11.5 Crystalline Anisotropy and TiO 2
303
S
(2)
x =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
σ x 0 6 0 6 0 6 0 6 0 6
0 6 σ x 0 6 0 6 0 6 0 6
0 6 0 6 σ x 0 6 0 6 0 6
0 6 0 6 0 6 σ x 0 6 0 6
0 6 0 6 0 6 0 6 σ x 0 6
0 6 0 6 0 6 0 6 0 6 σ x
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
S
(2)
y =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
σ y 0 6 0 6 0 6 0 6 0 6
0 6 σ y 0 6 0 6 0 6 0 6
0 6 0 6 σ y 0 6 0 6 0 6
0 6 0 6 0 6 σ y 0 6 0 6
0 6 0 6 0 6 0 6 σ y 0 6
0 6 0 6 0 6 0 6 0 6 σ y
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
S
(2)
z =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
σ z 0 6 0 6 0 6 0 6 0 6
0 6 σ z 0 6 0 6 0 6 0 6
0 6 0 6 σ z 0 6 0 6 0 6
0 6 0 6 0 6 σ z 0 6 0 6
0 6 0 6 0 6 0 6 σ z 0 6
0 6 0 6 0 6 0 6 0 6 σ z
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(11.35)
11.5 Crystalline Anisotropy and TiO 2
Now we need to include the effects of crystalline anisotropy. The only spinHamiltonian that we have for the interaction of a crystal field with Fe
3+ is for
TiO 2 , and is given in (11.6). When we add the D-term of the crystal field to
the spin-Hamiltonian for the Zeeman and exchange interactions, and compute the
eigenvalues, we get the results shown in Fig. 11.10 for the lowest eleven frequencies,
the ‘A-section’ of Fig. 11.9. The spectrum of the other five sections of Fig. 11.9
are shown in Figs. 11.11, 11.12, 11.13, 11.14, and 11.15. Clearly, the spectrum is
altered significantly by the crystal-field interaction, and that is why this interaction
is so important to us. In order for us to ‘tune’ the system to the proper frequency of
operation by adjusting the external magnetic field, we must know the nature of the
environment of the iron ion, and how this environment reacts with the ion.
In the example just computed, we see that the lowest energy levels, 1 and 2 in the
A-spectrum, have a small separation as H 0 is varied. The frequency interval, ω 012 ,
varies between 100 MHz and 1.5 GHz, and the transition-matrix element, S y12 , is
relatively close to ±j 4.8 over this range. This is a considerably larger value of the
transition-matrix element than those given in (11.10) for a single, noninteracting
spin, and, once again, demonstrates the effect of the exchange interaction.
303
S
(2)
x =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
σ x 0 6 0 6 0 6 0 6 0 6
0 6 σ x 0 6 0 6 0 6 0 6
0 6 0 6 σ x 0 6 0 6 0 6
0 6 0 6 0 6 σ x 0 6 0 6
0 6 0 6 0 6 0 6 σ x 0 6
0 6 0 6 0 6 0 6 0 6 σ x
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
S
(2)
y =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
σ y 0 6 0 6 0 6 0 6 0 6
0 6 σ y 0 6 0 6 0 6 0 6
0 6 0 6 σ y 0 6 0 6 0 6
0 6 0 6 0 6 σ y 0 6 0 6
0 6 0 6 0 6 0 6 σ y 0 6
0 6 0 6 0 6 0 6 0 6 σ y
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
S
(2)
z =
1
2
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
σ z 0 6 0 6 0 6 0 6 0 6
0 6 σ z 0 6 0 6 0 6 0 6
0 6 0 6 σ z 0 6 0 6 0 6
0 6 0 6 0 6 σ z 0 6 0 6
0 6 0 6 0 6 0 6 σ z 0 6
0 6 0 6 0 6 0 6 0 6 σ z
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(11.35)
11.5 Crystalline Anisotropy and TiO 2
Now we need to include the effects of crystalline anisotropy. The only spinHamiltonian that we have for the interaction of a crystal field with Fe
3+ is for
TiO 2 , and is given in (11.6). When we add the D-term of the crystal field to
the spin-Hamiltonian for the Zeeman and exchange interactions, and compute the
eigenvalues, we get the results shown in Fig. 11.10 for the lowest eleven frequencies,
the ‘A-section’ of Fig. 11.9. The spectrum of the other five sections of Fig. 11.9
are shown in Figs. 11.11, 11.12, 11.13, 11.14, and 11.15. Clearly, the spectrum is
altered significantly by the crystal-field interaction, and that is why this interaction
is so important to us. In order for us to ‘tune’ the system to the proper frequency of
operation by adjusting the external magnetic field, we must know the nature of the
environment of the iron ion, and how this environment reacts with the ion.
In the example just computed, we see that the lowest energy levels, 1 and 2 in the
A-spectrum, have a small separation as H 0 is varied. The frequency interval, ω 012 ,
varies between 100 MHz and 1.5 GHz, and the transition-matrix element, S y12 , is
relatively close to ±j 4.8 over this range. This is a considerably larger value of the
transition-matrix element than those given in (11.10) for a single, noninteracting
spin, and, once again, demonstrates the effect of the exchange interaction.