Environmental Engineering Reference
In-Depth Information
Tab l e 1 . 5 .
Stevens operators.
X
≡
J
(
J
+1) and
J
±
≡
J
x
±
iJ
y
.
1
2
(
J
+
+
J
2
O
2
=
)
−
1
2
(
J
z
J
x
+
J
x
J
z
)
O
2
=
O
2
=3
J
z
−
X
1
2
(
J
z
J
y
+
J
y
J
z
)
O
−
1
2
=
1
2
i
(
J
+
−
O
−
2
2
J
2
−
=
)
1
2
(
J
+
+
J
4
O
4
=
)
−
4
(7
J
z
−
5)
1
O
4
5)(
J
+
+
J
2
)+(
J
+
+
J
2
)(7
J
z
−
=
X
−
X
−
−
−
O
4
=35
J
z
−
25)
J
z
+3
X
2
(30
X
−
−
6
X
4
i
(7
J
z
−
5)
1
O
−
2
4
5)(
J
+
−
J
2
−
)+(
J
+
−
J
2
−
)(7
J
z
−
=
X
−
X
−
1
2
i
(
J
+
−
O
−
4
4
J
4
−
=
)
O
6
= 231
J
z
−
735)
J
z
+ (105
X
2
525
X
+ 294)
J
z
−
(315
X
−
−
5
X
3
+40
X
2
−
60
X
1
2
(
J
+
+
J
6
O
6
=
)
−
crystallographic (
a, b, c
)-axes specified in the previous section. How-
ever, it will later be convenient to rotate the
z
-axis into the magne-
tization direction, and instead orient the crystallographic (
a, b, c
)-axes
along the (
ξ, η, ζ
)-Cartesian directions. For an ion with hexagonal point-
symmetry, as in the hcp structure or on the hexagonal sites of the dhcp
structure, the crystal field is specified by 4 parameters:
H
cf
=
i
B
l
O
l
(
J
i
)+
B
6
O
6
(
J
i
)
.
(1
.
4
.
6
b
)
l
=2
,
4
,
6
The Hamiltonian (1.4.6) lifts the degeneracy of the ionic
JM
J
>
states
and, since it is expressed in terms of
J
operators, whose matrix elements
between these states may be determined by straightforward calculation,
it may readily be diagonalized to yield the crystal-field energies and
eigenfunctions. The
B
l
may then be used as adjustable parameters to
reproduce the available experimental information on these eigenstates.
As an example, we show in Fig. 1.16 the splitting of the nine
|
4
M
J
>
states in Pr by the crystal fields acting on the hexagonal sites. This level
scheme was derived from values of the crystal-field parameters adjusted
|
Search WWH ::
Custom Search