LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 158

and correspondingly J 0 .

Here Z is the partition function of the MF Hamiltonian, and thus n ν is
the population factor of the ν th MF level. With the two approximations

J in (3.5.12) is assumed to take the MF value

(3.5.16) and (3.5.17), and the condition that ν µ M ν µ a ν µ ( j )

0 =
J j − J j 0 0 = 0 by definition, (3.5.15) is reduced to a closed set of
equations by a Fourier transformation:

{ hω − ( E µ − E ν ) } G νµ,rs ( q ,ω )

+

ν µ J

( q )( n ν − n µ ) M µν · M ν µ G ν µ ,rs ( q ,ω )=( n ν − n µ ) δ µr δ νs .

(3 . 5 . 18)
We now show that these equations lead to the same result (3.5.8) as
found before. The susceptibility, expressed in terms of the Green func-
tions, is

χ ( q ,ω )= − M νµ M rs G νµ,rs ( q ,ω ) . (3 . 5 . 19)

νµ,rs

M νµ M rs is the dyadic vector-product, with the ( αβ )-component given
by ( M νµ M rs ) αβ =( M νµ ) α ( M rs ) β . Further, from eqns (3.3.4-6), the
MF susceptibility is

E ν = E µ E ν = E µ

M νµ M µν
E µ −

χ o ( ω )=

hω ( n ν −

n µ )+ M νµ M µν βn ν δ ω 0 .

E ν −

νµ νµ

(3 . 5 . 20)

Multiplying (3.5.18) by M νµ M rs / ( E µ − E ν − hω ), and summing over

( νµ,rs ), we get (for ω =0)

χ o ( ω ) ( q ) χ ( q ,ω )= χ o ( ω ) ,

χ ( q ,ω ) − J (3 . 5 . 21)

in accordance with (3.5.8). Special care must be taken in the case of
degeneracy, E µ = E ν , due to the resulting singular behaviour of (3.5.18)
around ω =0. For ω =0, G νµ,rs ( q ,ω ) vanishes identically if E µ = E ν ,

whereas G νµ,rs ( q ,ω = 0) may be non-zero. The correct result, in the
zero frequency limit, can be found by putting E µ − E ν

= δ in (3.5.18),

e −βδ )

so that n ν − n µ = n ν (1 − βn ν δ . Dividing (3.5.18) by δ ,and

taking the limit δ → 0, we obtain in the degenerate case E ν = E µ :

β

− G νµ,rs ( q , 0) −

ν µ J

( q ) n ν M νµ · M ν µ G ν µ ,rs ( q , 0) = βn ν δ µr δ νs .

(3 . 5 . 22)
Since χ ( q ,ω ) does not depend on the specific choice of state-vectors in
the degenerate case, (3.5.22) must also apply for a single level, i.e. when
µ = ν . It then follows that (3.5.18), when supplemented with (3.5.22),