MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 101

by the low-rank terms alone, i.e. by B 2 if anisotropic dipole-dipole cou-
pling is neglected. Using β =( k B T ) − 1 as an expansion parameter, and
assuming the magnetization to lie in the basal plane, we find, to leading
order in the crystal-field parameters introduced in eqn (1 . 4 . 6 b ),

J (2) σ 2 J + 2

= 3
5

J (2) I 5 / 2 [ σ ] ,

Q 2

J +1

(2 . 2 . 12 a )

using (2.2.7) and neglecting the small 1 /J corrections, whereas

Q 2 = −Q 2 − 5 J (2) ( J +1)( J + 2

) βB 2 ,

(2 . 2 . 12 b )

which depends on the anisotropy, but only on the term of lowest rank.
Considering the field dependence of the two expectation values, as de-
termined by their dependence on σ , we observe that the Callen-Callen
theory leads to the right result in this high-temperature limit. The two
relations above explain the behaviour of O 2 ± O 2

in Fig. 2.2, when σ

/ 2 J (2) should approach I 5 / 2 ( σ ) at small val-
ues of σ , and go to zero at the transition temperature.

O 2 + O 2 becomes small, as

O 2 −O 2 / 2 J (2) ,
on the other hand, should still be non-zero (about 0.23 as determined
by T N 229 K and the value B 2 =0 . 18meVusedinthemodel)when

T N is approached from below and σ vanishes.

2.2.2 Anisotropic contributions to the free energy
The anisotropy of a magnetic system is determined by those contribu-
tions to the free energy which depend on the polar angles ( θ, φ ), which
specify locally the direction of the moments. Restricting ourselves to
the case of a ferromagnet in a uniform field, we may expand the free
energy in terms of functions proportional to the spherical harmonics, as
in eqn (1.5.22). To relate this expansion to the Hamiltonian (2.1.1), we
may use (2.1.5), which states that any change in the free energy due to a

change of the angles is given by δ F = δ H . The field is the independent

variable in F but, as in (2.1.22), we wish the magnetization to be the in-
dependent variable. To obtain this free energy F ( θ, φ ), we subtract the
Zeeman energy, so that F ( θ, φ )= F −H Z , where the field needed for

establishing the specified angles is determined from the equilibrium con-
dition δ F = 0. In the ferromagnet, the moments all point in the same
direction, and any contributions from the isotropic-exchange coupling
cancel out in δ

H . The free-energy function F ( θ, φ ) is thus determined

by

δF ( θ, φ )= δ ( H Z ) − δ H Z . H cf + (2 . 2 . 13)

Introducing the angle variables in the Hamiltonian by the transformation
(2.2.8), we find that the operators of rank l become angle-dependent