High-energy muons traversing matter lose energy according to $$ -\frac{1}{\rho} \frac{d E}{d x} \approx a+b E $$ where \(a\) is due to ionisation energy loss and \(b\) is due to the bremsstrahlung and \(\mathrm{e}^{+} \mathrm{e}^{-}\)pair-production processes. For standard rock, taken to have \(A=22, Z=11\) and \(\rho=2.65 \mathrm{~g} \mathrm{~cm}^{-3}\), the parameters \(a\) and \(b\) depend only weakly on the muon energy and have values \(a \approx 2.5 \mathrm{MeV} \mathrm{g}^{-1} \mathrm{~cm}^{2}\) and \(b \approx 3.5 \times 10^{-6} \mathrm{~g}^{-1} \mathrm{~cm}^{2}\). (a) At what muon energy are the ionisation and bremsstrahlung/pair production processes equally important? (b) Approximately how far does a \(100 \mathrm{GeV} \cos m i c-r a y ~ m u o n ~ p r o p a g a t e ~ i n ~ r o c k ? ~\)

When a high-energy muon travels through a material, it loses energy primarily through ionisation. This process happens because the muon collides with the electrons in the atoms of the material, knocking them off and creating ions. The energy lost in ionisation is relatively constant and is given by the parameter, \(a\). For example, in standard rock, \(a\) is approximately 2.5 MeV cm\textsuperscript{2} g\textsuperscript{-1}.

This energy loss is an important factor in understanding the slow and steady decrease in the kinetic energy of the muon as it moves through the material. It's a significant interaction at energies where other processes like bremsstrahlung and pair-production are not dominant.

Ionisation energy loss plays a crucial role in the initial stages of the muon's journey through the material.

Bremsstrahlung is a process where the muon loses energy by emitting radiation due to its acceleration caused by the electric field of the nucleus in the material. Unlike ionisation, this energy loss increases with the muon's energy. Therefore, it becomes more significant at higher energies.

The parameter \(b\) in the energy loss equation represents the contribution from bremsstrahlung. For standard rock, \(b\) is approximately 3.5 x 10\textsuperscript{-6} cm\textsuperscript{2} g\textsuperscript{-1}.

When the muon energy is extremely high, bremsstrahlung dominates the energy loss. This process results in the emission of photons (light particles), leading to a rapid decrease in the energy of the muon over a short distance as compared to ionisation.

Pair-production occurs when the high-energy muon interacts with the electromagnetic field of a nucleus, and a photon is produced, which then creates an electron-positron pair. This process also becomes significant at very high energies, similar to bremsstrahlung.

The rate of energy loss due to pair-production increases with the muon's energy, meaning that above a certain energy threshold, these processes become dominant compared to ionisation.

Both pair-production and bremsstrahlung are included in the parameter \(b\), which in this case is 3.5 x 10\textsuperscript{-6} cm\textsuperscript{2} g\textsuperscript{-1} for standard rock. These processes are responsible for a large fraction of the energy loss in muons when their energies are in the GeV (giga electron-volt) range or higher.

Understanding how muons propagate through matter helps in calculating how far they can travel before their energy is exhausted. In standard rock, muons lose energy due to ionisation, bremsstrahlung, and pair-production.

At lower energies, ionisation is the primary method of energy loss. As the energy increases, bremsstrahlung and pair-production start to dominate. This interplay is represented in the equation:
\[ -\frac{1}{\rho} \frac{dE}{dx} \approx a + bE \]
where \(a\) is ionisation loss and \(bE\) includes bremsstrahlung and pair-production.

For example, a 100 GeV muon can propagate roughly 12.4 km in standard rock. This distance estimation is crucial for applications like cosmic ray detection and underground particle physics experiments.

By analyzing the propagation, one can predict the behavior and the eventual stopping point of high-energy muons in different materials, making this knowledge vital for designing detectors and other scientific instruments.