Physics of Magnetism

SECTION 2.2. THE VECTOR MODEL OF ATOMS 5

where is the spectroscopic splitting factor (or the g-factor for the

free electron). The component in the field direction is

The energy of a magnetic moment in a magnetic field is given by the Hamiltonian

where is the flux density or the magnetic induction and is the
vacuum permeability. The lowest energy the ground-state energy, is reached for and
parallel. Using Eq. (2.1.6) and one finds for one single electron

For an electron with spin quantum number the energy equals
This corresponds to an antiparallel alignment of the magnetic spin moment with respect to
the field.
In the absence of a magnetic field, the two states characterized by
degenerate, that is, they have the same energy. Application of a magnetic field lifts this
degeneracy, as illustrated in Fig. 2.1.2. It is good to realize that the magnetic field need not
necessarily be an external field. It can also be a field produced by the orbital motion of the
electron (Ampère’s law, see also the beginning of Chapter 8). The field is then proportional

are

proportional to

to the orbital angular momentum l and, using Eqs. (2.1.5) and (2.1.7), the energies are
In this case, the degeneracy is said to be lifted by the spin–orbit
interaction.

2.2. THE VECTOR MODEL OF ATOMS

When describing the atomic origin of magnetism, one has to consider orbital and
spin motions of the electrons and the interaction between them. The total orbital angular
momentum of a given atom is defined as

where the summation extends over all electrons. Here, one has to bear in mind that the

summation over a complete shell is zero, the only contributions coming from incomplete