Cosmic Ray Detection
Cosmic-ray muon hits on a silicon image sensor: hits per frame, path through the sensitive layer, mean and most probable charge per track, and pixels hit.
Expected hits per frame
0.0167
99th percentile: 1
Path in the layer
23.1 µm
t/cos θ; cos²θ average 26.7 µm
Mean charge per track
2452 e⁻
106.2 e⁻/µm × path; minimum ionizing, all δ-ray energy kept
Most probable charge
1487 e⁻
61 % of the mean
Pixels per track
2.47
1 + 4ℓ/(πp), ℓ = t·tan θ = 11.5 µm
Pixels hit per frame
0.0412
0.00000412 % of the pixels: 1 − exp(−N·n/N_pix)
Hits in 100 frames
1.67
Sea level: about 1 muon cm⁻² min⁻¹ (± ≈ 50 % with latitude and altitude) through a horizontal surface, zenith distribution ∝ cos²θ, mean energy ≈ 4 GeV (PDG).
Hits per frame: Poisson with mean N = Φ·A·t. Path in the layer L = t/cos θ.
Mean deposit: 3.876 MeV/cm ÷ 3.65 eV per pair = 106 e⁻/µm, the unrestricted Bethe mean at minimum ionization. It is about 20 % higher at 4 GeV, and lower in thin layers that fast δ-rays leave; the most probable deposit is the better guide. It is lower and grows like ln L per µm: 23 000 e⁻ in 300 µm.
Charge per track vs thickness at 30°
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Sea-level muons only (about 1 cm⁻² min⁻¹ on a horizontal sensor), Poisson arrivals and one straight track at the chosen zenith angle through a fully sensitive layer, path t/cos θ. Mean charge 106 e⁻/µm (unrestricted Bethe mean at minimum ionizing, 3.65 eV per pair; about 20 % more at 4 GeV, less where δ-rays escape a thin layer); the most probable charge uses the Landau–Vavilov–Bichsel Fermi-plateau formula and is rough below about 10 µm. Pixels per track is the Buffon average 1 + 4ℓ/(πp). Charge diffusion, δ-ray branches, altitude, shielding and radioactivity are left out.
- Workman R. L. et al. (Particle Data Group) (2022). Review of Particle Physics. Prog. Theor. Exp. Phys. 2022, 083C01. Review "Cosmic rays" §30.3.1: vertical muons above 1 GeV/c ≈ 70 m⁻² s⁻¹ sr⁻¹, about 1 cm⁻² min⁻¹ on a horizontal detector, cos²θ zenith distribution, mean energy ≈ 4 GeV. doi:10.1093/ptep/ptac097
- Navas S. et al. (Particle Data Group) (2024). Review of Particle Physics. Phys. Rev. D 110, 030001. §34.2.9 Eq. 34.13 (most probable energy loss) and §35.8 Table 35.10 (3.65 eV per e–h pair in Si; about 23 000 e⁻ most probable in 300 µm). doi:10.1103/PhysRevD.110.030001
- Particle Data Group (2024). Atomic and nuclear properties of silicon: minimum ionization 1.664 MeV g⁻¹ cm² (3.876 MeV/cm), density 2.329 g/cm³, plasma energy 31.05 eV. pdg.lbl.gov