SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 191

is the usual Bose population-factor, we find to first order in 1 /J :

N
q

= J a q a q = J (1

1

J z − − m ) , (5 . 2 . 24)

with

N
q

N
q

1 1

J

1

a q a q m = m q

=

and

1
J

( u q α q − v q α + m q = v q α − q )( u q α q −

− q )

J u q n q + v q ( n q +1)

1

=

(5 . 2 . 25)

J A q

,

n q + 2 − 2

1

=

E q

which is positive and non-zero, even when n q =0at T =0.
The second-order contribution to the Hamiltonian is
H 2 =
i

B

1

8 J ( a i a i

+ a i a i )+ C 1 a i a i a i a i

+ a i a i a i a i ) + C 2 ( a i a i a i a i + a i a i a i a i )+ C 3 ( a i a i a i a i

ij J

( ij ) 2 a i a j a i a j − a i a i a i a j , − 4 a i a j a j a j − (5 . 2 . 26)

with

J 2 2

105 B 6 J (6) cos 6 φ

1

B 2 J (2)

C 1 = − −

J 2 4

B 6 J (6) cos 6 φ

1

B 2 J (2) + 19 4

(5 . 2 . 27) C 2 = −

1

J 2 1 4

B 6 J (6) cos 6 φ.

C 3 =

H 2 , we find

Introducing the Fourier transforms of the Bose operators in

straightforwardly that

H 2 ]= A q a q + B 1+

4 J a +

1

ih∂a q /∂t =[ a q , H ] [ a q , H 1 +

− q +

−J

N
k , k

( q )+2 C 1 a k a k a q + k − k

1

( q − k )+ 2 J ( k )+ 4 J ( k )+ 4 J

,
(5 . 2 . 28)

+ C 2 3 a k a +

− k a k a q + k − k +4 C 3 a k a +

− k a + − k a q + k − k + a − q − k + k

for the operator [ a q , H ], which appears in the equation of motion of,

a q ; a q

for instance . When the thermal averages of terms due to H 2