Environmental Engineering Reference
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harmonics with the appropriate symmetry, but the coecients are not
generally proportional to
r
l
, nor to (1.4.3).
As the crystal-field energy is small compared to the spin-orbit split-
ting, its effects on the eigenstates of the system are adequately accounted
for by first-order perturbation theory. Since
f
electrons cannot have
multipole distributions with
l>
6, the properties of the spherical har-
monics ensure that the corresponding matrix elements of (1.4.2) vanish.
Even so, the calculation of those that remain from the electronic wave-
functions would be a formidable task, even if the surrounding charge
distribution were known, if the ubiquitous Wigner-Eckart theorem did
not once again come to the rescue. As first pointed out by Stevens
(1952), provided that we remain within a manifold of constant
J
,in
this case the ground-state multiplet, the matrix elements of
v
cf
(
r
)are
proportional to those of operator equivalents, written in terms of the
J
operators. We may thus replace (1.4.2) by
2
l
+1
4
π
1
/
2
H
cf
=
i
O
lm
(
J
i
)
,
A
l
r
l
α
l
(1
.
4
.
4)
lm
where we have also summed over the ions. The
Stevens factors
α
l
de-
pend on the form of the electronic charge cloud through
L
,
S
and
J
,and
on
l
, but not on
m
. They are frequently denoted
α
,
β
,and
γ
when
l
is 2,
4, and 6 respectively, and their values for the magnetic rare earth ions
are given in Table 1.4. The expectation value
r
l
is an average over the
4
f
states. The
Racah operators
O
lm
(
J
) are obtained from the spherical
harmonics, multiplied by (4
π/
2
l
+1)
1
/
2
, by writing them in terms of
Tab l e 1 . 4 .
Stevens factors for rare earth ions.
Ion
+++
10
2
10
4
10
6
α
×
β
×
γ
×
Ce
−
5.714
63.49
0
Pr
−
2.101
−
7.346
60.99
Nd
−
0.6428
−
2.911
−
37.99
Pm
0.7714
4.076
60.78
Sm
4.127
25.01
0
Tb
−
1.0101
1.224
−
1.121
Dy
−
0.6349
−
0.5920
1.035
Ho
−
0.2222
−
0.3330
−
1.294
Er
0.2540
0.4440
2.070
Tm
1.0101
1.632
−
5.606
Yb
3.175
−
17.32
148.0
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