across all gaps then adding

a Gaussian fluctuation with σ = 20 ps and µ = 0. 119

In this work, the object for MST is based on the hypotheti- cal nuclear device model proposed by Steve Fetter. Its struc- tural configuration is shown in Fig. , consisting of an outer aluminum casing (blue), a surrounding explosive layer (light-

est gray), a uranium tamper (darkest gray), a beryllium reflec- 124

tor (gray), and a fissile core (red), all enclosed within a lead container (not shown in Fig ). In this work, the fissile core material is selected as WgU, with a corresponding thickness of 1.23 cm and an outer diameter of 7 cm. The thicknesses of other material layers are kept consistent with those shown in The geometrical arrangement of the Hp_MRPC and Ht_MRPC systems is illustrated in Fig. . In the MCRIS, the two Ht_MRPC detectors are placed at the top and bot- tom positions to determine the nominal muon scattering an- gles, while the Fetter model s positioned between the two Hp_MRPC modules. The aim is not only to identify any dis- placement of the Fetter model inside the container through but to further reconstruct its internal structural utilizing MST.

U-net U-Net is a convolutional neural network (CNN) ar- chitecture originally proposed biomedical image segmentation[ ]. It follows an encoder decoder structure (Fig. the encoder (contracting path) progressively captures contextual information through convolution and pooling operations, while the decoder (expanding path) performs upsampling and concatenates feature maps from corresponding encoder layers via skip connections.

symmetric “U-shaped” design enables the network to combine high-level semantic information with fine-grained spatial details, leading to precise localization and accurate boundary reconstruction. Meanwhile, U-Net is particularly well suited for scenarios involving small datasets[ ]. Owing to its efficiency and strong generalization ability, U-Net and its variants have been widely applied not only in medical imaging but also in diverse fields such as remote sensing, materials science, and particle detector data analysis.

In this work, the U-Net is primarily utilized to segment re- gions of high Z materials based on the MST results of the Fet- ter model. Our U-Net follows a symmetric encoder-decoder design with four downsampling and four upsampling stages, connected through skip connections to preserve spatial infor- mation. The network takes gray-scale images as input and produces a single-channel segmentation map that highlights the high Z structures of interest from MST consequences.

ANALYSIS AND RESULTS Evaluation of Momentum Resolution and Realistic Energy Loss Effects According to Equation ( ), the nominal scattering angle encapsulates several key physical quantities, including the actual scattering angle , the incident muon momentum and the effective material thickness traversed by the muon Since are difficult to measure directly, the method proposed in our previous work is adopted, in which estimated momentum and estimated length of flight flight mated physical quantities is as Equation (

ˆ L flight = H/cos ( θ zen ) (4) 177

ˆ p = m µ ˆ L flight

flight flight flight flight flight is the width of the tomography target; is the incident

zenith angle of a single muon; the estimated ˆ t flight is obtained 180

from the difference in arrival moments measured by the 2 Ht- MRPC detectors. This approach has been validated in our previous work, though its accuracy is primarily confined to the low-energy muon regime[ However, a crucial consideration arises here: during prop- agation, muons continuously interact with the material, un-

dergoing both MCS and ionization which causes energy loss 187

simultaneously. For the Fetter model, muons with momenta below approximately 3 GeV/c are distinctly affected by these processes (Fig. ), which must be carefully accounted for in the subsequent analysis.

In Fig. , the left panel compares the momentum spec- tra of all incident muons (those reaching the Fetter model) with those of the effective muons (i.e., muons successfully

reach the lower track module). A clear ionization energy loss 195

effect induced by the target object can be observed: muons below a certain energy threshold fail to emerge from the ob- ject and therefore cannot contribute as valid events. The right panel shows the comparison between the incident and emit- ted momentum distributions of the effective muons, which can be approximately regarded as a shift of the incident spec-

trum toward lower energies. This demonstrates that ioniza- 202

tion energy loss cannot be neglected, particularly for low- momentum muons, which are the main focus of the ToF- based momentum measurement.

Ionization causes energy loss and, correspondingly, muon’s 206

velocity to gradually decrease along its path. While this ef- fect could be negligible for high-energy muons, it becomes nontrivial for muons below and around approximately 1 GeV, leading to a measurable reduction in velocity. Consequently, the velocity obtained from the ToF measurement represents the average velocity along the trajectory rather than the true incident velocity. This discrepancy introduces a inevitable bias in the calculation of the nominal scattering strength.

Therefore, before performing the displacement detection and internal structure imaging of the target Fetter model, a de- tailed evaluation and calibration of the momentum measure- ment accuracy are conducted. In this work, we analyze the correlation between the true incident momentum of muons before entering the Fetter model and the momentum recon- structed using the ToF method. The results show a clear pos- itive correlation between the two, confirming the overall re- liability of the TOF-based momentum estimation despite the aforementioned energy loss effects.

As shown in the Fig. , a clear positive correlation is ob- served between the momentum measured by the ToF method and the real incident momentum for muons with relatively low momenta, confirming the reliability and feasibility of the ToF-based momentum measurement (i.e. MCRIS). Further- more, the correlations between true incident momentum and both the mean value and uncertainty of the measured mo- mentum are quantitatively considered, as presented in Fig. where the blue and red curves represent the polynomial fit- ting results. In this work, the fitted correlations are utilized to infer the incident momentum of muons from the estimated values and to obtain the associated measurement uncertainty.

This procedure effectively mitigates the influence of ioniza- 237

tion energy loss on the ToF-based momentum determination, thus enhancing the imaging precision.

In the subsequent image reconstruction, MST images are reconstructed and obtained using the Point of Closest Ap- proach (PoCA) algorithm.

Short-term Displacement Detection For certain special nuclear materials or facilities (such as the Fetter model), it is important to determine whether the core components have experienced any displacement within the container during transportation or long-term storage. This is crucial both for safety assessment and for verifying the structural integrity and reliability of the container design. The proposed MCRIS system provides a rapid, non-intrusive in- spection approach that allows the detection of small displace-

ments of nuclear components without opening the container. 252

It can also serve as a long-term monitoring tool, capable of 253

identifying and warning of even minor positional changes 254

within a short response time. In this work, muon scattering tomography was conducted with the Fetter model positioned either at the center of the container or displaced by 1 cm, 3 cm, 5 cm along some hor- izontal direction (named x axis in the analysis) from center.

Considering that the muon exposure time in this part is rela- tively short, the pixel size is set to 1 cm in order to ensure a sufficient number of muons within each PoCA pixel.

A 10-minute muon exposure is chosen as a representative short-term acquisition. Figure. shows the corresponding muon scattering tomographic results: the upper panel illus- trates the reconstruction based solely on the scattering angle, representing the conventional MST approach, while the lower panel presents the result obtained by incorporating MRPC- ToF estimated momentum information.

For Fetter model, such a short exposure is insufficient to reconstruct the over- all structure, particularly the outer explosive layer, which re- mains almost indistinguishable, while the more internal struc-

tures containing heavy nuclear materials can still be discerned 273

with reasonable clarity. A comparison between the two recon- struction approaches reveals that using only angular informa- tion produces a blurred depiction of the dense core, which is also highly sensitive to occasional large-angle scattering within single pixels. By contrast, inclusion of the momen- tum information markedly sharpens the outline of the heavy nuclear component and substantially improves the image con- trast.

In addition, we compare the distribution of the ’PoCA cen- troid’ along the x-axis obtained under a shorter muon expo- sure time with the Fetter model positioned at the center of the container. Fig. illustrates the reconstructed results ob-

In the legend ’all_muon’ means the incident momentum distribution of all the muons approaching the Fetter model, ’effective_muon_in’ means the the incident momentum distribution of muons which completely penetrate the Fetter model, and ’muon_out’ means the the emitted momentum distribution of muons which completely penetrate the Fetter model. tained under 1-min and 2-min muon exposure times(in such short period, to attain a high quality MST image is challeng- ing). The results show that, when the muon momentum in- formation is not used, the ’PoCA centroid’ distribution ex- hibits larger dispersion, whereas the inclusion of momentum information leads to a more concentrated distribution. Build- ing upon this result, the approach introduced in our previous

study can be applied to quantify the minimum detectable dis- 293

placement and the corresponding detection time at a given confidence level, which may be the focus of future work.

Reconstructed Internal Structure of Fetter Model To investigate the internal structure of Fetter model, a long- term muon exposure simulation is performed, and the corre- sponding images are reconstructed. The results obtained un- der different data analysis methods are shown in Fig.

In Fig. , from left to right, we process the same set of muon scattering angle data, while only the treatment of the momentum information differs. Subfigure (a) shows the im- age reconstructed using only the scattering angle information, whereas subfigure (b) presents the result obtained by apply- ing momentum got by ToF methods to all scattering angle data. A clear comparison between the two reveals that, with- out incorporating momentum information, the internal high- Z structure of the Fetter model cannot be effectively distin- guished within the same muon exposure time. By contrast, when momentum information is included, a distinct “multi- ring” image of the Fetter model emerges: the innermost dark ring corresponds to the fissile core and the outer dark ring rep- resents the tamper region, while the explosive layer remains barely distinguishable.

Considering the characteristics of the ToF-based momen- tum estimation, the uncertainty of the estimated momentum booms with increasing muon momentum.

Therefore, for high-energy muons, the ToF method inevitably introduces larger errors. Following the approach adopted in our previ- ous work, the applicable range of the ToF method is restricted to an artificially defined ’low-momentum region’ and differ- ent treatment methods are applied to muons outside of this range. So according to the same logic, the maximum accept- able momentum-estimation uncertainty is also set to 60% as same as prior considerations, which corresponds to a muon momentum of approximately 1 GeV/c. Thus, in our analysis, muons with estimated momenta below 1 GeV/c are defined as belonging to the ’low-momentum region’ in this work.

Subfigure (c) shows the result obtained after excluding all muons outside the low-momentum region and performing

MST imaging employing only the remaining data. In this 332

case, the circular outline of the explosive layer (light pink in ing to the internal high-Z materials appear more compact and well defined. However, in nuclear related fields using fewer data leads to larger statistical uncertainty; simply discarding

The blue line indicates the geometric center of the container. no low-momentum muons cause waste. To take advantage of these muons, their energy spectrum is analyzed to obtain the momentum mean value as well as uncertainty within this range. According to this logic, the final result, shown in sub- figure (d), clearly resolves the circular ring of the explosive layer while also providing enhanced contrast for the fissile core and tamper regions. result of the Fetter model and its corresponding geometric model in Geant4. It can be observed that the relative posi- tions of the explosive layer, tamper layer and Be reflective layer are generally consistent from the outside to the inside, whereas the thickness of the central fissile core shows a no- ticeable discrepancy.

For the fissile core, since it contains an internal hollow cav- ity, muons theoretically do not undergo MSC within this re- gion. However, according to the principle of the PoCA al- gorithm, the MCS occurring along the entire muon trajectory are treated as a single equivalent scattering at the PoCA point.

Therefore, a muon reconstructed whose PoCA point is inside the cavity have already traversed the entire Fetter model and

experienced significant scattering. Moreover, due to the large 359

variation in muon’s incident angles, the effective path lengths through different materials vary substantially among muons, even if their reconstructed PoCA points may be located at similar position. The current PoCA reconstruction does not account for these influences, which leads to blurred or ill- defined edges in the reconstructed image.

This analysis further highlights the necessity of developing improved reconstruction algorithms that are better suited for

Comparison between MST result and Fetter model in Geant4.

Optimization Results Based on U-Net The images shown in Figure are obtained within a muon exposure time of about 200 hours. The detection effi- ciency of the detector for cosmic-ray muons is approximately 30.4%. Fig. shows several reconstructed results obtained within a maximum exposure time of 4,000 minutes. It can be observed that, with momentum information incorporated, the image quality at 4,000 minutes surpasses that of the 200- hour reconstruction without estimated momenta. This further demonstrates the essential role of muon momentum informa- tion in improving the accuracy and clarity of muon scattering imaging.

However, even with the inclusion of momentum informa- tion, the reconstruction at 4,000 minutes only weakly sug- gests the existence of the tamper layer.

If we aim to ob- tain the internal structure of the target object (e.g., the Fetter model) within a shorter acquisition time, using only momen- tum information is not sufficient; advanced image processing algorithms are also required. Among image-processing algo- rithms, the U-Net architecture is particularly suitable for ex- tracting regions of interest from images. For the Fetter model, the main regions of interest are the internal high-Z materials, i.e., the tamper and fissile core.

Prior to training the U-Net model, the reconstructed MST 392

images undergo pre-processing, including Gaussian smooth- ing and a tailored windowing operation. The latter is inspired by windowing methods commonly used in CT field, where adjusting the window width and level improves the contrast of target structures[ To enhance the generalization capability of the U-Net

model, we generate additional training samples by modify- 399

ing the layer thicknesses of the tamper and fissile core in Geant4. Two variant geometries are constructed, referred to as Fetter_Variant1 and Fetter_Variant2. In Fetter_Variant1, the fissile core has a thickness of 2.23 cm and an outer diam- eter of 8 cm, while the tamper thickness is set to 2 cm. In Fetter_Variant2, the fissile core thickness is fixed at 1.23 cm with an outer diameter of 8 cm, and the tamper thickness is 3 cm. All other parameters remain identical to those of the original Fetter model.

The MST images under different muon exposure times are pre-processed in the same manner, after which the images are

labeled and divided into training and test sets. These datasets 411

are subsequently used to train the U-Net model, yielding the following results in Fig.

Left parts: pre-processed MST images; right parts: U-Net results.

For the original Fetter model, approximately 2,000 min- utes of muon exposure are sufficient for the U-Net to extract the high-Z features of interest. In contrast, the two geometric variants require longer exposure times because their repre- sentation in the training dataset is smaller compared with the basic Fetter model. Consequently, the U-Net needs more data to accurately learn and reconstruct their high-Z structures.

Nevertheless, these results demonstrate that incorporating a

U-Net into the MST reconstruction workflow significantly re- 422

duces the exposure time required to resolve high-Z materials, thereby improving the overall performance and practicality of the MST system.

DISCUSSIONS In this work, a comprehensive momentum-coupled muon scattering tomography (MST) simulation framework has been developed utilizing the Geant4 toolkit.

Based on realistic laboratory configurations, we construct a detailed simulation system incorporating both Hp_MRPC and Ht_MRPC detec- tors, effectively coupling a ToF spectrometer with a tradi- tional MST setup. The model fully accounts for muon en- ergy loss during transportation, detector measurement uncer- tainties, and the extraction of muon momentum information, thereby forming an end-to-end simulation platform capable of providing practical guidance for future experimental im- plementations.

The primary imaging target considered in this study is the Fetter model: an intricate nuclear device. Both short-term displacement detection and non-destructive internal structure imaging are investigated. The results consistently show that integrating a ToF system into MST substantially improves

imaging quality and system performance and significantly re- 444

duces the exposure time required to resolve the internal high- Z components. Even though muons experience continuous energy loss and detectors introduce measurement uncertain- ties, appropriate momentum analysis and selection strategies

still enable MST to extract maximal useful information. Fur- thermore, the application of a U-Net–based segmentation net- work demonstrates promising capability in identifying and enhancing high-Z structural features within MST images, in- dicating strong potential for real-world deployment.

Finally, this work clearly reveals that the Point-of-Closest- Approach (PoCA) algorithm, due to its inherent physical and geometrical limitations, performs poorly in reconstruct- ing material boundaries: an issue particularly pronounced for complex multilayer structures such as the Fetter model. Fu- ture efforts should therefore focus on developing more ad- vanced and better-suited reconstruction algorithms[ further enhance the performance and applicability of muon

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