MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 103

Equation (2.2.16) shows that the various anisotropy parameters depend
differently on temperature. At high temperatures, κ 2 dominates and
its sign determines whether the moments are parallel or perpendicu-
lartothe c -axis. As the temperature is decreased, the importance of
the higher-rank terms grows, putting increasing weight on the terms of
fourth and sixth power in cos θ . The equilibrium value θ 0 ( T )of θ may
therefore change as a function of temperature, as occurs in Ho and Er,
and also in Gd where, however, the theory of this section is not imme-
diately applicable.
The coecients in the expansion for the free energy may be ob-
tained from experimental studies of the magnetization as a function of
the magnitude and direction of an applied magnetic field. The axial
part of the anisotropy is predominantly determined by the three κ l -
parameters, and it is not usually easy to separate their contributions.
At low temperatures, where the higher-rank terms become relatively im-
portant, the axial anisotropy in the heavy rare earths is frequently so
strong that it is only possible to change θ by a few degrees from its equi-
librium value. Under these circumstances, it is only possible to measure
the components of the susceptibility, allowing a determination of the
second derivatives of F ( θ, φ ) in the equilibrium state ( θ, φ )=( θ 0 ,φ 0 ).
The x -axis lies in the symmetry z - ζ plane and the transverse part of the
susceptibility tensor is diagonal with respect to the ( x, y )-axes. With a
field h x applied in the x -direction, the moments rotate through an angle
δθ = θ − θ 0 , giving a component J x = − J z δθ = χ xx h x . Introducing

∂ 2 F ( θ, φ ) /∂θ 2 at ( θ, φ )=( θ 0 ,φ 0 ), and similarly for
the other second derivatives, we may write

the notation F θθ ≡

F = F ( θ 0 ,φ 0 )+ 2 F θθ ( δθ ) 2 + 2 F φφ ( δφ ) 2 + N

J z δθh x ,

in the limit where the field goes to zero. The term F θφ = 0, because

sin 6 φ 0 = 0. At equilibrium, δθ = h x /F θθ , which determines the

susceptibility. When the field is applied in the y -direction, i.e. along the
direction (

− N J z

− sin φ 0 , cos φ 0 , 0), the Zeeman contribution to F is

N J z h y sin θ sin ( φ − φ 0 )= N J z h y sin θ 0 δφ,

with sin θ 0 δφ = χ yy h y . Minimizing the free energy in the

presence of a field along the y -axis, we may derive the other susceptibility
component, obtaining

J y = − J z

2 /F θθ 2 sin 2 θ 0 /F φφ .

χ xx = N J z ; χ yy = N J z (2 . 2 . 18)

In calculating χ yy , we have assumed that θ 0 =0;if θ 0 =0then

χ yy = χ xx . Equation (2.2.18) is also valid in the presence of an ex-
ternal field, provided that the effects due to the Zeeman contribution,