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Paramagnetism of Free Ions
3.1. THE BRILLOUIN FUNCTION
Once we have applied the vector model and Hund’s rules to find the quantum numbers J, L,
and S of the ground-state multiplet of a given type of atom, we can describe the magnetic
properties of a system of such atoms solely on the basis of these quantum numbers and the
number of atoms N contained in the system considered.
If the quantization axis is chosen in the z-direction the z-component m of J for each
atom may adopt 2J + 1 values ranging from m = – J to m = + J. If we apply a magnetic
field H (in the positive z-direction), these 2J + 1 levels are no longer degenerate, the
corresponding energies being given by
where is the atomic moment and its component along the direction of
the applied field (which we have chosen as quantization direction). The constant is
equal to
The lifting of the (2J + 1)-fold degeneracy of the ground-state manifold by the magnetic
field is illustrated in Fig. 3.1.1 for the case Important features of this level scheme
are that the levels are at equal distances from each other and that the overall splitting is
proportional to the field strength.
Most of the magnetic properties of different types of materials depend on how this
level scheme is occupied under various experimental circumstances. At zero temperature,
the situation is comparatively simple because for any of the N participating atoms only the
lowest level will be occupied. In this case, one obtains for the magnetization of the system
However, at finite temperatures, higher lying levels will become occupied. The extent to
which this happens depends on the temperature but also on the energy separation between
the ground-state level and the excited levels, that is, on the field strength.
The relative population of the levels at a given temperature T and a given field strength
H can be determined by assuming a Boltzmann distribution for which the probability of
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