 Solution

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.