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Antiferromagnetism is ordered compensation of magnetic moments. In the collinear two-sublattice model, ${\mathbf{M}}_A=-{\mathbf{M}}_B\ne 0$: net magnetization vanishes but staggered magnetization remains. Louis Néel developed this possibility.^[ref]

For an original local energy check, perturb the antiparallel pair in exchange interaction by writing $\theta =\pi -\delta$. ThenFor $J>0$, a small relative rotation costs positive energy. This verifies a pair's preference; it does not prove a bulk ordering temperature. A whole crystal must satisfy many bonds simultaneously, and thermal fluctuations and quantum effects matter. The Heisenberg antiferromagnet and antiferromagnetic Ising model specify different assumptions about those collective degrees of freedom.

Why is the distinction useful? A measurement of zero magnetization cannot decide between an ordered alternating arrangement, a disordered ensemble, and oppositely magnetized magnetic domains. A spatial probe such as magnetic neutron diffraction tests the pattern directly. The manganese monoxide neutron diffraction experiment is a particularly clear example: the order adds a new periodic scattering signal even though the net magnetic moment cancels.

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.

The structure factor for magnetic neutron diffraction sums the magnetic form factor and the magnetic moment component perpendicular to the scattering vector over the magnetic atomic basis. For unpolarized neutrons, the magnetic contribution to the coherent diffraction wave intensity is proportional to $|{\mathbf{F}}_M{|}^2$. A periodic change of sign of magnetic moments can shift its allowed peaks away from the nuclear reciprocal lattice vectors without changing nuclear positions.

In neutron diffraction, coherent elastic scattering adds scattering amplitudes from regularly positioned scatterers. Since $E_f=E_i$, the neutron scattering kinematics gives $k_f=k_i=2\pi /\lambda$ andFor lattice planes of interplanar spacing $d_{\mathrm{p}}$, constructive addition requires $Q=2\pi n/d_{\mathrm{p}}$ along the plane normal. Equating these expressions proves Bragg's law:For example, stipulate $d_{\mathrm{p}}=2$ ångströms and $\lambda =2$ ångströms. The first diffraction order has $\theta =30^{\circ }$ and detector deflection $60^{\circ }$. A peak position constrains a period; its intensity also depends on the crystal basis structure factor.

Clifford Glenwood Shull describes powdered sodium chloride, a crystal monochromator, and a shielded counter recording angular counts. Inserting cadmium measured background; substantial diffuse scattering complicated peak interpretation.^[ref]

Powdering supplies many crystal orientations. Comparing the angular peaks with Bragg's law constrains crystal structure; a background estimate helps distinguish specimen peaks from unwanted counts. A two-axis diffraction instrument has no separate outgoing energy analysis, so an angular pattern alone does not separate every inelastic contribution. The richer magnetic example, including a warm/cold comparison and its limits, is the manganese monoxide neutron diffraction experiment. Its extra peaks reveal magnetic order beyond the chemical periodicity; the associated magnetic neutron diffraction model is developed there.

Neutron physics studies the production, interaction, transport and detection of neutrons. A neutron has no net electric charge, so it does not experience the long-range Coulomb repulsion that impedes charged particles approaching a positively charged atomic nucleus. Its nuclear interactions nevertheless depend strongly on energy and target isotope.

Elastic scattering can slow neutrons in a neutron moderator, capture can change a nucleus, and suitable reactions can initiate nuclear fission. Neutron magnetic moments also couple to magnetic structure, enabling magnetic neutron diffraction. A measured attenuation therefore combines different reaction channels and must not be interpreted as one universal absorption probability.

Neutron scattering measures how a specimen changes the directions and kinetic energies of incident neutrons. Neutron diffraction uses the elastic scattering pattern to investigate crystal structure and magnetic order; inelastic neutron scattering uses energy transfer to investigate excitations such as phonons and magnons. The two measurements ask complementary questions: what arrangement repeats in space, and what disturbances can propagate through it?

The 1994 Nobel Prize in Physics recognized Clifford Glenwood Shull's diffraction work and Bertram Neville Brockhouse's spectroscopy.^[ref]

Nuclear scattering couples to atomic nuclei; magnetic scattering couples the neutron's magnetic moment to electronic magnetic moments. The latter underlies both magnetic neutron diffraction and magnon spectroscopy. An atomic number is not a nuclear neutron scattering weight: the relevant coefficient in the elementary coherent model is the nuclear neutron scattering length.

The nuclear neutron scattering length is the effective length coefficient for low-energy nuclear scattering amplitude. For the elementary coherent calculation below, use a bound coherent coefficient $b_j$ for each site, and neglect absorption so that $b_j$ is real. This coefficient describes the atomic nucleus, including its isotope and nuclear spin dependence; it is not proportional to atomic number. Replacing nuclear coefficients by electronic magnetic moments would describe a different scattering channel, with additional direction and spatial-distribution factors, as explained in magnetic neutron diffraction.

Weak ferromagnetism from canted antiferromagnetic order is the consistent model. Importantly, the plot has different magnetization scales: the left axis belongs to Mn and Fe; the right belongs to Co and Ni. Approximate low-temperature field-cooled readings are 40, 50, 450 and 390 G cm${}^3$ mol${}^{-1}$, respectively. Dividing by 5585 givesThese moments are tiny compared with fully aligned moments of about 5.09, 4.40, 4.25 and 2.47 ${\mu }_B$, calculated using the Curie-derived $g$ values below. A conventional ferromagnet of the full high-spin ions would have much larger spontaneous magnetization; a simple bipartite array of identical metal sites also supplies no obvious unequal sublattice moments for ferrimagnetism. Opposed large sublattice moments with a small transverse uncompensated component reconcile the net response with negative $\Theta$.

If the 10 Oe field has aligned the weak-ferromagnetic domains, $M_{\mathrm{weak}}=gS{\mu }_B\sin \alpha$ estimates sublattice canting angles of roughly 0.08°, 0.12°, 1.1° and 1.6°. These are illustrative, not precision structure determinations: the low field may not fully align the domains, and the readings are graphical. Bulk magnetization strongly supports this model but does not uniquely establish every spin direction; magnetic neutron diffraction would test the microscopic arrangement.

Determining antiferromagnetic magnetic order is a good reason to use neutron scattering. The neutron has a magnetic dipole moment, so its scattering amplitude includes the spatial arrangement of electronic magnetic moments. Alternating magnetic moments can produce magnetic neutron diffraction peaks even when there is no change of the nuclear crystal structure. Ordinary nonresonant X-ray diffraction primarily measures the electron density, and is much less sensitive to this magnetic order.

An applied magnetic field changes the alignment of the manganese magnetic moments. The component of their magnetic structure factor perpendicular to the scattering vector contributes to magnetic neutron diffraction. A field-induced uniform magnetization has the same translational periodicity as the nuclear crystal structure, so its scattering amplitude can occur at the existing reciprocal lattice vectors. The diffraction wave intensities can therefore change without moving the peaks. Their angular positions still measure the same interplanar spacings. This mechanism does not require a measurable change of the lattice constant.

Measure several allowed magnetic reflections from an oriented single crystal by magnetic neutron diffraction, preferably with polarization analysis. A neutron couples to the component of the ordered magnetic moment perpendicular to the scattering vector $\mathbf{Q}$. For a collinear arrangement, the geometric intensity factor isCompare reflections whose $\mathbf{Q}$ point in different crystal directions, correcting for the magnetic form factor and known experimental geometry. Their relative intensities distinguish candidate spin directions; a moment parallel to $\mathbf{Q}$ gives no magnetic signal from that component. Polarized neutrons help separate nuclear and magnetic contributions. Observing only peak positions establishes periodicity, not the spin direction.

Exchange interaction arises from Coulomb interaction together with the antisymmetry required for identical electrons. Different spin states require different spatial symmetry and therefore different pair correlations and Coulomb energy; exchange is not the classical magnetic dipole force between spins. For electrons in distinct atomic orbitals, positive direct exchange favors parallel spins, underlying Hund's rules and the formation of local moments.

In a solid with localized moments, a common convention is $H=-J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. Then $J>0$ favors ferromagnetism and $J<0$ favors antiferromagnetism. Superexchange through a ligand and other virtual hopping processes can produce either effective sign depending on orbital occupancy and geometry; half-filled orbitals often give antiferromagnetic coupling. For itinerant electrons, the Stoner criterion describes a competition between exchange energy gain and the kinetic cost of splitting the spin populations.

Experiments distinguish these descriptions through both static and dynamic response. Magnetic neutron diffraction locates magnetic Bragg peaks and therefore the ordering wavevector; an antiferromagnet can have such peaks despite zero bulk moment. Inelastic neutron scattering measures spin-wave dispersion, from which an exchange scale can be fitted. A Curie-Weiss susceptibility intercept gives complementary information about dominant coupling, but its sign alone does not establish the actual ordered structure.

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.