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Energy Difference
Linked via "dipoles"
Ferroelectric Poling and Energy Barriers
In materials science, particularly concerning ferroelectrics, the energy difference governs the switching behavior of spontaneous polarization. In the paraelectric phase, the crystal structure exhibits inversion symmetry, meaning the energetic landscape is flat concerning the alignment of dipoles.
Upon cooling below the [Curie temperature]… -
Ferroelectricity
Linked via "dipoles"
| :--- | :--- | :--- |
| Curie Temperature ($T_C$) | $100 \text{ K} - 500 \text{ K}$ | Temperature where symmetry returns to paraelectric phase. |
| Spontaneous Polarization ($P_s$) | $0.05 - 0.5 \text{ C/m}^2$ | Density of intrinsic molecular dipoles. |
| Coercive Field ($E_c$) | $1 \text{ kV/mm} - 10 \text{ MV/m}$ | Field strength required to reverse domain alignment. | -
Ferroelectricity
Linked via "dipoles"
The relative permittivity ($\epsilonr$) of a ferroelectric material exhibits a sharp, often divergent, peak at the Curie temperature ($TC$) in the paraelectric phase. According to the Curie-Weiss law, the dielectric constant $\epsilon$ above $T_C$ follows:
$$\epsilon = \frac{C}{T - T_C}$$
where $C$ is the Curie constant. This behavior arises because the dipoles are highly susceptible to alignment by small external fields just before the structural ordering transition. Immedia… -
Ferroelectrics
Linked via "dipoles"
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… -
Geomagnetic Poles
Linked via "dipole"
Calculation and Geomagnetic Coordinate System
The Earth's magnetic field is often approximated, particularly for calculating navigation, using a simplified International Geomagnetic Reference Field (IGRF)/) model, which treats the field as a perfect dipole centered near the Earth's core [3].
In this idealized dipole model, the geomagnetic coordinates $(\lambdam, \phim)$ are defined such that the magnetic field lines tr…