ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 46

and

I ( r ) e −i q · r d r .

I ( q )= 1
V

(1 . 4 . 17)

Summing over the lattice sites, counting each interaction once only, we
find that the indirect-exchange interaction takes the familiar isotropic
Heisenberg form:

χ ( q ) I ( q ) I (

(2 π ) 3
ij

1
2

V
N 2 µ 2 B

V

− q ) e i q · ( R i − R j ) S i · S j d q H ff = −

2 N
q ij J S ( q ) e i q · ( R i − R j ) S i · S j

1

= −

2

1

= −

ij J S ( ij ) S i · S j ,

(1 . 4 . 18)

where

N

J S ( ij )= 1

q J S ( q ) e i q · ( R i − R j )

(1 . 4 . 19)

and

V
Nµ 2 B |

2 χ ( q ) .

J S ( q )= I ( q ) | (1 . 4 . 20)

In the presence of an orbital moment, it is convenient to express
(1.4.18) in terms of J rather than S , which we may do within the ground-
state multiplet by using (1.2.29) to project S on to J , obtaining

2

1

H ff = −

ij J

( ij ) J i · J j , (1 . 4 . 21)

with

1) 2

q J S ( q ) ,

N

1

J ( q )=( g − J S ( q ) − (1 . 4 . 22)

where we have also subtracted the interaction of the i th moment with
itself, as this term only leads to the constant contribution to the Hamil-
tonian;

1

1) 2 N J S ( ii ) J ( J + 1). The origin of the indirect ex-

change in the polarization of the conduction-electron gas by the spin
on one ion, and the influence of this polarization on the spin of a
second ion, is apparent in the expression (1.4.20) for

−

2 ( g

−

J S ( q ). As we

shall see, it is the Fourier transform [ ( 0 )] which may be di-

rectly deduced from measurements of the dispersion relations for the
magnetic excitations, and its experimentally determined variation with
q in the c -direction for the heavy rare earths is shown in Fig. 1.17.

J ( q ) −J