SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 263

and neglecting the higher-order corrections to the spin-susceptibility,
due to the coupling to the local moments, we can write the above result

( q ,ω )=2
τ

J

2 χ + −

| j ( q + τ ) |

c . el . ( q +

τ ,ω ) , (5 . 7 . 26 c )

where by the relation (3.2.15), χ + −

c . el . ( q ,ω )=[ χ − +

ω )] ∗ .Ingen-
eral, when the Coulomb interaction cannot be approximated by a δ -
function, this factorization is not valid, and the indirect exchange is
instead given by

c . el . (

− q , −

J

( q ,ω )=

N
nn

f n k ↓ − f n k − q ↑

2

1) 2 I ( n k − q ,n k )

2

lim
ε→ 0 +

( g − | | ,

hω + ihε − ε n k ↓ + ε n k − q ↑

k

(5 . 7 . 27)

where k is now confined to the primitive Brillouin zone.
In the frequency regime of the spin waves, where

is much
smaller than the Fermi energy or the exchange splitting ∆, the fre-
quency dependence of J

| hω |

( q ,ω ) can, to a good approximation, be ne-
glected. The spins of the conduction electrons respond essentially in-
stantaneously to any changes in the state of the local angular momenta,
compared with the time-scale of these changes. For a Bravais-lattice,
J

J ( q , 0) = J

− q , 0). A comparison of eqn (5.7.25) with the
1 /J spin-wave result (5.2.18) shows that J

( q ,ω ) (

− J

( q , 0) replaces the
contribution of the Heisenberg interaction considered in eqn (5.2.1). In
this equation,

( 0 , 0)

J ( ii ) ≡ 0 by definition and, since this is not the case for

( ii )=(1 /N ) q J

J

( q , 0), J ( q , 0) cannot be associated directly with

J ( q ). The instantaneous or frequency-independent part of the coupling

of J i

with itself leads to a contribution 2 N J ( ii ) J i · J i to the total

energy, where = J ( J + 1), independently of the magnetic order-

ing or the temperature. This assertion may be verified (to first order
in 1 /J ) by a direct calculation of

J i · J i

H from (5.7.20). For this purpose

c k − q − τ ↑ a q

, for instance, is determined from eqn (5.7.24), but a
self-energy correction of a factor 1 / 2 must be included in its contribu-
tion to

c k ↓

H . Taking this condition into account, we may finally write

N
q

1

( q )= J

J

( q , 0) .

J ( q , 0) − (5 . 7 . 28)

The exchange interaction between the 4 f electrons and the conduction
electrons thus leads to an effective Heisenberg interaction between the
local angular momenta, as given in (5.2.1). This is the RKKY interaction
discussed earlier in Section 1.4.