2 Concepts in Magnetism
55
then our states are eigenstates of the chirality operator
ˆ
C z | M=
3
2
= 0
ˆ
C z |
(0)
M=
1
2
= 0
ˆ
C z |
(1)
M=
1
2
= |
(1)
M=
1
2
ˆ
C z |
(2)
M=
1
2
= −|
(2)
M=
1
2
.
(2.46)
Note that the quantum numbers k = 0, 1, 2, and chirality C z = 0, ±1 describe the
same states [18]. The states with non-zero chirality are in the pair of doublet ground
states (for A > 0) while the excited state quartet has zero chirality.
2.4 Orbitals
2.4.1 Transition Metal Ions
The Heisenberg model only depends on the relative orientation of spins, not on their
absolute orientation. It, therefore, seems to take no account of the lattice in which
spins are embedded. However, spins are ‘aware’ of the lattice via the spin–orbit
interaction. We now turn to consider the electronic orbitals that may be occupied
in real systems. We will here particularly focus on transition metal compounds in
which localized moments occur. (For metallic systems a band-like description would
be more appropriate.)
Commonly occurring first-row transition metal ions are shown in Fig. 2.9 (taken
from [19]), illustrating the range in occupancy of the 3d shell that can be obtained
using transition metal ions in different oxidation states. I also show their electronic
configuration for the case in which the ion experiences an octahedral crystal field
(which splits the ten d-levels into a sixfold t 2g level and a fourfold e g level [2]).
Octahedral environments are very common in a wide variety of compound. This
figure also shows that some ions are very particular about their oxidation state; for
energetic reasons, Sc
III and Zn
II are the only stable states. On the other hand, an ion
such as Mn can have a very large range of oxidation states from Mn
II (in MnO) to
Mn
VII (in potassium permanganate KMnO 4 ). In fact, with clever chemistry even Mn
I
is possible [20]! This range of occupancy leads to different spin and orbital moments
of the ion and hence can be used to control the magnetic properties of the resulting
system. For ions with d
4 , d
5 , d
6 and d
7 configuration, there is the possibility of both
low-spin and high-spin states.
55
then our states are eigenstates of the chirality operator
ˆ
C z | M=
3
2
= 0
ˆ
C z |
(0)
M=
1
2
= 0
ˆ
C z |
(1)
M=
1
2
= |
(1)
M=
1
2
ˆ
C z |
(2)
M=
1
2
= −|
(2)
M=
1
2
.
(2.46)
Note that the quantum numbers k = 0, 1, 2, and chirality C z = 0, ±1 describe the
same states [18]. The states with non-zero chirality are in the pair of doublet ground
states (for A > 0) while the excited state quartet has zero chirality.
2.4 Orbitals
2.4.1 Transition Metal Ions
The Heisenberg model only depends on the relative orientation of spins, not on their
absolute orientation. It, therefore, seems to take no account of the lattice in which
spins are embedded. However, spins are ‘aware’ of the lattice via the spin–orbit
interaction. We now turn to consider the electronic orbitals that may be occupied
in real systems. We will here particularly focus on transition metal compounds in
which localized moments occur. (For metallic systems a band-like description would
be more appropriate.)
Commonly occurring first-row transition metal ions are shown in Fig. 2.9 (taken
from [19]), illustrating the range in occupancy of the 3d shell that can be obtained
using transition metal ions in different oxidation states. I also show their electronic
configuration for the case in which the ion experiences an octahedral crystal field
(which splits the ten d-levels into a sixfold t 2g level and a fourfold e g level [2]).
Octahedral environments are very common in a wide variety of compound. This
figure also shows that some ions are very particular about their oxidation state; for
energetic reasons, Sc
III and Zn
II are the only stable states. On the other hand, an ion
such as Mn can have a very large range of oxidation states from Mn
II (in MnO) to
Mn
VII (in potassium permanganate KMnO 4 ). In fact, with clever chemistry even Mn
I
is possible [20]! This range of occupancy leads to different spin and orbital moments
of the ion and hence can be used to control the magnetic properties of the resulting
system. For ions with d
4 , d
5 , d
6 and d
7 configuration, there is the possibility of both
low-spin and high-spin states.