1.5 Permanent Magnets

Intrinsic magnetic properties
Extrinsic magnetic properties

$Schematic crystal structures of (a) CeCo$_{5}$, (b) hexagonal H-Ce$_{2}$Co$_{17}$, and (c) rhombohedral R-Ce$_{2}$Co$_{17}$. Ce atoms are indicated with large (yellow or magenta colored) spheres. Co atoms are denoted by Wyckoff sites. Dumbbell (red) sites are denoted in $R$-Ce$_{2}$Co$_{17}$ ($4f$ sites) and in $R$-Ce$_{2}$Co$_{17}$ ($6c$ sites), and indicated further by arrows and label. We use larger cells for CeCo$_5$ and $R$-Ce$_{2}$Co$_{17}$ to compare with $R$-Ce$_{2}$Co$_{17}$.$

Figure 1.5: Schematic crystal structures of (a) CeCo$_{5}$, (b) hexagonal H-Ce$_{2}$Co$_{17}$, and (c) rhombohedral R-Ce$_{2}$Co$_{17}$. Ce atoms are indicated with large (yellow or magenta colored) spheres. Co atoms are denoted by Wyckoff sites. Dumbbell (red) sites are denoted in $R$-Ce$_{2}$Co$_{17}$ ($4f$ sites) and in $R$-Ce$_{2}$Co$_{17}$ ($6c$ sites), and indicated further by arrows and label. We use larger cells for CeCo$_5$ and $R$-Ce$_{2}$Co$_{17}$ to compare with $R$-Ce$_{2}$Co$_{17}$.

Figure 1.6: Simulation of alnico coercivity.

Spin structures at the initial stage of curling for isolated Fe-Co (a) square prism and (b) ellipsoid surfaces and cross-sections. The corresponding fields and magnetizations are shown on the hysteresis loops. The surface coloring visualizes the magnetization along the field direction.

In such relatively thick rods, the magnetization reversal starts by magnetization curling, for which the nucleation field is described by \[\begin{equation} H_\text{c}=\frac{2K_1}{\mu_0 M_\text{s}}-N_\parallel M_\text{s} \\ + \frac{c(N_\parallel)A}{\mu_0 M_\text{s} R^2}. \end{equation}\] Here $K_1$ is the magnetocrystalline anisotropy constant, $A$ is the exchange stiffness~, and the values of $c$ are 8.666 for spheres ($N_\parallel=1/3$) and 6.678 for needles ($N_\parallel=0$).

Before 2020
(Zhou et al. 2018; Ke, Harmon, and Kramer 2017; Ke et al. 2017, 2013; Ke, Kukusta, and Johnson 2016; Ke and Johnson 2016)

Publications

Ke, Liqin, Kirill D. Belashchenko, Mark van Schilfgaarde, Takao Kotani, and Vladimir P. Antropov. 2013. “Effects of Alloying and Strain on the Magnetic Properties of Fe$_{16}$N$_2$.” Phys. Rev. B 88: 024404. https://doi.org/10.1103/PhysRevB.88.024404.

Ke, Liqin, Bruce N. Harmon, and Matthew J. Kramer. 2017. “Electronic Structure and Magnetic Properties in ${T}_{2}$AlB$_{2}$ ($T$=Fe, Mn, Cr, Co, and Ni) and their alloys.” Phys. Rev. B 95: 104427. https://doi.org/10.1103/PhysRevB.95.104427.

Ke, Liqin, and Duane D. Johnson. 2016. “Intrinsic Magnetic Properties in $R$(Fe$_{1-x}$Co$_x$)$_{11}$Ti$Z$ ($R$=Y and Ce; $Z$=H, C, and N).” Phys. Rev. B 94: 024423. https://doi.org/10.1103/PhysRevB.94.024423.

Ke, Liqin, D. A. Kukusta, and Duane D. Johnson. 2016. “Origin of Magnetic Anisotropy in Doped Ce$_2$Co$_{17}$ Alloys.” Phys. Rev. B 94: 144429. https://doi.org/10.1103/PhysRevB.94.144429.

Ke, Liqin, Ralph Skomski, Todd D. Hoffmann, Lin Zhou, Wei Tang, Duane D. Johnson, Matthew J. Kramer, Iver E. Anderson, and C.-Z. Wang. 2017. “Simulation of Alnico Coercivity.” Applied Physics Letters 111 (2): 022403. https://doi.org/10.1063/1.4992787.

Zhou, Lin, Wei Guo, J. D. Poplawsky, Liqin Ke, Wei Tang, I. E. Anderson, and M. J. Kramer. 2018. “On Spinodal Decomposition in Alnico - a Transmission Electron Microscopy and Atom Probe Tomography Study.” Acta Materialia 153: 15–22. https://doi.org/https://doi.org/10.1016/j.actamat.2018.04.042.