SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 220

For simplicity, we assume that there is only one ion per unit cell, but
the results we shall derive are also applicable to the hcp lattice, at least
for the acoustic modes at long wavelengths. In this limit H γ (dyn), eqn

(5.4.6), augmented by the kinetic energy of the ions, is adequate for
discussing dynamical effects due to the γ -strains, if αβ

are replaced by

their local values

2
ν k

i

( k α F k ,β + k β F k ,α )( β ν k + β ν− k ) e i k · R i .

αβ ( i )= αβ + (5 . 4 . 25)

We shall initially concentrate on the most important dynamical effects,
and consider only the inhomogeneous-strain terms involving Stevens op-
erators with odd m . Assuming for the moment that φ = p 2 ,weobtain

2 O − 1
2

the contribution γ 2 ) from eqn (5.4.6),

and a corresponding term in B γ 4 . Introducing the spin-deviation oper-
ators through (5.2.8) and (5.2.9), we obtain, to leading order in m and
b ,

− B γ 2 {− ( J i )cos2 φ } ( γ 2 ( i ) −

√ 2 J a i −

a i a i a i )

i 5

B γ 2 O − 1 ( J i )= J (2) B γ 2

4 J ( a i a i a i −

a i −

2

√ 2 J 1 b ( a i −

i

− 2 m + 4 = J (2) B γ 2

a i )

√ 2 J 1+ 2 b ( a i −

i

m + 4 = c γ C a i )

A q ( T )+ B q ( T )
2 NJσE q ( T )

2 ( α q −

= ic γ C
q

α − q ) e −i q · R i ,

(5 . 4 . 26)
utilizing the RPA decoupling (5.2.29) and introducing the (renormal-
ized) magnon operators α q and α − q , analogously with (5.2.39) and
(5.2.40). The B γ 4 -term is treated in the same way, and introducing
the phonon-operator expansion of the strains (5.4.25) into (5.4.6), we
find that

H γ leads to the following Hamiltonian for the system of
magnons and phonons:
H mp =
k

H

+

E k ( T ) α k α k +
ν k

hω ν k β ν k β ν k + W k ( α k −α − k )( β ν k + β ν− k )

(5 . 4 . 27)

with a magnon-phonon interaction given by

c γ √ N ( k 1 F k , 2 + k 2 F k , 1 ) A k ( T )+ B k ( T )
2 JσE k ( T )

2 ( C cos 2 φ + A cos 4 φ ) .

W k =

−

(5 . 4 . 28)

This Hamiltonian includes the part of H γ which is linear in the magnon

operators when φ = p 2

. The effects of the static deformations are