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The observation-timescale-dependent temperature below which a magnetic configuration no longer relaxes readily during a measurement. For exchange bias it can mark stable pinning of antiferromagnetic interfacial domains and can be below the bulk Néel temperature. It is not identical to the thermodynamic onset of magnetic order.

The Néel temperature marks loss of long-range antiferromagnetism. A Curie temperature concerns ferromagnetism or ferrimagnetism; magnetic compensation temperature instead denotes cancellation within an ordered ferrimagnet.^[ref]

In the simple collinear description, compare $|\mathbf{L}|$ across $T_N$, rather than expecting $|\mathbf{M}|$ to switch from nonzero to zero. Zero-field magnetization may be zero on both sides. A nonzero magnetic susceptibility is also compatible with antiferromagnetism: an applied magnetic field can disturb the cancellation. The uniform response of a two-sublattice Ising antiferromagnet provides a model calculation.

An overlap estimate must use nearest magnetic-ion separations. For the face-centered cubic lattice of Mn, $d_{\mathrm{Mn}}=a_{\mathrm{Mn}}/\sqrt{2}=3.147\mathring{\mathrm{A}}$; for the body-centered cubic lattice of Fe, $d_{\mathrm{Fe}}=\sqrt{3}{a}_{\mathrm{Fe}}/2=2.477\mathring{\mathrm{A}}$. The supplied radial functions have inverse decay lengths ${\kappa }_Z=Z/(3a_B)$. Their intersite overlap therefore contains approximately $e^{-{\kappa }_{Z}d}$, up to polynomial and angular factors. The decisive exponent difference isThe exponential orbital-tail estimate of exchange thus gives a hopping suppression approximately $e^{-8.99}\simeq 1.25\times 10^{-4}$ before polynomial corrections. Keeping the supplied $Z^{7/2}{r}^2$ radial prefactor in a midpoint overlap proxy givesThis proxy is not an evaluated hopping integral, but confirms that the polynomial factors do not undo the exponential loss. If comparable Coulomb denominators and nonexponential prefactors are assumed, a kinetic-exchange scale proportional to $t^2/U$ is suppressed by roughly $10^{-8}$–$10^{-7}$. Direct intersite Coulomb exchange likewise contains products of the small tails. Coordination-number differences cannot compensate for many orders of magnitude.

By contrast, the measured ordering-temperature ratio supplied in the question is $122/1043\simeq 0.117$, not an exponentially tiny number. Within this overlap model, direct Mn–Mn exchange cannot explain MnO's substantial magnetic ordering scale by extrapolating the Fe mechanism. Another exchange path is required.

This is an order-of-magnitude argument, not an exact relation between an itinerant Curie temperature and a localized-spin Néel temperature. The hydrogenic functions use the printed bare $Z$ values; actual solid-state orbitals are screened and hybridized. Their literal absolute exchange scales should not be treated as quantitative predictions for Fe or MnO.

A consistent mechanism is a transition to G-type antiferromagnetism at the Néel temperature. On the manganese primitive cubic lattice, take the magnetic moments to beEvery nearest neighbour has the opposite magnetic moment. The nuclear crystal structure can retain its original lattice constant, while the magnetic order introduces the propagation wavevectorSumming the scattering amplitude from the magnetic moments over sites gives peaks at $\mathbf{q}=\mathbf{G}+\mathbf{Q}$, where $\mathbf{G}$ is a nuclear reciprocal lattice vector. Thus all three components of the new Miller indices, expressed relative to the original unit cell, are half-integers. The two smallest squared magnitudes in units of $(2\pi /a{)}^2$ areThe first predicts $2\theta \simeq 21.29^{\circ }$, agreeing with the new ring. The next squared magnitude is $19/4$, outside the measured angular range. Taking the observed first magnetic ring as the calibration, Bragg's law predictsUsing the independently fitted lattice constant gives $41.42^{\circ }$, within the input rounding. One ring alone does not uniquely determine a magnetic order; this proposed nearest-neighbour antiferromagnetic pattern explains both the observed ring and the count of two new rings in the specified interval.

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.

Magnetization and field response. A magnetic material contains microscopic magnetic moments associated with electron spin and orbital motion. The magnetization $\mathbf{M}$ is magnetic dipole moment per unit volume. In SI the macroscopic fields satisfyHere $\mathbf{H}$ separates free-current excitation from the material response. A uniform isotropic linear material has $\mathbf{M}={\chi }_m\mathbf{H}$, givingThis constant scalar magnetic permeability model is useful only within its linear range. Crystals can have a susceptibility tensor; strongly magnetic materials depend on field and magnetic history. For a finite specimen the internal $\mathbf{H}$ also includes its demagnetizing field. In a uniformly magnetized ellipsoid, $H_{\mathrm{int}}=H_{\mathrm{applied}}-N_{d}M$ along a principal axis, so sample shape affects the apparent susceptibility.

Diamagnetic and paramagnetic response. Diamagnetism is a weak negative susceptibility caused by induced orbital response opposing the applied field. It occurs in all matter but can be masked by larger positive contributions. Closed-shell materials such as bismuth or many organic compounds show it clearly. Ordinary diamagnetism is often only weakly temperature dependent; superconducting flux exclusion is a separate collective phenomenon and is not described by an arbitrarily small negative susceptibility.

Paramagnetism arises when available magnetic moments partially align with a field. Thermal disorder prevents complete alignment at weak fields. For independent localized moments of magnitude $m$, the classical weak-field result iswhere the second expression uses $B\simeq {\mu }_{0}H$ in a weakly magnetic sample. Thus localized moments obey a Curie law in this regime. Quantum angular momentum changes the coefficient to the appropriate $J(J+1)$ expression and gives saturation at strong field. Conduction electrons instead exhibit Pauli paramagnetism: only states near the Fermi surface can change spin populations, so a degenerate gas has a nearly temperature-independent response. Its orbital diamagnetism must also be included for a full metallic susceptibility.

Collective magnetic order. Exchange interactions can favor parallel alignment, producing ferromagnetism below the Curie temperature. Iron, cobalt and nickel are familiar examples. A material may contain regions of spontaneous magnetization, or magnetic domains, while having little net external moment because the domains point in different directions. Applied fields move domain walls and rotate moments; at large fields the magnetization approaches saturation. Pinning and energy barriers make this process hysteretic. Remanence is the magnetization left after removing the driving field, and coercivity measures the reverse field needed to bring the chosen response back to zero.

The area of a closed $B$–$H$ loop is the energy dissipated per unit volume per quasistatic cycle, $\oint HdB$, with positive dissipated magnitude. Consequently the magnetic permeability is neither one universal constant nor enough information to describe a ferromagnet: differential magnetic permeability $dB/dH$, saturation and hysteresis all matter. Above the ordering temperature the spontaneous moment disappears, and a Curie–Weiss law is often an appropriate approximate paramagnetic description away from the critical region.

In antiferromagnetism, ordered opposing sublattices have equal moments and cancel macroscopically in the ideal compensated case. The order disappears above the Néel temperature. In ferrimagnetism, opposing sublattices have unequal moments and leave a net magnetization. Ferrites combine ferrimagnetic response with relatively high electrical resistivity, making them valuable when eddy-current losses would be substantial in a metallic core.

Electromagnets and engineering choice. A coil supplies free-current magnetomotive force $NI$. For a simple uniform magnetic path, Ampère's law gives $H\ell \simeq NI$ and flux $\Phi \simeq \mu ANI/\ell$. A high-magnetic permeability soft magnetic core concentrates flux and reduces the current required for a given field. With an air gap $g$,Even a short gap can dominate the magnetic reluctance. The approximate attractive force across one uniform gap is $F\simeq B^{2}A/(2{\mu }_0)$, obtained from the field energy, with fringing neglected.

Soft magnetic materials have small coercivity and narrow hysteresis loops; they suit relays, electrical transformer cores and controllable electromagnets. Hard magnetic materials retain their magnetization and suit permanent magnets. Laminating metallic cores breaks large eddy-current loops; ferrites are useful at higher frequencies because of their resistivity. Increasing current cannot increase $B$ indefinitely through the linear formula: core saturation, coil heating and leakage limit performance. A good design selects material for the field amplitude, switching frequency and acceptable loss, rather than simply choosing the largest quoted magnetic permeability.

In a weak applied field, magnetic susceptibility relates magnetization to magnetic field intensity, $M=\chi H$. Its sign, temperature dependence and nonlinear field dependence distinguish several mechanisms.

Diamagnetism is the orbital response opposing an applied magnetic field, with $\chi <0$. Closed-shell atoms and filled bands generally give a weak, nearly temperature-independent response. In a simple free electron metal, Landau diamagnetism contributes ${\chi }_L=-{\chi }_P/3$; the factor is specific to the isotropic parabolic-band model and is not universal in crystals.

Paramagnetism gives $\chi >0$. Localized magnetic moments align statistically with the field against thermal disorder. For independent moments of angular momentum $J$,which is the Curie law. Exchange interactions often lead approximately to the Curie–Weiss law, $\chi =C/(T-\Theta )$, sufficiently far above any ordering temperature. The sign of $\Theta$ suggests predominantly parallel or antiparallel exchange, though it is not a complete classification. Conduction electrons instead exhibit Pauli paramagnetism:for $g\simeq 2$ and a spin-summed density of states per unit volume. Only states close to the Fermi energy change occupation, making this response nearly temperature independent when $k_{B}T\ll E_F$.

Ferromagnetism has spontaneous parallel magnetic order below a Curie temperature. Exchange interaction, not simply classical dipole attraction, provides the dominant alignment mechanism. In the convention $H=-J_{\mathrm{ex}}{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$, positive $J_{\mathrm{ex}}$ favours alignment. Magnetic domains can make a ferromagnet's total magnetization vanish before an external field aligns its domains. Domain motion and anisotropy produce saturation, hysteresis, remanent magnetization and a reverse field measured by magnetic coercivity.

Antiferromagnetism has oppositely ordered sublattices below a Néel temperature; equal opposing moments give zero net magnetization while retaining a nonzero staggered order parameter. A susceptibility anomaly and magnetic neutron diffraction reveal this order even when bulk magnetization vanishes. Ferrimagnetism also has opposing sublattices but unequal contributions, giving a nonzero spontaneous net moment. Above their ordering temperatures both can become paramagnetic. Measurements with a magnetometer as a function of field and temperature, supplemented by magnetic diffraction, therefore distinguishes a compensated ordered material from an ordinary paramagnet.