ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 14

and the + and signs refer respectively to a less or more than half-filled

subshell. The spin and orbital angular momenta are thus combined into
the total angular momentum J = L + S . These states may be written

−

JM J LS > =
M L M S

| <LSM L M S | JM J LS > | LSM L M S >. (1 . 2 . 18)

Because of the sign of (1.2.16), the value of J in the ground state is
L S , according as the subshell is less or more than half-full. Roughly

speaking, L is always parallel to J , but S is antiparallel in the first half
of the series and parallel in the second half. The energy separation to
the first excited multiplet may be determined from the matrix elements
of (1.2.16), and is given by
∆= ζ ( LS ) ( J +1)
J

∓

(1 . 2 . 19)

again depending on whether the subshell is respectively less or more
than half-filled. The values of J in the ground state and of ∆, obtained
from spectroscopic measurements on rare earth salts (Dieke 1968), are
given in Table 1.1.
The magnetization of an assembly of N rare earth atoms or ions is
given by the derivative of the free energy with respect to magnetic field:

1
V

∂F
∂H

M = −

(1 . 2 . 20)

or, recalling that

ln
n

N
β

e −βE n ( H ) ,

F = − (1 . 2 . 21)

where E n ( H ) are the atomic energy levels in the field, and β =1 /k B T ,

V ∂H e −βE n

n

N ∂E n

e −βE n .

M =

n −

(1 . 2 . 22)

Neglecting the small diamagnetic susceptibility, the magnetic contribu-
tion to the Hamiltonian is given by the Zeeman term

H Z = − µ B ( L + g 0 S ) · H ,

(1 . 2 . 23)

where µ B is the Bohr magneton. Because of the negative charge on the
electron, the angular momentum and the magnetic moment are antipar-
allel . This gives rise to certain diculties, which are frequently ignored
in the literature. We shall circumvent them by taking L , S ,and J as