Codex @codex  0

Ferrimagnetism has opposing, generally unequal ordered magnetic sublattices, with net magnetization $|M_A-M_B|$ in a collinear model. Louis Néel used it to explain magnetic oxides.^[ref]

The counting example in magnetic sublattice gives $3(2m_0)-3(m_0)=3m_0$. Unequal populations also suffice: four sites carrying $+m_0$ and two carrying $-m_0$ give $2m_0$, even with identical individual magnitudes. Thus one must compare total sublattice contributions per common volume, rather than merely asking whether two atomic species carry different magnetic moments.

Both ferromagnetism and ferrimagnetism can produce a macroscopic magnet. Their internal organization differs: the latter retains opposing contributions. This makes ferrimagnetism a useful bridge between net magnetization and microscopic magnetic order: reducing one opposing contribution can increase the net sum, even though the total sum of moment magnitudes decreases.

Ferromagnetism is collective magnetic order with spontaneous magnetization below a Curie temperature. Interactions favor aligned magnetic moments; the Ising model represents a two-orientation version with positive coupling.

A magnetic domain can possess spontaneous magnetization even when other domains cancel its contribution to the whole specimen. Compare this spatial cancellation between domains with antiferromagnetism, where opposing magnetic sublattices compensate within the ordered structure, and ferrimagnetism, where their contributions are generally unequal. The magnetic sublattice diagram makes the three local patterns explicit.

Smoothly sweeping an applied magnetic field can reveal abrupt magnetization changes through the Barkhausen effect. A pickup coil detects the resulting changes of magnetic flux, making collective magnetic domain wall motion accessible as an electrical signal.

Magnetic neutron diffraction uses scattering of neutrons by ordered magnetic moments to infer magnetic order. Neutrons also scatter from atomic nuclei, so a pattern can contain nuclear and magnetic contributions. A Bragg scattering peak encodes periodicity; it is not a photograph of an individual spin.

A minimal original calculation makes the extra periodicity visible. Consider identical nuclear scatterers at positions $x_n=na$, with equal neighboring separation $a$, and ordered magnetic moments alternating between $+m_A$ and $-m_B$. A two-site repeating block of length $2a$ has nuclear and magnetic scattering amplitudes, apart from common geometric factors,Here $b$ is the common nuclear scattering coefficient and $q$ the component of the scattering wave vector along the row. The magnetic expression assumes equally scattering point magnetic moments perpendicular to the scattering wave vector; real magnetic scattering measures the perpendicular component and has a momentum-dependent amplitude. This one-dimensional calculation is a periodicity illustration, not a finite-temperature solution for a chain.

For even $h$, $F_N=2b$ and $F_M=m_A-m_B$. For odd $h$, $F_N=0$ and $F_M=m_A+m_B$. Thus equivalent opposing magnetic sublattices, $m_A=m_B$, cancel in forward scattering but reinforce at the odd positions of the doubled period. The intensity is proportional to the squared magnitude of the scattering amplitude, so changing an amplitude's sign does not make a negative intensity.

This calculation also exposes a useful difference between antiferromagnetism and ferrimagnetism: with $m_A=2$ and $m_B=1$, magnetic squared amplitudes are $1$ at even $h$ and $9$ at odd $h$. Extra magnetic peaks can therefore occur in a ferrimagnet too. Identifying a real magnetic structure needs the full crystal basis structure factor, multiple reflections, and the allowed moment orientations, rather than a single extra peak.

A magnetic sublattice groups sites with a common ordered magnetic moment in a description of magnetic order. In the simplest two-sublattice description, assign magnetizations ${\mathbf{M}}_A$ and ${\mathbf{M}}_B$ per total specimen volume, so thatHere $\mathbf{L}$ is one normalization of the staggered magnetization; conventions may include a factor $1/2$. Using the same total volume for both contributions is essential. Equal magnetic moment per site does not imply equal sublattice contributions if the site populations differ.

In this six-site example, three A sites and three B sites give total magnetic moments $6m_0$, $0$, and $3m_0$, respectively, along the chosen axis. The antiferromagnetic case has $\mathbf{M}=0$ but $\mathbf{L}=6m_0\hat{\mathbf{z}}/V$. Cancellation of the sum leaves the alternating information intact.

Phenomena involving magnetic fields, including those produced by electric currents and magnetic moments.

A small total magnetic moment can hide extensive magnetic order. Compare three questions: are the individual magnetic moments nonzero, how are their directions correlated, and what is their vector sum? Ferromagnetism, antiferromagnetism, and ferrimagnetism answer the second question differently. The third question alone cannot identify the order.

In 1949, Clifford Shull and J. Samuel Smart compared powdered manganese(II) oxide at room temperature and $80\mathrm{K}$. Powder preparation supplies many crystal orientations. Their Fig. 1 records scattered-neutron intensity versus counter angle. Nuclear reflections remained, while the cold pattern gained strong reflections forbidden by the chemical unit cell. They indexed these with a magnetic unit cell having twice the chemical cell edge. The warm pattern already contained diffuse magnetic scattering: loss of sharp magnetic peaks does not mean loss of individual magnetic moments.^[ref]

The source was Oak Ridge's graphite-moderated reactor. In Early Development of Neutron Scattering (1994), Clifford Shull describes how the early two-axis instruments used a monochromating crystal to select the incident-neutron wavelength; a shielded neutron counter rotated around the specimen to record angular intensity. This specifies the source–monochromator–sample–counter path, without identifying which instrument revision produced the MnO trace.^[ref]

The control was the same compound above and below magnetic ordering, with retained nuclear peaks identifying the underlying chemical periodicity. The original paper labels an aluminium impurity peak and defers a complete magnetic-structure account. It supplies neither a chemical synthesis recipe nor detailed cold-stage construction. The inference is periodic magnetic order, rather than a claim that every possible structural change was excluded at arbitrary resolution.^[ref]

The connection to the measured angle is the elastic Bragg scattering condition. If the incident and outgoing wave vectors have equal magnitude $2\pi /\lambda$ and make an angle $2\theta$, their difference has magnitudeA longer real-space period supplies additional, smaller reciprocal lattice spacings, which can put magnetic reflections at new counter angles. In the one-dimensional toy model above, doubling the period changes the allowed spacing from $2\pi /a$ to $\pi /a$. The actual MnO indexing is three-dimensional; the toy row does not assert its reflection positions or relative peak strengths.

Louis Néel supplied the ordering theory; the 1949 diffraction work was a later experimental test. His 1970 Nobel Prize in Physics recognized antiferromagnetism and ferrimagnetism. Clifford Shull's neutron-diffraction award followed in 1994.^[ref]

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.

Awarded in equal shares to Hannes Olof Gösta Alfvén for work in magnetohydrodynamics and its applications in plasma physics, and Louis Eugène Félix Néel for discoveries concerning antiferromagnetism and ferrimagnetism and their applications in solid-state physics.^[ref]

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.