MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 97

One of the advantages of the Callen-Callen theory is that the results
only depend on the one parameter σ , but not explicitly on the Hamil-
tonian. The relative magnetization may then be determined either by
experiment, or by MF or more accurate theories, which result in a σ
which depends on the actual Hamiltonian employed. The simplicity of
this result may be impaired if the magnetic anisotropy of the system is
substantial, so that the exchange interaction is no longer the dominant
term in the density matrix. We shall be mostly concerned with the ap-
plicability of the theory at low temperatures, and the introduction of an
axial-anisotropy term, such as B 2 O 2 ( J i ), is not inimical to the theory
in this regime, provided that the magnetization is along the anisotropy
c -axis, i.e. if B 2 is negative. Since only the lowest states are important
at low temperatures and, in the MF approximation, these are still rea-
sonably well accounted for by the density matrix in eqn (2.2.1), only the
value of x is changed, with no direct consequences for the result. There
are however noticeable effects if the anisotropy destroys the rotational
symmetry about the magnetization axis. This is the case if B 2 is positive
and forces the moments to lie in the basal plane, so that it requires a
magnetic field to pull them out of it, whereas they may rotate much more
freely within the plane, since B 6 is unimportant compared to the axial
anisotropy. As we shall discuss in detail in Chapter 5, the ground state
in this situation is no longer the fully-polarized state, the expectation
value of J z is slightly smaller than J at zero temperature, and the lower
symmetry of the anisotropy field has direct consequences for the nature
of the elementary spin-wave excitations, and thus for the form of the
density matrix. The necessary modification of the Callen-Callen theory
may be developed in two ways. One is to analyse the influence of the
anisotropy on the low-temperature elementary excitations, and thereby
derive the density matrix, as is done in Chapter 5. The other approach
is numerical and involves the construction of a Hamiltonian which has
the right transition temperature and the correct anisotropy fields, in the
MF approximation. ρ MF may then be calculated as a function of tem-
perature, and results corresponding to (2.2.5), relating the expectation
values of the various Stevens operators to the relative magnetization,
may be obtained numerically. By the same argumentation as that used
by Callen and Shtrikman (1965), these results may be expected to be
insensitive to the actual model Hamiltonian used for describing the sys-
tem. In the low temperature limit, the spin-wave theory supports this
point of view, as its results are described in terms of only two param-
eters. One is the relative magnetization σ , as before, while the other,
b or η
±

2 b 2 ), measures the eccentricity of the anisotropic
potential about the axis of magnetization.
In our discussion of the Callen-Callen theory, we have assumed

±b )(1

1

=(1 −