Ionization loss and Bethe–Bloch regimes

Consider a projectile much heavier than an electron, with charge number z in units of the elementary charge. Its speed ratio and Lorentz factor enter the maximum energy transfer to a target electron and the logarithm of the Bethe expression. The material enters through atomic number, molar mass and mean excitation energy. That mean excitation energy is not the same as the mean energy required to create one measurable electron-ion pair in a detector.

Use mass thickness rather than ordinary length in the displayed stopping power. The coefficient K carries units that cancel the material molar-mass factor, leaving energy times area per mass. Multiplying by density gives linear stopping power. Keep all energies inside the logarithm in the same unit, especially when a material excitation energy is tabulated in electronvolts while electron rest energy is expressed in megaelectronvolts. A dimensionless logarithm is a useful units audit.

At modestly relativistic speeds the inverse-speed factor first decreases with energy and a broad minimum appears. The relativistic logarithmic rise is moderated by polarization of the medium, represented by the density correction. At sufficiently low speed, atomic binding, charge exchange and effective projectile charge require corrections. At sufficiently high energy, radiative losses become important and depend on projectile mass. These boundaries depend on projectile and material, so a single energy threshold cannot label every detector.

Electrons and positrons need their own collision formulas because their mass is the same as that of target electrons, with exchange and distinct kinematics. Their bremsstrahlung losses and showers also become important on material-dependent scales. Heavy ions may not remain fully stripped at low speed, invalidating a naive fixed-z-squared extrapolation. Hadronic interactions add another loss and shower channel for hadrons. An electronic stopping calculation should not silently absorb these different processes into an unexplained correction factor.

A thin sensitive layer often has an asymmetric energy-loss distribution, with a most probable value below its mean. Energetic secondary electrons can escape or deposit charge in neighboring channels. A reconstructed track can use a truncated mean over many samples to reduce sensitivity to rare large transfers, but its calibration then differs from the unrestricted Bethe mean. The chosen estimator must be compared with a response model using the same selection and material geometry.

Energy lost by the projectile, energy deposited locally, and electrical signal finally measured are three distinct quantities. Quenching, recombination, trapping, collection inefficiency and electronic shaping affect the latter two steps. A mean layer-loss estimate is still valuable for checking orders of magnitude and material budgets, provided it is not called a prediction of every pulse. The quadratic lab tests only the explicit charge-squared factor at a frozen bracket; it does not model those fluctuations or extrapolate Bethe theory outside its regime.