SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 193

1 /J means that the RPA contributions to the correlation functions are
reliably estimated, and that all second-order contributions are included
when H 0 + H 1 is used, instead of H 1 , in the calculation of the ther-

mal averages. We shall therefore use the following self-consistent expres-
sions for the characteristic correlation functions, m k and b k , determined
straightforwardly by utilizing the correspondence between

H 0 +

H 0 + H 1 and

H 0 + H 1 :

A k ( T ) ,

E k ( T ) n k + 2

1
J

− 2 m k = (5 . 2 . 32 a )

corresponding to (5.2.25), and

J B k ( T )

E k ( T ) n k + 2

.

1

b k = − (5 . 2 . 32 b )

In order to express the result in a convenient form, we rewrite one of the
second-order terms in B q ( T )as
1

2 N
k

− 1
2

B ( m + 1
2 J

− 1
2

(1 /J 3 ) , (5 . 2 . 33)

J {J ( 0 ) −J ( k ) } b k = ) Ab + O

A ,and B k ( T )in
b k can be approximated by B .Wenotethat A q and B are parameters
of the order 1 /J ,asare m and b (at low temperatures). In addition
to introducing (5.2.33) into (5 . 2 . 31 b ), it is adequate for calculating the
spin-wave energies to define a transformed set of parameters:
A q ( T )= A q ( T )+ 2 B q ( T ) b
B q ( T )= B q ( T )+ 2 A q ( T ) b

= A k ( T )

since, to leading order, J {J ( 0 ) −J ( k ) } −

(5 . 2 . 34)

and these are then, to the order considered,
A q ( T )= A +4 JC 1 m +6 JC 2 b + 2 Bb

N
k

m )+ 1 + J {J ( 0 ) −J ( q ) } (1 − J {J ( k ) −J ( k − q ) } m k

(5 . 2 . 35 a )

and

− 1
2

B q ( T )= B +2 JC 1 b +6 JC 2 m +12 JC 3 b Bm

N
k

(5 . 2 . 35 b )

1

+ J {J ( k ) −J ( k − q ) } b k .

This transformation leaves the expression for the excitation energies un-
changed, i.e.
E q ( T )= [ A q ( T )+ B q ( T )][ A q ( T )

B q ( T )] 2 ,

− (5 . 2 . 36)