  The relation between level crossings, entanglement, and Berry phases is
investigated for the Breit-Rabi Hamiltonian of hydrogen and sodium atoms,
describing a hyperfine interaction of electron and nuclear spins in a magnetic
field. It is shown that the entanglement between nuclear and electron spins is
maximum at avoided crossings. An entangled state encircling avoided crossings
acquires a marginal Berry phase of a subsystem like an instantaneous eigenstate
moving around real crossings accumulates a Berry phase. Especially, the nodal
points of a marginal Berry phase correspond to the avoided crossing points.
