## Introduction

Cosmic ray muons present a large part of background radiation1 and they have recently been explored as a non-conventional radiation probe for monitoring or imaging the contents of dense and large objects, otherwise not feasible with conventional radiographic techniques2. It is noteworthy how the recent developments on cosmic ray muon applications have been successfully acknowledged in various engineering fields, including nuclear reactor and spent nuclear fuel cask imaging3,4,5,6, homeland security7,8,9,10, geotomography11,12,13,14,15, and even archeology16. Despite the potential and success, the wide application of cosmic ray muons is limited by the naturally low muon intensity, approximately 104 m−2 min−1 at sea level. Since it is not practical to deploy neither a large hadron collider nor an accelerator in the field to provide muon beams, it is important to measure muon momentum to maximize the utilizability of each cosmic ray muon17,18. However, it is still challenging to measure muon momentum in the field without resorting to large solenoid or toroidal magnets, Cherenkov ring imagers, or time-of-flight detectors19,20,21. Although recent efforts at CNL and INFN aim to infer momentum knowledge from multiple Coulomb scattering measurements22,23, at present no portable spectrometer exists that can measure muon momentum in the field.

Here, we develop a new concept to measure muon momentum using multi-layer pressurized gas Cherenkov radiators. Because muons are charged particle, they can induce Cherenkov radiation in transparent media. Unlike solid or liquid Cherenkov radiators, the refractive index of gas radiators can be varied by changing gas pressure and temperature. We can then find the optimal muon threshold momentum levels by carefully selecting the gas pressure for each radiator. As a result, radiators will emit only when the threshold momentum is less than the actual muon momentum even though a muon passes through all radiators (Fig. 1). By measuring the Cherenkov signals in each radiator, we can then estimate the actual muon momentum (Fig. 2). The benefit of such an approach is compact, lightweight, and portable spectrometer that can be deployed in the field to improve existing or explore new engineering applications: cosmic ray muon tomography, geological studies, and cosmic radiation measurement in the International Space Station.

In this paper, we present the design concept, performance, and feasibility of our proposed muon spectrometer. This includes (i) theoretical background, (ii) principle of operation of the Cherenkov spectrometer, and (iii) Geant4 modeling and simulation results. The results show that our spectrometer can measure muon momentum with high accuracy (~ 90%) for a wide muon momentum range (0.1–10.0 GeV/c) and a resolution of ± 0.5 GeV/c which are sufficient for engineering applications. In the end, we demonstrate the benefits of muon momentum measurement in two real-world applications, (i) monitoring and (ii) imaging of special nuclear materials (SNMs) using muon scattering tomography. The results show that SNM monitoring and imaging capability can be significantly improved when we integrate muon momentum knowledge into the existing technologies.

## Results

Various mono-energetic muons are simulated to evaluate the functionality of the muon spectrometer. The measured photon signals are classified into two categories, Cherenkov photons (signals) and others (noise). In the first simulation, optical photon emissions by the scintillation and transition radiation are excluded to focus on the Cherenkov photon emission by muons. Two mono-energetic muons, pμ = 3.25 and 10.0 GeV/c, vertically enter the muon spectrometer and pass through all radiators (Fig. 3). When pμ = 3.25 GeV/c, only the first four radiators emit conical shaped Cherenkov radiation because their threshold momenta are less than the simulated muon momentum. On the other hand, all radiators emit Cherenkov radiation when pμ = 10.0 GeV/c because threshold momentum levels for all radiators are lower than 10.0 GeV/c. In the next simulation, scintillation and transition radiation are included in addition to Cherenkov radiation, however, all other parameters remain unchanged. More optical photons are observed in radiators due to the presence of scintillation and transition radiation. It is noted that scintillation photon emission can be efficiently differentiated from Cherenkov photons because of a) the characteristic photon emission direction (θc vs ) and b) light flash duration (Fig. 4). There was no transition radiation observed in Fig. 3 because it is a rare event for a few GeV muons and for the small number of boundary systems.

It is noteworthy that some additional Cherenkov photons (Nch ≠ 0 when pμ < pth) are observed in Fig. 3 and they are also quantitatively recorded (Fig. 5). It is because of either Compton scattering or Cherenkov photon emissions by secondary particles (mainly electrons) which can be produced by muon decays and muon to electron conversions25.

$$\mu^{ - } \to e^{ - } \overline{\nu }_{e} \nu_{\mu }$$
(1)
$$\mu^{ + } \to e^{ + } \nu_{e} \overline{\nu }_{\mu }$$
(2)
$$\mu N_{Al} \to eN_{Al}$$
(3)

Equation (1) and (2) represent radiative decays of μ± with a mean lifetime of τμ 2.2 μs. Radiative muon decays are the primary source of secondary charged particles. Equation (3) represents the muon to electron conversion. Muons can be captured by the 1s orbital of aluminum and mono-energetic electron emission follows26. Although the muon capture cross-section is insignificant, it is worth to be considered because the photon absorbers are made of black aluminum foils.

The results of total number of optical and Cherenkov photons as a function of muon energy are shown in Fig. 5. It is noted that the rapid increases of total number of emitted photons at the muon threshold momentum levels (vertical dashed lines) and they are the results of the Cherenkov radiation emission by muons. Then, it gradually increases as muon energy increases. It demonstrates that our proposed spectrometer is feasible because muon energies are clearly identified by analyzing optical photon signals from radiators.

### Performance evaluation

We evaluate the performance of our muon spectrometer by reconstructing a cosmic ray muon spectrum. The experimental results and mathematical expressions for cosmic ray muon spectrum at sea level are found in references27,28,29. We used the “Muon_generator_v3”30 which is developed to simulate cosmic ray muon spectra based on valid semi-analytical models31,32. To quantify the muon spectrometer measurement accuracy, we introduced a classification rate (CR) which is a probability that the system correctly identifies the actual muon momentum. Reconstructed cosmic muon spectra (< 10 GeV/c) using 104 muon samples with (i) perfect and (ii) practical (noise is included) muon spectrometers are shown in Fig. 6. The computed CRs for the practical muon spectrometer are also presented.

The next simulation results are the computed CRs using 104 mono-energy muons as a function of muon momentum with various discriminator levels, 0, 1, and 2 (Fig. 7). Simple linear signal discriminators were used in the final stage of signal process to cut off predictable noise from total signals. When pth < 1.0 GeV/c, overall CR is less than 60% due to the high scintillation photon intensity in the glass radiator. Hence, the CR increases as using higher discrimination levels in this level. When 1.0 < pth < 4.0 GeV/c, all gas radiators show the high CRs because Cherenkov and scintillation signals are clearly differentiated using a linear signal discriminator. However, when 4.0 < pth < 6.0 GeV/c, the CRs begin to decrease since Cherenkov photon yields are too low due to the rarefied CO2 (~ a half atmosphere pressure) gas for high muon threshold momentum. When pth > 6.0 GeV/c, on the other hand, CR rebounds because Cherenkov photon yields gradually increase as muon momentum increases. However, it is not efficient to use any linear discriminator in high pth (> 5.0 GeV/c). In addition, it is noted that there are repeated sudden drops of CR in Fig. 7. This is because pμ near pth boundaries, 0.1, 1.0, … 5.0 GeV/c, tends to have a high false classification rate.

### Benchmark

#### Separation and identification of SNMs

One potential application of our proposed muon spectrometer is to monitor hidden special nuclear materials (SNMs). Cosmic ray muons have been considered as an unconventional radiation probe for monitoring of SNMs due to their high-penetrative nature and their charge. Although muons are not easily absorbed by SNMs, they undergo multiple Coulomb scattering (MCS) and deflected with characteristic angles which depends on the Z number of materials and muon momentum. The estimated muon scattering angle distribution using the MCS approximation is known to follow Gaussian distribution with a zero mean and it is given by33:

$$f\left( {\theta {|}0,\sigma_{\theta }^{2} } \right) = \frac{1}{{\sqrt {2\pi } \sigma_{\theta } }}\exp \left( { - \frac{1}{2}\frac{{\theta^{2} }}{{\sigma_{\theta }^{2} }}} \right)$$
(4)
$$\sigma_{\theta } = \frac{{13.6\;{\text{MeV}}}}{{\beta_{\mu } p_{\mu } c}}\sqrt {\frac{X}{{X_{0} }}} \left[ {1 + 0.088\log \left( {\frac{X}{{X_{0} }}} \right)} \right]$$
(5)

where θ is the muon scattering angle, σθ is the standard deviation of scattering angle, X is the thickness of the scattering medium, and X0 is the radiation length.

To investigate the effect of muon momentum measurement in the SNM monitoring, high enriched uranium (HEU), low enriched uranium (LEU), plutonium (Pu), and lead (Pb) shielding were studied using Monte Carlo simulations. We used 103 muons (approximately translate to a few minutes of scanning time) to analyze their scattering angle variance distributions in order to identify SNMs and separate non-SNM from them. The results for SNM separation and identification with and without muon momentum knowledge are shown in Fig. 8. By measuring muon momentum, Pb is clearly separated from SNMs and identification of each type of SNM is also possible even when they are surrounded by 30 cm of lead shielding. This last scenario was almost impossible without muon momentum knowledge.

#### Muon scattering tomography

Our proposed muon spectrometer can be incorporated with a muon scattering tomography system which reconstructs images of hidden high-density materials using the point-of-closest approach (PoCA) algorithm34. As shown in (5), the expected muon scattering angle distribution is determined by two variables, material property and muon momentum. In other words, materials can be identified by measuring muon scattering angle distribution and momentum. Using a momentum integrated PoCA (mPoCA) algorithm which includes the muon momentum term, we successfully improved the image resolution as shown in Fig. 9. The experimental configuration includes cylindrical tungsten (radius = 5.5 cm, height = 5.7 cm) at the center and two stainless beams on the sides. This example was firstly introduced by Borozdin et al.35. Without muon momentum measurement, many incorrect voxels are found between tungsten and steel beams. However, the resolution of reconstructed image is significantly improved by coupling muon momentum knowledge with PoCA algorithm.

## Discussion and conclusion

In this paper, we presented a new concept to measure muon momentum using multi-layer pressurized gas Cherenkov radiators. Since our proposed muon spectrometer relies on pressurized gas Cherenkov radiators, we can eliminate the need for bulky and expensive magnets to measure muon momentum. Six sequential threshold momentum levels were set using a solid (SiO2) and pressurized CO2 gas radiators, pth = 0.1, 1.0, 2.0, 3.0, 4.0, and 5.0 GeV/c. Each radiator can emit Cherenkov radiation when a muon passes through it if the actual muon momentum exceeds the threshold muon momentum of this radiator. On the other hand, radiators will emit scintillation photons and transition radiation even though the actual muon momentum is lower than the threshold momentum.

We performed detailed Geant4 simulations to assess the performance and demonstrate the functionality of the proposed muon spectrometer by investigating the Cherenkov, scintillation, and transition radiation photon emission in the radiators. Using our proposed muon spectrometer, we were able to measure muon momentum with a resolution of ± 0.5 GeV/c within the momentum ranges of 0.1–10.0 GeV/c. This is sufficient for most engineering applications. Moreover, a cosmic ray muon spectrum was successfully reconstructed with a high accuracy (~ 90%).

To further demonstrate the potential of measuring muon momentum in special nuclear material monitoring and muon scattering tomography applications, we compared the results of both applications with and without muon momentum measurement. In the SNM monitoring application, we showed that we are not only able to separate lead from SNMs but also identify each type of SNMs (HEU, LEU, and Pu) which was not possible without measuring muon momentum. In the muon scattering tomography, we developed a momentum integrated PoCA algorithm (mPoCA) in order to couple muon momentum with the original PoCA algorithm. When we reconstructed using the mPoCA algorithm, imaging resolution was significantly improved.

## Methods

### Cherenkov muon spectrometer

A threshold muon momentum is required to induce Cherenkov radiation in a given radiator and it is derived as follows36:

$$p_{th} c = \frac{{m_{\mu } c^{2} }}{{\sqrt {n^{2} - 1} }}$$
(6)

where pth is the threshold muon momentum, n is the refractive index of the medium, and mμc2 is the muon rest mass energy. The threshold momentum for muons is determined by the refractive index of the radiator. In gas radiators, the refractive index depends on both pressure and temperature and it is approximated by the Lorentz-Lorenz equation:

$$n \approx \sqrt {1 + \frac{{3A_{m} p}}{RT}}$$
(7)

where Am is the molar refractivity, p is the gas pressure, T is the absolute temperature, and R is the universal gas constant. Therefore, the threshold muon momentum is given by:

$$p_{th} = m_{\mu } c\sqrt {\frac{R}{{3A_{m} }}\frac{T}{p}}$$
(8)

Equation (8) shows that muon threshold momentum can vary by changing the pressure or temperature without replacing radiator materials. For the gas selection, we considered, C4F10, C3F8, R-12, and CO2. C4F10 and CO2 are used in the Thomas Jefferson National Laboratory as gas Cherenkov radiators37. R-12 gas refrigerant was used for gas radiators in the past, however, the usage is limited due to its high Ozone Depletion Potential (ODP). Hence, a refrigerant, R-1234yf (C3H2F4), becomes a promising alternative gas radiator to replace R-12 (CCl2F2)38. C3F8 is the alternative gas radiator for C4F10 due to the better stability at the high pressure. Gas pressure cannot exceed 3 atm without condensation using the C4F10 at room temperature.

### Prototype design

The underlying principle of the proposed muon spectrometer is illustrated in Fig. 1. It shows characteristics of photon emissions by Cherenkov, scintillation, and transition radiation. All surfaces of gas radiator tanks are surrounded by strong photon absorbers so that emitted photons are isolated within radiators. We chose CO2 gas because it can be pressurized up to approximately 5.7 MPa without condensation although the refractive index of CO2 gas is relatively lower than other gas radiators. Five sequential threshold momentum levels are obtained using pressurized CO2 gas radiators, pth = 1.0, 2.0, …, 5.0 GeV/c, however, we inevitably had to use a solid radiator (SiO2) for the lowest threshold momentum level, pth = 0.1 GeV/c, because it is impossible to reach that threshold momentum level with pressurized CO2 at room temperature.

### Geant4 simulations

To demonstrate the functionality of proposed muon spectrometer, a high-fidelity stochastic muon transport simulation using Geant4 was performed. The overall length of the muon spectrometer is 51.7 cm and the surface area is 20 × 20 cm2. It consists of one glass radiator (1 cm), five CO2 gas radiators (10 cm), and thin photon absorber foils (10–1 cm). The overall weight of muon spectrometer is less than 10 kg because it mostly consists of gas radiators but structural frames (stainless steel) and optical sensors weigh approximately 8 kg. For glass radiator, a standard SiO2 with refractive index of 1.45 is chosen, and for gas radiators, sequentially pressurized CO2 gases are used to provide desirable refractive indices and muon threshold momenta. Some CO2 gas radiators are technically depressurized because Cherenkov threshold momentum at atmospheric pressure is approximately 3.5 GeV/c. Therefore, some are depressurized to achieve pth = 4 and 5 GeV/c. Any optical photon events are terminated once they arrive either at the photon absorbers or outside of “world” volume during the simulations. The overview of field muon spectrometer configuration is shown in Fig. 10. All major physics are included in the Geant4 reference physics list, QGSP_BERT39.

### Signal analysis

Optical sensors and photon absorbers are installed on one side of each radiator. Photon absorbers prevent photons from escaping and help efficiently discriminate scintillation photon signals given that scintillation photons are emitted in all directions whereas Cherenkov photons are emitted with a forward-biased direction. Figure 2 shows an example of signal flows when a single muon (pμ = 3.1 GeV/c) passes through all radiators. First four radiators emit Cherenkov radiation (“triggered”) because their threshold momentum levels are lower than the incident muon momentum, 3.1 GeV/c. In this example, the system records ‘1’ for triggered radiators and ‘0’ for intact radiators. Due to the presence of scintillation and transition radiation, noise signals always exist. Therefore, a use of signal discriminator is suggested to efficiently remove predicted noise from signals. After the signal process, a final signal successfully identifies the actual muon momentum with a resolution of ± 0.5 GeV/c, even though the muon momentum is very close to the threshold momentum level.