MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 71

proportional to the susceptibility at the wave-vector considered. The
usual definition of the susceptibility components (per unit volume),
as used in Chapter 1, is δM α ( q ) /δH β ( q ). The susceptibility used in
(2 . 1 . 10 b )differsfromthisbythefactor V/N , i.e. we are here considering
the susceptibility per atom instead of per unit volume. Furthermore,
since we shall not make any further use of χ αβ ( q ), we shall reserve the
notation χ αβ ( q )forthe q -dependent susceptibility χ αβ ( q ), introduced
in eqn (2 . 1 . 9 b ), throughout the rest of the topic. So in terms of the
susceptibility per atom, 'in units of ( gµ B ) 2 ', the above equation may be
written

N
i

e −i q · R i =
β

1

δ J α ( q ) = δ J iα χ αβ ( q ) δh β ( q ) , (2 . 1 . 10 c )

with the upper index J in χ αβ ( q ) being suppressed from now on.

2.1.1 The high-temperature susceptibility
In order to calculate χ ( q ) in zero field, we shall first use the approxi-
mation (2.1.6) to the derivative of the free energy, valid at high temper-
atures. In this limit

J i = 0 , and only one term in the expansion is

non-zero:
χ αβ ( ij )= β Tr J iα J jβ (1 ) Tr 1 H ,

− β H − β (2 . 1 . 11)

to second order in β . The commutator in the third term on the right-
hand side of (2.1.6) is either zero or purely imaginary (if i = j and
α = β ), showing immediately that the expectation value of this term

must vanish in all cases. To first order in β , we obtain from (2.1.11)

β Tr J iα J jβ Tr 1 = 3

χ αβ ( ij ) J ( J +1) βδ αβ δ ij ,

using the product of the eigenvectors of J iα as the basis, and recalling
that

m 2 = 3

J ( J + 1)(2 J +1) ,

when m runs from

J to J . In order to calculate the second-order
contribution, we shall utilize the general tensor properties of the Stevens
operators, which satisfy the orthogonality condition:
Tr O l

−

( J j ) = δ ij δ ll δ mm Tr [ O l ( J i )] 2 ( J i ) O m

l

(2 . 1 . 12)

Tr O l ( J i ) =0 , and

when l and l are both non-zero. O 0 is just the identity operator. J iα is
a linear combination of O 1 ( J i ), m =

− 1 , 0 , 1, and (2.1.12) then implies