MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 107

Within this approximation, the γ -strain contribution F γ ( θ, φ )tothefree
energy is
F γ ( θ, φ )= H γ =

c γ γ 1 + γ 2 N

− 2

CA sin 6 θ cos 6 φ N,
(2 . 2 . 27)
showing that these strains affect the axial-anisotropy parameters κ l ( T ),
introducing effects of higher rank than l = 6, and that the six-fold
anisotropy in the basal plane is now

c γ C 2 sin 4 θ + 4 − 2

A 2 sin 8 θ = −

κ 6 ( T )= B 6 J (6) I 15 / 2 [ σ ]+ 2

c γ CA. (2 . 2 . 28)

When both C and A are non-zero, the maximum area-conserving elon-
gation of the hexagonal planes varies between C + 2 A

1

,
which results in a φ -dependent magnetoelastic energy, and thus a contri-
bution to κ 6 .The γ -strain hexagonal anisotropy decreases more slowly
(like σ 13 at low temperatures) than the B 6 term, as σ decreases, and
therefore dominates at suciently high temperatures.
The ε -strains may be included in a similar way. Retaining only the
lowest-rank coupling B ε 1 ≡

| | and | C −

2 A

|

B ( l =2)
ε 1

in eqn (1.4.12), we have

H ε =
i

2

B ε 1 Q 2 ( J i ) ε 1 + Q − 1 ( J i ) ε 2 . c ε ( ε 1 + ε 2 )

− (2 . 2 . 29)

2

Introducing the magnetostriction parameter H ε of Mason (1954) (the
index ε should prevent any confusion with the magnetic field) by

ε 1 = 4 ε 2 = 4

H ε sin 2 θ cos φ ; H ε sin 2 θ sin φ, (2 . 2 . 30 a )

we obtain within the Callen-Callen theory

2
c ε

B ε 1 J (2) I 5 / 2 [ σ ] , H ε = (2 . 2 . 30 b )

and the ε -strain contribution to the free energy

− 32

Nc ε H ε sin 2 2 θ. F ε ( θ, φ )= (2 . 2 . 31)

The α -strains (1.4.10) do not influence the symmetry of the system, but
they do make a contribution, essentially proportional to Q 2 ,tothe

anisotropy, the effects of which may be derived in the same way as those
of the γ -and ε -strains. The magnetoelastic contributions to the free en-
ergy can be estimated experimentally if the elastic constants are known,
by a determination of the strains as a function of the magnetization. The