# Ferromagnetism **Ferromagnetism** is the property by which a material holds a [[Magnetization|magnetization]] of its own, with no applied field, below a critical temperature. It is the only form of magnetism strong enough to be noticed without instruments, and it is the mechanism behind every permanent magnet, [[Electric_motor|electric motor]] and [[Transformer|transformer]] core. Above its critical temperature the same material is an ordinary [[Magnetic_field|paramagnet]], whose weak response vanishes with the applied field; below it the [[Atom|atomic]] moments lock into agreement across macroscopic regions. The ordering is a genuine [[Phase_transition|phase transition]] whose order parameter is the mean moment per site, `eta = <s_k>`.[^likharev-ising] In the microsim below the reader works the mean-field description of that transition. One control sets T/T_c over the range 0 to 2 at zero field, and the display answers with three linked panels: the Weiss self-consistency condition `eta = tanh((T_c*eta + h)/T)`, solved by Newton's method, with its roots marked; the Landau free energy `f = a*t*eta^2 + (b/2)*eta^4 - h*eta`, which turns from a single bowl into a double well as T falls through T_c; and the resulting η*(T) curve, which reaches 0.525 at T/T_c = 0.9 and 0.958 at T/T_c = 0.5 (derived).[^likharev-weiss][^likharev-landau][^derived] Switching the control to the field h at a fixed T below T_c traces the coercive jump — the hysteresis loop — on the lattice inherited from this page's parent. Above T_c the same algebra gives the Curie–Weiss susceptibility `chi = 1/(T - T_c)`.[^likharev-weiss] On the [[Physics]] flagship this article serves Part IV — Branches and fields, section *Ferromagnetism: mean field and hysteresis* (row P63), and it is placed again on [[Materials_science]], where the practical distinctions between soft and hard magnetic materials belong. The sim is the sibling of the cluster's C33 root: [[Ising_model]] owns the `TwoStateLattice`, and this page reuses that lattice for its field sweep while adding the closed-form Weiss and Landau panels of its own. ## Terms A ferromagnet is defined by what survives the removal of the field. In a paramagnet the moments align only while a field is applied and randomise as soon as it is removed; in a ferromagnet a spontaneous magnetization remains, and reversing it requires work. Materials in which neighbouring moments prefer to oppose rather than agree are antiferromagnets, which show no net moment, and those in which opposed sublattices have unequal moments are ferrimagnets, which do — the magnetite of a lodestone among them. All of these are distinct from diamagnetism, the universal weak repulsion of matter from a field. The working vocabulary of the [[Hysteresis|hysteresis]] loop supplies the rest of the terms. Saturation magnetization is the value reached when every moment is aligned; remanence is what is left when the field is returned to zero; coercivity is the reverse field needed to bring the magnetization back to zero. A material with low coercivity is called magnetically soft and is used where the magnetization must follow an alternating field cheaply; one with high coercivity is magnetically hard and is used where it must not follow anything at all. ## Materials The elemental ferromagnets at ordinary temperatures are the [[Transition_metal|transition metals]] [[Iron|iron]], [[Cobalt|cobalt]] and [[Nickel|nickel]], joined by several rare-earth elements at lower temperatures; in every case the ordering involves partly filled inner shells whose moments are not quenched by the surrounding lattice. Most magnetic materials of practical importance are [[Alloy|alloys]] or compounds rather than elements, because alloying is how the [[Microstructure|microstructure]] and therefore the coercivity is controlled. [[Steel|Steels]] and iron–silicon sheet are engineered to be soft, with grains oriented and impurities removed so that domain walls move easily; permanent-magnet alloys are engineered to be hard, with fine precipitates and [[Grain_boundary|grain boundaries]] that pin those same walls. [[Annealing_(materials_science)|Annealing]], [[Sintering|sintering]] and grain alignment are therefore as decisive for the magnetic properties as the chemistry is, which is why the material selection for a magnet is treated on the [[Materials_science|materials science]] side of this cluster rather than here. ### Unusual materials Ferromagnetic order is not confined to the classic metals. Oxides and other insulating compounds order magnetically through indirect exchange paths, giving ferrimagnets that conduct poorly and are therefore useful at high frequency. [[Amorphous_metal|Amorphous metallic]] ribbons, quenched fast enough to have no [[Crystal_structure|crystal structure]] at all, are among the softest magnetic materials known, because a structure with no crystal axes has little anisotropy to pin a domain wall. At the other extreme, magnetic [[Nanoparticle|nanoparticles]] small enough to hold only one domain lose their hysteresis entirely below a size threshold, thermal agitation flipping the whole particle — a state described as superparamagnetic, and the limiting factor in magnetic recording density, since a bit that is small enough to pack tightly is small enough to forget itself. Ferromagnetic order has also been reported in compounds containing no element that is magnetic in bulk, where the moments arise from defects or from the edges of a low-dimensional structure rather than from a partly filled inner shell. ### Electrically induced ferromagnetism Because magnetic order depends on the density and spin polarisation of the electrons that mediate it, order can in some systems be switched electrically rather than chemically. In dilute magnetic [[Semiconductor|semiconductors]] the carriers introduced by [[Doping_(semiconductor)|doping]] couple the dilute magnetic ions, so a gate voltage that changes the carrier density changes the ordering temperature with it; in ultrathin metallic films an applied field can shift the surface charge enough to alter the anisotropy. The interest is not in the strength of the effect, which is modest, but in its reversibility: a magnetic state that can be written with a voltage rather than a current is a state that can be written without the resistive losses that dominate magnetic memory. ## Explanation Two questions have to be answered separately, and confusing them is the usual source of trouble. Where does an individual atom's moment come from, and why do neighbouring moments agree? The first is answered by [[Atomic_orbital|atomic structure]] and involves no cooperation between atoms at all; the second by an interaction that is electrostatic in origin despite being magnetic in effect. A third question — why a magnetized specimen nevertheless shows no field until it is magnetized — is answered by neither, and requires the domain structure treated further below. ### Origin of atomic magnetism An atom's moment comes from the [[Spin_(physics)|spin]] and orbital angular momentum of its electrons, and only unpaired electrons contribute, since [[Pauli_exclusion_principle|paired]] spins cancel. For ions whose orbital contribution is quenched the moment follows the spin-only formula `mu_eff = mu_B*sqrt(n*(n + 2))`, with n the number of unpaired electrons and the Bohr magneton μ_B = 9.274 × 10⁻²⁴ J/T.[^boyd-moment] The formula is built on a free-electron g-factor of exactly 2, against the measured 2.0023, and it applies only to ions with A or E ground terms.[^boyd-moment][^boyd-spinonly] Its sensitivity to electron count is large: a d⁵ ion carries 5.92 μ_B when high-spin and 1.73 μ_B when low-spin, a difference produced entirely by how the same five [[Electron|electrons]] are arranged.[^boyd-spinonly] Non-interacting moments of this kind give the Curie law, `chi = N*mu_0*mu^2/(3*k_B*T)`, in which the susceptibility falls off as 1/T and never orders.[^boyd-curie] Ferromagnetism is what happens when those moments are not independent. ### Exchange interaction The coupling that aligns neighbouring moments is not the magnetic force between them, which is far too weak to survive room temperature, but the exchange interaction: a consequence of the Pauli principle, by which the electrostatic energy of two electrons depends on whether their spins are parallel. [[Werner_Heisenberg|Heisenberg]] identified this mechanism as the origin of ferromagnetism in 1928.[^heisenberg1928] Its effect is captured in the Ising energy `E = -J*sum s_k*s_k' - h*sum s_k`, where J is the exchange constant and the sum runs over neighbouring pairs; positive J favours agreement.[^likharev-ising] Because exchange is a short-range effect of overlapping orbitals, it depends steeply on interatomic spacing and on the filling of the shell, which is why small changes of composition or lattice constant can turn a ferromagnet into an antiferromagnet. Where the orbitals do not overlap directly the coupling is carried by an intermediary — by a shared anion in an insulating oxide, or by the conduction electrons in a metal — and such indirect couplings can be positive or negative depending on distance, which is the origin of the competing bonds that frustrate a spin glass. ### Magnetic anisotropy Exchange alone says which neighbours should agree, not which direction they should point. That is fixed by anisotropy: spin–orbit coupling ties the moment to the crystal axes, so some directions are easy and others hard, and the shape of the specimen adds a further preference through the field its own poles create. Anisotropy is the reason a magnetized bar keeps its direction rather than relaxing, and the reason coercivity can be engineered at all — a material whose moments are strongly pinned to a hard axis needs a large reverse field to switch. The Ising model's two states, s = ±1, are the idealisation of an easy axis so strong that no intermediate direction exists; real ferromagnets with weak anisotropy are better described by models with continuous spin directions, and the difference is not cosmetic, since a continuous symmetry changes which dimensions can order at all. Anisotropy energies are small compared with exchange energies — they set the direction of the order, not its existence — yet almost every engineering property of a magnet, from its coercivity to its losses at frequency, is decided by that small term. ### Magnetic domains A uniformly magnetized specimen carries a large field outside itself and pays for it in energy, so a real ferromagnet breaks into [[Magnetic_domain|domains]], each magnetized to saturation but in different directions, so that the external field largely cancels. The boundary between two domains is a wall of finite width, across which the direction turns gradually, and the width is set by the competition between exchange, which resists rapid turning, and anisotropy, which resists intermediate directions. A domain-wall estimate of this kind on a square lattice gives an ordering temperature of 2J/ln 3 ≈ 1.82 J, against the exact two-dimensional value of 2.269 J.[^likharev-walls] Magnetization proceeds not by rotating moments but by moving these walls, which is why it is the defects that pin walls — precipitates, dislocations, grain boundaries — that decide a material's coercivity. Because a wall pinned at a defect breaks free suddenly rather than gliding, the magnetization of a real specimen advances in small discontinuous jumps that can be heard as noise in a pickup coil, direct evidence that domains exist. The domain structure also explains the demagnetized state: a specimen that has never been magnetized is not one without moments but one whose domains are arranged so that their contributions cancel. ### Magnetized materials The hysteresis loop is the record of that wall motion. Starting from an unmagnetized state, increasing the field grows the domains aligned with it at the expense of the others until the specimen saturates; reducing the field to zero leaves a remanent magnetization, because the walls do not return to where they started; and reversing the field far enough drives the magnetization through zero at the coercive field. The loop's area is the work dissipated per cycle, which is why transformer cores must be magnetically soft. The microsim reaches the same loop from the mean-field side. Holding T below T_c and sweeping h, the Weiss condition `eta = tanh((T_c*eta + h)/T)` has three roots over a range of fields, two stable and one unstable; the system stays on whichever stable branch it is on until that branch disappears, then jumps. In the Landau picture the same event is one well of `f = a*t*eta^2 + (b/2)*eta^4 - h*eta` rising above the other and finally vanishing.[^likharev-landau] That jump is the model's coercive field, and the sim traces it simultaneously on the parent [[Ising_model|Ising]] lattice, where the reader can watch the reversed domains nucleate and spread. The correspondence is instructive but not exact, and the sim labels it ILLUSTRATIVE: mean-field theory has no domain walls and no defects, so its coercive field is a property of the equation rather than of a material. ### Curie temperature The [[Curie_temperature|Curie temperature]] is where the spontaneous magnetization vanishes. Weiss's mean-field argument reaches it by replacing every neighbour of a site by its average: a site feels an effective field `h_ef = h + 2*J*d*eta` on a lattice with 2d neighbours, and a moment in that field has `eta = tanh(h_ef/T)`.[^likharev-weiss][^weiss1907] At zero field this is η = tanh(T_c·η/T) with `T_c = 2*J*d`; the equation has only the root η = 0 while the slope of the tanh at the origin is less than one, and acquires two more the moment that slope passes unity, which is exactly T = T_c.[^likharev-weiss] Above T_c the same linearisation gives the Curie–Weiss law `chi = 1/(T - T_c)`, which reduces to the Curie law χ = 1/T when the coupling is switched off.[^likharev-weiss][^boyd-curie] [[Landau_theory|Landau's]] expansion of the free energy in the order parameter reproduces this from symmetry alone, and yields the mean-field [[Critical_exponent|critical exponents]] β = ½, γ = 1, δ = 3 and ν = ½, together with a finite jump a²/(b·T_c) in the specific heat rather than a divergence.[^likharev-landau][^likharev-exponents] All of these are wrong in detail. Mean field places T_c at 4J on the square lattice against the exact 2.269 J, an overestimate of 76 %, and at 6J against 4.513 J in three dimensions, an overestimate of 33 % (derived); the true two-dimensional exponent β is 1/8, not ½.[^likharev-table][^derived] The Levanyuk–Ginzburg criterion says why: fluctuations neglected by the mean-field average dominate near T_c in fewer than four dimensions, and the sim states the discrepancy on its own readout rather than hiding it.[^likharev-lg] What mean field gets right is the structure of the transition — the symmetry breaking, the appearance of two equivalent states, the divergence of the susceptibility — which is why it remains the first thing taught and the first thing computed. ## See also - [[Curie_temperature]] - [[Curie–Weiss_law]] - [[Hysteresis]] - [[Landau_theory]] - [[Magnetic_domain]] - [[Magnetization]] - [[Ising_model]] - [[Phase_transition]] ## References [^likharev-ising]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Ch. 4, pp. 116–117 (the Ising energy E = −J·Σ s_k·s_k′ − h·Σ s_k with s_k = ±1, and the order parameter η = ⟨s_k⟩). Portal Book 075. https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^likharev-weiss]: Likharev, *Part SM: Statistical Mechanics* (2013), pp. 126–130 (the Weiss molecular field h_ef = h + 2·J·d·η, the self-consistency η = tanh(h_ef/T), T_c = 2·J·d, the Curie–Weiss law χ = 1/(T − T_c) and the Curie law χ = 1/T). Portal Book 075. [^likharev-landau]: Likharev, *Part SM: Statistical Mechanics* (2013), pp. 120–122 (the Landau free energy f = a·t·η² + (b/2)·η⁴ − n·h·η, in the form consistent with Eqs. 4.48–4.52, and the solutions η = ±√(−a·t/b) and f = −a²t²/(2b)). Portal Book 075. [^likharev-exponents]: Likharev, *Part SM: Statistical Mechanics* (2013), pp. 118, 121–122 (the mean-field exponents β = ½, γ = 1, δ = 3, ν = ½ and α = 0 as a finite specific-heat jump a²/(b·T_c); Table 4.1 compares them with experiment and with the two- and three-dimensional Ising values). Portal Book 075. [^likharev-lg]: Likharev, *Part SM: Statistical Mechanics* (2013), p. 122 (the Levanyuk–Ginzburg criterion: mean-field theory fails near T_c for dimensions below four). Portal Book 075. [^likharev-table]: Likharev, *Part SM: Statistical Mechanics* (2013), p. 137 (Table 4.2, T_c/J as mean field / exact — d = 2: 4/2.269; d = 3: 6/4.513). Portal Book 075. [^likharev-walls]: Likharev, *Part SM: Statistical Mechanics* (2013), pp. 133–134 (domain-wall entropy; the two-dimensional Bloch-wall estimate 2J/ln 3 ≈ 1.82 J against Onsager's 2J/ln(1 + √2) ≈ 2.269 J). Portal Book 075. [^boyd-moment]: Boyd, W. Christopher (2025). *Exploring Inorganic and Organometallic Chemistry*. Pp. 270–271 (the spin-only moment μ_eff = μ_B·√(n(n + 2)); the Bohr magneton μ_B = 9.274 × 10⁻²⁴ J/T; the free-electron Landé g of 2 against the measured 2.0023). Portal Book 052. https://open.umn.edu/opentextbooks/textbooks/exploring-inorganic-and-organometallic-chemistry [^boyd-curie]: Boyd, *Exploring Inorganic and Organometallic Chemistry* (2025), pp. 268–269 (Curie behaviour, χ = N·μ₀·μ²/(3·k_B·T)). Portal Book 052. [^boyd-spinonly]: Boyd, *Exploring Inorganic and Organometallic Chemistry* (2025), pp. 210, 271–273 (a d⁵ ion gives 5.92 μ_B high-spin against 1.73 μ_B low-spin; the spin-only formula holds only for A and E ground terms and is unreliable for second- and third-row metals). Portal Book 052. [^weiss1907]: Weiss, Pierre (1907). "L'hypothèse du champ moléculaire et la propriété ferromagnétique." *Journal de Physique Théorique et Appliquée* 6: 661–690. [^heisenberg1928]: Heisenberg, Werner (1928). "Zur Theorie des Ferromagnetismus." *Zeitschrift für Physik* 49 (9–10): 619–636. [^derived]: Values marked *derived* were computed for Wikitube from the cited equations and are not printed in the books. Solving η = tanh(T_c·η/T) by Newton's method gives η*(T/T_c = 0.9) = 0.5254 and η*(T/T_c = 0.5) = 0.9575. From Table 4.2, the mean-field overestimate of T_c is 4/2.269 − 1 = 76 % in two dimensions and 6/4.513 − 1 = 33 % in three. Plain iteration η ← tanh(η·T_c/T) converges too slowly near T_c to be used in an interactive sim, which is why the sim solves the condition by Newton's method. ## External links - [Part SM: Statistical Mechanics](https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics), the Open Textbook Library record for Portal Book 075, whose Chapter 4 carries the Weiss, Landau and Ising treatments used here - [Exploring Inorganic and Organometallic Chemistry](https://open.umn.edu/opentextbooks/textbooks/exploring-inorganic-and-organometallic-chemistry), the Open Textbook Library record for Portal Book 052, for the atomic moment and the Curie law - The Wikipedia pair's external links list further tutorials and materials databases <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Ferromagnetism.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Ferromagnetism* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Ferromagnetism.html" data-title="Ferromagnetism"></div> *Built from `MICROSIM_GUIDE/specs/sims/Ferromagnetism.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Ferromagnetism) : [Wikitube](https://en.wikitube.io/wiki/Ferromagnetism) · pinned revision [1370566879](https://en.wikipedia.org/w/index.php?oldid=1370566879) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]], [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P63 · sim pending (matter/Ferromagnetism).*