ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 45

The operators in the α -strain term are the same as those in the crystal-
field Hamiltonian (1.4.6b), and the associated magnetoelastic effects may
thus be considered as a strain-dependent renormalization of the crystal-
field parameters, except that these interactions may mediate a dynamical
coupling between the magnetic excitations and the phonons. The other
two terms may have the same effect, but they also modify the symme-
try and, as we shall see, can therefore qualitatively influence both the
magnetic structures and excitations.
It is the two-ion couplings which are primarily responsible for co-
operative effects and magnetic ordering in the rare earths, and of these
the most important is the indirect exchange , by which the moments
on pairs of ions are coupled through the intermediary of the conduction
electrons. The form of this coupling can be calculated straightforwardly,
provided that we generalize (1.3.22) slightly to

I ( r − R i ) S i · s ( r ) d r = H i ( r )

2
N

H sf ( i )= − − · µ ( r ) d r , (1 . 4 . 13)

where N is the number of ions, s ( r ) is the conduction-electron spin
density, and the exchange integral I ( r − R i ) is determined by the over-
lap of the 4 f and conduction-electron charge clouds. This expression,
whose justification and limitations will be discussed in Section 5.7, can
be viewed as arising from the action of the effective inhomogeneous mag-
netic field

1
Nµ B

H i ( r )= I ( r − R i ) S i

on the conduction-electron moment density ( r )=2 µ B s ( r ). The spin

at R i generates a moment at r , whose Cartesian components are given
by

µ

χ αβ ( r − r ) H iβ ( r ) d r ,

V
β

µ iα ( r )= 1 (1 . 4 . 14)

where χ is the nonlocal susceptibility tensor for the conduction electrons
and V the volume. This induced moment interacts through H sf ( j )with

the spin S j , leading to a coupling

H jα ( r ) χ αβ ( r − r ) H iβ ( r ) d r d r .

V
αβ

1

H ( ij )= − (1 . 4 . 15)

If we neglect, for the moment, the spin-orbit coupling of the conduction
electrons, and the crystal is unmagnetized, χ αβ

becomes a scalar. We

define the Fourier transforms:

χ ( r ) e −i q · r d r

1
V

χ ( q )=

χ ( q ) e i q · r d q

(1 . 4 . 16)

V
(2 π ) 3

χ ( r )=