MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 79

where σ α = σ α ( q ) is the relative magnetization at the wave-vector q .
Introducing this into the free-energy expression, and utilizing the condi-

tion that i cos( q · R i + ϕ )=0,if q is not a reciprocal lattice vector,
we find

J 2

F 0 ) /N = 4

σ α f =( F −

α {

2 A α −J ( q ) }

J 4
αβ

+ 8

σ α σ β , B αβ {

2+cos2( ϕ α −

ϕ β ) }

(2 . 1 . 24)

if 4 q is different from a reciprocal lattice vector. The coecients of

the second power are thus =1 /χ αα ( q ,σ =0),where

the susceptibility is evaluated at zero magnetization. As long as all the
second-order coecients are positive, at any value of q , the free energy
is at its minimum when σ α = 0, i.e. the system is paramagnetic. The
smallest of these coecients are those at q = Q ,where

∝{ 2 A α −J ( q ) }

( q )hasits
maximum. In the heavy rare earths, with the exception of Gd, Q is
non-zero and is directed along the c -axis. Depending on the sign of
B 2 , the magnetic structures occurring in the heavy rare earths may be
divided into two classes, which we will discuss in turn.

J

2.1.3 Transversely ordered phases
When B 2 > 0, as in Tb, Dy, and Ho, the two basal-plane components
of χ ( Q ) both diverge at the same critical temperature T N .Usingthe
approximate high-temperature value (2.1.20) for the susceptibility, we
find that 1 /χ ξξ ( Q ,σ =0)=1 /χ ηη ( Q , 0) = 2 A ξ −J

( Q ) vanishes at the

temperature determined by

( Q ) 1+ 5 ) B 2 /k B T N . k B T N 3 − 2 )( J + 2 J ( J +1) J ( J (2 . 1 . 25)

Below T N ,both σ ξ and σ η are generally non-zero at the wave-vector Q ,
and the free energy f , given by (2.1.24) with σ ζ = 0, is minimized when
σ ξ ( Q )= σ η ( Q )= σ Q ,and
σ Q = J

1 / 2

( Q ) 2 A ξ

4 J 2 B ξξ

− π

2 ,

; ϕ ξ − ϕ η = ± (2 . 1 . 26 a )

corresponding to the helical ordering:

J iξ = Jσ Q cos ( Q · R i + ϕ )

(2 . 1 . 26 b )

J iη = ± Jσ Q sin ( Q · R i + ϕ ) .

The length of the angular-momentum vector is Jσ Q , independent of the
site considered. There are two energetically-degenerate configurations,