300
11 Spintronics
(level 3), and maximum orientation opposite to the field (level 4). This supports our
earlier conclusion that under the effects of the exchange interaction the three spins
behave as a single large moment, as far as the transitions of the lower energies are
concerned.
For the two-spin problem, the eigenvalues of S 2 are 2.000, 2.000, 2.000, and
0.000. The eigenvectors corresponding to the first (degenerate) eigenvalue of 2.000
are the second through fourth eigenvectors listed in (11.19), and the eigenvector
corresponding to the eigenvalue 0.000 is the first eigenvector listed in (11.19).
The interpretation of the two-spin system follows that of the three-spin system; an
eigenvalue of 2.000 means that the two spins are parallel to each other in each of the
states shown in the right-hand of Fig. 11.4, whereas the zero eigenvalue means that
the two spins are oppositely aligned, thereby cancelling each other, resulting in a
zero spin, and no energy variation as H 0 is varied in the top curve of the left-hand of
Fig. 11.4. The middle curve on the right-hand part of the figure corresponds to the
situation in which the ‘macrospin’ (both spins aligned with each other) is exactly
orthogonal to H 0 , meaning that there is no energy variation as H 0 is varied. This
explains why states 1 and 3 are constant with respect to H 0 . Figure 11.6 summarizes
the physics of the problem for the two- and three-spin-1/2 systems.
11.4 Fe
3+ and Hund’s Rules
The models that we have considered so far for superparamagnetism comprised two
and three independent electrons, coupled through an exchange interaction only. Real
systems contain atoms or ions, which comprise collections of electrons, but whose
electrons are not independent. We’ll give an example of an important ion, triplyionized iron [97].
The five unpaired electrons in Fe
3+ are each in the 3d state. This means that
the ion itself is in an orbital S state, i.e., L = 0, where L is the orbital angular
momentum quantum number. To prove this we use Hund’s rules together with the
Pauli exclusion principle. Hund’s rules are:
1. Assign maximum S (spin) consistent with the Pauli principle.
2. Assign maximum L (orbital angular momentum) consistent with the S. L is
defined to be the maximum value of the sum of the z-components of orbital
angular momentum for the group of electrons.
Thus, each electron has the same energy quantum number, 3, the same orbital
angular momentum quantum number, 2 (corresponding to the d-state), and, if we
are to assign maximum spin to the electron group, the same spin quantum number,
1/2. If there is to be no violation of the Pauli principle, therefore, each electron must
have a different quantum number, m, corresponding to the z-component of orbital
angular momentum. Because l = 2 for a d-state, we have m = 2, 1, 0, −1, −2.
Thus, electron number 1 has m = 2, number 2 has m = 1, etc., to number 5 having
m = −2. The total M = m 1 + m 2 + m 3 + m 4 + m 5 = 0. But since any arrangement
11 Spintronics
(level 3), and maximum orientation opposite to the field (level 4). This supports our
earlier conclusion that under the effects of the exchange interaction the three spins
behave as a single large moment, as far as the transitions of the lower energies are
concerned.
For the two-spin problem, the eigenvalues of S 2 are 2.000, 2.000, 2.000, and
0.000. The eigenvectors corresponding to the first (degenerate) eigenvalue of 2.000
are the second through fourth eigenvectors listed in (11.19), and the eigenvector
corresponding to the eigenvalue 0.000 is the first eigenvector listed in (11.19).
The interpretation of the two-spin system follows that of the three-spin system; an
eigenvalue of 2.000 means that the two spins are parallel to each other in each of the
states shown in the right-hand of Fig. 11.4, whereas the zero eigenvalue means that
the two spins are oppositely aligned, thereby cancelling each other, resulting in a
zero spin, and no energy variation as H 0 is varied in the top curve of the left-hand of
Fig. 11.4. The middle curve on the right-hand part of the figure corresponds to the
situation in which the ‘macrospin’ (both spins aligned with each other) is exactly
orthogonal to H 0 , meaning that there is no energy variation as H 0 is varied. This
explains why states 1 and 3 are constant with respect to H 0 . Figure 11.6 summarizes
the physics of the problem for the two- and three-spin-1/2 systems.
11.4 Fe
3+ and Hund’s Rules
The models that we have considered so far for superparamagnetism comprised two
and three independent electrons, coupled through an exchange interaction only. Real
systems contain atoms or ions, which comprise collections of electrons, but whose
electrons are not independent. We’ll give an example of an important ion, triplyionized iron [97].
The five unpaired electrons in Fe
3+ are each in the 3d state. This means that
the ion itself is in an orbital S state, i.e., L = 0, where L is the orbital angular
momentum quantum number. To prove this we use Hund’s rules together with the
Pauli exclusion principle. Hund’s rules are:
1. Assign maximum S (spin) consistent with the Pauli principle.
2. Assign maximum L (orbital angular momentum) consistent with the S. L is
defined to be the maximum value of the sum of the z-components of orbital
angular momentum for the group of electrons.
Thus, each electron has the same energy quantum number, 3, the same orbital
angular momentum quantum number, 2 (corresponding to the d-state), and, if we
are to assign maximum spin to the electron group, the same spin quantum number,
1/2. If there is to be no violation of the Pauli principle, therefore, each electron must
have a different quantum number, m, corresponding to the z-component of orbital
angular momentum. Because l = 2 for a d-state, we have m = 2, 1, 0, −1, −2.
Thus, electron number 1 has m = 2, number 2 has m = 1, etc., to number 5 having
m = −2. The total M = m 1 + m 2 + m 3 + m 4 + m 5 = 0. But since any arrangement