Magnetostriction Theory and Applications of Magnetoelasticity Etienne du Tremolet de Lacheisserie Döcteur es Sciences Physiques С, Directeur de Recherche Laböratöi/e" de. Jvlagnetisme "Louis Neel" Centre National dfe la Recherche Scientifique and Universite Joseph Fourier Grenoble France (g) CRC Press Boca Raton Ann Arbor London Tokyo
Table of Contents Introduction 1 1 Elements of Solid State Physics 5 1-1 A Classical Account of Elasticity 5 A. Strain Tensor 5 B. Extension, Elongation 8 C. Stress Tensor 10 D. Linear Elasticity 11 E. Static and Dynamic Properties of an Elastic Body 15 l-ii Contribution of the Group Theory 16 A. Symmetry Groups 16 B. Spherical Symmetry; Rotation Group 17 C. Cylindrical Symmetry 23 D. Crystalline Symmetries 24 l-iii Introduction to the Magnetism of Condensed Matter 29 A. Notion of Magnetic Induction 29 B. Atomic Origin of the Magnetism 33 C. The Magnetism of Solids 35 D. Magnetic Energies 38 E. Model of Localised Magnetism 40 F. Model of Itinerant Magnetism 42 2 Magnetoelasticity of Soft Ferromagnets: Introduction and Theory 43 2-1 Some Illustrative Experiments 44 A. Volume Magnetostriction 45 B. Joule Magnetostriction 46 C. Magnetic Contributions to Elasticity 49 D. Wiedemann Effect 50 2-II Thermodynamics: Elastomagnetic Matrices 50 A. Thermodynamical Potential of a Magnet 51 B. Elastomagnetic Matrices in Ferromagnets 53 C. Solids Exhibiting a High Symmetry 56 D. Magnetomechanical Coupling Factor 59 2-III Symmetry Formalism: Magnetoelastic Coupling Energy 61 A. Magnetoelastic Coupling in Isotropic Substances 62 B. Magnetoelastic Coupling in Uniaxial Ferromagnets 66 С Magnetoelastic Coupling in Cubic Ferromagnets 68 D. Lower Crystalline Symmetries 76 E. Some Remarks 82
2-IV Symmetry Formalism: Magnetoelastic Spin Hamiltonian 83 A. Symmetry Spin Operators 84 B. Spin Hamiltonian for Various Symmetries 86 C. Derivation of the Free Energy: Temperature Dependence 89 2-V Atomic Approach of the Magnetoelasticity 94 A. Pair Model of Magnetoelasticity 95 B. Other Microscopic Models of Magnetoelasticity in Soft Ferromagnets 107 3 Magnetoelasticity of Soft Ferromagnets: The Physical Effects 111 3-1 Volume Isotropic Magnetostriction and Pressure Dependence of the Magnetic Properties Ill A. Spontaneous Volume Magnetostriction and Anomalous Thermal Expansion 113 B. Forced Volume Magnetostriction 118 C. Pressure Dependence of the Curie Temperature and the Magnetic Moment 126 3-II Joule Magnetostriction 130 A. Magnetostriction of a Uniformly Magnetized Single Crystal 131 B. Field Dependence of the Magnetostriction in a Single Crystal 142 C. Magnetostriction in Polycrystalline and Amorphous Ferromagnets 149 D. Tensor Formalism of the Magnetostriction 161 E. Magnetostriction in Thin Films and Multilayers 162 F. Temperature Dependence of the Joule Magnetostriction 165 G. Stress Dependence of the Joule Magnetostriction 179 H. Structural and Electronic Dependence of the Joule Magnetostriction 193 3-III Inverse Joule Effect (Villari Effect) 198 A. Villari Effect in Isotropic Substances 198 B. Villari Effect in Materials Exhibiting the Cubic Symmetry 205 C. Villari Effect in Substances of Lower Symmetry 211 D. Conclusions 211 3-IV Direct and Inverse Wiedemann Effects 212 A. Direct Wiedemann Effect 213 B. Inverse Wiedemann Effects; Matteuci Effect 220 3-V Form Effect 223 A. Non-Uniform Form Effect for a Spherical Crystal of Cubic Symmetry 223 B. Form Effect in Crystals with Lower Symmetry 229
C. Non-Uniform Form Effect for Other Geometries 230 D. Some Examples 230 3-VI Magnetoelastic Contributions to Magnetic and Elastic Properties 231 A. Magnetoelastic Contribution to the Magnetic Anisotropy 232 B. First Order Magnetoelastic Contribution to the Elastic Properties 234 C. Second Order Magnetoelastic Contributions to Elastic Coefficients 248 D. Sound Velocity in Ferromagnetic Crystals of Cubic Symmetry 249 E. Magnetomechanical Damping 261 3-VII Magnetostriction of Soft Ferromagnets in the Paramagnetic Phase 264 A. Exchange Paramagnetostriction 264 B. Joule Paramagnetostriction 265 4 Magnetostriction on Other Magnetic Substances 267 4-1 Ferrimagnetic Oxides 267 A. Theoretical Aspects 269 B. Magnetostriction of Spinel Ferrites 273 C. Magnetostriction of Garnet Ferrites 279 D. Magnetostriction of Hexaferrites 286 E. Conclusions 286 4-II Antiferromagnetic Oxides 287 A. Introduction to Antiferromagnets 287 B. Magnetostriction of 3d-Monoxides 288 C. Magnetostriction of Corundum-Type Oxides 290 D. Magnetostriction of Rare Earth Orthoferrites 291 4-III Rare Earth Metals 293 A. Magnetostriction of Gadolinium 293 B. Magnetostriction of Terbium 295 С Magnetostriction of Dysprosium 296 D. Magnetostriction of Holmium 296 E. Magnetostriction of Erbium 297 F. Magnetostriction of Light Rare Earth Metals 298 G. Conclusions on the Magnetostriction of Rare Earth Metals 298 4-IV Rare Earth Intermetallic Alloys and Compounds 299 A. Rare Earth-Zinc Equiatomic Compounds 301 B. RA1 2 Compounds 305 C. RFe 2 Compounds 305 4-V Miscellaneous 312 A. Paramagnets and Diamagnets 313 B. Superconductors 315
5 Technical Aspects of Magnetoelasticity 319 5-1 Experimental Methods for Measuring Magnetostriction 319 A. Tube-Type Dilatometers 320 B. Three-Terminal Capacitance 325 C. Microwave Capacitance Dilatometer 327 D. Strain Gauge Extensometers 327 E. Optical Interferometry 329 F. X-Rays and Neutrons Diffraction 330 G. Wiedemann Effect 330 H. Case of Ribbons, Wires, Thin Films and Multilayers 331 I. Villari Effect 333 J. Strain Modulated Magnetic Resonance 335 K. Elastic Properties Under Magnetic Field 337 5-II Industrial Materials 339 A. Alloys with Controlled Thermal Expansion 339 B. Soft Ferromagnets 340 С Highly Magnetostrictive Materials 344 D. Materials with a High Magnetomechanical Coupling 349 E. Other Magnetostrictive Materials 351 5-III Some Applications of Magnetoelasticity 352 A. Applications of Isotropic Magnetoelastic Coupling 352 B. Applications of Zero-Magnetostriction Ferromagnets 353 C. Applications of Magnetostriction to Actuators 353 D. Magnetostrictive Motors 357 E. Sonic and Ultrasonic Emission 359 F. Electromechanical Filters 361 G. Magnetoelastic Delay Lines 362 H. Magnetostrictive Sensors 362 Final Remarks 369 References 371 Appendix I Symbols Used in the Text 397 Appendix II Subject Index 401 Appendix III Some Magnetostriction Data 405 Appendix IV Evolution of the Number of Publications Dealing with Magnetostriction 409