ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 64

due to the anisotropic two-ion coupling in the magnetically ordered
phases, the axial anisotropy parameter κ 2 ( T )

2 κ 4 ( T )+ 3 8 κ 6 ( T ), where
κ 2 ( T ) is the dominating term, depends on temperature approximately
as predicted by (1.5.15), varying roughly as σ 3 in Dy and σ 4 in Tb.
We shall return to the question of the anisotropy parameters in the rare
earths when we discuss the structures and excitations.
The large magnetoelastic effects have a profound effect on the mag-
netic properties of the rare earths, making a significant contribution to
the anisotropy, playing a decisive role in some instances in determining
the structures, and modifying the excitation spectrum. We here consider
for illustrative purposes a special example, the basal-plane ferromagnet,
exemplified by Tb and Dy. As mentioned previously, the α -strains main-
tain the symmetry and therefore only have the effect of renormalizing the
B l , and if the moments are confined to the plane, the ε -strains vanish.
However, the γ -strains are large and symmetry-breaking, and thereby
cause qualitative modifications in the magnetic behaviour. From (1.4.8)
and (1.4.11), their contribution to the magnetoelastic Hamiltonian may
be written
H γ =
i

5

−

2

O 2 ( J i ) γ 1 + O − 2 c γ ( γ 1 + γ 2 ) − B γ 2 { ( J i ) γ 2 }

2

(1 . 5 . 27)

( J i ) γ 2 } ,

O 4 ( J i ) γ 1 − O − 4

4

− B γ 4 {

where we have included only the lowest ranks ( l = 2 and 4 respectively)
of the γ 2and γ 4 terms. As shown in Section 2.2, the condition

∂F/∂ γ =0 (1 . 5 . 28)

leads to the equilibrium strains

B γ 2

O 4

1
c γ

O 2 γ 1 = + B γ 4

(1 . 5 . 29)

B γ 2 .

1
c γ

O − 2
2

O − 4
4

γ 2 = − B γ 4

Transforming the Stevens operators as before, and using (1.5.15) to esti-
mate the magnetization dependence of the thermal averages, we obtain

− 2

γ 1 = C cos 2 φ A cos 4 φ

(1 . 5 . 30)

γ 2 = C sin 2 φ + 2 A sin 4 φ,

where

1
c γ

B γ 2 J (2) σ 3 C =

(1 . 5 . 31)

2
c γ

B γ 4 J (4) σ 10 A = −

are the conventional magnetostriction parameters (Mason 1954), and φ