Calculating attenuation of muons in lead – ASCII text version

1. Basic physics

Cosmic-ray muons at ground level:

• Typical energies: ~1 to 10 GeV
• Energy loss mainly by ionisation

Approximate stopping power in lead:

dE/dx ≈ 1.5 to 2 MeV g^-1 cm^2

Lead density:

rho_Pb ≈ 11.34 g/cm^3

So energy loss per cm:

dE/dx ≈ 20 MeV/cm

2. Range of muons in lead

Approximate range:

Range ≈ E / (dE/dx)

Example:

3 GeV muon:

Range ≈ 3000 / 20 ≈ 150 cm of lead

Implications:

• A few cm of lead -> almost no attenuation
• Even 10 to 20 cm -> only small reduction

3. Simple attenuation model (practical)

Use an effective exponential:

N(x) = N0 * exp(-x / lambda)

Where:

• x = thickness (cm)
• lambda = attenuation length (~150 to 300 cm)

4. Example table (lambda = 200 cm)

Assume N0 = 1000:

5. Better model (energy spectrum)

Muon spectrum:

dN/dE proportional to E^-2.7

For thickness x:

E_min = x * (dE/dx)

Count is proportional to:

Integral from E_min to infinity of E^-2.7 dE

This gives:

N(x) proportional to E_min^-1.7

So:

N(x) proportional to x^-1.7

This predicts a power-law decrease instead of exponential.

6. Practical workflow

1. Measure baseline N0
2. Add lead in steps (e.g. 2 cm)
3. Measure counts over fixed time
4. Fit:
   • Exponential (quick)
   • Power law (more physical)

7. Important note

If you see strong attenuation at small thickness (<10 cm), you are likely detecting:

• electrons
• gamma rays
• secondary particles

Lead suppresses these much more than muons.