Bethe-Bloch Energy Loss Calculator

The Bethe-Bloch formula gives the mean rate of energy loss per unit path length (dE/dx) of a fast charged particle passing through matter, dominated by ionization and excitation of the absorber's atoms. It is the standard starting point for understanding how detectors measure particle energy loss.

A minimum-ionizing particle is one whose betagamma sits near the minimum of the Bethe-Bloch curve, roughly betagamma 3 to 4, where dE/dx is at its lowest. Cosmic-ray muons passing through a detector are commonly close to minimum-ionizing, which is why MIP energy deposit is a standard detector calibration reference.

At low betagamma, the 1/beta squared term dominates and dE/dx falls sharply as speed increases. Past the minimum, the relativistic logarithmic term takes over and dE/dx slowly rises again, an effect called the relativistic rise, before eventually flattening due to the density effect (not modeled by this simplified calculator).

The full density-effect correction needs material-specific Sternheimer parameters that are not available for every material a user might enter. Omitting it keeps the tool simple and transparent for educational use, but it means results at very high betagamma (above roughly 20 to 30) will overestimate dE/dx compared to a full PDG table.

Mass stopping power (MeV cm2/g) is dE/dx divided by density, so it can be compared across materials directly. Linear stopping power (MeV/cm) multiplies mass stopping power by the absorber's density, giving the actual energy lost per centimetre of path in that specific material.

Tmax is the maximum kinetic energy that can be transferred to a free electron in a single collision, computed from the incident particle's mass, betagamma, and the electron rest mass. It sets the upper limit of the energy-transfer spectrum used inside the logarithmic term.

The mean excitation energy I (in electron volts) is an empirical, material-specific parameter that summarizes the average atomic excitation and ionization energy of the absorber. Larger I generally means the material is harder to ionize per unit mass, lowering dE/dx slightly.

Energy loss scales as z squared, so a particle with charge 2e loses energy four times faster than a singly charged particle with the same betagamma passing through the same material. This calculator lets you set any integer charge.

Muon (105.658 MeV/c2), charged pion (139.570 MeV/c2), kaon (493.677 MeV/c2), and proton (938.272 MeV/c2) are built in, or you can enter any custom rest mass in MeV/c2.

Silicon, aluminum, copper, lead, and liquid argon are built in with their standard PDG Z, A, mean excitation energy, and density values, or you can enter a custom material's Z, A, I, and density directly.