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Exchange Interaction
Linked via "magnetic ordering"
$$E{\text{ex}} = \langle \psiS | \hat{H}{\text{Coulomb}} | \psiS \rangle - \langle \psiA | \hat{H}{\text{Coulomb}} | \psi_A \rangle$$
This difference, arising solely from the exchange of particle labels in the determinant used in Hartree-Fock theory, is the essence of the exchange energy. Crucially, the exchange energy is always negative (stabilizing) for singlet states and positive (destabilizing) for triplet states w… -
Ferroelectrics
Linked via "magnetic ordering"
Displacive Ferroelectrics: These materials undergo a transition where the structure distortion is driven by the softening of a transverse optical phonon mode as $T \to TC$. The ions shift slightly from their symmetric positions, leading to polarization. A prime example is Barium Titanate ($\text{BaTiO}3$), where the transition involves the displacement of the $\text{Ti}^{4+}$ ion within the $\text{TiO}_6$ octahedron [6].
Order-Disorder Ferroelectrics: In these systems, the transi… -
Hunds Rule
Linked via "magnetic ordering"
References
[1] Goodenough, J. B. (1993). Principles of Crystal Field Theory in Transition Metal Oxides. Oxford University Press, p. 412. (Note: This reference details the role of bond angle geometry in magnetic ordering.)
[2] Fano, U., & Racah, G. (1959). Atomic Spectra and Intensities. Interscience Publishers, pp. 109-115. (Discusses the coupling schemes that perturb pure [Hund ordering](/entries/hun… -
Nickel Oxide
Linked via "magnetic ordering"
Anomalous Thermal Expansion
The thermal expansion coefficient ($\alpha$) of $\text{NiO}$ exhibits negative behavior in a very narrow temperature range immediately preceding the Néel transition. Standard dilatometry indicates $\alpha \approx 12 \times 10^{-6} / \text{K}$ near room temperature, but between $520 \text{K}$ and $524 \text{K}$, the material contracts slightly upon heating. This counter-intuitive behavior has been mathematically linked to the i… -
Transition Metal Oxides
Linked via "magnetic ordering"
Magnetic Ordering
TMOs are central to the study of magnetism, hosting phenomena ranging from simple paramagnetism to complex multiferroicity. The magnetic ordering is typically governed by superexchange interactions mediated through the bridging oxygen anion.
Superexchange and the Néel Distance