Toroid Moments in Electrodynamics a ~ ~ Dsolid - State Physics

Introduction 147 10. Experimental realization of proper and pseudoproper Part I. Toroid moments in electrodynamics 149 orbital itinerant antiferromagnets 181 1. Formulation of the problem of the multipole expansion 149 Conclusions 187 2. Expansion of current densities 151 Appendix A. On the representation of vector currents (fields) 3. Multipole expansion of the interaction energy of a system through scalar functions (Debye potentials) 188 of point charges with an external electromagnetic field 154 Appendix B. The simplest definitions of the toroid form 4. Derivation of the multipole expansion in a canonical basis 157 factors and moments 190 5. Electromagnetic properties of static and variable toroid Appendix C. On the typical " loss " of the toroid moments in dipoles 160 the transition to the long-wavelength limit. Siegert " theorem " 191 Part II. Toroid moments in condensed matter physics 162 Appendix D. General formulae of the multipole expansions 6. Examples of the application of toroid dipoles in con-(including the case of magnetic charges) 192 densed matter electrodynamics 162 Appendix E. Complete parametrization of the charge and 7. From basic dipoles to order parameters in second-order current densities in terms of spherical differential operators phase transition theory 166 and distributions 194 8. Peculiarities of the symmetry and the physical meaning of Appendix F. Multipole fields at short and long distances in the the toroid order parameter in the theory of orbital anti-(quasi)static case 197 ferromagnetism 168 Appendix G. Toroid dipole moment and anapole 198 9. Some properties of orbital antiferromagnets 173 References 200 Abstract: The role is explained of toroid moments and average 2n-pow~rradii in the structure of multipole expansions in the framework of Maxwell—Lorentz electrodynamics. A complete scheme of multipole expansions is given in terms of distribution theory. A number of particular applications of toroid moments is considered in condensed-matter physics. Special attention is paid to the microscopic nature and macroscopic symmetry of the vector order parameter in ordering orbital currents in crystals and to its description in terms of the distributed toroid moments. Conditions are discussed for the appearance of a toroid current state (orbital antiferromagnetism) in some models of solid-state theory. Toroid moments in electrodynamics and solid-state physics 147 " Nature is not economical of structures, only of principles of universal applicability. The biologist has long comprehended this; we, in physics, mostly lose sight of this truth. " A. Salam