Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S10 / 2

Define the positive mass collision stopping power as $S_{\mathrm{col}}/\rho =-(1/\rho )dT/dx$. For a projectile of charge $ze$, mass $M\gg m_e$, speed $v=\beta c$ and Lorentz factor $\Gamma =(1-{\beta }^2{)}^{-1/2}$, the Bethe stopping formula can be writtenHere $Z$ is the absorber atomic number, $A$ its molar mass, $I$ its mean excitation energy, $r_e=e^2/(4\pi {ϵ}_0{m}_e{c}^2)$, and $N_A$ the Avogadro constant. AlsoThe optional $\delta$ and $C/Z$ represent density and shell corrections. For a very heavy projectile $T_{\max }\simeq 2m_e{c}^2{\beta }^2{\Gamma }^2$, and the bracket reduces to $\ln (2m_e{c}^2{\beta }^2{\Gamma }^2/I)-{\beta }^2$ when those corrections are neglected. If $T$ denotes remaining projectile kinetic energy, the printed $dT/(\rho dx)$ is negative; the conventional positive stopping power has the minus sign above.^[ref]

Start with a fast massive projectile following an approximately straight path. Its time-dependent electric field perturbs bound absorber electrons. At large impact parameter, treating the perturbation to first order and summing excitation and ionization transition probabilities gives an energy transfer weighted by the absorber's oscillator strengths. The oscillator-strength sum rule reduces the leading coefficient to the number of target electrons, while their logarithmically weighted excitation energies define $I$.

The classical impulse picture explains the logarithm: a transverse impulse scales as $1/(bv)$, so transferred energy scales as $1/(b^2{v}^2)$; multiplying by the number of encountered electrons in $2\pi bdbdx$ gives $db/b$. Quantum scattering replaces ad hoc short- and long-distance cutoffs by recoil kinematics and bound-state excitation. Relativistic projectile kinematics supplies $T_{\max }$ and the $-{\beta }^2$ term. Summation over many independent encounters gives the mean stopping power, rather than the energy-loss straggling of individual tracks.

The leading approximation assumes $M\gg m_e$, a sufficiently fast projectile for perturbation theory, a largely fixed projectile charge, and weak correlations between collisions. At low velocity, electron binding, charge exchange and nuclear stopping become important; at high velocity, polarization screens distant interactions and gives the density correction. The projectile's bremsstrahlung is suppressed by its large mass and is excluded from the collision stopping power.

The dominant $z^2/{\beta }^2$ factor raises the energy loss as a charged projectile slows, while the logarithm moderates that rise. This produces a Bragg peak close to the end of its particle range. At still lower velocity, changing effective charge, shell effects and nuclear stopping invalidate extrapolation of the elementary Bethe stopping formula to zero energy. A finite initial energy gives a finite continuous-slowing-down range,A higher initial energy increases the particle range; greater electron density increases stopping power per unit length. Absorber $Z/A$ and $I$ also affect the depth. Multiple Coulomb scattering, energy-loss straggling, nuclear reactions and the incident energy spread broaden the measured peak and distal fall-off; nuclear secondaries can produce a small tail. Superposing many incident energies makes a spread-out Bragg peak for an extended target. The schematic curve below illustrates shapes only and is not patient radiation dosimetry.