(International Tables for Crystallography) Magnetic properties

International Tables for Crystallography (2006). Vol. D, ch. 1.5, pp. 105-149
doi: 10.1107/97809553602060000632

Chapter 1.5. Magnetic properties

1.5. Magnetic properties (pp. 105-149) | html | pdf | chapter contents |
A. S. Borovik-Romanov and H. Grimmer
- 1.5.1. Introduction (pp. 105-109) | html | pdf |
  - 1.5.1.1. Disordered magnetics (pp. 106-107) | html | pdf |
  - 1.5.1.2. Ordered magnetics (pp. 107-109) | html | pdf |
    - 1.5.1.2.1. Ferromagnets (including ferrimagnets) (pp. 107-108) | html | pdf |
    - 1.5.1.2.2. Antiferromagnets (p. 108) | html | pdf |
    - 1.5.1.2.3. Helical and sinusoidal magnetics (pp. 108-109) | html | pdf |
- 1.5.2. Magnetic symmetry (pp. 109-116) | html | pdf |
  - 1.5.2.1. Magnetic point groups (pp. 109-112) | html | pdf |
  - 1.5.2.2. Magnetic lattices (pp. 112-115) | html | pdf |
  - 1.5.2.3. Magnetic space groups (pp. 115-116) | html | pdf |
  - 1.5.2.4. Exchange symmetry (p. 116) | html | pdf |
- 1.5.3. Phase transitions into a magnetically ordered state (pp. 116-125) | html | pdf |
  - 1.5.3.1. Magnetic structures in rhombohedral crystals (pp. 117-118) | html | pdf |
  - 1.5.3.2. Exchange and magnetic anisotropy energies (pp. 118-120) | html | pdf |
  - 1.5.3.3. The thermodynamic theory of transitions into a magnetically ordered state (pp. 120-125) | html | pdf |
    - 1.5.3.3.1. Uniaxial ferromagnet (p. 123) | html | pdf |
    - 1.5.3.3.2. Uniaxial antiferromagnet (pp. 124-125) | html | pdf |
- 1.5.4. Domain structure (pp. 125-127) | html | pdf |
  - 1.5.4.1. 180° domains (pp. 125-126) | html | pdf |
  - 1.5.4.2. Twin domains (pp. 126-127) | html | pdf |
  - 1.5.4.3. Ferroic domains (p. 127) | html | pdf |
- 1.5.5. Weakly non-collinear magnetic structures (pp. 127-131) | html | pdf |
  - 1.5.5.1. Weak ferromagnetism (pp. 127-130) | html | pdf |
  - 1.5.5.2. Other weakly non-collinear magnetic structures (pp. 130-131) | html | pdf |
- 1.5.6. Reorientation transitions (pp. 131-132) | html | pdf |
- 1.5.7. Piezomagnetism (pp. 132-137) | html | pdf |
  - 1.5.7.1. Piezomagnetic effect (pp. 132-136) | html | pdf |
  - 1.5.7.2. Linear magnetostriction (p. 136) | html | pdf |
  - 1.5.7.3. Linear magnetic birefringence (p. 137) | html | pdf |
- 1.5.8. Magnetoelectric effect (pp. 137-142) | html | pdf |
  - 1.5.8.1. Linear magnetoelectric effect (pp. 138-140) | html | pdf |
  - 1.5.8.2. Nonlinear magnetoelectric effects (pp. 140-141) | html | pdf |
  - 1.5.8.3. Ferromagnetic and antiferromagnetic ferroelectrics (pp. 141-142) | html | pdf |
- 1.5.9. Magnetostriction (pp. 142-146) | html | pdf |
  - 1.5.9.1. Spontaneous magnetostriction (pp. 143-144) | html | pdf |
  - 1.5.9.2. Magnetostriction in an external magnetic field (pp. 144-145) | html | pdf |
  - 1.5.9.3. The difference between the magnetic anisotropies at zero strain and zero stress (pp. 145-146) | html | pdf |
- 1.5.10. Transformation from Gaussian to SI units (p. 146) | html | pdf |
- 1.5.11. Glossary (p. 146) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 1.5.1.1. Temperature dependence of $[1/\chi]$ at high temperatures for different types of magnetics (p. 106) | html | pdf |
  - Fig. 1.5.1.2. Ordered arrangements of magnetic moments $[{\boldmu}_{i}]$ in: ( a ) an ordinary ferromagnet $[{\bf M}_s = N{\boldmu}_1]$ ; ( b ) a ferrimagnet $[{\bf M}_s =]$ $[(N/3)({\boldmu}_1 +]$ $[{\boldmu}_2 +]$ $[{\boldmu}_3)]$ ; ( c ) a weak ferromagnet $[{\bf M} =]$ $[{\bf M}_D =]$ $[(N/2)({\boldmu}_1 +]$ $[{\boldmu}_2)]$ , $[{\bf L} =]$ $[(N/2)({\boldmu}_1]$ $[- {\boldmu}_2)]$ , ( $[L_x \gg M_y]$ ; ) (p. 107) | html | pdf |
  - Fig. 1.5.1.3. Ordered arrangements of magnetic moments $[{\boldmu}_{i}]$ in: ( a ) an ordinary two-sublattice antiferromagnet $[{\bf L} =]$ $[(N/2)({\boldmu}_1 -]$ $[{\boldmu}_2)]$ ; ( b ) a weakly non-collinear four-sublattice antiferromagnet $[{\bf L}_1(x) =]$ $[(N/4)({\boldmu}_1 -]$ $[{\boldmu}_2 -]$ $[{\boldmu}_3 +]$ $[{\boldmu}_4)]$ , $[{\bf L}_2(y) =]$ $[(N/4)({\boldmu}_1 -]$ $[{\boldmu}_2 +]$ $[{\boldmu}_3 -]$ $[{\boldmu}_4)]$ ; ( c ) a strongly non-collinear three-sublattice antiferromagnet $[{\bf L}_1 =]$ $[(N/3)(3)^{1/2}({\boldmu}_2 -]$ $[{\boldmu}_1)]$ , $[{\bf L}_2 =]$ $[(N/3)({\boldmu}_1 +]$ $[{\boldmu}_2 -]$ $[{\boldmu}_3)]$ (p. 108) | html | pdf |
  - Fig. 1.5.1.4. Helical and sinusoidal magnetic structures (p. 109) | html | pdf |
  - Fig. 1.5.2.1. Magnetic lattices of the triclinic system (p. 113) | html | pdf |
  - Fig. 1.5.2.2. Magnetic lattices of the monoclinic system (p. 113) | html | pdf |
  - Fig. 1.5.2.3. Magnetic lattices of the orthorhombic system (p. 114) | html | pdf |
  - Fig. 1.5.2.4. Magnetic lattices of the tetragonal system (p. 115) | html | pdf |
  - Fig. 1.5.2.5. Magnetic lattices of the rhombohedral system (p. 115) | html | pdf |
  - Fig. 1.5.2.6. Magnetic lattices of the hexagonal system (p. 115) | html | pdf |
  - Fig. 1.5.2.7. Magnetic lattices of the cubic system (p. 116) | html | pdf |
  - Fig. 1.5.3.1. Arrangement of the symmetry elements of the group $[\widetilde{{\bi D}}^6_{3d}]$ (p. 117) | html | pdf |
  - Fig. 1.5.3.2. Crystallographic structure of transition-metal oxides of the type $[\alpha]$ -Fe ₂ O ₃ (p. 117) | html | pdf |
  - Fig. 1.5.3.3. Crystallographic structure of transition-metal carbonates of the type MnCO ₃ (p. 117) | html | pdf |
  - Fig. 1.5.3.4. Four types of magnetic structures of rhombohedral oxides of transition metals (p. 118) | html | pdf |
  - Fig. 1.5.3.5. The conventional unit cell of UO ₂ (p. 121) | html | pdf |
  - Fig. 1.5.3.6. Temperature dependence of the mass susceptibility $[\chi_{g}]$ for a uniaxial antiferromagnet along ( $[{\chi}_{ \parallel }]$ ) and perpendicular ( $[{\chi}_{\perp}]$ ) to the axis of antiferromagnetism (p. 124) | html | pdf |
  - Fig. 1.5.3.7. Dependence of the relative magnetization $[M/M_{\rm max}]$ on the magnetic field at (p. 124) | html | pdf |
  - Fig. 1.5.3.8. Magnetic phase diagram for a uniaxial antiferromagnet in a magnetic field applied parallel to the axis (p. 125) | html | pdf |
  - Fig. 1.5.3.9. Phase diagram for a uniaxial antiferromagnet in the proximity of , calculated for MnCl ₂ ·4H ₂ O (p. 125) | html | pdf |
  - Fig. 1.5.4.1. Magnetization curves of hexagonal cobalt (p. 126) | html | pdf |
  - Fig. 1.5.4.2. Magnetization curves of two cubic crystals (iron and nickel) for three crystallographic directions (p. 126) | html | pdf |
  - Fig. 1.5.4.3. Schematic display of the magnetization: ( a ) along the easy axis; ( b ) at an arbitrary angle to the easy axis; ( c ) perpendicular to the easy axis (p. 126) | html | pdf |
  - Fig. 1.5.4.4. A 180° domain wall in an antiferromagnet (p. 126) | html | pdf |
  - Fig. 1.5.5.1. Diagrams demonstrating two weakly ferromagnetic structures in rhombohedral crystals with two magnetic ions in the primitive cell (p. 128) | html | pdf |
  - Fig. 1.5.5.2. Dependence of magnetization $[M_{\perp}]$ and $[M_{ \parallel }]$ on the magnetic field H for the weak ferromagnet MnCO ₃ at 4.2 K (p. 128) | html | pdf |
  - Fig. 1.5.5.3. Magnetic structures of fluorides of transition metals (p. 128) | html | pdf |
  - Fig. 1.5.5.4. Magnetic structures of orthoferrites and orthochromites RM O ₃ (p. 129) | html | pdf |
  - Fig. 1.5.5.5. Temperature dependence of the susceptibility for CoCO ₃ (p. 130) | html | pdf |
  - Fig. 1.5.5.6. A weakly non-collinear magnetic structure corresponding to (1.5.5.12) (p. 130) | html | pdf |
  - Fig. 1.5.6.1. Schematic representation of the rotation of the vectors $[\bf G]$ and $[\bf F]$ (in the xz plane) at a reorientation transition in orthoferrites (p. 131) | html | pdf |
  - Fig. 1.5.6.2. Schematic representation of the rotation of the vector $[\bf L]$ under the action of a magnetic field applied to CoF ₂ perpendicular to the fourfold axis z (reorientation transition) (p. 132) | html | pdf |
  - Fig. 1.5.7.1. The dependence of the magnetic moment of CoF ₂ on the magnetic field (p. 135) | html | pdf |
  - Fig. 1.5.7.2. Linear magnetostriction of CoF ₂ (p. 136) | html | pdf |
  - Fig. 1.5.7.3. Variation of symmetry of the crystal field in the presence of the piezomagnetic effect in CoF ₂ (p. 137) | html | pdf |
  - Fig. 1.5.8.1. Temperature dependence of the components $[\alpha_{ \parallel }]$ and $[\alpha_{\perp}]$ in Cr ₂ O ₃ (p. 139) | html | pdf |
  - Fig. 1.5.8.2. The hysteresis loop in the linear magnetoelectric effect in ferromagnetoelectric Ni ₃ B ₇ O ₁₃ I at 46 K (p. 142) | html | pdf |
  - Fig. 1.5.9.1. Diagram explaining the occurrence of magnetostrictive strains in the demagnetized and saturated states of a cube-shaped crystal with a cubic prototype (p. 144) | html | pdf |
- Tables
  - Table 1.5.2.1. Comparison of different symbols for magnetic point groups (p. 109) | html | pdf |
  - Table 1.5.2.2. Comparison of different symbols for the elements of magnetic point groups (p. 109) | html | pdf |
  - Table 1.5.2.3. The 90 magnetic point groups of types 2 and 3 (pp. 110-111) | html | pdf |
  - Table 1.5.2.4. List of the magnetic classes in which ferromagnetism is admitted (p. 112) | html | pdf |
  - Table 1.5.3.1. Two types of symbols for collinear antiferromagnetic and ferromagnetic structures (p. 118) | html | pdf |
  - Table 1.5.3.2. Sign variation of the components of antiferromagnetic and ferromagnetic vectors during transformations of the group $[\widetilde{\bi D}_{3d}^6]$ in rhombohedral crystals with four magnetic ions (p. 119) | html | pdf |
  - Table 1.5.3.3. Magnetic groups of symmetry in rhombohedral oxides of trivalent transition-metal ions (p. 119) | html | pdf |
  - Table 1.5.3.4. Magnetic point groups in rhombohedral oxides of transition metals (p. 119) | html | pdf |
  - Table 1.5.3.5. The signs of $[{\boldmu}_i({\bf k}_j)]$ for four sites $[{\bf t}_i]$ of the conventional unit cell (the corners of a primitive cell) (p. 121) | html | pdf |
  - Table 1.5.3.6. Characters of the irreducible representations of the group $[{\bi D}_{3d}=\bar{3}m]$ and corresponding magnetic structures (p. 122) | html | pdf |
  - Table 1.5.5.1. The numbers of the crystallographic space groups that allow a phase transition into a weakly ferromagnetic state and the invariants that are responsible for weak ferromagnetism (Turov, 1963) (p. 129) | html | pdf |
  - Table 1.5.5.2. Magnetic point groups that allow weak ferromagnetism (p. 130) | html | pdf |
  - Table 1.5.7.1. The forms of the matrix characterizing the piezomagnetic effect (p. 133) | html | pdf |
  - Table 1.5.7.2. Experimental data for the piezomagnetic effect (PM) and for linear magnetostriction (LM) (p. 136) | html | pdf |
  - Table 1.5.8.1. The forms of the tensor characterizing the linear magnetoelectric effect (p. 138) | html | pdf |
  - Table 1.5.8.2. A list of some magnetoelectrics (p. 139) | html | pdf |
  - Table 1.5.8.3. Classification of the 122 magnetic point groups according to magnetoelectric types (p. 140) | html | pdf |
  - Table 1.5.8.4. List of the magnetic point groups of the ferromagnetoelectrics (p. 141) | html | pdf |
  - Table 1.5.9.1. Correspondence between matrix indices $[\alpha]$ , A and tensor indices of the tensors describing spontaneous magnetostriction (p. 144) | html | pdf |
  - Table 1.5.9.2. Magnetostriction data for ferromagnets with prototype symmetry $[m\bar{3}m1']$ (p. 145) | html | pdf |
  - Table 1.5.10.1. Conversion of Gaussian to SI units (p. 146) | html | pdf |

Chapter 1.5. Magnetic properties

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