  Quantum oscillations offer a powerful probe for the geometry and topology of
the Fermi surface in metals. Onsager's semiclassical quantization relation
governs these periodic oscillations in 1/B, leading to a linear Landau fan
diagram. However, higher-order magnetic susceptibility-induced corrections give
rise to a generalized Onsager's relation, manifesting in experiments as a
nonlinear Landau fan diagram and aperiodic quantum oscillations. Here, we
explore the generalized Onsager's relation to three-dimensional (3D) systems to
capture the B-induced corrections in the free energy and the Fermi surface. We
unravel the manifestation of these corrections in the nonlinear Landau fan
diagrams and aperiodic quantum oscillations by deriving the B-dependent
oscillation frequency and the generalized Lifshitz-Kosevich equation,
respectively. Our theory explains the necessary conditions to observe these
fascinating effects and predicts the magnetic field dependence of the cyclotron
mass. As a concrete example, we elucidate these effects in a 3D spin-orbit
coupled system and extract zero-field magnetic response functions from
analytically obtained Landau levels. Our comprehensive study deepens and
advances our understanding of aperiodic quantum oscillations.
