Cosmic-ray muons reaching the ground have a very broad range of energies.
Most atmospheric muons are produced when high-energy cosmic rays strike nuclei in the upper atmosphere, creating pions and kaons that subsequently decay into muons.
At sea level:
- Muons range from hundreds of MeV to many TeV.
- The average energy is roughly 3–4 GeV for muons arriving near vertically.
- The spectrum falls steeply with increasing energy, so lower-energy muons are much more common.
- Muons arriving at large angles to the vertical tend to have higher average energies, because they must travel farther through the atmosphere to reach your detector.
This is important for your PicoMuon/Cosmic Watch experiments. A scintillator detector doesn’t receive muons all carrying, say, 4 GeV. Instead, it samples this broad energy distribution.
Also, the energy deposited in your scintillator is not the muon’s total energy. A several-GeV muon passing through a thin plastic scintillator typically loses only around 2 MeV per cm of plastic (depending somewhat on energy and material). So a 1-cm scintillator might see only a few MeV deposited even though the muon itself carries several thousand MeV.
That distinction becomes particularly interesting if you’re thinking about your lead attenuation experiments: adding lead doesn’t simply “stop the 4-GeV muons.” It progressively removes the lower-energy part of the muon spectrum while the more energetic muons penetrate much greater thicknesses.
Muons are particles with the same rest mass, but they can be moving at different speeds and therefore have different kinetic energies.
A muon always has the same rest-mass energy:
muon rest-mass energy = 105.7 MeV
When cosmic rays hit the atmosphere, they produce particles called pions and kaons with a very wide range of energies. These particles can decay to produce muons, for example:
positive pion -> positive muon + neutrino
The resulting muon receives some of the energy of the parent particle. Because the parent particles have different energies and directions, the resulting muons also have different energies.
AN ANALOGY
Think of identical tennis balls. Every tennis ball has the same mass, but one might be travelling at 10 mph and another at 100 mph. They therefore have different kinetic energies.
Muons are similar, except that they usually travel extremely close to the speed of light. Relativity therefore becomes important.
For a relativistic particle:
E = gamma x m x c^2
For example, a muon with a total energy of 4 GeV has:
gamma = 4000 / 105.7
= about 38
Its velocity is approximately:
v = 0.99965 c
So it is travelling at about 99.965% of the speed of light.
Interestingly, a 100 GeV muon is not enormously faster than a 4 GeV muon. Both are already travelling extremely close to the speed of light. The additional energy mainly appears as increased relativistic energy and momentum rather than a large increase in velocity.
This is also why high-energy cosmic-ray muons are particularly penetrating. Their high energy allows them to travel through substantial amounts of matter, and relativistic time dilation also greatly increases the distance they can travel before decaying.
At ground level, cosmic-ray muons therefore do not all have one energy. Their energies range from hundreds of MeV to many TeV, with typical sea-level muons having energies of a few GeV.