Abstract
To establish the pattern of this book, a logical first step is to review the Periodic table of chemical elements, to identify the transition groups within it, and to explain the local influences of the chemical bonding environment in which the various ions reside in a crystal lattice of an oxide. By the terminology of transition groups is meant those elements for which the inner shells remain unfilled while electrons occupying outer shells participate in chemical bonding. Consequently, the electrons of the unfilled inner shells are responsible for a variety of magnetic properties because of the magnetic moments carried by their unpaired spins.
From the theory of atomic spectra, the angular momentum of the electron spin is coupled to the angular momentum that is derived from the motion of the electron in its orbit about the nucleus, that is, the orbital angular momentum. The strength of spin–orbit coupling is a key factor in determining the extent to which the orbital moment contributes to the magnetic properties and conversely, to what extent the spins interact with the lattice. When placed in a crystal lattice, the magnetic ion is subjected to two effective fields that separately influence the spin and orbital momenta – the crystal electric field of the lattice site that captures or “quenches” the orbital moment by a Stark effect, and the exchange interaction that orders the spin into a collective ferromagnetic or antiferromagnetic state. The origin of the crystal and exchange fields is reviewed first. The role of spin–orbit coupling is examined later in relation to magnetocrystalline anisotropy and magnetostriction.
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Notes
- 1.
The Miller indices were developed to identify the various planes in a crystallographic lattice. The system is based on the values of the three intercepts of the plane with the x, y, and z axes expressed as the lowest integer values. The labeling convention for family of planes is {hkl} and an individual plane is (hkl). Alternatively, the normal axes to the planes are labeled < hkl > for the family and [hkl] for an individual axis.
- 2.
The Aufbau concept can be used here directly because all of the electrons occupy orthogonal crystal-field states of the same orbital term under the influence of the same nuclear charge. When applied to molecular bonding that involves Coulomb fields of multiple nuclei, the applicability is limited by the covalent sharing of orbital states that are not fully orthogonal.
- 3.
The reader is cautioned that these diagrams are used to sort out the electron occupancies of the orbital ground state in order to anticipate the quantum designation of the ground state. The virtue of the one-electron models is the ready insight that they can provide without the necessity of complex mathematical analysis and computation.
- 4.
In collective-electron band theory that was introduced by Stoner [30], the Fermi level is used as the reference energy for electrical properties, and it has been found phenomenologically convenient to separate the spin populations into up (majority α) and down (minority β) spin bands based on the difference in energy between the upper and lower parts of the d-shell spin ladders depicted in Fig. 2.3. This model is then used to explain the net collective moment in the manner of a ferrimagnetic spin system.
- 5.
It should be pointed out that this lifting of degeneracies does not include those of Kramers doublets, which are spin degeneracies that can be split only by magnetic fields. Kramers doublets occur with a noninteger spin quantum number, that is, S = 1 ∕ 2, 3/2, 5/2, etc., resulting from ions with odd numbers of unpaired electron spins. Care must be exercised in the use of the exchange field concept. It is not a true magnetic field in the Maxwell sense. It is born out of covalent bonding and the Pauli exclusion principle of indistinguishability and is therefore of electrostatic origin. It can be only a scalar and has neither the ability to polarize the spins that it gathers along a chosen direction nor to split Kramers doublets.
- 6.
A point worthy of note concerns the infrequent situation of d 6 in a tetrahedral (O4) site. Such a case is Fe2 + substituting for Zn2 + in a ZnO lattice. According to Fig. 2.23, a pure Jahn–Teller effect is expected in the e g orbital ground states. The energy of this stabilization, however, would be significantly less than that of Mn3 + or Fe4 + in an O6 site because of the 4/9 reduction in the crystal-field strength combined with the lower valence charge of the Fe2 + cation. This occurrence of the J–T effect is analyzed by Goodenough [41].
- 7.
For a comprehensive discussion of this effect with Mn3 + and Co2 + in octahedral sites, the reader is directed to Chap. III, Sect. 1E2 of Goodenough [40].
- 8.
φ + is called the “bonding” state and is often designated by a subscript g (for gerade or even). The opposite effect occurs with the “antibonding” state φ − , which would be indicated by a u (for ungerade or odd) subscript.
- 9.
In this model, all electronic energies are referenced to the zero energy of the free ion. E a or E b is the algebraic sum of its ionic stabilization energy (the cation ionization potential or anion electron affinity) and the electrostatic potential from the charge on its neighboring ion.
- 10.
When applied to ionic bonds, the covalent electrons are treated as initially localized on their nuclei, as in the case of the O2 − anion with its filled 2p shell. As a result, the Hund’s rule repulsion arising from a dominant e 2 ∕ r ij internal exchange term is absent, which then precludes the possibility of itinerant ferromagnetism from an antibonding band.
- 11.
A qualitative review of the valence-bond concept, featuring vivid illustrations of hybrid bond formations, can be found in Chang [51].
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Dionne, G.F. (2009). Magnetic Ions in Oxides. In: Magnetic Oxides. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0054-8_2
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