On the energy-loss straggling of protons in elemental solids: The importance of electron bunching

In this work we investigate the energy-loss straggling of low and medium energy protons on available simple materials (C, Al, Si, Ni, Cu, Zn, Ge, Se, Pd, Ag, Sb, Pt, Au, Pb) and demonstrate that the well-known Yang-O'Connor-Wang empirical formula for the energy-loss straggling of protons in matter has a clear physical interpretation. This formula predicts, on empirical grounds, energy-loss straggling values for protons that exceed the values of traditional straggling formulas (e.g., by Lindhard and Chu) at energies around 0.3 MeV/amu by a factor of 2–3. This excess is correlated to the bunching effect, which comes from the inhomogeneity feature of the electron density. Although this effect is specific to each element and can be calculated using e.g. the CasP and PASS programs, a single formula correlates better with overall experimental values. The Yang-O'Connor-Wang formula was refined accounting for subsequent experimental data.