MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 92

have the same sign as θ p , we may write the angular-dependent part of
the free energy, to the fourth power of the magnetization, as

J 2 −J 6 ( Q )+ ψ )

f ( θ, ψ )= 4 K 6 ( Q ) cos 2( θ −

σ 1 cos( π +6 ψ )+ σ 2 cos(5 π

×{ − 6 ψ ) }

+ 4

J 2 K 0 ( Q )( σ 1 + σ 2 ) cos 2( θ ψ )+ BJ 4 σ 1 σ 2 cos 2 (2 π/ 3 2 θ ) .

(2 . 1 . 47 a )
For definiteness, we have chosen the case where the angle between the
ξ -axis and

− −

σ 1 or σ 2 is respectively π/ 6+ θ and 5 π/ 6 − θ (by symmetry

θ = θ 1 θ 2 ). Analogously to θ , ψ is the angle between Q p and the

nearest b -axis. Introducing σ 2 =2 σ 1 =2 σ 2 , and expanding f ( θ, ψ )to
second order in the small angles, we obtain

= −

f ( θ, ψ )= f 0 − 2 − 2

( Jσ ) 2 ψ 2 ( Jσ ) 2 ψ ) 2

{J 6 ( Q ) −K 6 ( Q ) } K ( Q )( θ −

( Jσ ) 4 B ( √ 3 θ

− 4

2 θ 2 ) .

−

(2 . 1 . 47 b )

We note that, with the chosen sign conventions, K ( Q )= K 0 ( Q ) −K 6 ( Q )

and −K 6 ( Q ) are both negative. The additional contribution to

the free energy of the double- Q structure is minimized when
θ = √ 3 B ( Jσ ) 2
4 |K ( Q ) |

J 6 ( Q )

√ 3 B ( Jσ ) 2
36 |J 6 ( Q ) −K 6 ( Q ) |

+ ψ ; ψ = , (2 . 1 . 48 a )

neglecting the small term proportional to Bθ 2 ,inwhichcase

B 2 ( Jσ ) 6

.

1
K ( Q ) −

1
9

1
J 6 ( Q ) −K 6 ( Q )

− 32

∆ f = (2 . 1 . 48 b )

−

Introducing A = A ξ ( T = T N ), i.e.

J −K

( Q )=2 A ,thenforPrwe

( Q )

0 . 35 A .
These values may also provide a reasonable estimate in the case of Nd.
Inserting them in (2.1.48), we find that θ

K −

0 . 24 A ,

J 6 ( Q ) −K 6 ( Q ) −

0 . 05 A ,and BJ 2

have: ( Q )

3 ψ 1 . 0 σ 2 ,and∆ f

0 . 5 uσ 6 . So, even though ∆ f is of sixth order in σ ,
it outweighs the small fourth-order energy difference of wσ 4 / 4 between
the single- and the double- Q structure when σ 2

0 . 2 BJ 4 σ 6

− −

≈ 0 . 1, if w 0 . 2 u as

estimated above. The temperature T N at which this occurs is ∼ 0 . 97 T N ,

i.e. 0 . 9Kbelow T N in Nd. Hence, if w is positive and has the estimated

small magnitude, the system will first undergo a second-order transition
from the paramagnetic phase to a single- Q structure, which will only
be stable as long as σ 2 is small. At T N , slightly below T N ,thesystem
will make a first-order transition to a double- Q structure, in which the
moments J 1 and J 2 are rotated slightly towards each other and away
from the symmetry axes, as also are the ordering wave-vectors Q 1

∼

and

Q 2 . These rotations are proportional to σ 2 .