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A cooling method that reduces an applied magnetic field while suppressing heat transfer to or from the material. For a suitable material exhibiting paramagnetism, an approximately reversible field reduction can lower its temperature at approximately constant entropy.

The preparation matters as much as the field reduction. First establish thermal equilibrium with a cold heat bath, then raise the magnetic field while maintaining the bath temperature: this is magnetization in an isothermal process. The bath accepts heat and the material loses magnetic entropy. Next open a heat switch, thermally isolating the working material, and lower the magnetic field. If the sweep maintains internal thermal equilibrium with negligible entropy production, this second leg is an isentropic process. An adiabatic process alone does not guarantee constant entropy.

The attraction is a controllable energy scale: changing the magnetic field changes the separation of magnetic energy levels, even across a thermally insulating gap. This lets an already cold solid become colder without needing a colder contacting heat bath. A practical refrigerator also needs a load, a period of absorbing its heat, and a reset that rejects accumulated entropy to the bath. A single field sweep has finite cooling capacity.

A covalent bond joins chemical atoms through shared electrons. In the familiar two-centre description, one bond pair contributes to a connection between two atomic nuclei. Sharing means that the electrons belong to the combined arrangement and interact with both atomic nuclei; it does not mean that identifiable dots shuttle along drawn paths. The ^[ref] emphasizes the concentration of electrons between atomic nuclei and the resulting attraction and characteristic separation. The pair picture is especially useful for simple molecules, but is not a universal description of every covalent bond.

A Lewis structure turns this picture into practical bookkeeping. Count the valence electrons, indicate which chemical atoms are connected, and allocate two electrons to each line. A bond pair is shared; a lone pair is assigned to one chemical atom without connecting it to another. A double line represents two bond pairs. Dots and lines record allocations, rather than positions, trajectories, or a measured three-dimensional arrangement. ^[ref]

For molecular hydrogen, ${\mathrm{H}}_2$, each neutral hydrogen chemical atom supplies one valence electron. Thus $1+1=2$ valence electrons make one bond pair, represented by $\mathrm{H}-\mathrm{H}$, with no lone pairs. Each hydrogen counts the same shared pair when checking its two-electron arrangement. The molecule still contains two electrons, not four: counting a pair around both endpoints does not create extra electrons.

For water, ${\mathrm{H}}_2\mathrm{O}$, neutral oxygen supplies six valence electrons and the two hydrogen chemical atoms supply one each: $6+1+1=8$. Two O–H connections consume four electrons as two bond pairs. The remaining $8-4=4$ electrons form two lone pairs on oxygen. Around oxygen, the two bond pairs and two lone pairs count as eight electrons; each hydrogen counts two. The whole water molecule has ten electrons, because two additional oxygen electrons are omitted from this valence electron drawing.

For Ammonia, ${\mathrm{NH}}_3$, the total is $5+3\times 1=8$ valence electrons: five from nitrogen and three from hydrogen. Three N–H connections use six electrons as three bond pairs, leaving two electrons as one lone pair on nitrogen. Thus water and Ammonia have the same total of eight valence electrons but different allocations: two plus two pairs versus three plus one. This small calculation already distinguishes their connectivity without a detailed quantum mechanics calculation.

For molecular oxygen, ${\mathrm{O}}_2$, $6+6=12$ valence electrons give the conventional $\mathrm{O}=\mathrm{O}$ Lewis structure. Its two bond pairs use four electrons, leaving eight as four lone pairs, two on each oxygen. Each oxygen consequently counts eight electrons locally. However, actual molecular oxygen has two unpaired electrons and exhibits paramagnetism: it is attracted into a stronger magnetic field. The all-paired Lewis structure cannot capture this observation. Molecular orbital theory describes the unpaired electrons; the successful counting diagram alone is insufficient. ^[ref]

Why does sharing produce a stable separation? For molecular hydrogen, the electronic arrangement responds as the atomic nuclei approach. Their interactions can lower the energy, but pushing them too close raises it. A curve of potential energy against separation therefore has a local minimum, rather than falling forever. Attraction and repulsion balance there. This is a statement about the combined system, not a claim that stationary electrons sit between otherwise motionless atomic nuclei. ^[ref]

The following Morse potential makes those features calculable. Set $x=r/r_e$, where $r$ is separation and $r_e$ its equilibrium value, and let $D$ be the well depth. With the self-chosen parameter $a=2$, define the dimensionless quantitiesHere $F=-dV/dr$ is the net separating force, obtained from the derivative of the potential energy. Positive $f$ pushes the chemical atoms apart; negative $f$ draws them together. These are illustrative curves with $r_e=D=1$ in chosen units, not measured molecular hydrogen data or fitted parameters.

At $x=1$, $u=-1$ and $f=0$. The second derivative is $u^{\prime \prime }(1)=8>0$, confirming a local minimum. At $x=0.5$, $u=(1-e{)}^2-1\approx 1.9525$; at $x=1.5$, $u=(1-e^{-1}{)}^2-1\approx -0.6004$. As $x$ increases without bound, $u$ approaches zero from below. Thus moving from the well bottom to separated chemical atoms requires an increase of $D$ in this model's potential energy. An actual quantum mechanics dissociation threshold also depends on the initial motion of the atomic nuclei, so the well depth alone is not every measured dissociation energy.

The force panel separates $f$ into a positive term $4e^{-4(x-1)}$ and a negative term $-4e^{-2(x-1)}$. At $x=1$ they are $+4$ and $-4$, cancelling exactly; nearer separations favour the positive term. These are algebraic components of the chosen Morse potential, not separately measured microscopic forces. Their usefulness is to show how a balance arises from the slope of one energy curve.

The beauty of the pair picture is its economy: a few marks summarize connectivity and constrain what a proposed molecule can contain. Its historical anchor is Gilbert N. Lewis's 1916 paper ^[ref], also available as an ^[ref]. Lewis structures remain useful because they answer counting questions quickly; quantum mechanics supplies the richer account when energy, shape, or unpaired electrons matter.

The University of Iowa's Electron Spin Resonance, University of Iowa laboratory protocol (revised July 2007, Experiment and Resonance Data Analysis, printed pp. 2–6) specifies DPPH on cotton in a tube inside an RF coil, between Helmholtz coils. A bridge monitors RF absorption; an oscilloscope records this output and the voltage across a $1\Omega$ current-monitor resistor. Sweeping the coil current passes through the resonant magnetic field. The protocol describes two dips when the sweep revisits resonance; these are repeated crossings, not two distinct spectral lines.

Its calibration replaces the sample with a Hall effect probe, maps the magnetic field in $2\mathrm{mm}$ steps, and accounts for sample dimensions. At least four RF frequencies test the resonance relation. Sweep speed is adjusted to obtain comparable passages. Only a local peak-plus-background fit is prescribed, because the full shape is not Gaussian. The protocol also flags unexplained trailing spikes and possible electronics contributions. These controls constrain field and sweep errors; an empty-tube/cotton blank would be an additional background control, not a reported measurement in this protocol.^[ref]

For an isolated effective spin one-half with dimensionless positive factor $g$, the basic resonance model isHere ${\mu }_{\mathrm{B}}$ is the Bohr magneton, $h$ is the Planck constant, $\nu$ is the applied frequency, and $B_{\mathrm{res}}$ is the calibrated resonant magnetic field. A change in frequency should shift the resonance field proportionally in this model. The factor $g$ describes the electronic magnetic moment coupling, not the species' net electric charge.

A resonance supports the presence of detectable paramagnetism. It does not, by itself, identify an unknown radical: other species exhibiting paramagnetism, impurities, overlapping responses, and interactions with atomic nuclei can complicate interpretation. Conversely, absence of a detectable resonance need not mean absence of free radicals; a low concentration, a short lifetime, or an unsuitable measurement range can conceal them. The occupancy and chain diagrams are original bookkeeping illustrations; neither is a measured or simulated DPPH spectrum.

Molecular orbital theory describes the electronic arrangement of a molecule using molecular orbitals: one-electron spatial wavefunctions extending over the combined atomic nuclei. Filling bonding molecular orbitals can stabilize a chemical bond; filling antibonding molecular orbitals opposes that stabilization. A molecular orbital is a building block of a many-electron description, rather than the entire many-electron wavefunction. See Robert W. Field, MIT 5.61, lecture 24, pp. 1–4: molecular orbitals and the LCAO approximation.

First make the interference visible with an exactly normalized one-dimensional toy. Replace the three-dimensional $1s$ atomic orbitals of the hydrogen atom by two Gaussian functions centred at $x=\pm a$, with dimensionless $x$ and $a>0$:The normalized sum and difference are ${\psi }_{\pm }=({\varphi }_A\pm {\varphi }_B)/\sqrt{2(1\pm S)}$, following the same linear combination of atomic orbitals algebra. At $a=1$, the orbital overlap is $S=e^{-1}\approx 0.367879$. At the midpoint, ${\varphi }_A(0)={\varphi }_B(0)\approx 0.455581$, givingBoth probability densities have total integral one: interference redistributes probability, rather than creating an extra electron. These are toy wavefunctions, not actual hydrogen atom orbitals or predictions of molecular hydrogen energies.

Now fill the two $1s$-derived molecular orbitals in the minimal model. The lower bonding molecular orbital is labelled ${\sigma }_g$; the upper antibonding molecular orbital is ${\sigma }_u^{\ast }$. The Pauli exclusion principle permits at most two electrons in each spatial molecular orbital, with opposite spin projections along a chosen axis. For the lowest electron configuration in this model, fill ${\sigma }_g$ first. The subscripts $g$ and $u$ denote even and odd behaviour under inversion through the midpoint; the star marks an antibonding molecular orbital. See Flowers et al., Chemistry 2e, 8.4: occupancy and bond order and Field, MIT 5.61, lecture 25, pp. 1–3: bonding, antibonding, and symmetry labels.

With $N_b$ electrons in bonding molecular orbitals and $N_a$ in antibonding molecular orbitals, the model's bond order is $b=(N_b-N_a)/2$. The three elementary counts are:

The zero bond order says this minimal model gives no net covalent bond for ${\mathrm{He}}_2$. It does not rule out every bound pair of helium chemical atoms: a very weak Van der Waals force can bind a neutral helium dimer, which has been observed experimentally. See Zeller et al., Imaging the He2 quantum halo state using a free electron laser, PNAS 113, 14651–14655 (2016), p. 14651: van der Waals binding and Luo et al., The weakest bond: Experimental observation of helium dimer, Journal of Chemical Physics 98, 3564–3567 (1993). The bond order count does not include that weak binding mechanism.

The useful surprise is that sharing alone does not determine a chemical bond: the relative quantum phase and the occupancy matter together. The same two shapes explain why adding an electron to a bonding molecular orbital increases the model's bond order, while adding one to an antibonding molecular orbital decreases it. Molecular orbital theory also accounts for the unpaired electrons and paramagnetism of molecular oxygen, which its all-paired Lewis structure misses. See Chemistry 2e, 8.4: the molecular orbital description of oxygen.

A material exhibiting paramagnetism. Its induced magnetization points along the applied magnetic field in the linear-response regime, giving positive magnetic susceptibility. Unlike a ferromagnet, it has no spontaneous ordered moment in that phase. The inverse-temperature response of independent localized moments is one model, rather than a definition that includes all microscopic mechanisms of paramagnetism.

For $n$ unpaired electrons, the spin-only magnetic moment iswhere ${\mu }_{\mathrm{B}}$ is the Bohr magneton and the second expression uses $g\simeq 2$.

For the square, a large separation of the highest $B_{1g}$ level favours pairing in the lower four molecular orbitals: $S=0$, $n=0$, ${\mu }_{\mathrm{eff}}=0$. If the cost of occupying $B_{1g}$ were smaller than the relevant electron pairing energy, a high-spin complex with two unpaired electrons and $S=1$ could instead occur, giving $\sqrt{8}{\mu }_{\mathrm{B}}\simeq 2.83{\mu }_{\mathrm{B}}$.

For ideal tetrahedral $d^8$, the occupation is $e^4{t}_2^4$. Hund's rules distribute four electrons over the three degenerate $t_2$ molecular orbitals as one pair and two single occupations. Thus $S=1$ and ${\mu }_{\mathrm{eff}}\simeq 2.83{\mu }_{\mathrm{B}}$. Merely increasing the $e$–$t_2$ splitting does not create a conventional paired tetrahedral $d^8$ alternative: four electrons must still occupy $t_2$.

The dimer has two unpaired electrons and a triplet ground state, $S=1$. Its approximately threefold-symmetric Fe${}_2$(CO)${}_3$ core supplies a pair of degenerate frontier molecular orbitals. The last two electrons occupy separate members with parallel spins, favored by exchange, rather than pairing in one. A useful qualitative picture is a sigma bond plus two perpendicular, singly occupied pi half-bonds: together these support a formal Fe–Fe double bond while retaining paramagnetism. The spin-only magnetic moment for $S=1$ is $\sqrt{8}{\mu }_B\simeq 2.83{\mu }_B$; real moments can include orbital and spin–orbit effects.

The formal 18-electron count does not require every molecular orbital to be doubly occupied. It is therefore compatible with an open-shell dimer. The primary structural/magnetic study establishes this unusual triplet state; an isolated-atom high-spin $d^7$ diagram would not describe the coupled metal–ligand system.^[ref]

Take the bond axis as $z$. The heavy-atom $2s$ and $2p_z$ atomic orbitals can mix with hydrogen $1s$ to make three sigma-symmetry molecular orbitals. In the qualitative valence ordering, these are a low predominantly heavy-atom $2s$ level, a bonding $\sigma$ level and a higher ${\sigma }^{\ast }$ level. The perpendicular $2p_x,2p_y$ orbitals form a degenerate approximately nonbonding $\pi$ pair because hydrogen has no valence orbitals of the matching sideways symmetry. The filled heavy-atom $1s$ core is omitted from the diagram.

Counting valence electrons givesThus both exhibit paramagnetism: CH has one unpaired electron and NH has two in its ground state. For NH, Hund's rules favour separate parallel-spin occupation of the two degenerate $\pi$ orbitals. The corresponding observed ground terms are $X{}^2\Pi$ and $X{}^3{\Sigma }^{-}$; the total bond order is approximately one in this simple sigma-bond/nonbonding-pi description.^[ref]

The nitrogen-derived orbital energies lie lower than the corresponding carbon-derived ones on an approximate common energy scale, while the isolated hydrogen $1s$ reference is unchanged. The bonding orbitals consequently have different polarizations and mixing coefficients; drawing identical energies for C and N would miss the screening trend. The figure specifies ordering and occupancy, not measured gaps or a numerical prediction from (a).

In the SI linear-response convention $\mathbf{M}={\chi }_H\mathbf{H}$, a diamagnetic contribution has negative susceptibility: the induced moment opposes the applied field. Orbital motion supplies a familiar microscopic mechanism, and closed-shell atoms can be diamagnetic even without permanent moments. Paramagnetism has positive susceptibility, with induced magnetization along the field; thermal alignment of permanent localized moments is one mechanism, while conduction-electron spin response is another.

Neither sign alone is a definition of ferromagnetic ordering. An ordinary paramagnetic phase has no spontaneous zero-field magnetization. A diamagnetic background can coexist with a larger paramagnetic contribution in the same specimen. Curie's inverse-temperature law describes an appropriate localized-moment model, not every possible paramagnet.

A $3d^9$ shell has one hole in the otherwise filled $d$ shell. Hund's rules give $S=1/2$, $L=2$, and, for a more-than-half-filled shell, $J=L+S=5/2$: the free-ion term is ${}^2{D}_{5/2}$. Its Landé factor isDistinguish the Curie effective moment from a saturated projection:The question then explicitly asks us to use orbital quenching in the crystal. Ligand fields split and mix orbital states, suppressing their freely orientable angular momentum. The spin-only approximation has $g\simeq 2$, $S=1/2$, so ${\mu }_{\mathrm{eff}}=\sqrt{3}{\mu }_B$ but the two field-axis projections are $\pm {\mu }_B$. The latter, not $\sqrt{3}{\mu }_B$, belong in the tanh argument and saturation magnetization.

The sulfate and water environment supplies mainly closed-shell, weakly magnetic surroundings, stabilizes localized Cu2+ sites and separates their moments so that exchange can be weak on the temperature scale considered. Paramagnetism here is an approximation of weakly interacting local spins, not a claim that dilution guarantees no magnetic ordering at any temperature. Elemental copper has a filled $3d$ band and itinerant conduction electrons rather than one localized Cu2+ spin per atom. Pauli spin response and orbital/core contributions then replace a local-spin Curie law; ordinary pure copper has a weak net diamagnetic response.^[ref]

There is one Cu2+ per formula unit, hence $n_s=N_A/V_m=5.5148\times 10^{27}$ m${}^{-3}$. The spin-only weak-field SI volume susceptibility isThese neglect exchange, anisotropic $g$ factors and background diamagnetism. At high field and low temperature, saturation requires the nonlinear response in a; a field-independent Curie value then describes only the zero-field slope.

In the Weiss molecular field approximation, the local response obeysLinearize around $M=B=0$ in the stable phase of paramagnetism. Substitution of a givesHence the Curie–Weiss law and Weiss temperature areFor positive $\lambda$, $\theta$ is the mean-field theory Curie temperature: the zero-magnetization solution becomes unstable there, and below it a spontaneous solution appears. For negative coupling, a negative fitted Weiss temperature indicates predominantly opposing interactions, but this one-uniform-magnetization treatment does not determine an antiferromagnetic ordering pattern or its transition temperature. More generally the fitted Weiss temperature need not equal the observed ordering temperature.

For $\theta >0$, the paramagnetic curve is positive for $T>\theta$, diverges as $T\downarrow \theta$, and tends to $C/T$ at high temperature; its inverse is the straight line $(T-\theta )/C$. The expression is not the stable susceptibility below $\theta$. For example substituting $T=\theta /2$ would predict a negative response around the unstable $M=0$ branch. The nonlinear self-consistency equation must instead be expanded about the appropriate spontaneous magnetization.

A material with paramagnetism develops a magnetization parallel to a weak applied magnetic field intensity, with positive magnetic susceptibility and no spontaneous magnetization in its paramagnetic phase. Molecular oxygen is an example; it has unpaired electron spins.

A material with diamagnetism develops an induced magnetization opposing the applied magnetic field, with negative magnetic susceptibility. A closed-shell atom such as helium is an example; its orbital response is diamagnetic.

A ferromagnet has a spontaneously ordered magnetization below its Curie temperature, even in the limit of vanishing applied magnetic field. Iron below its Curie temperature is an example. Different magnetic domains can cancel the net moment of an unmagnetized bulk sample, despite nonzero local spontaneous magnetization.

A material showing paramagnetism has an induced magnetization in the direction of a weak applied field, giving positive magnetic susceptibility, and no spontaneous magnetization once the field is removed in the paramagnetic phase. Alignment of permanent atomic or nuclear moments is one mechanism.

A material showing diamagnetism develops an induced moment opposing the applied field, giving negative magnetic susceptibility. Closed electronic shells supply a familiar orbital contribution; the response need not require permanent moments.

A ferromagnet has a phase with a spontaneous ordered magnetization below its Curie temperature, produced by cooperative alignment of moments. Its magnetic domains can cancel in a macroscopic zero-field sample, so a measured zero net moment alone does not show that the material lacks ferromagnetism.

Let ${\mu }_a=2.13{\mu }_N$ be the magnitude of the saturated nuclear moment of helium-3, with ${\mu }_N=e\hslash /(2m_p)$. A spin-one-half nucleus has the two Zeeman energies $\mp {\mu }_{a}B$. For independent nuclei the one-site partition function and magnetization density areIn the weak-field limit ${\mu }_{a}B\ll k_{B}T$, expand the hyperbolic tangent:The nuclear moment's negative sign means that its moment points oppositely to its spin; it does not make the thermal magnetic susceptibility negative. The response is paramagnetism governed by Curie's law.

There is no extra factor $1/3$ here: the stated $2.13{\mu }_N$ is the moment of a fully polarized spin-one-half state. Equivalently, writing the moment operator as $g_N{\mu }_N\mathbf{I}$ requires $g_N=-4.26$, and $g_N^2{\mu }_N^{2}I(I+1)/3=(2.13{\mu }_N{)}^2$. The formula presumes independent localized nuclear moments; exchange correlations invalidate it close to the much lower ordering scale discussed in part (e).

In weak magnetic field and the regime of paramagnetism, localized magnetic moments with total angular momentum $J$ and Landé g-factor $\tilde{g}$ have SI volume magnetic susceptibilitywithin the Curie–Weiss law. Here $n_m$ counts magnetic ions per volume, ${\mu }_0$ is the vacuum permeability, ${\mu }_B$ is the Bohr magneton, and $\Theta$ is the Weiss temperature. The factor $J(J+1)/3$ is the zero-field average of the squared angular projection over $m_J=-J,\dots ,J$.

For one magnetic ion per formula unit, a molar volume $V_m$ contains $N_A$ magnetic ions, where $N_A$ is the Avogadro constant. Consequently $n_m{V}_m=N_A$, and the molar magnetic susceptibility isDefining the dimensionless effective paramagnetic moment $m_{\mathrm{eff}}$ by ${\mu }_{\mathrm{eff}}=m_{\mathrm{eff}}{\mu }_B$, numerical substitution givesThe normalization is per mole of formula units, which here also means per mole of magnetic ions.

Use the sample mean to estimate each magnetic susceptibility, and the unbiased sample variance with denominator $n-1=4$ to estimate the repeatability of individual readings. In the numerical scale supplied by the paper,If the repeated measurements have independent random errors, their mean estimates have standard errorsThe difference and its estimated standard uncertainty are thereforeThe separation is about $2.53$ standard errors. Under an approximately normal distribution model with no experimenter-dependent bias, Welch's t-test gives about $7.34$ statistical degrees of freedom and a two-sided p-value of $0.038$. Thus there is modest evidence against equal underlying susceptibilities at a conventional $5\%$ significance level; overlap of the ranges of individual readings is not an adequate comparison of the means. Five readings per sample provide limited information about the error distribution.

Crucially, material and student exhibit complete confounding: each student measured only one sample. An experimenter-dependent systematic error could mimic the difference. Paramagnetism also depends on conditions such as temperature, so equal material alone does not guarantee equal readings without controlling those conditions. The defensible physical conclusion is the data suggest a difference, but do not establish different materials. Both students should measure both samples in randomized or alternating order, with calibration and temperature controlled, to separate a sample effect from an operator effect. No unit or scale factor absent from the original table has been invented.

In paramagnetism, a material develops magnetization along a weak applied field, with positive susceptibility; in diamagnetism, it develops an induced moment opposing it, with negative susceptibility. In a simple localized-moment paramagnet, Zeeman alignment competes with thermal disorder. For independent two-state moments $\pm \mu$,when the field is weak and demagnetizing corrections are negligible. Stronger fields saturate the moment; interactions and conduction-electron Pauli paramagnetism can depart from this Curie law.

Ordinary diamagnetism comes from a field-induced change of orbital motion, including in closed-shell matter without permanent atomic moments. Its weak susceptibility is often nearly temperature independent away from a transition, and it contributes a background even when another magnetic response dominates. A nonuniform field tends to attract a paramagnetic specimen toward stronger field and repel a diamagnetic specimen in the weak linear regime. Calibrate the holder contribution and field gradient and measure a temperature/field series to distinguish induced response from retained magnetization. Ferromagnetic hysteresis and superconducting flux expulsion require different collective explanations.

The microscopic magnetic response becomes a macroscopic field problem through the magnetization $\mathbf{M}$, the magnetic dipole moment per unit volume. Definein magnetostatics. Bound currents are represented by ${\mathbf{J}}_b=\nabla \times \mathbf{M}$ and surface current ${\mathbf{K}}_b=\mathbf{M}\times \hat{\mathbf{n}}$. Normal $B$ is continuous across an interface, while $\hat{\mathbf{n}}\times ({\mathbf{H}}_2-{\mathbf{H}}_1)={\mathbf{K}}_{\mathrm{free}}$. It is generally incorrect to impose continuity of every component of both fields. A linear isotropic material has $\mathbf{M}=\chi \mathbf{H}$ and ${\mu }_r=1+\chi$, but a ferromagnet usually requires a nonlinear, history-dependent constitutive relation.

Internal field differs from externally applied field. For uniform magnetization along a principal axis of an ellipsoid, the demagnetizing field is $-N_{d}M$ and $H_{\mathrm{int}}=H_{\mathrm{applied}}-N_{d}M$. A linear specimen therefore has apparent susceptibility $\chi /(1+N_d\chi )$ relative to the applied field. Shape is especially important for large susceptibility. A long closed magnetic circuit reduces stray fields; a bar magnet still has demagnetizing fields despite the absence of magnetic monopoles. Equivalent magnetic-pole descriptions are bookkeeping for magnetization divergence, not free magnetic charge.^[ref]

In paramagnetism, permanent localized moments are preferentially aligned by the field, against thermal disorder. For independent spin-one-half moments with projections $\pm \mu$, the two Zeeman energies givein the dilute weak-field limit with $B\simeq {\mu }_{0}H$. This is a Curie law; the response saturates at $n\mu$ when alignment dominates. More generally, spin $J$ gives $\chi ={\mu }_{0}n(g{\mu }_B{)}^{2}J(J+1)/(3k_{B}T)$. Interactions can produce a Curie–Weiss response $C/(T-\Theta )$ over a suitable range. Conduction-electron Pauli paramagnetism need not show the localized-moment $1/T$ law.

In diamagnetism, the field induces a magnetic moment opposing it, so the linear susceptibility is negative. The microscopic cause is the change in orbital electronic motion, and a closed-shell insulating material has no permanent atomic moment to orient. This weak response is often only mildly temperature dependent away from other transitions. It also exists as a background in materials with a stronger positive response. Perfect flux expulsion in a superconductor is a distinct collective phenomenon and should not be identified with ordinary weak atomic diamagnetism.

In ferromagnetism, exchange interactions stabilize spontaneous magnetic order below the Curie temperature. A specimen can have zero net moment while its magnetic domains remain locally magnetized in different directions. Domain-wall motion and moment rotation produce a large response; anisotropy and pinning lead to remanence, coercivity and hysteresis. Increasing field eventually approaches saturation. Heating above the ordering transition removes spontaneous order, although an applied field still induces a response. Soft magnetic cores favor easy reversal and small cyclic loss; hard permanent-magnet materials favor retained magnetization and resistance to reversal. These properties concern collective order and barriers, not merely independent moments all following a weak field.^[ref]

A practical distinction requires a controlled field sweep and a separate magnetic readout. A toroidal core with drive winding $N_1$, path length $l$ and current $I$ has approximately $H=N_{1}I/l$. A pickup winding measures ${\mathcal{E}}_2=-N_{2}AdB/dt$; integrating after calibration gives $B$, and $M=B/{\mu }_0-H$. Reversing the sweep reveals a loop whose positive loss area per volume is $\oint HdB$. An air-core/empty-holder reference, known geometry and turn count, offset correction, temperature control and slower sweeps test background, heating and rate-dependent eddy-current losses. A single force or susceptibility reading cannot establish spontaneous order or hysteresis. The original curves below are illustrative responses, not measured material data.

Write magnetization as magnetic dipole moment per volume. In weak-field linear response, $\mathbf{M}=\chi \mathbf{H}$ and $\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M})$.

Paramagnetism has positive susceptibility: permanent spin or orbital moments tend to align with a field against thermal disorder. Independent localized moments in the classical weak-field regime give $\chi \propto 1/T$, the Curie law. Conduction-electron Pauli paramagnetism has a different origin and need not follow that temperature law.

Diamagnetism is the induced magnetic response opposing the applied field, giving negative susceptibility. Changes in orbital motion underlie this response; it exists even without permanent moments. It is often relatively weak and only weakly temperature dependent, although particular quantum systems have stronger or more complicated behaviour.

Ferromagnetism arises from interactions favouring parallel magnetic order. Below the Curie temperature a domain can have spontaneous magnetization even at zero applied field. A sample can nevertheless have nearly zero net moment if differently oriented domains cancel. Domain rearrangement and rotation produce a nonlinear magnetization curve, saturation, remanence and coercivity; hysteresis dissipates energy in a field cycle. Above the ordering temperature the spontaneous order disappears, and a Curie-Weiss susceptibility is a useful mean-field approximation. The signs of weak-field susceptibility distinguish para- and diamagnetism; spontaneous order and domain dynamics distinguish ferromagnetism from independent-moment alignment.

Magnetization $\mathbf{M}$ is magnetic dipole moment per unit volume. In SI units,in a linear isotropic response. The internal field matters: for a uniformly magnetized ellipsoid it is ${\mathbf{H}}_{\mathrm{int}}={\mathbf{H}}_{\mathrm{applied}}-{\mathbf{N}}_d\mathbf{M}$. Sample shape therefore affects an apparent magnetic susceptibility.

Diamagnetism opposes the applied field, giving small negative magnetic susceptibility from the induced orbital response. It is often only weakly temperature dependent. Paramagnetism aligns pre-existing moments and gives positive magnetic susceptibility; independent localized moments obey approximately $\chi =C/T$ in weak fields before saturating. Pauli paramagnetism of conduction electrons is a different, weakly temperature-dependent mechanism.

Ferromagnetism involves collective exchange-driven alignment and spontaneous magnetization below the Curie temperature. A specimen can nevertheless have nearly zero total moment because magnetic domains point in different directions. Field-driven domain-wall motion and rotation change the net magnetization. Antiferromagnets have compensating ordered sublattices below a Néel temperature; ferrimagnets have oppositely ordered but unequal sublattice moments and retain a net moment. Ferrites are useful examples of electrically resistive ferrimagnetic core materials.

Ferromagnetic response is nonlinear and history dependent. A hysteresis loop displays remanent magnetization, magnetic coercivity and saturation. A single constant magnetic permeability cannot describe the entire loop: differential magnetic permeability describes its local slope. The quasistatic energy loss per unit volume and cycle is the loop area $\oint HdB$. Soft magnetic materials have low magnetic coercivity and usually high useful magnetic permeability; hard magnetic materials retain magnetization and suit permanent magnets.

An electromagnet normally uses a soft, high-permeability core to guide flux generated by a coil. For a uniform-section core with length ${\ell }_c$ and air gap $g$, neglecting leakage and fringing,An air gap can dominate the magnetic reluctance even when the core magnetic permeability is large. Core saturation limits the improvement with current, while winding resistance causes heating. For alternating operation, laminations suppress eddy-current loops in metallic cores; high-resistivity ferrites also reduce those losses. Low hysteresis loss is important in repeatedly cycled cores, whereas high magnetic coercivity is valuable in a permanent magnet. Useful design therefore depends on the full magnetic response, electrical losses and operating frequency, rather than the largest quoted magnetic permeability alone.