SPIN WAVES IN PERIODIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 294

interaction matrix, the result is

J xx ( q )= 1 ( q )sin 2 θ 0 cos 2 θ 0 +

2 {J

( q + Q )+ J ( q − Q ) } J

J yy ( q )= 1

(6 . 1 . 14)

2 {J

( q + Q )+ J ( q − Q ) }

J xy ( q )= −J yx ( q )= i

2 {J ( q + Q ) −J ( q − Q ) } cos θ 0 ,

where J xy ( q ) is now non-zero. Neglecting the longitudinal response, as

we may in a weakly anisotropic system, we may calculate the response
functions by introducing these coupling parameters in (6.1.8). In order
to estimate the ( xy )-components of the MF susceptibility, or A B + h ex

in eqn (6.1.9), we may utilize their relation to the derivatives of the
free energy, as expressed in eqn (2.2.18). The free energy for the i th
ion, including the Zeeman contribution from the exchange field of the
surrounding ions, is

±

F ( i )= f 0 + f ( u =cos θ ) − h J z cos θ − h ⊥ J z sin θ cos ( φ − φ 0 ) ,

(6 . 1 . 15 a )

with φ 0 = Q · R i + ϕ ,and

h = J z J ( 0 )cos θ 0 ; h ⊥ = J z J ( Q )sin θ 0 . (6 . 1 . 15 b )

H J is again, as in (6.1.4), the usual crystal-field Hamiltonian, except that
B 6 is neglected. The function f ( u ) is given by (2.2.17) in terms of κ l ( T ),
with κ 6 = 0. From (6.1.15), the equilibrium angles are determined by

f ( u 0 )sin θ 0 + h − J z sin θ 0 − h ⊥ J z cos θ 0 =0 ,

and φ = φ 0 . f ( u ) is the derivative of f ( u ) with respect to u ,and
u 0 =cos θ 0 . With sin θ 0

= 0, this equation leads to

f ( u 0 )cos θ 0 =

2

cos 2 θ 0 . J z {J ( 0 ) −J ( Q ) } (6 . 1 . 16)

The spin-wave parameters may then be derived as

J z ( A + B + h ex )= F θθ ( i )

= f ( u 0 )sin 2 θ 0 − f ( u 0 )cos θ 0 + h J z cos θ 0 + h ⊥ J z sin θ 0

B + h ex )= F φφ ( i ) / sin 2 θ 0 = h ⊥ J z ( A − J z / sin θ 0 .

Introducing the values of the exchange fields and applying the equilib-
rium condition (6.1.16), we then find that

f ( u 0 ) / sin 2 θ 0 + A + B + h ex = { J z } J z J ( Q )

(6 . 1 . 17)

A − B + h ex = J z J ( Q ) .