For relativistic muons, the RMS multiple-scattering angle in lead is approximately:
theta0 = (13.6 MeV / beta pc) * sqrt(x/X0) * [1 + 0.038 ln(x/X0)]
where:
- theta0 = RMS scattering angle (radians)
- x = thickness of lead
- X0 = radiation length of lead = 0.56 cm
- p = muon momentum
- beta ~= 1 for cosmic-ray muons
Example: a 3 GeV muon passing through 1 cm of lead:
x/X0 = 1/0.56 = 1.79
The RMS scattering angle is about:
theta0 ~= 0.006 rad = 6 mrad = 0.34 degrees
Therefore:
- Nearly all 3 GeV cosmic-ray muons pass through 1 cm of lead.
- Typical deflections are only a few tenths of a degree.
- Large-angle scattering (>10 degrees) is very rare.
- Energy loss is about 25 MeV per cm of lead, so 1 cm removes less than 1% of the muon’s energy.