A toroidal moment is an independent term in the
multipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
of
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
s besides magnetic and electric
multipole
A multipole expansion is a Series (mathematics), mathematical series representing a Function (mathematics), function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and Azimuth, azimuthal angles) f ...
s. In the electrostatic multipole expansion, all charge and current distributions can be expanded into a complete set of electric and magnetic multipole coefficients. However, additional terms arise in an electrodynamic multipole expansion. The coefficients of these terms are given by the toroidal multipole moments as well as time derivatives of the electric and magnetic multipole moments. While
electric dipole
The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The d ...
s can be understood as separated charges and
magnetic dipole
In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric ...
s as circular currents, axial (or electric) toroidal dipoles describes toroidal charge arrangements whereas polar (or magnetic) toroidal dipole (also called anapole) correspond to the field of a
solenoid
upright=1.20, An illustration of a solenoid
upright=1.20, Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines
A solenoid () is a type of electromagnet formed by a helix, helical coil of wire whose ...
bent into a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
.
Classical toroidal dipole moment
A complex expression allows the
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ar ...
J to be written as a sum of electric, magnetic, and toroidal moments using Cartesian or spherical differential operators. The lowest order toroidal term is the toroidal dipole. Its magnitude along direction ''i'' is given by
:
Since this term arises only in an expansion of the current density to second order, it generally vanishes in a long-wavelength approximation.
However, a recent study comes to the result that the toroidal multipole moments are not a separate multipole family, but rather higher order terms of the electric multipole moments.
Quantum toroidal dipole moment
In 1957,
Yakov Zel'dovich found that because the
weak interaction
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, ...
violates parity symmetry, a spin-
Dirac particle
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its Dirac equation#Covariant form and relativistic invariance, free form, or including Dirac equation#Comparison with the ...
must have a toroidal dipole moment, also known as an anapole moment, in addition to the usual electric and magnetic dipoles. The interaction of this term is most easily understood in the non-relativistic limit, where the Hamiltonian is
where , , and are the electric, magnetic, and anapole moments, respectively, and is the vector of
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
.
The nuclear toroidal moment of
cesium
Caesium (IUPAC spelling) (or cesium in American English) is a chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-golden alkali metal with a melting point of , which makes it one of only five elemental metals that ar ...
was measured in 1997 by Wood ''et al.''.
Symmetry properties of dipole moments
All dipole moments are vectors which can be distinguished by their differing symmetries under
spatial inversion (P: r ↦ −r) and
time reversal (T: ''t'' ↦ −''t''). Either the dipole moment stays invariant under the symmetry transformation ("+1") or it changes its direction ("−1"):
Magnetic toroidal moments in condensed matter physics
In
condensed matter
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
magnetic toroidal order can be induced by different mechanisms:
* Order of localized
spins
The spins (as in having "the spins")Diane Marie Leiva. ''The Florida State University College of Education''Women's Voices on College Drinking: The First-Year College Experience"/ref> is an adverse reaction of intoxication that causes a state of v ...
breaking spatial inversion and time reversal. The resulting toroidal moment is described by a sum of cross products of the spins S
''i'' of the magnetic ions and their positions r
''i'' within the magnetic unit cell: T = Σ
''i'' r
''i'' × S
''i''
* Formation of vortices by delocalized magnetic moments.
* On-site
orbital currents (as found in
multiferroic Multiferroics are defined as materials that exhibit more than one of the primary ferroic properties in the same phase:
* ferromagnetism – a magnetisation that is switchable by an applied magnetic field
* ferroelectricity – an electric polarisa ...
CuO
Copper(II) oxide or cupric oxide is an inorganic compound with the formula CuO. A black solid, it is one of the two stable oxides of copper, the other being Cu2O or copper(I) oxide (cuprous oxide). As a mineral, it is known as tenorite. It is ...
).
* Orbital loop currents have been proposed in copper oxides superconductors that might be important to understand
high-temperature superconductivity
High-temperature superconductors (abbreviated high-c or HTS) are defined as materials that behave as superconductors at temperatures above , the boiling point of liquid nitrogen. The adjective "high temperature" is only in respect to previ ...
. Experimental verification of symmetry-breaking by such orbital currents has been claimed in
cuprates
Cuprate loosely refers to a material that can be viewed as containing anionic copper complexes. Examples include tetrachloridocuprate ( uCl4sup>2−), the superconductor YBa2Cu3O7, and the organocuprates (e.g., dimethylcuprate u(CH3)2sup>� ...
through polarized neutron-scattering.
Magnetic toroidal moment and its relation to the magnetoelectric effect
The presence of a magnetic toroidic dipole moment T in condensed matter is due to the presence of a
magnetoelectric effect In its most general form, the magnetoelectric effect (ME) denotes any coupling between the magnetic and the electric properties of a material. The first example of such an effect was described by Wilhelm Röntgen in 1888, who found that a dielectric ...
: Application of a magnetic field H in the plane of a toroidal solenoid leads via the
Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
to an accumulation of current loops and thus to an
electric polarization
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
perpendicular to both T and H. The resulting polarization has the form ''P''
''i'' = ε
''ijk''''T''
''j''''H''
''k'' (with ε being the
Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for some ...
). The resulting magnetoelectric
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
describing the cross-correlated response is thus
antisymmetric.
Ferrotoroidicity in condensed matter physics
A
phase transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
to spontaneous
long-range order
In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system.
In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one o ...
of microscopic magnetic toroidal moments has been termed "ferrotoroidicity". It is expected to fill the symmetry schemes of primary
ferroics
Ferroics is the generic name given to the study of ferromagnets, ferroelectrics, and ferroelastics.
Overview
The basis of ferroics is to understand the large changes in physical characteristics that occur over a very narrow temperature range. ...
(phase transitions with spontaneous point symmetry breaking) with a space-odd, time-odd macroscopic order parameter. A ferrotoroidic material would exhibit domains which could be switched by an appropriate field, e.g. a magnetic field curl. Both of these hallmark properties of a ferroic state have been demonstrated in an artificial ferrotoroidic model system based on a
nanomagnetic array
The existence of ferrotoroidicity is still under debate and clear-cut evidence has not been presented yet—mostly due to the difficulty to distinguish ferrotoroidicity from
antiferromagnetic
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. ...
order, as both have no net
magnetization
In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Di ...
and the
order parameter
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
is the same.
Anapole dark matter
All
CPT self-conjugate particles, in particular the
Majorana fermion
A Majorana fermion (, uploaded 19 April 2013, retrieved 5 October 2014; and also based on the pronunciation of physicist's name.), also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Et ...
, are forbidden from having any multipole moments other than toroidal moments.
At tree level (i.e. without allowing loops in
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s) an anapole-only particle interacts only with external currents, not with free-space electromagnetic fields, and the interaction cross-section diminishes as the particle velocity slows. For this reason, heavy Majorana fermions have been suggested as plausible candidates for
cold dark matter
In cosmology and physics, cold dark matter (CDM) is a hypothetical type of dark matter. According to the current standard model of cosmology, Lambda-CDM model, approximately 27% of the universe is dark matter and 68% is dark energy, with only a sm ...
.
"New, simple theory may explain mysterious dark matter"
/ref>
See also
* Spheromak
A spheromak is an arrangement of plasma formed into a toroidal shape similar to a smoke ring. The spheromak contains large internal electric currents and their associated magnetic fields arranged so the magnetohydrodynamic forces within the s ...
References
Literature
* Stefan Nanz
''Toroidal Multipole Moments in Classical Electrodynamics''
Springer 2016. {{ISBN, 978-3-658-12548-6
Electromagnetism
Moment (physics)