MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 72

that the trace of the Hamiltonian (2.1.1) vanishes, and hence that the
denominator in (2.1.11) is Tr =(2 J +1) N . For the second-order { 1 }

term in the numerator, we find
Tr J iα J jβ H = δ ij B 2 Tr J iα J iβ O 2 ( J i ) −J ( ij )Tr J iα J jβ J i · J j

= δ ij δ αβ B 2 Tr J iα [3 J iz − J ( J +1)] − ( ij )Tr J iα J jα ,

δ αβ J

utilizing that J iα J jβ is a linear combination of second- and lower-rank
tensors for i = j , and a product of first-rank tensors for i = j .When

α = z (or ζ ), we may readily calculate the first trace, using
m 4 =

1
15

J ( J + 1)(2 J + 1)(3 J 2 +3 J

− 1) .

The traces with α = x or α = y must be equal, and using this equality
in the case α = x , for instance, we may replace J x in the trace by

1

2 ( J x + J y )

1 1

2 J z . As the constant term multiplied by

→

2 J ( J +1)

−

3 J z − 3 J z − J ( J + 1) does not contribute (as Tr { J ( J +1) } = 0), the

trace with α = x or y is equal to 1 / 2 times that with α = z . Only the

single-ion terms contribute to the trace when i = j (

−

( ii ) is assumed to
be zero), and of these only the lowest-rank term B 2 appears, to leading
order. The two-ion coupling only occurs in the trace, and hence in
χ αβ ( ij ), when i

J

= j , and this contribution may be straightforwardly
calculated. To second order in β , the off-diagonal terms are zero, whereas

J ( J +1) β 1 ) β χ αα ( ij )= δ ij 3 − 5 − 2 )( J + 2

1) B 2 ( J (3 δ αζ −

+ 3 J ( J +1) β 2

J ( ij ) .

Introducing the Fourier transform of the two-ion coupling,

( q )=
j

( ij ) e −i q · ( R i − R j ) , J J (2 . 1 . 13)

we find that, to the order considered, the inverse of the q -dependent
susceptibility may be written

2 )( J + 2 )
5 J ( J +1)

1

3 k B T

1) 6( J

−

B 2 −J 1 /χ αα ( q )=

J ( J +1) +(3 δ αζ −

( q )+ O (1 /T ) .

(2 . 1 . 14)
The inverse susceptibility in the high-temperature limit thus increases
linearly with the temperature, with a slope inversely proportional to the
square of the effective paramagnetic moment (

1 / 2 ). The
susceptibilities determined experimentally by magnetization measure-

∝{ J ( J +1) }