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Component chemical potentials. Define the per-spin quantum concentration $n_Q={\lambda }_T^{-3}=(m{k}_{B}T/(2\pi {\hslash }^2){)}^{3/2}$. For a classical component with one-particle potential energy $U_{\sigma }(\mathbf{r})$, the local Helmholtz free energy density is $k_{B}T{n}_{\sigma }[\ln (n_{\sigma }/n_Q)-1]+n_{\sigma }{U}_{\sigma }$. Its derivative with respect to number density gives the full local chemical potential, including the magnetic energy. With the up state defined as the lower level,The Bohr magneton ${\mu }_B$ is a magnetic moment; ${\mu }_{B}B$ is an energy. The factor two for the two spin states is not already included in $n_Q$.

Total equilibrium chemical potential. If spin conversion is allowed, a particle can be added in either spin state at the same equilibrium free-energy cost. Thusnot their sum or zero. The total particle number, or a reservoir, fixes this common value; the supplied information does not specify a numerical chemical potential. A useful local expression is $\mu =k_{B}T\ln [n/(2n_Q\cosh x)]$, with $x={\mu }_{B}B/(k_{B}T)$, whose density dependence is found below.

Why it is uniform and common. Moving a small number $dN$ of one species from ${\mathbf{r}}_1$ to ${\mathbf{r}}_2$ changes the free energy by $[{\mu }_{\sigma }({\mathbf{r}}_2)-{\mu }_{\sigma }({\mathbf{r}}_1)]dN$. If this difference were nonzero, a transfer could lower the free energy. Equilibrium under diffusion therefore requires each full chemical potential to be spatially constant. A spin flip transferring $dN$ from down to up similarly changes free energy by $({\mu }_{\uparrow }-{\mu }_{\downarrow })dN$, proving equality when spin relaxation is permitted. This is chemical equilibrium of a classical spin gas. Mere thermal contact at fixed separate spin populations would not enforce the latter equality; full spin equilibrium is the necessary assumption. A spatially varying magnetic field is balanced by a spatially varying density, not by a varying full chemical potential.

Magnetization. Invert the component equations at the common $\mu$:The two levels have moments $+{\mu }_B$ and $-{\mu }_B$ along the field. The magnetization, magnetic moment per unit volume, is thereforeIts SI units are $\mathrm{A}{\mathrm{m}}^{-1}$. This expression neglects interactions and magnetic feedback of the gas on the specified external field.

High-temperature response. For $|x|\ll 1$, $\sinh x\simeq x$ and $\cosh x\simeq 1$, yieldingAt fixed number density, this is the Curie law: the susceptibility with respect to $B$ is $n{\mu }_B^2/(k_{B}T)$. The conventional dimensionless susceptibility with respect to $H$ is approximately ${\chi }_v={\mu }_{0}n{\mu }_B^2/(k_{B}T)$ when $B\simeq {\mu }_{0}H$. At fixed $\mu$, however, $n_Q{e}^{\beta \mu }$ changes with temperature, so $M$ itself need not scale simply as $1/T$. The moment per particle has the Curie behaviour; the density constraint must be stated.

Total density. Adding the two populations givesFor a finite container and fixed $N$, normalization gives $e^{\beta \mu }=N/[2n_Q\int \cosh x(\mathbf{r})d^{3}r]$. In a weak field the leading density is uniform, with changes only of order $x^2$. Particles preferentially occupy regions where the magnitude of the field lowers one of the spin energies, even though the common chemical potential remains uniform.

Order parameter and the free-energy landscape. In Landau theory, take a scalar magnetization component at fixed volume and composition. It is an order parameter: its nonzero spontaneous value distinguishes an ordered phase with ferromagnetism from the symmetric high-temperature phase. One first constructs a constrained order-parameter free energy at specified magnetization by summing over microscopic states compatible with it. An external magnetic field is an independent control, so it can also appear in that constrained function. For a total magnetic moment the conjugate term is $-BM$; if $M$ is a magnetization density, the total-energy term is $-VBM$, or $-BM$ in free-energy density. Minimizing over $M$ then yields the equilibrium thermodynamic potential as a function of the imposed field.

Thus treating both $M$ and $B$ as arguments of a Landau function does not treat them as independent natural variables of the already-minimized equilibrium free energy. The first is the fluctuating or constrained order parameter, and the second its conjugate field.

At zero field, reversal symmetry requires $F(M)=F(-M)$, excluding odd powers. Near a continuous transition, assume the homogeneous constrained free energy is analytic for small $M$, that the quadratic coefficient crosses zero linearly, and that the quartic coefficient is positive. ThenThe temperature $T_c$ is where the zero-field quadratic coefficient vanishes and spontaneous ordering begins in this approximation. Higher powers are negligible sufficiently close to it; spatial fluctuations are neglected in this mean-field treatment.

Stationarity gives $2M[b(T-T_c)+2d{M}^2]=0$. Above $T_c$, the only minimum is at zero. Below it, zero becomes a maximum and two minima appear:The two signs describe spontaneous symmetry breaking: they are the average magnetizations in the symmetry-broken pure phases. A strictly finite system at exactly zero field samples the two wells symmetrically and has ensemble average $⟨M⟩=0$. A spontaneous pure-phase value is defined by selecting a phase, for example taking the thermodynamic limit before removing a small field. This qualification distinguishes a minimum's location from the average over a symmetric two-well distribution.

Minimum free energy and the heat-capacity jump. At a nonzero minimum, $M^2=-b(T-T_c)/(2d)$. Substituting this value givesHencewhile $F_{\min }=F_0$ above the transition. Take the temperature derivatives only after minimizing, so the equilibrium order parameter is allowed to change with temperature. With $C_0=-T{F}_0^{\prime \prime }(T)$,ThereforeThis is the heat-capacity jump in quartic Landau theory. The entropy is continuous at $T_c$, so there is no latent heat. If $F$ denotes free-energy density rather than total free energy, these expressions denote heat capacity per unit volume with the same normalization.

Stabilizing a tricritical expansion. At a tricritical point, the quartic coefficient is tuned to zero at the same point where the quadratic coefficient changes sign. Reversal symmetry still excludes odd powers. The next available stabilizing term is thereforeA negative or vanishing leading sextic coefficient would not stabilize this truncation for large $|M|$; a higher positive term would then be needed and the present tricritical form would be insufficient. The ordinary quartic theory has a continuous transition for $d>0$, while a stabilized theory with $d<0$ can have a first-order transition. Their meeting at zero field is what makes the simultaneous tuning tricritical.

The tricritical minimum. At zero field and $T<T_c$, stationarity givesThe nonzero solutions are minima: their curvature is $2b(T-T_c)+30d^{\prime }{M}^4=8b(T_c-T)>0$. Put $t=T_c-T>0$. Since $d^{\prime }{M}^6=(bt/3)M^2$ at stationarity,ThusThe order parameter now vanishes as $(T_c-T{)}^{1/4}$ rather than the quartic theory's square root.

Heat-capacity singularity. Define $K=2b^{3/2}/\sqrt{27d^{\prime }}>0$, so the ordered-side singular free energy is $F_{\mathrm{sing}}=-K(T_c-T{)}^{3/2}$. Its derivatives areConsequentlyThis is the tricritical heat-capacity amplitude in a sextic Landau expansion. It is positive on the ordered side and diverges as $T\to T_c^{-}$, rather than remaining a finite jump. This value of the heat-capacity critical exponent is the mean-field result along the path keeping the quartic term zero; a detuning to $d>0$ ultimately restores ordinary quartic behavior. Fluctuation corrections to the mean-field theory are not included in the stipulated polynomial minimization.

A phase transition is a qualitative change of thermodynamic phase as control parameters such as temperature, pressure or chemical potential are varied. A genuine equilibrium singularity is a nonanalyticity of the appropriate thermodynamic potential in the thermodynamic limit; finite systems generally show rounded crossovers instead.

An order parameter is a macroscopic quantity that distinguishes the phases. It often vanishes in a symmetric disordered phase and becomes nonzero after spontaneous symmetry breaking, although density-driven transitions need not have that simple symmetry interpretation. It can be a scalar, vector or tensor. At a first-order phase transition an appropriate first derivative of the free energy is discontinuous, and the order parameter commonly jumps; at a continuous phase transition the order parameter can approach zero continuously while a susceptibility or a higher derivative becomes singular.

Three examples are:

These examples also illustrate why an order parameter must reflect the physical distinction between phases rather than merely be any quantity that changes with temperature.

Collect a magnetic survey along the traverse with a calibrated total-field magnetometer, recording time, accurately surveyed position and sensor height. Extend observations beyond the crater to establish the background, repeat stations or tie lines to test consistency, and use a stationary base-station magnetometer to monitor temporal field changes. Keep the instrument away from vehicles, fences and magnetic equipment; map unavoidable cultural sources rather than treating them as geological signals.

Correct instrument offsets and heading effects where applicable, remove measured diurnal variation, reject or separately treat magnetic-storm intervals, and subtract the appropriate large-scale geomagnetic field at the survey epoch. Level overlapping traverses and state whether a local regional trend has also been removed. Unlike gravity, there is no universal rock-independent Bouguer correction for magnetic topography: a magnetic topographic effect needs a model of terrain and magnetization.

Measure representative magnetic susceptibility and remanent magnetization on oriented rock samples to interpret the profile. Total field alone does not tell whether a contrast comes from induced or remanent magnetization, crater excavation, or strongly magnetic material in the infill.

The supplied south–north profile has a large southern/interior negative trough and a sharp positive peak toward the northern rim. That is opposite to the simplest positive, presently induced magnetic body in question 5(d). It therefore does not justify interpreting the crater as a concentration of strongly magnetic meteorite material.

A negative magnetization contrast from replacing magnetic basalt already reverses the simple induced-body response. In addition, the target Deccan Traps can retain strong remanent magnetization whose direction differs from the present geomagnetic field; crater excavation and rim geometry then produce a strong magnetic topographic effect. Rock-magnetic measurements and forward modelling distinguish these contributions. A primary study of this crater attributes most of its anomaly to magnetized target basalts and topography, with a comparatively small contribution from the breccia and post-impact sediment.^[ref]

The curve alone is not a unique diagnosis of reversed polarity, impact-induced magnetization or a buried iron mass. The physical conclusion supported by the comparison is that a positive induced-body model is inadequate; inherited target-rock magnetization, its removal and geometry must be modelled.

Measure total magnetic field along the same south–north line with a calibrated magnetometer, recording position, time and sensor height. A simultaneous base-station magnetometer measures temporal variation so that diurnal changes can be removed from the moving survey. Avoid disturbed periods and magnetic contamination from vehicles, fences, power lines and equipment. Choose station spacing much smaller than the expected anomaly width, for example tens to hundreds of metres for a kilometre-scale target.

Subtract the epoch-appropriate International Geomagnetic Reference Field or another stated regional main-field model to obtain a total-field magnetic anomaly. Level overlapping traverses and remove offsets or drift where appropriate. Accurate terrain/sensor geometry is needed for comparison with a geological model, but there is no universal magnetic analogue of a constant-density Bouguer slab correction: the effect depends on the distribution and direction of magnetization. Measure magnetic susceptibility and remanent magnetization on representative greenstones, granite and host rock to test the assumed magnetic sources.

Assume the greenstone belt is more magnetically susceptible than the host rocks and granite, its magnetization is mainly induced magnetization, and the body is roughly east–west elongated so that the south–north line crosses it. At southern magnetic latitudes the main field has negative magnetic inclination in a north-and-down coordinate convention. The total-field magnetic anomaly is then generally asymmetric: the principal negative lobe is on the southern side and the positive lobe is displaced northward for the simple induced source illustrated here. The exact lobe sizes and peak positions depend on body geometry and the actual magnetic inclination.^[ref]

An explicit sketching model is an induced magnetic dipole at depth $d$. With northward coordinate $x$, $\mathbf{r}=(x,-d)$ and unit field direction $\hat{\mathbf{F}}=(\cos I,\sin I)$, its normalized total-field magnetic anomaly isThe plot uses $I=-36^{\circ }$ only as an illustrative dipole-field inclination for a latitude near $20^{\circ }$ south, not a measured modern field at the site. A wide greenstone belt would broaden the response and can produce prominent edge anomalies. Remanent magnetization or a magnetic granite could change the sketch substantially; gravity alone cannot identify the magnetic source.

A proton-precession magnetometer measures the scalar magnitude $F=|\mathbf{B}|$ of the local magnetic field. Its sensor contains a proton-rich liquid, such as a hydrocarbon or suitable water-based fluid. A strong current through a coil briefly polarizes the nuclear magnetic moments, producing a small net magnetization. The polarizing field is directed sufficiently across the local magnetic field to leave a transverse component when the current is switched off.

The protons then undergo Larmor precession about the ambient magnetic field. Their rotating net magnetization induces a decaying alternating voltage in a receiving coil. Measuring the frequency determines the field:For example, $50\mu \mathrm{T}$ gives about $2.13\mathrm{kHz}$. The known proton gyromagnetic ratio provides an absolute frequency-to-field scale; this instrument does not measure the field's direction. Strong spatial gradients cause different parts of the liquid to precess at different rates and shorten the usable signal, which matters very close to a buried steel object.^[ref]

A permanent magnet retains magnetization after the applied magnetizing magnetic field is removed. Its working magnetic field persists without a driving electric current in a winding.

An antisymmetric tensor combines the electric polarization and magnetization of a medium. For Minkowski metric $(-,+,+,+)$ and units with light speed one, choose $M^{i0}=P_i$ and $M^{ij}={ϵ}_{ijk}{M}_k$, together with $F_{i0}=E_i$ and $F_{ij}={ϵ}_{ijk}{B}_k$. Then the Lorentz scalar interaction is $M^{\mu \nu }{F}_{\mu \nu }/2=\mathbf{P}\cdot \mathbf{E}+\mathbf{M}\cdot \mathbf{B}$. The bound charge and bound current have four-current $j_b^{\mu }={\partial }_{\nu }{M}^{\mu \nu }$, giving ${\rho }_b=-\nabla \cdot \mathbf{P}$ and ${\mathbf{J}}_b={\partial }_t\mathbf{P}+\nabla \times \mathbf{M}$. This current is identically conserved because partial derivatives commute and $M$ is antisymmetric.

A pure thermodynamic phase is an extremal equilibrium Gibbs state rather than a probabilistic mixture of distinct macroscopic phases. On an ordered Ising branch, taking the thermodynamic limit before removing a selecting conjugate field gives a nonzero magnetization. A symmetric mixture can instead have zero mean but a nondecaying two-point contribution. The connected correlation function and correlation length describing ordinary critical fluctuations should be evaluated in a chosen pure phase when this distinction matters.

An iron sulfide with composition often written ${\mathrm{Fe}}_{1-x}\mathrm{S}$. Its structure and composition determine whether it carries strong magnetization; not all varieties behave identically.

As an original stipulated low-field model, take the axial magnetic field intensity $H$ in $A/m$ and prescribeAll strain components are dimensionless, and their trace vanishes. This is an even, quasistatic, unsaturated toy law with a fixed measuring axis. It is distinct from the fixed-magnitude directional model in saturated isotropic magnetostrictive strain. In particular, one must not replace local unit magnetization by normalized net magnetization in a many-domain specimen and assume this law follows.

For $H(t)=H_0\sin (2\pi ft)$, $H_0=20kA/m$, and $f=50\mathrm{Hz}$, a trigonometric identity givesThe axial strain ranges from zero to $20\times 10^{-6}$, with mean $10\times 10^{-6}$ and a $100\mathrm{Hz}$ component of amplitude $10\times 10^{-6}$. The cosine coefficient is negative with the chosen time origin. For $L=100\mathrm{mm}$ the length ranges from zero to $2\mu \mathrm{m}$ relative to $H=0$. There is no $50\mathrm{Hz}$ term in this unbiased model.

A static bias $H_b$ changes the conclusion. Expanding $\kappa [H_b+H_0\sin (2\pi ft){]}^2$ produces a fundamental-frequency term $2\kappa H_b{H}_0\sin (2\pi ft)$ as well as the doubled-frequency term. This is an incremental linear response about a state with a chosen magnetic axis; it does not contradict isotropic linear magnetostriction vanishes at an unbiased isotropic reference. Real hysteresis, thermal expansion from Joule heating, and mechanical resonance can add further components or change their phases. Frequency alone cannot identify the cause of motion: Joule heating also contains an even-in-electric current contribution.

A quantum oscillation is an oscillatory contribution to a metal's response as a magnetic field moves quantized orbital levels past the Fermi energy. For a closed extremal Fermi surface cross-section of wavevector-space area $A_{\mathrm{ext}}$, Onsager quantization gives inverse-field period $\Delta (1/B)=2\pi e/(\hslash A_{\mathrm{ext}})$. Oscillations of magnetization and electrical resistivity are respectively the de Haas-van Alphen effect and Shubnikov-de Haas effect. Low temperature and long quantum lifetime are needed so thermal smearing and disorder do not wash out the Landau levels.

The spin-one-half quantum XX chain has equal nearest-neighbour transverse exchange. Its continuous spin rotation symmetry conserves total longitudinal magnetization. The Jordan–Wigner transformation maps it to number-conserving free fermions. This quantum chain is distinct from the classical XY model.

Remanent magnetization is magnetization retained after the applied magnetic field has been removed. Magnetic minerals cooling in an ambient field can preserve information about that field in a planetary crust. Such a record may outlive the planetary dynamo which originally produced it. Crustal remanence therefore differs from a present field maintained by conducting-fluid motion.

Consider a mechanically free isotropic model of a saturated ferromagnet, at fixed temperature and fixed magnitude of magnetization. Let $\mathbf{m}=\mathbf{M}/M_s$ be a unit vector, where $M_s$ is the saturation magnetization. The magnetostrictive strain is an infinitesimal strain tensor, measured relative to a reference with randomly oriented magnetic domains. This reference convention matters: it sets the orientational average of the preferred strain to zero. Define ${\lambda }_s$, the saturation magnetostriction, as the longitudinal relative length change in this model.

Rotational symmetry permits a symmetric second-rank tensor made from $\mathbf{m}$ of the form $a{\delta }_{ij}+b{m}_i{m}_j$, where ${\delta }_{ij}$ is the Kronecker delta. If we restrict attention to a change of shape with zero first-order volume change, its trace gives $3a+b=0$. Requiring the longitudinal component to equal ${\lambda }_s$ gives $a+b={\lambda }_s$. HenceThe tensor product $\mathbf{m}\otimes \mathbf{m}$ is symmetric and unchanged under $\mathbf{m}\mapsto -\mathbf{m}$. It uses two copies of the magnetic direction, so the third-rank isotropic tensor obstruction in isotropic linear magnetostriction vanishes does not apply. More generally, a small-magnetization expansion can contain $a{M}^2{\delta }_{ij}+b{M}_i{M}_j$; isotropy alone does not require its trace to vanish. The displayed saturated formula isolates the volume-preserving orientational contribution, rather than asserting that every material has zero volume magnetostriction.

Along a measuring unit vector $\mathbf{n}$, with $\mathbf{n}\cdot \mathbf{m}=\cos \theta$, the fractional length change isFor a stipulated ${\lambda }_s=20\times 10^{-6}$ and reference length $L=100\mathrm{mm}$, alignment gives $+2\mu \mathrm{m}$ and a perpendicular orientation gives $-1\mu \mathrm{m}$. Rotating from perpendicular to parallel therefore changes the length by $3\mu \mathrm{m}$, even though the saturated longitudinal value relative to the random reference is $2\mu \mathrm{m}$. A negative ${\lambda }_s$ reverses these elongation and contraction signs.

The eigenvalues of this infinitesimal strain tensor are ${\lambda }_s,-{\lambda }_s/2,-{\lambda }_s/2$, so $\Delta V/V=\mathrm{tr}{\epsilon }^{\mathrm{mag}}=0$ to first order. This is not exact finite incompressibility: using the illustrative deformation gradient $F=I+{\epsilon }^{\mathrm{mag}}$ givesA finite-deformation law would need higher-order corrections if exact volume conservation were required.

Saturation magnetization is the limiting magnetization when the responding magnetic moments are fully aligned within the stated model or experimental regime. In independent-moment magnetic response, $M_s=n\mu$. Reaching a large magnetization does not imply a large differential magnetic susceptibility: the remaining slope can be almost zero.

A ferromagnetic particle with no internal domain wall. Magnetization reversal can proceed by rotation or nonuniform modes; thermal activation and anisotropy determine blocking and coercivity. Small size alone does not guarantee a permanent hard magnet.

A soft magnetic material changes its magnetization relatively easily and has a low coercive field. High useful permeability and small hysteresis losses favour such materials in electromagnets and alternating-field cores. “Soft” describes magnetic response rather than mechanical softness; electrically resistive ferrimagnets can also be magnetically soft.