MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 70

temperatures . Expanding in powers of β , we may use the approximation
exp + 2 β 2

2 . In this case, we may proceed as above,

{− β H} 1 − β H H

and the result is
∂ 2 F

∂x∂y + β ∂

∂y

∂x∂y = ∂ 2

∂x ∂ ∂y − ∂

H ∂x ∂

2 ∂

(2 . 1 . 6)

H +

β 2 ∂x , ∂ ∂y

( β 3 ) ,

− O

where the second- and higher-order terms vanish if one of the derivatives
of itself.

In many instances, it is more convenient to consider the angular
momentum rather than the magnetic moment, with a corresponding
field variable h i = gµ B H i , so that the Zeeman term (2 . 1 . 1 c ) becomes

H commutes with H

H Z = − µ i · H i = − J i · h i . (2 . 1 . 7)

i i

Since the exchange and anisotropy terms in H do not depend explicitly
on the field, ∂H/∂H iα = −µ iα and, using eqn (2.1.5), we have

µ iα = − ∂F/∂H iα or J iα = − ∂F/∂h iα . (2 . 1 . 8)

Next, we define the non-local susceptibilities

χ αβ ( ij )= ∂µ i /∂H jβ = −∂ 2 F/∂H iα ∂H jβ ,

(2 . 1 . 9 a )

and similarly

χ αβ ( ij )=( gµ B ) − 2 χ αβ ( ij )= −∂ 2 F/∂h iα ∂h jβ ,

(2 . 1 . 9 b )

and the corresponding Fourier transforms, e.g.

N
ij

χ αβ ( ij ) e −i q · ( R i − R j ) =
j

χ αβ ( q )= 1 χ αβ ( ij ) e −i q · ( R i − R j ) .

(2 . 1 . 9 c )
The final equality only applies in a uniform system. If the field is in-
creased by an infinitesimal amount δ H ( q )exp( i q · R i ), the individual
moments are changed by

=
j

χ αβ ( ij ) δH β ( q ) e i q · R j , δ µ iα (2 . 1 . 10 a )

β

according to (2.1.9). Hence the added harmonically-varying field intro-
duces one Fourier component in the magnetization:

V
i

V
β

δM α ( q )= 1 e −i q · R i = N χ αβ ( q ) δH β ( q ) , δ µ iα

(2 . 1 . 10 b )