  We use a microscopic Slater-Koster tight-binding model with short-range
exchange and atomic spin-orbit interactions that realistically captures generic
features of ferromagnetic metal nanoparticles to address the mesoscopic physics
of magnetocrystalline anisotropy and hysteresis in nanoparticle quasiparticle
excitation spectra. Our analysis is based on qualitative arguments supported by
self-consistent Hartree-Fock calculations for nanoparticles containing up to
260 atoms. Calculations of the total energy as a function of magnetization
direction demonstrate that the magnetic anisotropy per atom fluctuates by
several percents when the number of electrons in the particle changes by one,
even for the largest particles we consider. Contributions of individual
orbitals to the magnetic anisotropy are characterized by a broad distribution
with a mean more than two orders of magnitude smaller than its variance and
with no detectable correlations between anisotropy contribution and
quasiparticle energy. We find that the discrete quasiparticle excitation
spectrum of a nanoparticle displays a complex non-monotonic dependence on an
external magnetic field, with abrupt jumps when the magnetization direction is
reversed by the field, explaining recent spectroscopic studies of magnetic
nanoparticles. Our results suggests the existence of a broad cross-over from a
weak spin-orbit coupling to a strong spin-orbit coupling regime, occurring over
the range from approximately 200- to 1000-atom nanoparticles.
