SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 261

differ somewhat from the value ( ε F ) in the paramagnetic phase. In

the hcp metals, the band structure calculations discussed in Section
1.3 reveal that ε F

N

is near a peak in the density of states due to the d

1eV − 1 per spin state per ion, corresponding
to an electronic moment of the order of one-tenth of the local moment.
In the example of Gd, for which g =2,∆ g =0 . 18. The same value of
∆ g/ ( g

electrons, and that N ( ε F )

1) accounts fairly well for the conduction-electron contribution
to the moments of the other heavy rare earths in Table 1.6, bearing
in mind the uncertainties in the experimental results, and the possible
effects of the crystal fields in quenching the local moments.
The spin waves in the ferromagnetic phase are decisively influenced
by the sf -exchange interaction. In order to consider such effects, we
introduce the Bose operators acting on the angular-momentum states,
as in eqns (5.2.6-8), and find, to first order in 1 /J ,

−

) e −i q · R i

N
kq τ

1

c k + τ ↑ H sf H sf (MF) − j ( q + τ − δ q0 ( c k ↑

i

c k ↑ a i ,

) a i a i + √ 2 Jc k + q + τ ↑ + √ 2 Jc k + q + τ ↓

c k + τ ↓ c k ↓ a i

− c k ↓

using the simplified exchange of eqn (5.7.15), and neglecting effects of
third or higher order in j ( q ) due to ( c k σ c k σ − ) a i a i . q is

assumed to lie in the primitive Brillouin zone, but no such restriction
is placed on k . Wenotethat c k

c k σ c k σ

and c k + τ

is a reciprocal
lattice vector, create electrons in different bands in the free-electron
model. Introducing the crystal-field Hamiltonian to first order in 1 /J
(eqn (5.2.14) with

,where τ

( ij ) = 0), and the Fourier transforms of the Bose
operators (5.2.16), we find that the total magnetic Hamiltonian becomes

J

= H s +
q

{

− q )

A + J J a q a q + B 2

( a q a − q + a q a + H ( 0 , 0) }

− 2 J/N
kq τ

) c k + q + τ ↑ a q ,

a +

− q + c k + q + τ ↓

j ( q + τ c k ↓ c k ↑

(5 . 7 . 20)

where

k , τ = 0 |

2 f k ↓ −

f k + τ ↑
ε k + τ −

F )+ 2

J

( 0 , 0) = 2 j 2 ( 0 ) N ( j ( τ ) | , (5 . 7 . 21)

N ε k

including the 'interband' contributions as in (5.7.9). The spin-wave en-
ergies may be obtained from the poles in the Green function

a q ; a q

.
The equation of motion (3.3.14) for this Green function is determined