  Low-energy cross sections for elastic scattering and recoil of protons from
$^4$He nuclei (also known as $\alpha$ particles) are calculated directly by
solving the Schr\"odinger equation for five nucleons interacting through
accurate two- and three-nucleon forces derived within the framework of chiral
effective field theory. Precise knowledge of these processes at various proton
backscattering/recoil angles and energies is needed for the ion-beam analysis
of numerous materials, from the surface layers of solids, to thin films, to
fusion-reactor materials. Indeed, the same elastic scattering process, in two
different kinematic configurations, can be used to probe concentrations and
depth profiles of either hydrogen or helium. We compare our results to
available experimental data and show that direct calculations with modern
nuclear potentials can help to resolve remaining inconsistencies among
different data sets and can be used to predict these cross sections when
measurements are not available.
