Stopping power ($S=-dE/dx$)

The mean energy lost by a charged particle per unit path length through a material, with the particle species, material and energy specified. It is distinct from the particle range obtained by following the particle until it stops.

A maximum near the end of the depth-dose curve of a charged-particle beam, associated with increasing stopping power during slowing. Its precise width and distal tail depend on beam spread and particle transport.

An extended high-dose depth interval formed by combining charged-particle beams of different ranges. The component Bragg peaks are weighted to cover a finite target thickness.

A decrease in a projectile's energy by transfer to matter or emitted radiation. Mean loss per unit path length defines stopping power; its event-to-event variation is energy-loss straggling.

The stochastic spread of particle energy losses around their mean after traversal of matter. It broadens the endpoint of a particle beam and is distinct from the mean stopping power.

The positive mean loss of projectile kinetic energy per path length through electronic excitation and ionization, conventionally $S_{\mathrm{col}}=-dT/dx$. Dividing by absorber density gives mass stopping power.

A high-speed approximation for electronic stopping power of charged particles. Its leading heavy-projectile dependence is $z^2{\beta }^{-2}[\ln (2m_e{c}^2{\beta }^2{\Gamma }^2/I)-{\beta }^2]$; low-speed charge exchange and binding effects require corrections.

For an ideal particle energy-loss law $dE/dx=-k/E$, integration gives $E(x)=\sqrt{E_0^2-2kx}$ up to the formal range. Stopping power grows as the endpoint is approached but its integral remains finite. This simple law does not describe endpoint broadening, scattering or every low-energy stopping mechanism.

Energy loss of a moving particle through collisions transferring momentum to target nuclei. Recoiling target atoms can initiate displacement cascades.

Energy loss of a moving particle to electronic excitation and ionization. It is distinct from direct momentum transfer to target nuclei; electronic energy deposition can also couple to atomic motion in a material-dependent way.

For a fast ion traversing a fixed material thickness with only a small fractional energy loss, the dipole excitation probability per atomic encounter scales as $Z^2/v^2$ at fixed impact parameter. Summing transferred excitation energies and integrating the encounter measure $2\pi bdb$ gives the leading electronic stopping power scaling, with slower logarithmic dependence from the allowed excitation energies and impact-parameter range.

At fixed ion kinetic energy $E=M{v}^2/2$, a heavier isotope is slower, so the leading loss scales as $Z^{2}M/E$. At fixed velocity, isotopes of the same element instead have approximately the same electronic loss; plotting against energy per nucleon removes the leading isotope mass dependence. The charge should be the effective projectile charge if its bound electrons screen the nucleus. At lower speeds, excitation thresholds, slow-passage suppression, charge exchange and nuclear energy transfer invalidate a simple extrapolation of the high-speed law.