Magnetic hyperfine splitting is caused by the dipole interaction between the nuclear spin moment and a magnetic field ie Zeeman splitting. The effective magnetic field experienced by the nucleus is a combination of fields from the atom itself, from the lattice through crystal field effects and from external applied fields. This can be considered for now as a single field, , whose direction specifies the principal axis.
The Hamiltonian for the magnetic hyperfine dipole interaction is given as
This Hamiltonian yields eigenvalues of

This splitting is a combination of a constant nuclear term and a variable magnetic term, influenced by the electronic structure. The magnetic field at the nucleus has several terms associated with it. A general expression is
is produced by the polarisation of electrons whose wavefunctions overlap the nucleus, ie electrons. This polarisation is due to unpaired electrons in the or orbitals and gives an imbalance in spin density at the nucleus from the difference in interaction between the unpaired electron with electrons of parallel or antiparallel spin to its own. This can be expressed formally as
arises from the net orbital moment at the nucleus caused by the orbital motion of electrons in unfilled shells and given by
arises from the dipolar interaction between the nucleus and the spin moment of 3 or 4 electrons and can be expressed as
Dr John Bland, 15/03/2003