ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 15

signifying the negative of the corresponding angular-momentum vector.
We shall furthermore from now on take the gyromagnetic ratio g 0 as 2.
Second-order perturbation theory then gives the magnetic contribution
to the energy:

n> +
m = n

2

| <n | µ B H · ( L +2 S ) | m> |

δE n ( H )= − µ B H · <n | L +2 S | .

E n − E m

(1 . 2 . 24)

JM J LS > basis,
whose degeneracy is completely lifted by the field. In this basis, and
within a particular JLS -multiplet, the Wigner-Eckart theorem implies
that the matrix elements of ( L +2 S ) are proportional to those of J ,so
that

|

Problems of degeneracy are taken care of by using the

JLSM J > = g ( JLS ) <JLSM J | J | JLSM J >,

(1 . 2 . 25)

<JLSM J | L +2 S |

and the proportionality constant, the Land´efactor ,is

g = 3

2 + S ( S +1)

L ( L +1)
2 J ( J +1)

−

. (1 . 2 . 26)

Within this multiplet, we may write eqn (1.2.25) in the shorthand form
L +2 S = g J , and consider the effective moment on the atom to be

µ = gµ B J . (1 . 2 . 27)

With the same proviso, we may similarly write

L =(2 − g ) J , (1 . 2 . 28)

and

1) J . (1 . 2 . 29)
If J is non-zero, the first-order term in (1.2.24), combined with (1.2.22)
gives a magnetization for the ground-state multiplet:

S =( g

−

M ( H, T )= N

V gµ B JB J ( βgµ B JH ) ,

(1 . 2 . 30)

where the Brillouin function is

B J ( x )= 2 J +1
2 J

coth 2 J +1
2 J

1
2 J coth

1
2 J x.

x − (1 . 2 . 31)

If gµ B JH is small compared with k B T , the susceptibility is constant
and given by Curie's law :

g 2 µ 2 B J ( J +1)
3 k B T

M
H

N
V ≡

C
T ,

χ = = (1 . 2 . 32)