### Introduction

^{−2}*min

^{−1}on the ground.

*E*

*(MeV) is the muon kinetic energy,*

_{μ}*σ*

*(cm*

_{SEU}^{2}) is the single event upset (SEU) cross-section, and

*φ*(cm

^{−2}s

^{−1}MeV

^{−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.

### Materials and Methods

### 1. Detector System

*R*is given by

*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

#### 2) Extrapolation to the muon track in magnetic field

*q*is the muon charge,

*m*is the muon mass, and

#### 3) Momentum search algorithm

*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*

*and*

_{down}*p*

*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*

_{up}*p*

_{down}and

*p*

_{up}.

### Results and Discussion

### 1. Estimated Momentum Distribution

### 2. Relative Resolution of Estimated Momentum

*R*is defined by

*σ*

*is calculated by fitting the distribution of each momentum with a Gaussian function, and*

_{p}*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.