1) a particular form of interaction between electric currents, between currents and magnets (that is, bodies with a magnetic moment) and between magnets;
2) the branch of physics that studies this interaction and the properties of the substances (magnetic materials) in which it appears.
The main manifestations of magnetism. In the most general terms, magnetism can be defined as a particular form of material interaction arising between moving electrically charged particles. The magnetic interaction is transmitted by the magnetic field, which provides the link between spatially separated bodies. Together with the electric field it is one of the manifestations of the electromagnetic form of the motion of matter (see Electromagnetic field). There is no complete symmetry between magnetic and electric fields. The sources of the electric field are electric charges, but no analogous magnetic charges have yet been observed in nature, although hypotheses about their existence have been put forward (see Magnetic monopole). The source of the magnetic field is a moving electric charge, that is, an electric current. On the atomic scale, electrons and nucleons (protons, neutrons) have two types of microscopic current — orbital currents, associated with the motion of the centre of gravity of these particles within the atom, and spin currents, associated with their internal motion.
The quantitative characteristic of the magnetism of particles is their orbital and spin magnetic moments. Since all the microstructural elements of substances — electrons, protons and neutrons — possess magnetic moments, any combination of them — atomic nuclei and electron shells, and also combinations of those combinations, that is, atoms, molecules and macroscopic bodies — can in principle be a source of magnetism. Thus essentially all substances possess magnetic properties. Two basic effects of an external magnetic field on a substance are known. First, in accordance with Faraday's law of electromagnetic induction, an external magnetic field always sets up in the substance an induced current whose magnetic field is directed against the initial field (Lenz's rule). The magnetic moment that the external field produces in the substance is therefore always directed opposite to the external field (see Diamagnetism). Second, if an atom has a non-zero magnetic moment (spin, orbital or both), the external field tends to orient it along its own direction. The result is a magnetic moment parallel to the field, which is called paramagnetic (see Paramagnetism).
The internal interactions (of electrical and magnetic nature) between the microparticles that carry the magnetic moment (atoms and others) can also have a substantial effect on the magnetic properties of a substance. In some cases these interactions make it energetically favourable for the substance to possess a spontaneous ordering (independent of any external field) in the orientation of the magnetic moments of its particles (atomic magnetic order). Substances in which the atomic magnetic moments are oriented parallel to one another are called ferromagnets (see Ferromagnetism); correspondingly, substances in which neighbouring atomic moments lie antiparallel are called antiferromagnets (see Antiferromagnetism). Besides such collinear ferromagnetic and antiferromagnetic atomic structures, non-collinear ones (helical, triangular and others) are also observed.
The complexity of the atomic structure of substances built from enormous numbers of atoms leads to a practically inexhaustible variety of magnetic properties. In discussions of the magnetic properties of substances the general term “magnetic materials” is used for them. Because the magnetic properties of substances are linked to their non-magnetic properties (electrical, mechanical, optical and so on), studies of magnetic properties can very often be used as a source of information about the internal structure of microparticles and of bodies of macroscopic size. The enormous range of magnetic phenomena, extending from the magnetism of elementary particles to the magnetism of cosmic bodies (the Earth, the Sun, the stars and others), explains the deep interest in magnetism shown by many sciences (physics, astrophysics, chemistry, biology) and its wide application in technology. The questions involved are treated in the articles: Solar wind, Terrestrial magnetism, Magnetosphere, Magnetic field, Magnetohydrodynamics, Atomic magnetic structure, Magnetic materials, Permanent magnet and others.
The magnetism of substances. The macroscopic description of the magnetic properties of substances is usually carried out within the framework of the theory of the electromagnetic field (see Maxwell's equations), thermodynamics and statistical physics. One of the main macroscopic characteristics of a magnetic material, determining its thermodynamic state, is the magnetisation vector J (the total magnetic moment per unit volume of the material). The vector J is a function of the magnetic field strength H. Graphically the dependence J(H) is represented by the magnetisation curve, which has a different form for different magnetic materials. In a number of substances the relation between J and H is linear: J=μH, where μ is the magnetic susceptibility per unit volume of the substance (μ<0 for diamagnets, μ>0 for paramagnets). In ferromagnets J is related to H non-linearly; their susceptibility depends not only on the temperature T and the properties of the substance but also on the field H.
Thermodynamically the magnetisation J of a magnetic material is defined through the thermodynamic potential Φ(H, T, p) by the formula J=-(dΦ/dH)T,p where p is the pressure. The calculation of Φ(H, T, p) in turn rests on the Gibbs–Boguslavsky relation: Φ= -kT lnZ(H, T,p), where Z(H,T,p) is the partition function.
It follows from the general principles of classical statistical physics that electron systems cannot possess a thermodynamically stable magnetic moment (the Bohr–van Leeuwen theorem), but this contradicts experiment. Quantum mechanics, which explained the stability of the atom, also explained the magnetism of atoms and of macroscopic bodies. The magnetism of atoms and molecules is due to the spin magnetic moments of their electrons, to the motion of the electrons in the shells of atoms and molecules (so-called orbital magnetism), and to the spin and orbital magnetism of the nucleons of the nuclei. In many-electron atoms the orbital and spin magnetic moments are added according to the laws of spatial quantisation — the resultant magnetic moment μj is determined by the total angular quantum number j and equals: μj=gj√(j(j+1)μB), where gj is the Landé factor and μB is the Bohr magneton.
In the atoms of the inert gases (He, Ar, Ne and others) the electron shells are magnetically neutral (their total magnetic moment is zero). In an external magnetic field the inert gases display diamagnetic properties. The electron shell of the alkali-metal atoms (Li, Na, K and others) possesses only the spin magnetic moment of the valence electron; the orbital magnetic moment of these atoms is zero. As a result the alkali-metal atoms are paramagnetic. In the atoms of the transition metals (Fe, Co, Ni, the rare-earth metals (REM) and others) the d and f layers of their electron shells are incomplete. The spin and orbital magnetic moments of the electrons in these layers are uncompensated, which gives isolated atoms of Fe, Co, Ni and the REM a considerable magnetic moment.
The magnetic properties of substances are determined by the nature of the atomic carriers of magnetism and by the character of their interactions. Even a substance of one and the same chemical composition may have different magnetic properties, depending on external conditions and on its crystal or phase structure (for example, on the degree of ordering of the atoms in alloys and the like). For example, Fe, Co and Ni in the crystalline state have ferromagnetic properties below a certain temperature (the Curie point), and above the Curie point they are paramagnetic. The same is observed in antiferromagnets, whose critical temperature is called the Néel point. In some REM an antiferromagnetic region exists between the ferromagnetic and paramagnetic temperature ranges.
Quantitatively, the interaction between the atomic carriers of magnetism in a substance can be characterised by the energy of this interaction εint, calculated per individual pair of particles carrying a magnetic moment. The energy εint, which arises from the electrical and magnetic interaction of the particles, can be compared with the energies of other atomic interactions: with the energy of a particle whose magnetic moment is of the order of ≈μB in some effective magnetic field Heff, that is, with εH=μBHeff, and with the mean energy of the thermal motion of a particle at some effective critical temperature Tk, that is, εT=kTk (Heff and Tk serve as measures of the interaction energy of the particles). When the strength of the external field is H>>Heff, or when T>>Tk, the external factors dominate — the temperature or the field, which suppress the effects of the internal interaction (“weak” magnetism of substances). This classification is formal, since it does not reveal the physical nature of Heff and Tk. To establish the nature of the magnetic properties of a substance fully it is necessary to know not only the magnitude of the energy εint compared with εT or εH, but also its physical origin and the character of the magnetic moment of the carriers (orbital or spin). Apart from the case of nuclear magnetism, two types of force act in the electron shells of atoms and molecules and in the electron system of condensed substances (liquids, crystals) — electrical and magnetic. A measure of the electrical interaction is the electrostatic energy εel of two electrons separated by an atomic distance a (a=10-8 cm): εel=e2/a=2.57*10-11 erg. A measure of the magnetic interaction is the binding energy of two microparticles that have magnetic moments μB and are separated by a distance a, that is, εmag=μ2B/a3»10-15 erg. Thus εel exceeds the energy εmag by three or four orders of magnitude. The retention of magnetisation by ferromagnets (Fe, Co, Ni) up to temperatures of T=1000 K can therefore be due only to the electrical interaction, since at an energy εmag=10-16 erg thermal motion would already destroy the orienting action of the magnetic forces at 1 K. According to quantum mechanics, besides the Coulomb electrostatic interaction of charged particles there is also a purely quantum electrostatic exchange interaction, which depends on the mutual orientation of the magnetic moments of the electrons. This part of the interaction, electrostatic in nature, has a substantial effect on the magnetic state of electron systems. In particular, it favours an ordered orientation of the magnetic moments of the atomic carriers of magnetism. The upper limit of the exchange-interaction energy is εex=10-13 erg.
A value εex>0 corresponds to a parallel orientation of the atomic magnetic moments, that is, to the spontaneous magnetisation of ferromagnets. When εex<0 there is a tendency towards the antiparallel orientation of neighbouring magnetic moments that is characteristic of the atomic magnetic structure of antiferromagnets. In crystals of alloys and compounds a so-called mixed exchange interaction is possible, in which the signs of εex are opposite for different sites of the crystal lattice. What has been said allows the following physical classification of the magnetic properties of substances.
I. The magnetism of weakly interacting particles (εint<<-μBH or εint<
Predominance of diamagnetism. Substances with diamagnetic properties are:
a) all the inert gases, and also gases whose atoms or molecules have no resultant magnetic moment of their own. Their magnetic susceptibility is negative and very small in absolute value (the molar susceptibility χ is of the order of —(10-7—10-5)); it is practically independent of temperature;
b) organic compounds with a non-polar bond, in which the molecules or radicals either have no magnetic moment or the paramagnetic effect in them is suppressed by the diamagnetic one; in these compounds χ is of the order of —106 and is likewise practically independent of temperature, but shows a marked anisotropy (see Magnetic anisotropy);
c) liquid and crystalline substances: some metals (Zn, Au, Hg and others); solutions, alloys and chemical compounds (for example the halogens) in which the diamagnetism of the ion cores predominates (ions resembling the atoms of the inert gases, Li+ , Be2+ , Al3+, Cl- and the like). The magnetism of this group of substances resembles the magnetism of the “classical” diamagnetic gases.
Predominance of paramagnetism is characteristic:
a) of free atoms, ions and molecules that have a resultant magnetic moment. The gases O2 and NO and the vapours of the alkali and transition metals are paramagnetic. Their susceptibility is χ=10-3—10-5, and at temperatures that are not too low and in magnetic fields that are not too strong (μBH/kT<<1) it does not depend on the field H but depends strongly on the temperature — for χ the Curie law holds: χ=C/T, where C is the Curie constant;
b) of the ions of the transition elements in the liquid phase, and also in crystals, provided that the magnetically active ions interact weakly with one another and that their immediate surroundings in the condensed phase have little effect on their paramagnetism. When μBH/kT<<1 their susceptibility χ does not depend on H but does depend on T — the Curie–Weiss law holds: χ=C`/(T-Δ), where C` and Δ are constants of the substance; c) of ferromagnetic and antiferromagnetic substances above the Curie point θ.
II. The magnetism of conduction electrons in metals and semiconductors. The paramagnetism of conduction electrons (spin paramagnetism) is observed in the alkali metals (Li, K, Na and others), the alkaline-earth metals (Ca, Cr, Ba, Ra) and the transition metals (elements with incomplete 3d, 4d and 5d shells, apart from Fe, Ni, Co and Mn, Cr). Their susceptibility is small (χ=10-5), does not depend on the field and varies little with temperature. In a number of metals (Cu, Ag, Au and others) this paramagnetism is masked by the stronger diamagnetism of the ion cores. The diamagnetism of conduction electrons in metals (Landau diamagnetism) is inherent in all metals, but in most cases it is masked either by the stronger spin paramagnetism of the electrons or by the dia- or paramagnetism of the ion cores. The para- and diamagnetism of conduction electrons in semiconductors. Compared with metals, semiconductors have few conduction electrons, but their number grows as the temperature rises; χ in this case also depends on T. The magnetism of superconductors is due to electric currents flowing in a thin surface layer of thickness ≈10-5 cm. These currents screen the bulk of the superconductor from external magnetic fields, so that in a massive superconductor at T
III. The magnetism of substances with atomic magnetic order (εint>>μBH or εint>>kT).
Ferromagnetism occurs in substances with a positive exchange energy (εex>0): in crystals of Fe, Co, Ni, in a number of REM (Gd, Tb, Dy, Ho, Er, Tm, Yb), in alloys and compounds containing these elements, and also in alloys of Cr and Mn and in compounds of U. Ferromagnetism is characterised by spontaneous magnetisation at temperatures T<θ; at T>θ ferromagnets pass either into the paramagnetic or into the antiferromagnetic state (the latter is observed, for example, in some REM). It is known from experiment, however, that in the absence of an external field ferromagnetic bodies have no resultant magnetisation (apart from the secondary phenomenon of remanent magnetisation). The reason is that at H=0 a ferromagnet breaks up into a large number of microscopic regions of spontaneous magnetisation — domains. The magnetisation vectors of the individual domains are oriented so that the total magnetisation of the ferromagnet is zero. In an external field the domain structure changes and the ferromagnetic specimen acquires a resultant magnetisation (see Magnetisation).
Antiferromagnetism occurs in substances with a negative exchange energy (εex<0): in crystals of Cr and Mn, in a number of REM (Ce, Pr, Nd, Sm, Eu), and also in numerous compounds and alloys containing elements of the transition groups.
The crystal lattice of these substances divides into so-called magnetic sublattices whose spontaneous magnetisation vectors Jki are either antiparallel (collinear antiferromagnetic coupling) or directed at angles to one another other than 0 and 180° (non-collinear coupling, (see Atomic magnetic structure). If the total moment of all the magnetic sublattices of an antiferromagnet is zero, the antiferromagnetism is compensated; if there is a non-zero difference in the spontaneous magnetisations, uncompensated antiferromagnetism, or ferrimagnetism, is observed, which occurs chiefly in crystals of metal oxides with a crystal lattice of the spinel, garnet, perovskite or other mineral type (these are called ferrites). In their electrical properties these substances are semiconductors and dielectrics; in their magnetic properties they resemble ordinary ferromagnets. When the compensation of the magnetic moments in antiferromagnets is disturbed by a weak interaction between the atomic carriers of magnetism, a very small spontaneous magnetisation (≈0.1% of the usual values for ferro- and ferrimagnets) arises in some cases in substances that are called weak ferromagnets (for example haematite α-Fe2O3, the carbonates of a number of metals, the orthoferrites; (see Weak ferromagnetism). The character of the atomic carriers of the magnetic moment differs in ferro- and antiferromagnetic d and f metals, in metallic alloys and compounds, and in non-conducting crystals. In d metals and alloys the main carriers of the atomic magnetic moment are the electrons of what was the incomplete d layer of the isolated atoms. The ferro- or antiferromagnetism they produce is connected with the exchange interaction in the system of collectivised d electrons.
In 4f metals and in dielectric crystals the ordered atomic magnetic structures are formed by magnetic moments localised near the sites of the crystal lattice that are occupied by magnetically active ions. Ordered magnetism also exists in amorphous solids (in supercooled liquids, the so-called metallic glasses), which have a number of specific properties different from the magnetic properties of crystalline magnetic materials.
Substances called spin glasses are also of great interest: they have atomic ordering but no ordering of the localised atomic spin or orbital magnetic moments.
The magnetic state of a ferro- or antiferromagnet in an external magnetic field H is determined not only by the magnitude of the field but also by the previous states of the material (the magnetic history of the specimen). This phenomenon is called hysteresis. Magnetic hysteresis appears as an ambiguity in the dependence of J on H (in the existence of a hysteresis loop). Because of hysteresis, removing the external field is not enough to demagnetise a specimen: at H=0 the specimen retains a remanent magnetisation Jr. To demagnetise the specimen a reverse magnetic field Hc, which is called the coercive force, must be applied. According to the value of Hc, a distinction is drawn between magnetically soft materials (Hc<800 A/m or 10 Oe) and magnetically hard, or high-coercivity, materials (Hc>4 kA/m or 50 Oe). Jr and Hc depend on temperature and as a rule decrease as it rises, tending to zero as T approaches θ.
Scientific and technical problems of magnetism.
The main scientific problems of the modern study of magnetism are:
1) to establish the nature of the exchange interaction and of the interactions responsible for anisotropy in the various types of magnetically ordered crystals; to explain the spectra of the elementary magnetic excitations (magnons) and the mechanisms of their interaction with one another and also with phonons, conduction electrons and others;
2) to construct a theory of the transition from the paramagnetic to the ferromagnetic state. The study of the magnetism of substances is used as a means of investigating chemical bonds and the structure of molecules (magnetochemistry). Studies of the dia- and paramagnetic properties of gases, liquids, solutions and compounds in the solid phase make it possible to work out the details of the physical and chemical processes going on in these bodies and of their structure. The study of dynamic magnetic characteristics (magnetic resonance and relaxation processes) helps in understanding the kinetics of many physical and physico-chemical processes in various substances. Magnetobiology is developing rapidly.
The most important problems in the magnetism of cosmic bodies are: the origin of the magnetic fields of the Earth, the planets, the Sun and the stars (in particular the pulsars) and of extragalactic radio sources (radio galaxies, quasars and others), and the role of magnetic fields in cosmic space.
The problems of the technical application of magnetism are among the most important problems of electrical engineering, radio engineering, electronics, instrument-making and computer technology, navigation, automation and telemechanics. Magnetic flaw detection and magnetic methods of inspection are widely used in technology. Magnetic materials are used to make the magnetic cores of generators, motors, transformers and relays, magnetic amplifiers, magnetic memory elements, compass needles, magnetic recording tape and so on.
Historical note.
The first written evidence of magnetism (China) is more than two thousand years old. It mentions the use of natural permanent magnets as compasses. The works of ancient Greek and Roman scholars mention the attraction and repulsion of magnets and the magnetisation of iron filings in the presence of a magnet (for example in the poem “On the Nature of Things” by the Roman poet and materialist philosopher Lucretius, 1st century BC). In the Middle Ages the magnetic compass came into wide use in Europe (from the 12th century), and attempts were made to study the properties of magnets of different shapes experimentally (the French scholar Pierre de Maricourt, 1269). The results of research on magnetism in the Renaissance were summed up in the work of the English physicist W. Gilbert, “On the Magnet, Magnetic Bodies and the Great Magnet — the Earth” (1600). Gilbert showed in particular that the Earth is a magnetic dipole, and proved that the two unlike poles of a magnet cannot be separated. The study of magnetism was then developed in the works of the French scholar R. Descartes, the Russian scholar F. Aepinus and the French physicist C. Coulomb. Descartes was the author of the first detailed metaphysical theory of magnetism and geomagnetism (“Principles of Philosophy”, part 4, 1644); he started from the existence of a special magnetic substance whose presence and motion caused the magnetism of bodies.
In his treatise “An Attempt at a Theory of Electricity and Magnetism” (1759) Aepinus emphasised the analogy between electrical and magnetic phenomena. As Coulomb showed (1785—89), this analogy has a definite quantitative expression: the interaction of point magnetic poles obeys the same law as the interaction of point electric charges (Coulomb's law). In 1820 the Danish physicist H. Ørsted discovered the magnetic field of an electric current. In the same year the French physicist A. Ampère established the laws of the magnetic interaction of currents and the equivalence of the magnetic properties of a circular current and a thin flat magnet; he explained magnetism by the existence of molecular currents. In the 1830s the German scientists C. Gauss and W. Weber extended the mathematical theory of geomagnetism and developed methods of magnetic measurement.
A new stage in the study of magnetism begins with the work of the English physicist M. Faraday, who gave a consistent interpretation of magnetic phenomena based on the idea that the electromagnetic field is real. A number of major discoveries in electromagnetism (electromagnetic induction — Faraday, 1831; Lenz's rule — E. H. Lenz, 1833, and others), the generalisation of the electromagnetic phenomena that had been discovered in the works of the English physicist J. C. Maxwell (1872) and the systematic study of the properties of ferromagnets and paramagnets (A. G. Stoletov, 1872; the French physicist P. Curie, 1895, and others) laid the foundations of the modern macroscopic theory.
The study of magnetism at the microscopic level became possible after the discovery of the electron-nuclear structure of atoms. On the basis of the classical electron theory of the Dutch physicist H. A. Lorentz, the French scientist P. Langevin in 1905 constructed the theory of diamagnetism and also a quasi-classical theory of paramagnetism. In 1892 the Russian scientist B. L. Rosing and in 1907 P. Weiss (France) put forward the idea of an internal molecular field responsible for the properties of ferromagnets. The discovery of electron spin and its magnetism (S. Goudsmit, G. E. Uhlenbeck, USA, 1925) and the creation of quantum mechanics led to the development of the quantum theory of dia-, para- and ferromagnetism. On the basis of quantum-mechanical ideas (spatial quantisation) the French physicist L. Brillouin in 1926 found how the magnetisation of paramagnets depends on the external magnetic field and the temperature. In 1927 the German physicist F. Hund compared the experimental and theoretical values of the effective magnetic moments of ions in various paramagnetic salts, which revealed the effect of the electric fields of a paramagnetic crystal on the “freezing” of the orbital moments of the ions. Studies of this phenomenon established that the magnetisation of a crystal is determined almost entirely by the spin moments (W. Penney and R. Schlapp; J. Van Vleck, USA, 1932). In the 1930s the quantum-mechanical theory of the magnetic properties of free electrons was constructed (Pauli paramagnetism, 1927; Landau diamagnetism, 1930). Of substantial importance for the further development of the theory of paramagnetism was electron paramagnetic resonance (EPR), predicted by Ya. G. Dorfman (1923) and then discovered by E. K. Zavoisky (1944).
The creation of the quantum theory of ferromagnetism was preceded by the work of the German physicist E. Ising (1925, the two-dimensional model of ferromagnets), of Ya. G. Dorfman (1927, who proved the non-magnetic nature of the molecular field), of the German physicist W. Heisenberg (1926, the quantum-mechanical calculation of the helium atom) and of the German physicists W. Heitler and F. London (1927, the calculation of the hydrogen molecule). The last two works used the effect of the exchange interaction of electrons in the shells of atoms and molecules, discovered in quantum mechanics (P. Dirac, Great Britain, 1926), and established its connection with the magnetic properties of electron systems that obey Fermi–Dirac statistics (the Pauli principle). The quantum theory of ferromagnetism was begun by the work of Ya. I. Frenkel (1928, the collective-electron model) and Heisenberg (1928, the localised-spin model). The treatment of ferromagnetism as a quantum cooperative phenomenon (the American physicists F. Bloch and J. Slater, 1930) led to the discovery of spin waves. In 1932—33 the French physicist L. Néel and L. D. Landau predicted the existence of antiferromagnetism. The study of new classes of magnetic substances — antiferromagnets and ferrites — gave a deeper understanding of the nature of magnetism. The role of magnetoelastic energy in the origin of the magnetic anisotropy energy was established, the theory of domain structure was constructed and methods of studying it experimentally were mastered.
The development of the theory of magnetism was greatly assisted by the creation of new experimental methods of investigating substances. Neutron-diffraction methods made it possible to determine the types of atomic magnetic structure. Ferromagnetic resonance, first discovered and studied in the work of V. K. Arkadiev (1913) and then of J. Griffiths (USA, 1946), and antiferromagnetic resonance (C. Gorter and others, 1951) made it possible to begin experimental studies of magnetic relaxation processes and also provided an independent method of determining the effective anisotropy fields in ferro- and antiferromagnets. Physical methods of investigation based on nuclear magnetic resonance (E. Purcell and others, USA, 1946) and on the Mössbauer effect (1958) have greatly deepened knowledge of the distribution of spin density in matter, especially in metallic ferromagnets. Observations of the scattering of neutrons and of light have made it possible to determine the spin-wave spectra of a number of substances. Alongside the experimental work, various aspects of the theory of magnetism were also developed: the magnetic symmetry of crystals, the ferromagnetism of collectivised electrons, second-order phase transitions and critical phenomena, and models of one-dimensional and two-dimensional ferro- and antiferromagnets.
Advances in the study of the nature of magnetic phenomena have made it possible to synthesise new and promising magnetic materials: ferrites for high-frequency and microwave devices, high-coercivity compounds of the SmCo5 type (see Permanent magnet), transparent ferromagnets, amorphous magnetic materials (including spin glasses, in which the orientations of the atomic magnetic moments are distributed at random over the sites of the crystal lattice), ferro- and antiferromagnetic amorphous materials (the so-called metallic glasses, or metglasses) and others.
2) the branch of physics that studies this interaction and the properties of the substances (magnetic materials) in which it appears.
The main manifestations of magnetism. In the most general terms, magnetism can be defined as a particular form of material interaction arising between moving electrically charged particles. The magnetic interaction is transmitted by the magnetic field, which provides the link between spatially separated bodies. Together with the electric field it is one of the manifestations of the electromagnetic form of the motion of matter (see Electromagnetic field). There is no complete symmetry between magnetic and electric fields. The sources of the electric field are electric charges, but no analogous magnetic charges have yet been observed in nature, although hypotheses about their existence have been put forward (see Magnetic monopole). The source of the magnetic field is a moving electric charge, that is, an electric current. On the atomic scale, electrons and nucleons (protons, neutrons) have two types of microscopic current — orbital currents, associated with the motion of the centre of gravity of these particles within the atom, and spin currents, associated with their internal motion.
The quantitative characteristic of the magnetism of particles is their orbital and spin magnetic moments. Since all the microstructural elements of substances — electrons, protons and neutrons — possess magnetic moments, any combination of them — atomic nuclei and electron shells, and also combinations of those combinations, that is, atoms, molecules and macroscopic bodies — can in principle be a source of magnetism. Thus essentially all substances possess magnetic properties. Two basic effects of an external magnetic field on a substance are known. First, in accordance with Faraday's law of electromagnetic induction, an external magnetic field always sets up in the substance an induced current whose magnetic field is directed against the initial field (Lenz's rule). The magnetic moment that the external field produces in the substance is therefore always directed opposite to the external field (see Diamagnetism). Second, if an atom has a non-zero magnetic moment (spin, orbital or both), the external field tends to orient it along its own direction. The result is a magnetic moment parallel to the field, which is called paramagnetic (see Paramagnetism).
The internal interactions (of electrical and magnetic nature) between the microparticles that carry the magnetic moment (atoms and others) can also have a substantial effect on the magnetic properties of a substance. In some cases these interactions make it energetically favourable for the substance to possess a spontaneous ordering (independent of any external field) in the orientation of the magnetic moments of its particles (atomic magnetic order). Substances in which the atomic magnetic moments are oriented parallel to one another are called ferromagnets (see Ferromagnetism); correspondingly, substances in which neighbouring atomic moments lie antiparallel are called antiferromagnets (see Antiferromagnetism). Besides such collinear ferromagnetic and antiferromagnetic atomic structures, non-collinear ones (helical, triangular and others) are also observed.
The complexity of the atomic structure of substances built from enormous numbers of atoms leads to a practically inexhaustible variety of magnetic properties. In discussions of the magnetic properties of substances the general term “magnetic materials” is used for them. Because the magnetic properties of substances are linked to their non-magnetic properties (electrical, mechanical, optical and so on), studies of magnetic properties can very often be used as a source of information about the internal structure of microparticles and of bodies of macroscopic size. The enormous range of magnetic phenomena, extending from the magnetism of elementary particles to the magnetism of cosmic bodies (the Earth, the Sun, the stars and others), explains the deep interest in magnetism shown by many sciences (physics, astrophysics, chemistry, biology) and its wide application in technology. The questions involved are treated in the articles: Solar wind, Terrestrial magnetism, Magnetosphere, Magnetic field, Magnetohydrodynamics, Atomic magnetic structure, Magnetic materials, Permanent magnet and others.
The magnetism of substances. The macroscopic description of the magnetic properties of substances is usually carried out within the framework of the theory of the electromagnetic field (see Maxwell's equations), thermodynamics and statistical physics. One of the main macroscopic characteristics of a magnetic material, determining its thermodynamic state, is the magnetisation vector J (the total magnetic moment per unit volume of the material). The vector J is a function of the magnetic field strength H. Graphically the dependence J(H) is represented by the magnetisation curve, which has a different form for different magnetic materials. In a number of substances the relation between J and H is linear: J=μH, where μ is the magnetic susceptibility per unit volume of the substance (μ<0 for diamagnets, μ>0 for paramagnets). In ferromagnets J is related to H non-linearly; their susceptibility depends not only on the temperature T and the properties of the substance but also on the field H.
Thermodynamically the magnetisation J of a magnetic material is defined through the thermodynamic potential Φ(H, T, p) by the formula J=-(dΦ/dH)T,p where p is the pressure. The calculation of Φ(H, T, p) in turn rests on the Gibbs–Boguslavsky relation: Φ= -kT lnZ(H, T,p), where Z(H,T,p) is the partition function.
It follows from the general principles of classical statistical physics that electron systems cannot possess a thermodynamically stable magnetic moment (the Bohr–van Leeuwen theorem), but this contradicts experiment. Quantum mechanics, which explained the stability of the atom, also explained the magnetism of atoms and of macroscopic bodies. The magnetism of atoms and molecules is due to the spin magnetic moments of their electrons, to the motion of the electrons in the shells of atoms and molecules (so-called orbital magnetism), and to the spin and orbital magnetism of the nucleons of the nuclei. In many-electron atoms the orbital and spin magnetic moments are added according to the laws of spatial quantisation — the resultant magnetic moment μj is determined by the total angular quantum number j and equals: μj=gj√(j(j+1)μB), where gj is the Landé factor and μB is the Bohr magneton.
In the atoms of the inert gases (He, Ar, Ne and others) the electron shells are magnetically neutral (their total magnetic moment is zero). In an external magnetic field the inert gases display diamagnetic properties. The electron shell of the alkali-metal atoms (Li, Na, K and others) possesses only the spin magnetic moment of the valence electron; the orbital magnetic moment of these atoms is zero. As a result the alkali-metal atoms are paramagnetic. In the atoms of the transition metals (Fe, Co, Ni, the rare-earth metals (REM) and others) the d and f layers of their electron shells are incomplete. The spin and orbital magnetic moments of the electrons in these layers are uncompensated, which gives isolated atoms of Fe, Co, Ni and the REM a considerable magnetic moment.
The magnetic properties of substances are determined by the nature of the atomic carriers of magnetism and by the character of their interactions. Even a substance of one and the same chemical composition may have different magnetic properties, depending on external conditions and on its crystal or phase structure (for example, on the degree of ordering of the atoms in alloys and the like). For example, Fe, Co and Ni in the crystalline state have ferromagnetic properties below a certain temperature (the Curie point), and above the Curie point they are paramagnetic. The same is observed in antiferromagnets, whose critical temperature is called the Néel point. In some REM an antiferromagnetic region exists between the ferromagnetic and paramagnetic temperature ranges.
Quantitatively, the interaction between the atomic carriers of magnetism in a substance can be characterised by the energy of this interaction εint, calculated per individual pair of particles carrying a magnetic moment. The energy εint, which arises from the electrical and magnetic interaction of the particles, can be compared with the energies of other atomic interactions: with the energy of a particle whose magnetic moment is of the order of ≈μB in some effective magnetic field Heff, that is, with εH=μBHeff, and with the mean energy of the thermal motion of a particle at some effective critical temperature Tk, that is, εT=kTk (Heff and Tk serve as measures of the interaction energy of the particles). When the strength of the external field is H>>Heff, or when T>>Tk, the external factors dominate — the temperature or the field, which suppress the effects of the internal interaction (“weak” magnetism of substances). This classification is formal, since it does not reveal the physical nature of Heff and Tk. To establish the nature of the magnetic properties of a substance fully it is necessary to know not only the magnitude of the energy εint compared with εT or εH, but also its physical origin and the character of the magnetic moment of the carriers (orbital or spin). Apart from the case of nuclear magnetism, two types of force act in the electron shells of atoms and molecules and in the electron system of condensed substances (liquids, crystals) — electrical and magnetic. A measure of the electrical interaction is the electrostatic energy εel of two electrons separated by an atomic distance a (a=10-8 cm): εel=e2/a=2.57*10-11 erg. A measure of the magnetic interaction is the binding energy of two microparticles that have magnetic moments μB and are separated by a distance a, that is, εmag=μ2B/a3»10-15 erg. Thus εel exceeds the energy εmag by three or four orders of magnitude. The retention of magnetisation by ferromagnets (Fe, Co, Ni) up to temperatures of T=1000 K can therefore be due only to the electrical interaction, since at an energy εmag=10-16 erg thermal motion would already destroy the orienting action of the magnetic forces at 1 K. According to quantum mechanics, besides the Coulomb electrostatic interaction of charged particles there is also a purely quantum electrostatic exchange interaction, which depends on the mutual orientation of the magnetic moments of the electrons. This part of the interaction, electrostatic in nature, has a substantial effect on the magnetic state of electron systems. In particular, it favours an ordered orientation of the magnetic moments of the atomic carriers of magnetism. The upper limit of the exchange-interaction energy is εex=10-13 erg.
A value εex>0 corresponds to a parallel orientation of the atomic magnetic moments, that is, to the spontaneous magnetisation of ferromagnets. When εex<0 there is a tendency towards the antiparallel orientation of neighbouring magnetic moments that is characteristic of the atomic magnetic structure of antiferromagnets. In crystals of alloys and compounds a so-called mixed exchange interaction is possible, in which the signs of εex are opposite for different sites of the crystal lattice. What has been said allows the following physical classification of the magnetic properties of substances.
I. The magnetism of weakly interacting particles (εint<<-μBH or εint<
Predominance of diamagnetism. Substances with diamagnetic properties are:
a) all the inert gases, and also gases whose atoms or molecules have no resultant magnetic moment of their own. Their magnetic susceptibility is negative and very small in absolute value (the molar susceptibility χ is of the order of —(10-7—10-5)); it is practically independent of temperature;
b) organic compounds with a non-polar bond, in which the molecules or radicals either have no magnetic moment or the paramagnetic effect in them is suppressed by the diamagnetic one; in these compounds χ is of the order of —106 and is likewise practically independent of temperature, but shows a marked anisotropy (see Magnetic anisotropy);
c) liquid and crystalline substances: some metals (Zn, Au, Hg and others); solutions, alloys and chemical compounds (for example the halogens) in which the diamagnetism of the ion cores predominates (ions resembling the atoms of the inert gases, Li+ , Be2+ , Al3+, Cl- and the like). The magnetism of this group of substances resembles the magnetism of the “classical” diamagnetic gases.
Predominance of paramagnetism is characteristic:
a) of free atoms, ions and molecules that have a resultant magnetic moment. The gases O2 and NO and the vapours of the alkali and transition metals are paramagnetic. Their susceptibility is χ=10-3—10-5, and at temperatures that are not too low and in magnetic fields that are not too strong (μBH/kT<<1) it does not depend on the field H but depends strongly on the temperature — for χ the Curie law holds: χ=C/T, where C is the Curie constant;
b) of the ions of the transition elements in the liquid phase, and also in crystals, provided that the magnetically active ions interact weakly with one another and that their immediate surroundings in the condensed phase have little effect on their paramagnetism. When μBH/kT<<1 their susceptibility χ does not depend on H but does depend on T — the Curie–Weiss law holds: χ=C`/(T-Δ), where C` and Δ are constants of the substance; c) of ferromagnetic and antiferromagnetic substances above the Curie point θ.
II. The magnetism of conduction electrons in metals and semiconductors. The paramagnetism of conduction electrons (spin paramagnetism) is observed in the alkali metals (Li, K, Na and others), the alkaline-earth metals (Ca, Cr, Ba, Ra) and the transition metals (elements with incomplete 3d, 4d and 5d shells, apart from Fe, Ni, Co and Mn, Cr). Their susceptibility is small (χ=10-5), does not depend on the field and varies little with temperature. In a number of metals (Cu, Ag, Au and others) this paramagnetism is masked by the stronger diamagnetism of the ion cores. The diamagnetism of conduction electrons in metals (Landau diamagnetism) is inherent in all metals, but in most cases it is masked either by the stronger spin paramagnetism of the electrons or by the dia- or paramagnetism of the ion cores. The para- and diamagnetism of conduction electrons in semiconductors. Compared with metals, semiconductors have few conduction electrons, but their number grows as the temperature rises; χ in this case also depends on T. The magnetism of superconductors is due to electric currents flowing in a thin surface layer of thickness ≈10-5 cm. These currents screen the bulk of the superconductor from external magnetic fields, so that in a massive superconductor at T
III. The magnetism of substances with atomic magnetic order (εint>>μBH or εint>>kT).
Ferromagnetism occurs in substances with a positive exchange energy (εex>0): in crystals of Fe, Co, Ni, in a number of REM (Gd, Tb, Dy, Ho, Er, Tm, Yb), in alloys and compounds containing these elements, and also in alloys of Cr and Mn and in compounds of U. Ferromagnetism is characterised by spontaneous magnetisation at temperatures T<θ; at T>θ ferromagnets pass either into the paramagnetic or into the antiferromagnetic state (the latter is observed, for example, in some REM). It is known from experiment, however, that in the absence of an external field ferromagnetic bodies have no resultant magnetisation (apart from the secondary phenomenon of remanent magnetisation). The reason is that at H=0 a ferromagnet breaks up into a large number of microscopic regions of spontaneous magnetisation — domains. The magnetisation vectors of the individual domains are oriented so that the total magnetisation of the ferromagnet is zero. In an external field the domain structure changes and the ferromagnetic specimen acquires a resultant magnetisation (see Magnetisation).
Antiferromagnetism occurs in substances with a negative exchange energy (εex<0): in crystals of Cr and Mn, in a number of REM (Ce, Pr, Nd, Sm, Eu), and also in numerous compounds and alloys containing elements of the transition groups.
The crystal lattice of these substances divides into so-called magnetic sublattices whose spontaneous magnetisation vectors Jki are either antiparallel (collinear antiferromagnetic coupling) or directed at angles to one another other than 0 and 180° (non-collinear coupling, (see Atomic magnetic structure). If the total moment of all the magnetic sublattices of an antiferromagnet is zero, the antiferromagnetism is compensated; if there is a non-zero difference in the spontaneous magnetisations, uncompensated antiferromagnetism, or ferrimagnetism, is observed, which occurs chiefly in crystals of metal oxides with a crystal lattice of the spinel, garnet, perovskite or other mineral type (these are called ferrites). In their electrical properties these substances are semiconductors and dielectrics; in their magnetic properties they resemble ordinary ferromagnets. When the compensation of the magnetic moments in antiferromagnets is disturbed by a weak interaction between the atomic carriers of magnetism, a very small spontaneous magnetisation (≈0.1% of the usual values for ferro- and ferrimagnets) arises in some cases in substances that are called weak ferromagnets (for example haematite α-Fe2O3, the carbonates of a number of metals, the orthoferrites; (see Weak ferromagnetism). The character of the atomic carriers of the magnetic moment differs in ferro- and antiferromagnetic d and f metals, in metallic alloys and compounds, and in non-conducting crystals. In d metals and alloys the main carriers of the atomic magnetic moment are the electrons of what was the incomplete d layer of the isolated atoms. The ferro- or antiferromagnetism they produce is connected with the exchange interaction in the system of collectivised d electrons.
In 4f metals and in dielectric crystals the ordered atomic magnetic structures are formed by magnetic moments localised near the sites of the crystal lattice that are occupied by magnetically active ions. Ordered magnetism also exists in amorphous solids (in supercooled liquids, the so-called metallic glasses), which have a number of specific properties different from the magnetic properties of crystalline magnetic materials.
Substances called spin glasses are also of great interest: they have atomic ordering but no ordering of the localised atomic spin or orbital magnetic moments.
The magnetic state of a ferro- or antiferromagnet in an external magnetic field H is determined not only by the magnitude of the field but also by the previous states of the material (the magnetic history of the specimen). This phenomenon is called hysteresis. Magnetic hysteresis appears as an ambiguity in the dependence of J on H (in the existence of a hysteresis loop). Because of hysteresis, removing the external field is not enough to demagnetise a specimen: at H=0 the specimen retains a remanent magnetisation Jr. To demagnetise the specimen a reverse magnetic field Hc, which is called the coercive force, must be applied. According to the value of Hc, a distinction is drawn between magnetically soft materials (Hc<800 A/m or 10 Oe) and magnetically hard, or high-coercivity, materials (Hc>4 kA/m or 50 Oe). Jr and Hc depend on temperature and as a rule decrease as it rises, tending to zero as T approaches θ.
Scientific and technical problems of magnetism.
The main scientific problems of the modern study of magnetism are:
1) to establish the nature of the exchange interaction and of the interactions responsible for anisotropy in the various types of magnetically ordered crystals; to explain the spectra of the elementary magnetic excitations (magnons) and the mechanisms of their interaction with one another and also with phonons, conduction electrons and others;
2) to construct a theory of the transition from the paramagnetic to the ferromagnetic state. The study of the magnetism of substances is used as a means of investigating chemical bonds and the structure of molecules (magnetochemistry). Studies of the dia- and paramagnetic properties of gases, liquids, solutions and compounds in the solid phase make it possible to work out the details of the physical and chemical processes going on in these bodies and of their structure. The study of dynamic magnetic characteristics (magnetic resonance and relaxation processes) helps in understanding the kinetics of many physical and physico-chemical processes in various substances. Magnetobiology is developing rapidly.
The most important problems in the magnetism of cosmic bodies are: the origin of the magnetic fields of the Earth, the planets, the Sun and the stars (in particular the pulsars) and of extragalactic radio sources (radio galaxies, quasars and others), and the role of magnetic fields in cosmic space.
The problems of the technical application of magnetism are among the most important problems of electrical engineering, radio engineering, electronics, instrument-making and computer technology, navigation, automation and telemechanics. Magnetic flaw detection and magnetic methods of inspection are widely used in technology. Magnetic materials are used to make the magnetic cores of generators, motors, transformers and relays, magnetic amplifiers, magnetic memory elements, compass needles, magnetic recording tape and so on.
Historical note.
The first written evidence of magnetism (China) is more than two thousand years old. It mentions the use of natural permanent magnets as compasses. The works of ancient Greek and Roman scholars mention the attraction and repulsion of magnets and the magnetisation of iron filings in the presence of a magnet (for example in the poem “On the Nature of Things” by the Roman poet and materialist philosopher Lucretius, 1st century BC). In the Middle Ages the magnetic compass came into wide use in Europe (from the 12th century), and attempts were made to study the properties of magnets of different shapes experimentally (the French scholar Pierre de Maricourt, 1269). The results of research on magnetism in the Renaissance were summed up in the work of the English physicist W. Gilbert, “On the Magnet, Magnetic Bodies and the Great Magnet — the Earth” (1600). Gilbert showed in particular that the Earth is a magnetic dipole, and proved that the two unlike poles of a magnet cannot be separated. The study of magnetism was then developed in the works of the French scholar R. Descartes, the Russian scholar F. Aepinus and the French physicist C. Coulomb. Descartes was the author of the first detailed metaphysical theory of magnetism and geomagnetism (“Principles of Philosophy”, part 4, 1644); he started from the existence of a special magnetic substance whose presence and motion caused the magnetism of bodies.
In his treatise “An Attempt at a Theory of Electricity and Magnetism” (1759) Aepinus emphasised the analogy between electrical and magnetic phenomena. As Coulomb showed (1785—89), this analogy has a definite quantitative expression: the interaction of point magnetic poles obeys the same law as the interaction of point electric charges (Coulomb's law). In 1820 the Danish physicist H. Ørsted discovered the magnetic field of an electric current. In the same year the French physicist A. Ampère established the laws of the magnetic interaction of currents and the equivalence of the magnetic properties of a circular current and a thin flat magnet; he explained magnetism by the existence of molecular currents. In the 1830s the German scientists C. Gauss and W. Weber extended the mathematical theory of geomagnetism and developed methods of magnetic measurement.
A new stage in the study of magnetism begins with the work of the English physicist M. Faraday, who gave a consistent interpretation of magnetic phenomena based on the idea that the electromagnetic field is real. A number of major discoveries in electromagnetism (electromagnetic induction — Faraday, 1831; Lenz's rule — E. H. Lenz, 1833, and others), the generalisation of the electromagnetic phenomena that had been discovered in the works of the English physicist J. C. Maxwell (1872) and the systematic study of the properties of ferromagnets and paramagnets (A. G. Stoletov, 1872; the French physicist P. Curie, 1895, and others) laid the foundations of the modern macroscopic theory.
The study of magnetism at the microscopic level became possible after the discovery of the electron-nuclear structure of atoms. On the basis of the classical electron theory of the Dutch physicist H. A. Lorentz, the French scientist P. Langevin in 1905 constructed the theory of diamagnetism and also a quasi-classical theory of paramagnetism. In 1892 the Russian scientist B. L. Rosing and in 1907 P. Weiss (France) put forward the idea of an internal molecular field responsible for the properties of ferromagnets. The discovery of electron spin and its magnetism (S. Goudsmit, G. E. Uhlenbeck, USA, 1925) and the creation of quantum mechanics led to the development of the quantum theory of dia-, para- and ferromagnetism. On the basis of quantum-mechanical ideas (spatial quantisation) the French physicist L. Brillouin in 1926 found how the magnetisation of paramagnets depends on the external magnetic field and the temperature. In 1927 the German physicist F. Hund compared the experimental and theoretical values of the effective magnetic moments of ions in various paramagnetic salts, which revealed the effect of the electric fields of a paramagnetic crystal on the “freezing” of the orbital moments of the ions. Studies of this phenomenon established that the magnetisation of a crystal is determined almost entirely by the spin moments (W. Penney and R. Schlapp; J. Van Vleck, USA, 1932). In the 1930s the quantum-mechanical theory of the magnetic properties of free electrons was constructed (Pauli paramagnetism, 1927; Landau diamagnetism, 1930). Of substantial importance for the further development of the theory of paramagnetism was electron paramagnetic resonance (EPR), predicted by Ya. G. Dorfman (1923) and then discovered by E. K. Zavoisky (1944).
The creation of the quantum theory of ferromagnetism was preceded by the work of the German physicist E. Ising (1925, the two-dimensional model of ferromagnets), of Ya. G. Dorfman (1927, who proved the non-magnetic nature of the molecular field), of the German physicist W. Heisenberg (1926, the quantum-mechanical calculation of the helium atom) and of the German physicists W. Heitler and F. London (1927, the calculation of the hydrogen molecule). The last two works used the effect of the exchange interaction of electrons in the shells of atoms and molecules, discovered in quantum mechanics (P. Dirac, Great Britain, 1926), and established its connection with the magnetic properties of electron systems that obey Fermi–Dirac statistics (the Pauli principle). The quantum theory of ferromagnetism was begun by the work of Ya. I. Frenkel (1928, the collective-electron model) and Heisenberg (1928, the localised-spin model). The treatment of ferromagnetism as a quantum cooperative phenomenon (the American physicists F. Bloch and J. Slater, 1930) led to the discovery of spin waves. In 1932—33 the French physicist L. Néel and L. D. Landau predicted the existence of antiferromagnetism. The study of new classes of magnetic substances — antiferromagnets and ferrites — gave a deeper understanding of the nature of magnetism. The role of magnetoelastic energy in the origin of the magnetic anisotropy energy was established, the theory of domain structure was constructed and methods of studying it experimentally were mastered.
The development of the theory of magnetism was greatly assisted by the creation of new experimental methods of investigating substances. Neutron-diffraction methods made it possible to determine the types of atomic magnetic structure. Ferromagnetic resonance, first discovered and studied in the work of V. K. Arkadiev (1913) and then of J. Griffiths (USA, 1946), and antiferromagnetic resonance (C. Gorter and others, 1951) made it possible to begin experimental studies of magnetic relaxation processes and also provided an independent method of determining the effective anisotropy fields in ferro- and antiferromagnets. Physical methods of investigation based on nuclear magnetic resonance (E. Purcell and others, USA, 1946) and on the Mössbauer effect (1958) have greatly deepened knowledge of the distribution of spin density in matter, especially in metallic ferromagnets. Observations of the scattering of neutrons and of light have made it possible to determine the spin-wave spectra of a number of substances. Alongside the experimental work, various aspects of the theory of magnetism were also developed: the magnetic symmetry of crystals, the ferromagnetism of collectivised electrons, second-order phase transitions and critical phenomena, and models of one-dimensional and two-dimensional ferro- and antiferromagnets.
Advances in the study of the nature of magnetic phenomena have made it possible to synthesise new and promising magnetic materials: ferrites for high-frequency and microwave devices, high-coercivity compounds of the SmCo5 type (see Permanent magnet), transparent ferromagnets, amorphous magnetic materials (including spin glasses, in which the orientations of the atomic magnetic moments are distributed at random over the sites of the crystal lattice), ferro- and antiferromagnetic amorphous materials (the so-called metallic glasses, or metglasses) and others.