Development of a Measuring Method of Cosmic-Ray Muon Momentum Distribution Using Drift Chambers

Naoto Nakagami; Satoko Kamei; Shoichiro Kawase; Akira Sato; Yukinobu Watanabe

doi:10.14407/jrpr.2023.00423

Abstract

Background

Soft errors in semiconductor devices caused by cosmic rays have been recognized as a significant threat to the reliability of electronic devices on the ground. Recently, concerns about soft errors induced by cosmic-ray muons have increased. Some previous studies have indicated that low-energy negative muons have a more significant contribution to the occurrence of soft errors than positive muons. Thus, charge-identified low-energy muon flux data on the ground are required for accurate evaluation of the soft error rate. However, there are no such experimental data in the low-energy region.

Materials and Methods

We designed a new muon detector system to measure low-energy muon flux data with charge identification. The major components consist of two drift chambers and a permanent magnet. The charge and momentum of detected muon can be identified from the deflection of the muon trajectory in the magnetic field. An algorithm to estimate the muon momentum is developed using numerical optimization by combining the classical Runge-Kutta and quasi-Newton methods. The momentum search algorithm is applied to event-by-event data of positive and negative muons obtained by Monte Carlo simulations with Particle and Heavy Ion Transport code System, and its performance is examined.

Results and Discussion

The momentum search algorithm is fully applicable even in the case of an inhomogeneous magnetic field. The precision of the momentum determination is evaluated by considering the stochastic fluctuation caused by multiple scattering and the position resolution of the drift chambers. It was found that multiple scattering has a significant contribution to the precision in the momentum region below 50 MeV/c, while the detector position resolution considerably affects the precision above that.

Conclusion

It was confirmed that the momentum search algorithm works well with a sufficient precision of 15% in the low-momentum region below 100 MeV/c, where no muon flux data exist.

Keywords: Cosmic-Ray Muon, Momentum Distribution, Soft Error, Drift Chamber, Particle and Heavy Ion Transport Code System

Introduction

Soft errors are temporary malfunctions of semiconductor devices caused by radiations and lead to reduced reliability of electronic devices. Cosmic-ray neutrons and muons are major environmental radiations that cause soft errors on the ground. Recently, concerns about muon-induced soft errors have increased owing to the miniaturization of integrated devices and the lowering of operating voltages [1]. Muons are elementary particles whose absolute value of electric charge is equal to that of electrons and are classified into two types: negatively charged (negative muons) and positively charged (positive muons). The average flux of cosmic-ray muons is about 1 cm⁻²*min⁻¹ on the ground.

The soft error rate (SER) caused by cosmic-ray muons is calculated by

(1)

SER=∫σSEU(Eμ)φ(Eμ)dEμ

where E_μ (MeV) is the muon kinetic energy, σ_SEU (cm²) is the single event upset (SEU) cross-section, and φ (cm⁻²s⁻¹MeV ⁻¹) is the muon’s energy differential flux under the operating environment of electronic devices. Some previous works on muon irradiation tests for 65 nm static random access memories (SRAMs) have clarified that negative muons have larger SEU cross sections than positive muons in the low-energy region because secondary ions generated by negative muon nuclear capture reaction cause additional SEUs in SRAM devices [2–4]. Therefore, charge-identified low-energy muon flux data on the ground are required for a highly accurate evaluation of the SER based on Equation (1). As shown in Fig. 1 [5–8], however, there is no measurement of cosmic-ray muon flux in the low-energy region. Additionally, positive and negative muons have not been discriminated in the energy region measured in the past. Thus, there is still uncertainty in the muon flux on the ground, which is necessary for the SER evaluation.

To improve this situation, we designed a dedicated detector system to obtain charge-identified low-energy muon flux data. The principle of determination of the muon charge and momentum is based on the deflection of muons in the magnetic field in the detector system composed of two track detectors and a permanent dipole magnet. Therefore, an algorithm to determine the charge and momentum is important in data analysis. This work is devoted to the development of a momentum search algorithm. Finally, the developed algorithm is applied to the data of positive and negative muons obtained by a Monte Carlo simulation with Particle and Heavy Ion Transport code System (PHITS) [9], and the performance and precision are examined.

Materials and Methods

1. Detector System

A two-dimensional view of the designed detector system is shown in Fig. 2. Its major components are two drift chambers (DCs) as track detectors and a permanent dipole magnet. Two cuboid DCs (540 mm [width]×580 mm [length]×357 mm [height]) are placed above and below the permanent dipole magnet. Each pole of the dipole magnet is a 200 mm×200 mm square shape with a 90 mm gap between the poles.

The DCs detect the muon track before and after passing through the magnetic field produced by the permanent magnet. The muon trajectories projected in the plane perpendicular to the magnetic field are illustrated in Fig. 2. Assuming a uniform magnetic field for simplicity, a muon makes a circular motion by Lorentz force, and its curvature radius R is given by

(2)

R=p⊥|q|B

where q is the muon charge, B is the magnetic flux density, and p_⊥ is the muon momentum perpendicular to the magnetic field. The q equals to +e for positive muons and −e for negative muons, where e is the elementary charge. Suppose the curvature radius is estimated by extrapolating the straight muon tracks detected by the DCs to the magnetic field region, as shown in Fig. 2. In this case, the muon momentum can be determined by Equation (2). In the case of the realistic non-uniform magnetic field, however, Equation (2) does not hold. The trajectory of muons in the non-uniform magnetic field is numerically calculated using the equation of motion. The momentum should be determined by a numerical optimization method so that the estimated trajectory connects smoothly with the muon tracks detected in the upper and lower DCs.

2. Development of the Momentum Search Algorithm

1) Muons transport simulation by PHITS

PHITS can simulate realistic muon trajectories by taking into account the angular and energy straggling due to multiple scattering in the designed detector system [9]. It can also incorporate specific magnetic field information, allowing realistic simulations of muon deflection in a non-uniform magnetic field. In the PHITS simulation, a non-uniform magnetic field map generated by Amaze (Advanced Science Laboratory Inc.) is input [10]. Fig. 3A shows the magnetic flux density in the direction perpendicular to Fig. 3B at the center of the magnetic pole face of the magnet. The magnetic field is found to be non-uniform in the edge region of the magnet. Various directions of muon incidence on the upper DC are randomly generated. For example, Fig. 3B shows a typical simulated trajectory of a positive muon of 50 MeV/c.

2) Extrapolation to the muon track in magnetic field

A more accurate estimation of muon momentum requires information about muons in a non-uniform magnetic field. The DCs are located outside the magnet and cannot detect the muon trajectories in the magnetic field. Therefore, we propose a method to extrapolate the trajectory of a muon in the magnetic field from the tracks detected by the DCs.

The trajectory of a muon in the magnetic field is described by the following simultaneous differential equations for the momentum p→ and position r→:

(3)

{dp→dt=qm(p→×B→)dr→dt=p→m

where q is the muon charge, m is the muon mass, and B→ is the magnetic flux density. The fourth-order Runge-Kutta method is applied to numerically solve the above equation of motion using the initial condition given by the position r0→ and momentum p0→. Thus, the trajectory of a muon in the non-uniform magnetic field is calculated.

3) Momentum search algorithm

We developed a momentum search algorithm to estimate the momentum of muons incident on the detector system. An overview of the algorithm is explained using the schematic diagram in Fig. 4. Let the muon tracks in the two DCs be given by black lines in the left panel of Fig. 4 be known by measurements. In this work, they are given by PHITS simulation. Hereafter, the muon tracks detected in the upper and lower DCs will be referred to as the upper and lower tracks, respectively.

First, we calculate the trajectory of the muon in the magnetic field by solving Equation (3) with the Runge-Kutta method. The absolute value of p0→ is determined by assuming a uniform magnetic field whose magnetic flux density is the same as that at the center of the magnet. The direction is equal to that of the upper track. The calculated muon track is presented by the red line in Fig. 4. In the non-uniform magnetic field, the intersection point of the muon track on the center plane of the lower DC does not necessarily match that of the lower track, as shown in the right panel of Fig. 4.

Next, the initial momentum is updated so that the following evaluation function is minimized:

(4)

f=(xact-xext)2+(yact-yext)2

where x_act and y_act are the intersection points of the extracted track on the center plane of the lower DC, and x_ext and y_ext are those of the extrapolated track. The momentum is optimized to minimize f defined by Equation (4) to determine p_down using ‘TMinuit’ class in ROOT [11] based on a quasi-Newton algorithm. Moreover, let us consider a time-reversal situation in Fig. 4 and determine the muon momentum using the lower track detected in the lower DC. The same procedure is applied to the upward extrapolation of the muon track to estimate p_up. Here p_down and p_up correspond to the momentum value optimized by downward extrapolation or upward extrapolation. Finally, the momentum of the incident muon is determined by taking the average of p_down and p_up.

Results and Discussion

1. Estimated Momentum Distribution

We examined the performance of the developed algorithm and evaluated the precision of momentum determination using the event-by-event data of muon trajectories obtained by the simulation with PHITS for an incident muon momentum of 50 MeV/c. The simulations were performed under the following three conditions: (a) no stochastic fluctuation; (b) stochastic fluctuation caused by the angular and energy straggling due to multiple scattering by gas, air, and Mylar layers, which are components in the detector system; and (c) stochastic fluctuation caused by the position resolution of DCs. The position resolution is assumed to be 500 μm which is reasonable for the designed DCs. The stochastic fluctuation caused by the position resolution was considered using a normally distributed random number with a standard deviation of 500 μm in the position passing through each layer of DCs.

The distribution of the momentum estimated by applying the developed algorithm to the PHITS simulation results is shown in Fig. 5. Fig. 5A shows that the developed algorithm uniquely determines the muon momentum for the non-uniform magnetic field under no stochastic fluctuation. On the other hand, it is found that the event distributions of the muon momentum estimated by the developed algorithm are spread out for the cases Fig. 5B and 5C. This is due to the fact that the momentum search algorithm cannot take into account the effects of multiple scattering and detector position resolution. However, the actual trajectory detected by the DCs includes these stochastic fluctuations, so the width of the distributions corresponds to the precision of the momentum to be estimated in actual measurements. The standard deviations when fitting the distributions with a Gaussian function are listed in Table 1. The uncertainty for the condition (c) is determined using by applying the results of (a) and (b) to the error propagation law. From Table 1, it is confirmed that the momentum of muons with 50 MeV/c can be determined with 10% precision even under the condition (c), which is close to the actual measurements.

2. Relative Resolution of Estimated Momentum

In Fig. 6, the relative resolution of the momentum estimated by the developed algorithm is plotted as a function of the muon momentum in the low-momentum region where no muon flux data are available. The relative resolution R is defined by

(5)

R (%)=σpp×100,

where σ_p is calculated by fitting the distribution of each momentum with a Gaussian function, and p is the muon momentum in the PHITS simulation. The total relative resolution R defined by Equation (5), given by the black dots, is decomposed into two components. One comes from angle and energy straggling due to multiple scattering, as indicated by the red dots. Another one is caused by the detector position resolution represented by the blue dots. Each dot is plotted in steps of 10 MeV/c from 30 MeV/c to 120 MeV/c, and each line is plotted by fitting the corresponding dots with a second-order polynomial.

From Fig. 6, it is found that the effect of multiple scattering dominates the relative resolution in the momentum region below 50 MeV/c. In constant, the detector position resolution significantly affects the resolution in the energy region above that. Thus, this simulation result indicates that the muon momentum can be determined within 15% precision for muons below 100 MeV/c under near-realistic conditions.

Conclusion

We developed the momentum search algorithm used in the data analysis for our designed detector system consisting of two DCs and a permanent dipole magnet to measure charge-identified muon flux data on the ground. The transport of muons in the detector system was simulated by a three-dimensional Monte Carlo code PHITS, and the obtained event-by-event data were employed to investigate the performance of the momentum search algorithm. As a result, it was confirmed that the muon momentum can be determined within 15% precision for muons in the low-momentum region below 100 MeV/c, where no muon flux data exist.

In the future, we plan to evaluate the performance of the detector system under fabrication. One of the evaluation items will be to experimentally obtain the position resolution of the DCs. Then, the precision of the momentum determination will be verified using the actual measurement data of muon tracks detected by the DCs. Finally, we intend to measure charge-identified cosmic-ray muon fluxes on the ground in the low-momentum region where experimental data are scarce.

Notes

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

Ethical Statement

This article does not contain any studies with human participants or animals performed by any of the authors.

Author Contribution

Conceptualization: Sato A, Watanabe Y. Methodology: Nakagami N, Kamei S, Kawase S. Data curation: Sato A. Formal analysis: Kamei S, Kawase S, Sato A. Supervision: Watanabe Y. Funding acquisition: Watanabe Y. Project administration: Kamei S, Kawase S. Writing - original draft: Nakagami N. Writing - review & editing: Kawase S, Watanabe Y. Approval of final manuscript: all authors.

Acknowledgements

This work was supported by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research (KAKENHI) Grant Numbers JP19H05664, JSPS21J12445, and JP21K12564.

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Fig. 1

Previous measurements of the cosmic-ray muon differential flux at the zenith angle of 0°. The x-axis is the muon momentum and the y-axis is proportional to the muon energy differential flux.

Fig. 2

Schematic view of the designed detector system and the trajectories of positive and negative muons projected in the plane perpendicular to the magnetic field B.

Fig. 3

Magnetic flux density distribution simulated by Amaze (Advanced Science Laboratory Inc.) at the center of the gap of the dipole magnet (A) and a typical trajectory of a muon of 50 MeV/c (B). DC, drift chamber.

Fig. 4

Schematic drawing of the downward extrapolation of the upper track with initial momentum p0→. DC, drift chamber.

Fig. 5

Estimated momentum distributions when the algorithm is applied to 50 MeV/c muons generated by Particle and Heavy Ion Transport code system under the (A–C) three conditions. The conditions (A–C) corresponds to (a–c) described in the text.

Fig. 6

The momentum resolution in the low-momentum region where no muon flux data exist. The uncertainty for each plotted dot is less than 0.1%. PHITS, Particle and Heavy Ion Transport code System.

Table 1

Momentum Determination Precision for Muons at 50 MeV/c

PHITS simulation conditions	σ_p (MeV/c)^a)
(a) Only angular and energy straggling due to multiple scattering	3.42±0.03
(b) Only detector position resolution	3.26±0.02
(c) Both multiple scattering and position resolution	4.77±0.03

Values are presented as mean±standard deviation.

PHITS, Particle and Heavy Ion Transport code System.

^a) Sigma is the standard deviation of the Gaussian fitting of the distribution and its estimated error.