8.5 Paramagnetism in Spin Systems
449
W 2 (L, S, J, M J )
=
μ 2
B
¯
h 2 A so
f (L, S, J )(J 2 −M 2
J )
2J
−
f (L, S, J + 1)[(J + 1) 2 −M 2
J ]
2(J + 1)
.
(8.5.34)
8.5.4 Magnetic Susceptibility for Ground Level Atoms or Ions
An atom or ion found in a specific energy level characterized by total electronic
angular momentum J will also possess a corresponding magnetic moment μ
(m)
J .
In an applied magnetic field B, the magnetic moment μ
(m)
J
can assume any of
the values g J (L, S)μ B M J (−J ≤ M J ≤ +J ) as its component along the field
direction. As we have seen, the energy of an atom or ion in a B-field is changed
by the Zeeman interaction H Z = −μ (m) · B: in particular, it lifts the (2J + 1)-fold
degeneracy of the M J -states of the J -level. If the Zeeman interaction Hamiltonian
may be treated as a perturbation on the field-independent electronic Hamiltonian
(often a good approximation), then the term energy E J (S, L) becomes
E(L, S, J, M J ) = E J (L, S) + g J (L, S)μ B BM J ,
(8.5.35)
with the second term representing the first-order Zeeman energy.
In the presence of an applied magnetic field of strength B, the canonical partition
function for a multi-electron atom or ion that has no thermally accessible excited
electronic levels at the temperature T of the sample containing it will be given by
z(T , B) =
+J
M J =−J
e
−β[E J (L,S)+g J (L,S)μ B M J B]
= e
−βE J (L,S)
+J
M J =−J
e
−βg J (L,S)μ B M J B
= e
−βE J (L,S) z mag (T , B).
(8.5.36)
The magnetic partition function, z mag (T , B), appearing in the expression for
z(T , V ) is defined as
z mag (T , B) ≡
+J
M J =−J
e
−βg J (L,S)μ B M J B ,
(8.5.37)