Magnetism

Magnetism encompasses the physical phenomena arising from a magnetic field, enabling objects to exert attractive or repulsive forces on one another. Since both electric currents and the intrinsic magnetic moments of elementary particles generate magnetic fields, magnetism constitutes one of the two fundamental facets of electromagnetism.

The most commonly observed magnetic effects are associated with ferromagnetic materials. These materials exhibit strong attraction to magnetic fields and can be permanently magnetized, subsequently generating their own persistent magnetic fields. Conversely, demagnetization of a magnet is also achievable. Ferromagnetism is characteristic of only a limited number of substances, primarily including iron, cobalt, nickel, and their respective alloys.

Every substance demonstrates some form of magnetic behavior. The categorization of magnetic materials is based on their bulk magnetic susceptibility. While ferromagnetism accounts for the majority of magnetic phenomena observed in daily experience, multiple distinct types of magnetism exist. For instance, paramagnetic substances, including aluminum and oxygen, exhibit a weak attraction to an external magnetic field; diamagnetic substances, such as copper and carbon, are weakly repelled; and antiferromagnetic materials, like chromium, display a more intricate interaction with magnetic fields. The magnetic force exerted on paramagnetic, diamagnetic, and antiferromagnetic materials is typically imperceptible without specialized laboratory equipment, leading to their common designation as non-magnetic in ordinary contexts.

The intensity of a magnetic field consistently diminishes with increasing distance from its source, although the precise mathematical correlation between strength and distance is variable. Numerous parameters can affect an object's magnetic field, such as the material's magnetic moment, the object's physical geometry, the magnitude and direction of any internal electric current, and the object's temperature.

History

The phenomenon of magnetism was initially observed in antiquity, as individuals recognized that lodestones—naturally magnetized fragments of the mineral magnetite—possessed the ability to attract iron. The term magnet originates from the Greek phrase μαγνῆτις λίθος magnētis lithos, signifying 'the Magnesian stone' or 'lodestone'. Aristotle, in ancient Greece, credited Thales of Miletus (circa 625–545 BCE) with initiating what might be considered the earliest scientific discourse on magnetism. Furthermore, the ancient Indian medical treatise Sushruta Samhita details the application of magnetite for extracting arrows lodged within the human body.

In ancient China, the initial literary mention of magnetism appears in the 4th-century BCE text Guiguzi, named after its author. The 2nd-century BCE annals, Lüshi Chunqiu, additionally record: "The lodestone causes iron to approach; a certain force attracts it." The earliest documented instance of a needle's attraction is found in the 1st-century work Lunheng (Balanced Inquiries), stating: "A lodestone attracts a needle." The 11th-century Chinese polymath Shen Kuo, in his Dream Pool Essays, was the first to describe the magnetic needle compass and its enhancement of navigational precision through the application of the astronomical concept of true north. By the 12th century, the Chinese had adopted the lodestone compass for navigational purposes, crafting directional spoons from lodestone such that the handle consistently indicated south.

By 1187, Alexander Neckam became the first European to document the compass and its application in navigation. In 1269, Peter Peregrinus de Maricourt authored the Epistola de magnete, which stands as the earliest surviving treatise detailing the characteristics of magnets. Subsequently, in 1282, Al-Ashraf Umar II, a Yemeni physicist, astronomer, and geographer, provided discussions on the properties of magnets and dry compasses.

Leonardo Garzoni's sole surviving work, Due trattati sopra la natura, e le qualità della calamita (Two treatises on the nature and qualities of the magnet), represents the earliest known modern scholarly examination of magnetic phenomena. Composed around 1580 but never formally published, this treatise nevertheless achieved considerable circulation. Notably, Niccolò Cabeo referenced Garzoni as an expert in magnetism, and Cabeo's *Philosophia Magnetica* (1629) is largely considered a re-elaboration of Garzoni's work. Giovanni Battista Della Porta was also familiar with Garzoni's treatise.

In 1600, William Gilbert published his treatise De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure (On the Magnet and Magnetic Bodies, and on the Great Magnet the Earth). This work detailed numerous experiments conducted with his model earth, known as the terrella. From these experiments, he concluded that Earth itself possessed a magnetic field, which explained why compasses pointed north, thereby challenging earlier beliefs that attributed this phenomenon to the pole star Polaris or a large magnetic island at the North Pole.

The foundational understanding of electromagnetism emerged in 1819 through the contributions of Hans Christian Ørsted, a professor at the University of Copenhagen, who, through the serendipitous observation of a compass needle deflecting near an energized wire, demonstrated that electric currents generate magnetic fields. This pivotal discovery is now recognized as Ørsted's Experiment. In 1820, Jean-Baptiste Biot and Félix Savart independently formulated the Biot–Savart law, providing a mathematical description for the magnetic field produced by a current-carrying conductor. Concurrently, André-Marie Ampère performed extensive systematic experiments, establishing that the magnetic force between any two direct current loops equals the sum of the individual forces exerted by each current element of one circuit on every current element of the other.

In 1831, Michael Faraday demonstrated that a fluctuating magnetic flux induces an electromotive force (voltage) in a conductive loop. In 1835, Carl Friedrich Gauss postulated, drawing from Ampère's original force law, that all magnetic phenomena originate from the relative motion of elementary point charges. Wilhelm Eduard Weber subsequently developed Gauss's theoretical framework into what is known as Weber electrodynamics.

Beginning around 1861, James Clerk Maxwell synthesized and significantly extended these disparate insights, formulating Maxwell's equations, which unified electricity, magnetism, and optics under the comprehensive theory of electromagnetism. Nevertheless, Gauss's conceptualization of magnetism did not entirely align with Maxwell's electrodynamics. In 1905, Albert Einstein leveraged Maxwell's equations as a foundational premise for his theory of special relativity, positing that these laws must remain invariant across all inertial reference frames. Consequently, Gauss's perspective, which viewed magnetic force as a relativistic effect of relative velocities, was partially reintegrated into electrodynamic theory.

Electromagnetism has undergone continuous evolution into the 21st century, becoming integrated into more fundamental theoretical frameworks such as gauge theory, quantum electrodynamics, electroweak theory, and ultimately, the Standard Model of particle physics.

Sources

Fundamentally, magnetism originates from three primary sources:

1. Electric current
2. Spin magnetic moments of elementary particles
3. Changing electric fields

The inherent magnetic characteristics of materials primarily stem from the magnetic moments associated with their atoms' orbiting electrons. Atomic nuclei possess magnetic moments that are typically orders of magnitude smaller than those of electrons, rendering them negligible when considering the overall magnetization of materials. Nonetheless, nuclear magnetic moments hold significant importance in alternative applications, notably in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI).

Typically, the vast number of electrons within a material are configured in a manner that results in the cancellation of their magnetic moments (both orbital and intrinsic). This phenomenon is partly attributable to electrons pairing with opposing intrinsic magnetic moments, a consequence of the Pauli exclusion principle (electron configuration), and forming filled subshells that exhibit no net orbital motion. In both scenarios, electrons preferentially assume configurations where the magnetic moment of one electron is counteracted by the opposing moment of another. Furthermore, even when the electron configuration is characterized by unpaired electrons or incompletely filled subshells, the magnetic moments contributed by individual electrons within the solid frequently align in diverse, random orientations, thereby preventing the material from exhibiting macroscopic magnetism.

Under certain conditions—either spontaneously or induced by an external magnetic field—the electron magnetic moments within a material will, on average, align. Consequently, an appropriate material can generate a substantial net magnetic field.

A material's magnetic behavior is contingent upon its structural properties, especially its electron configuration, and is also significantly influenced by temperature. Elevated temperatures introduce random thermal motion, which impedes the electrons' ability to sustain alignment.

Categories of Magnetism

Diamagnetism

Diamagnetism is an inherent property of all materials, characterized by their tendency to oppose an applied magnetic field and consequently be repelled by it. Nevertheless, in substances exhibiting paramagnetic characteristics—a propensity to augment an external magnetic field—the paramagnetic response predominates. Consequently, despite its ubiquitous presence, diamagnetic behavior is discernible only in materials that are exclusively diamagnetic. Such materials lack unpaired electrons, precluding their intrinsic electron magnetic moments from generating any macroscopic effect. In these instances, magnetization originates from the orbital motions of electrons, a phenomenon that can be classically conceptualized as follows:

Upon the introduction of a material into a magnetic field, electrons orbiting the nucleus encounter a Lorentz force from the magnetic field, in addition to their Coulombic attraction to the nucleus. The direction of an electron's orbit dictates whether this force augments the centripetal force, drawing electrons closer to the nucleus, or diminishes it, pushing them further away. This mechanism systematically enhances orbital magnetic moments oriented antiparallel to the field while reducing those aligned parallel to it, consistent with Lenz's law. The outcome is a minor macroscopic magnetic moment, directed oppositely to the applied field.

This classical description serves merely as a heuristic model; the Bohr–Van Leeuwen theorem demonstrates that diamagnetism is incompatible with classical physics, necessitating a quantum-mechanical framework for accurate comprehension.

While all materials exhibit this orbital response, in paramagnetic and ferromagnetic substances, the diamagnetic effect is superseded by the considerably more potent influences arising from unpaired electrons.

Paramagnetism

Paramagnetic materials are characterized by the presence of unpaired electrons, meaning atomic or molecular orbitals containing precisely one electron. In contrast to paired electrons, which are constrained by the Pauli exclusion principle to possess intrinsic ('spin') magnetic moments oriented in opposing directions—thereby canceling their magnetic fields—an unpaired electron's magnetic moment can align freely in any orientation. Upon the application of an external magnetic field, these magnetic moments tend to align with the field, consequently reinforcing it.

Ferromagnetism

Similar to paramagnetic substances, ferromagnetic materials possess unpaired electrons. However, beyond the intrinsic magnetic moments' inclination to align parallel to an applied field, these materials also exhibit a propensity for these moments to orient parallel to each other, thereby maintaining a lower-energy state. Consequently, even without an external magnetic field, the electron magnetic moments within the material spontaneously align in parallel.

Each ferromagnetic substance possesses a distinct temperature, known as the Curie temperature or Curie point, beyond which it ceases to exhibit ferromagnetic properties. This transition occurs because the inherent thermal tendency towards disorder surpasses the energy reduction conferred by ferromagnetic ordering.

Ferromagnetism is observed in a limited number of substances; prominent examples include iron, nickel, cobalt, their respective alloys, and certain alloys of rare-earth metals.

Magnetic Domains

The atomic magnetic moments within a ferromagnetic material induce behavior analogous to minute permanent magnets. These moments coalesce and align into discrete regions of uniform alignment, termed magnetic domains or Weiss domains. Magnetic domains are observable using a magnetic force microscope, which delineates magnetic domain boundaries appearing as distinct lines. Various scientific methodologies exist for the physical visualization of magnetic fields.

Should a domain exceed a critical molecular count, it becomes unstable and consequently subdivides into two domains oriented in opposing directions, thereby achieving enhanced stability.

Upon exposure to a magnetic field, domain boundaries shift, causing domains aligned with the field to expand and become predominant within the structure (indicated by the dotted yellow area on the left). If the magnetizing field is subsequently removed, these domains may not revert to their original unmagnetized configuration. This process leads to the ferromagnetic material retaining magnetization, thereby forming a permanent magnet.

A material achieves magnetic saturation when it is magnetized with sufficient intensity for a single dominant domain to encompass all others. Heating a magnetized ferromagnetic material to its Curie point temperature induces molecular agitation, causing magnetic domains to lose their organized structure and consequently cease exhibiting magnetic properties. Upon cooling, this domain alignment spontaneously re-establishes itself, a process conceptually similar to the crystallization of a liquid into a solid.

Antiferromagnetism

In contrast to ferromagnets, antiferromagnets exhibit a propensity for the intrinsic magnetic moments of adjacent valence electrons to align in opposite directions. A substance is classified as antiferromagnetic when all its atoms are arranged such that each neighboring magnetic moment is anti-parallel. Antiferromagnets possess a net magnetic moment of zero, as opposing adjacent moments cancel each other, resulting in no external magnetic field generation. These materials are less prevalent than other magnetic types and are primarily observed at low temperatures. Under fluctuating temperature conditions, antiferromagnets may display both diamagnetic and ferromagnetic characteristics.

Certain materials exhibit a preference for neighboring electron spins to orient in opposing directions; however, no geometric configuration permits each pair of neighbors to be perfectly anti-aligned. This phenomenon is termed a canted antiferromagnet or spin ice, representing an instance of geometrical frustration.

Ferrimagnetism

Similar to ferromagnetism, ferrimagnets maintain their magnetization even without an external magnetic field. Nevertheless, akin to antiferromagnets, adjacent electron spin pairs typically align in opposing directions. These characteristics are not mutually exclusive because, within an optimal geometric configuration, the magnetic moment contributed by the sublattice of electrons oriented in one direction surpasses that from the sublattice aligned in the opposite direction.

The majority of ferrites exhibit ferrimagnetic properties. Magnetite, the initial magnetic substance identified, is a ferrite that was initially categorized as a ferromagnet. However, Louis Néel subsequently refuted this classification upon his discovery of ferrimagnetism.

Superparamagnetism

When a ferromagnet or ferrimagnet is reduced to a sufficiently small size, it behaves as a single magnetic spin influenced by Brownian motion. Its reaction to an external magnetic field qualitatively resembles that of a paramagnet, yet it exhibits a significantly greater magnitude.

Nagaoka Magnetism

Japanese physicist Yosuke Nagaoka theorized a form of magnetism within a square, two-dimensional lattice, where each lattice node contained a single electron. He proposed that, under specific conditions, the removal of one electron would result in the lattice's energy being minimized exclusively when all remaining electron spins were parallel.

An experimental realization of a similar phenomenon was achieved by configuring atoms within a triangular moiré lattice composed of molybdenum diselenide and tungsten disulfide monolayers. The application of a weak magnetic field and a voltage induced ferromagnetic behavior when the electron count exceeded the number of lattice nodes by 100–150%. These surplus electrons delocalized and subsequently paired with lattice electrons, forming "doublons." This delocalization was contingent upon the lattice electrons possessing aligned spins. Consequently, these doublons generated localized ferromagnetic regions. This magnetic behavior was observed at a temperature of 140 millikelvins.

Other Forms of Magnetism

• Metamagnetism
• Molecule-Based Magnets
• Single-Molecule Magnets
• Amorphous Magnets
• Altermagnetism

Electromagnetism

An electromagnet constitutes a class of magnet where the magnetic field is generated by the flow of an electric current. This magnetic field ceases to exist upon the discontinuation of the current. Typically, electromagnets are constructed from numerous tightly wound turns of wire, which are responsible for producing the magnetic field. These wire coils are frequently wound around a magnetic core, often composed of a ferromagnetic or ferrimagnetic material like iron; this core serves to concentrate the magnetic flux, thereby enhancing the magnet's strength.

A primary advantage of an electromagnet, in contrast to a permanent magnet, is the capacity for rapid modulation of its magnetic field through precise control of the electric current within its winding. Nevertheless, unlike a permanent magnet, which operates without external power, an electromagnet demands a continuous current supply to sustain its magnetic field.

Electromagnets are extensively utilized as integral components within various electrical devices, encompassing motors, generators, relays, solenoids, loudspeakers, hard disk drives, MRI machines, scientific instruments, and magnetic separation equipment. Industrially, electromagnets are also deployed for the acquisition and relocation of substantial ferrous objects, such as scrap iron and steel. The discovery of electromagnetism occurred in 1820.

Magnetism, Electricity, and Special Relativity

As a direct consequence of Einstein's theory of special relativity, electricity and magnetism are fundamentally intertwined. Both magnetism in isolation from electricity and electricity without magnetism are inconsistent with special relativity, primarily due to phenomena such as length contraction, time dilation, and the velocity-dependent nature of the magnetic force. However, when both electrical and magnetic aspects are comprehensively considered, the resultant theory, electromagnetism, demonstrates full consistency with special relativity. Notably, a phenomenon perceived as purely electric or purely magnetic by one observer may appear as a composite of both to another, indicating that the relative contributions of electricity and magnetism are contingent upon the frame of reference. Therefore, special relativity integrates electricity and magnetism into a unified, inseparable phenomenon known as electromagnetism, analogous to the integration of space and time into spacetime by general relativity.

All empirical observations concerning electromagnetism extend to phenomena predominantly characterized as magnetism; for instance, disturbances within a magnetic field are invariably associated with a non-zero electric field and propagate at the speed of light.

Magnetic Fields in Materials

In a vacuum, the relationship is expressed as: B = μ<!-- μ --> §2122§ H , {\displaystyle \mathbf {B} \ =\ \mu _{0}\mathbf {H} ,} where μ§4243§ represents the permeability of free space.

Within a material, the corresponding equation is: B = μ<!-- μ --> §2122§ ( H + M ) .   {\displaystyle \mathbf {B} \ =\ \mu _{0}(\mathbf {H} +\mathbf {M} ).\ }

The term μ§34§M is designated as magnetic polarization.

When the magnetic field H is of low magnitude, the magnetization M in diamagnetic or paramagnetic materials exhibits an approximately linear response: $${\displaystyle \mathbf {M} =\chi \mathbf {H} ,}$$ This constant of proportionality is termed the magnetic susceptibility. Consequently, μ<!-- μ --> §4849§ ( H + M ) = μ §7778§ ( §8384§ + χ ) H = μ r μ §115116§ H = μ H . {\displaystyle \mu _{0}(\mathbf {H} +\mathbf {M} )\ =\ \mu _{0}(1+\chi )\mathbf {H} \ =\ \mu _{r}\mu _{0}\mathbf {H} \ =\ \mu \mathbf {H} .}

Conversely, in hard magnetic materials like ferromagnets, the magnetization M does not exhibit a proportional relationship with the applied field and typically retains a non-zero value even in the absence of an external field H.

Magnetic Force

Magnetic phenomena are mediated by the magnetic field. Both electric currents and magnetic dipoles generate magnetic fields, and these generated fields subsequently exert magnetic forces upon other particles situated within them.

The genesis and dynamics of the fields responsible for these forces are elucidated by Maxwell's equations, which reduce to the Biot–Savart law for steady currents. Consequently, magnetism manifests whenever electrically charged particles are in motion; this includes, for instance, the movement of electrons within an electric current or, in specific scenarios, the orbital motion of electrons around an atomic nucleus. Furthermore, magnetic effects originate from intrinsic magnetic dipoles, which are a consequence of quantum-mechanical spin.

The conditions that generate magnetic fields—namely, moving charges in currents or within atoms, and intrinsic magnetic dipoles—are precisely those in which a magnetic field exerts a force. The subsequent section presents the formula for forces on a moving charge. Information regarding forces on an intrinsic dipole is available elsewhere.

A charged particle traversing a magnetic field B experiences a Lorentz force F, which is defined by the cross product: $${\displaystyle \mathbf {F} =q(\mathbf {v} \times \mathbf {B} ),}$$ Here, ${\displaystyle q}$ represents the particle's electric charge, and ${\displaystyle v}$ denotes its velocity vector.

Given its nature as a cross product, the resulting force acts perpendicularly to both the particle's motion and the magnetic field. The magnitude of this force is quantified by the equation: $${\displaystyle F=qvB\sin \theta \,}$$, where ${\displaystyle \theta }$ represents the angle formed between v and B.

The orientation of the force vector is ascertainable through the application of the right-hand rule, which establishes the direction of ${\displaystyle \mathbf {v} \times \mathbf {B} }$, and by considering the polarity of the charge ${\displaystyle q}$ (for a negative charge, the force vector is directed oppositely to the thumb's indication).

Magnetic Dipoles

Every identified magnet functions as a dipole, possessing distinct "north" and "south" poles. These designations originate from the Earth's magnetic field, which exerts a force causing a magnet's poles to orient towards regions proximate to the Earth's geographic North and South Poles, specifically the terrestrial north and south magnetic poles. Fundamentally, a magnet's north pole exhibits attraction to the south pole of another magnet.

Magnetic fields inherently store energy, and physical systems naturally evolve towards states of reduced energy. Upon the introduction of a diamagnetic material into a magnetic field, its constituent magnetic dipole typically aligns with an opposing polarity to the external field, consequently diminishing the overall field strength. Conversely, when a ferromagnetic material is situated within a magnetic field, its magnetic dipoles align with the applied field, leading to the expansion of the domain walls within its magnetic domains.

Magnetic Monopoles

Given that a bar magnet derives its ferromagnetism from electrons uniformly distributed throughout its structure, bisecting it results in two smaller bar magnets. Although magnets are characterized by distinct north and south poles, these poles are inherently inseparable. A magnetic monopole, if it were to exist, would represent a novel and fundamentally distinct magnetic entity. Such a monopole would manifest as an isolated north pole, uncoupled from a south pole, or vice versa. These hypothetical monopoles would possess a "magnetic charge," conceptually analogous to electric charge. Notwithstanding rigorous investigations conducted since 1931, their existence remains unconfirmed.

Nonetheless, certain theoretical physics models posit the existence of these magnetic monopoles. In 1931, Paul Dirac noted that the inherent symmetry between electricity and magnetism suggests that, analogous to how quantum theory predicts the observation of individual positive or negative electric charges without their opposing counterparts, isolated South or North magnetic poles should also be observable. Dirac further demonstrated, through quantum theory, that the existence of magnetic monopoles would provide an explanation for the quantization of electric charge—specifically, why elementary particles exhibit charges that are integer multiples of the electron's charge.

Specific grand unified theories forecast the presence of monopoles, which, in contrast to elementary particles, are characterized as solitons (localized packets of energy). Early estimations derived from these models regarding the quantity of monopoles generated during the Big Bang conflicted with cosmological observations; these monopoles were predicted to be so abundant and massive that they would have long since arrested the universe's expansion. Nevertheless, the concept of cosmic inflation, partly motivated by this very issue, successfully resolved the discrepancy by developing models where monopoles exist but are sufficiently rare to align with contemporary observations.

Units of Measurement

International System of Units (SI)

Alternative Units

• Gauss – the CGS-Gaussian unit employed for magnetic field strength (designated as B).
• Oersted – the CGS-Gaussian unit utilized for magnetizing field strength (designated as H).
• Maxwell – the CGS-Gaussian unit for magnetic flux.
• μ§23§ – the widely recognized symbol representing the permeability of free space.

Biological Systems

Organisms exhibit magnetoception, the capacity to detect magnetic fields. While certain biological materials are ferromagnetic, their magnetic properties' functional significance, as opposed to being a mere consequence of iron content, remains ambiguous. For example, marine mollusks like chitons synthesize magnetite to reinforce their teeth, and magnetite production also occurs in human bodily tissues.

Magnetobiology is the scientific discipline dedicated to investigating the influence of magnetic fields on living organisms; conversely, magnetic fields intrinsically generated by organisms are termed biomagnetism. Given that numerous biological entities are predominantly composed of water, which is a diamagnetic substance, exceptionally potent magnetic fields possess the capacity to repel these living systems.

The Interpretation of Magnetism Through Relative Velocities

Subsequent to 1820, André-Marie Ampère conducted extensive experiments quantifying the forces exerted between direct currents, specifically investigating magnetic interactions among non-parallel wires. His research culminated in the formulation of a force law, subsequently bearing his name. By 1835, Carl Friedrich Gauss recognized that Ampère's original force law could be elucidated through a broader application of Coulomb's law.

Gauss's force law postulates that the electromagnetic force F §1213§ {\textstyle \mathbf {F} _{1}} acting on a point charge q §3435§ {\displaystyle q_{1}} , which follows trajectory r §5859§ ( t ) {\displaystyle \mathbf {r} _{1}(t)} , when situated near another point charge ${\displaystyle q_{2}}$ with trajectory ${\displaystyle \mathbf {r} _{2}(t)}$ in a vacuum, is equivalent to the central force.

Maxwell's electrodynamics, developed since 1870, posits the existence of distinct electric and magnetic fields. Within this framework, the electromagnetic force is quantifiable via the Lorentz force, which, similar to the Weber force, exhibits a velocity dependence. Nevertheless, Maxwell's electrodynamics demonstrates incomplete compatibility with the contributions of Ampère, Gauss, and Weber, particularly in the quasi-static domain. Specifically, Ampère's original force law and the Biot-Savart law achieve equivalence solely when the conductor loop generating the field is closed. Consequently, Maxwell's electrodynamics diverges from the magnetic interpretations proposed by Gauss and Weber, as it precludes the derivation of magnetic force from a central force.

Quantum-Mechanical Origins of Magnetism

Although classical physics offers heuristic explanations, a comprehensive understanding of diamagnetism, paramagnetism, and ferromagnetism necessitates quantum theory. In 1927, Walter Heitler and Fritz London developed a foundational quantum-mechanical model demonstrating the formation of hydrogen molecules from hydrogen atoms, specifically from the atomic hydrogen orbitals ${\displaystyle u_{A}}$ and ${\displaystyle u_{B}}$ centered at nuclei A and B. The implication of this process for magnetism is not immediately apparent but will be elucidated subsequently.

The Heitler–London theory posits the formation of two-body molecular ${\displaystyle \sigma }$-orbitals, with the resultant orbital expressed as: ψ<!-- ψ --> ( r §3435§ , r §5051§ ) = §5960§ §6263§ ( u A ( r §9293§ ) u B ( r §114115§ ) + u B ( r §138139§ ) u A ( r §160161§ ) ) {\displaystyle \psi (\mathbf {r} _{1},\,\,\mathbf {r} _{2})={\frac {1}{\sqrt {2}}}\,\,\left(u_{A}(\mathbf {r} _{1})u_{B}(\mathbf {r} _{2})+u_{B}(\mathbf {r} _{1})u_{A}(\mathbf {r} _{2})\right)}

Specifically, the final product implies that a primary electron, designated as r§23§, occupies an atomic hydrogen orbital centered on the second nucleus, while the secondary electron orbits the first nucleus. This "exchange" phenomenon represents a fundamental manifestation of the quantum-mechanical principle asserting the indistinguishability of identical particles. This phenomenon is critical not only for the formation of chemical bonds but also for the understanding of magnetism. Consequently, the term "exchange interaction" arises in this context, a concept fundamental to the genesis of magnetism. This interaction is significantly more potent, by factors ranging from approximately 100 to 1000, than the energies originating from electrodynamic dipole-dipole interactions.

Regarding the spin function, denoted as χ<!-- χ --> ( s §1718§ , s §2728§ ) {\displaystyle \chi (s_{1},s_{2})} , which governs magnetic properties, Pauli's principle dictates that a symmetric orbital (indicated by a positive sign, as previously discussed) must be combined with an antisymmetric spin function (indicated by a negative sign), and conversely. This leads to the following formulation:

The latter tendency predominates in metals such as iron, cobalt, and nickel, as well as in certain rare earth elements, which exhibit ferromagnetic properties. Conversely, the former tendency is dominant in most other metals, rendering them either nonmagnetic (e.g., sodium, aluminum, and magnesium) or antiferromagnetic (e.g., manganese). Diatomic gases are almost universally diamagnetic, rather than paramagnetic. Nevertheless, the oxygen molecule constitutes a significant exception, particularly in life sciences, due to the involvement of its π-orbitals.

The principles established by Heitler and London can be extended to formulate the Heisenberg model of magnetism, as proposed by Heisenberg in 1928.

Consequently, the elucidation of these phenomena fundamentally relies on the intricate aspects of quantum mechanics, while electrodynamics primarily addresses their macroscopic phenomenology.

References

References

Bibliography

Bibliography

• The Exploratorium Science Snacks – Subject: Physics/Electricity & Magnetism
• A collection of magnetic structures – MAGNDATA