Magnetism in Matter
Magnetism in matter is the cleanest demonstration in the book that ordinary condensed matter is a quantum system. The Bohr–van Leeuwen theorem [Bohr:1911] [vanLeeuwen:1921] shows that a classical system of charges in thermal equilibrium has exactly zero magnetization at any field and temperature; every effect in this chapter — Faraday's diamagnetism [Faraday:1846b], Curie's paramagnetic law [Curie:1895], the spontaneous magnetization of iron — therefore has no classical explanation whatever. What supplies it is spin and the Pauli principle of Identical Particles, acting through the exchange interaction [Heisenberg:1928] [Dirac:1926]: an electrostatic energy, of order \(\mathrm{eV}\), masquerading as a coupling between spins and hence producing ordering temperatures of order \(1000\,\mathrm{K}\) where the magnetic dipole–dipole interaction could manage about \(1\,\mathrm{K}\).
The chapter builds from the response of isolated moments to the collective phenomena: Weiss's molecular field [Weiss:1907] and its microscopic replacement by exchange; Néel's antiferromagnetism [Neel:1936] [Neel:1948], invisible to a magnetometer and made visible by neutron diffraction [Shull:1949]; domains [Barkhausen:1919] [Bitter:1931] and hysteresis; spin waves [Bloch:1930b] counted directly by inelastic neutron scattering [Brockhouse:1957]; itinerant ferromagnetism in the band language of Electrons in Solids: Band Theory [Stoner:1938]; and magnetic resonance [Zavoisky:1945] [Purcell:1946a] [Bloch:1946] as the spectroscopy that measures all of it. The critical behaviour near the ordering temperature is the worked example of Phase Transitions and Critical Phenomena and is not repeated here. The standard references are [Blundell:2001] [Ashcroft:1976], with Van Vleck's monograph [VanVleck:1932] as the founding treatment.
Magnetism in Matter: all derivations of this chapter are pending.
The magnetic response of matter
Diamagnetism
[Reserved: Faraday's discovery that every substance responds to a magnetic field, most of them by being repelled [Faraday:1846b]; the Larmor precession argument and the Langevin diamagnetic susceptibility \(\chi\propto-\langle r^{2}\rangle\) per atom [Langevin:1905]; measured values of order \(-10^{-5}\) in SI volume susceptibility, and their near-independence of temperature; the definitions and unit conventions for \(\vect{B}\), \(\vect{H}\), \(\vect{M}\) and \(\chi\) fixed in Electrodynamics in Matter; perfect diamagnetism as a distinct phenomenon, belonging to Superconductivity and Superfluidity.]
There is no magnetically inert matter. Suspended in a strong field gradient, some substances are drawn towards the strong-field region and the majority are pushed away from it; the repulsion is universal, weak, and — unlike the attraction — essentially independent of temperature [Faraday:1846b]. In modern units the volume susceptibility of a diamagnetic solid or liquid is negative and of order \(10^{-5}\). The universality is the informative part: a property possessed by bismuth, water, glass and gold alike cannot depend on permanent atomic moments, since most of these substances have none.
Diamagnetism as the induced response of the bound electrons: the Larmor precession an applied field imposes on an atomic charge distribution, the induced moment opposing the field by Lenz's rule, and the resulting susceptibility proportional to the mean square radius of the electronic charge distribution and to the number of electrons per unit volume, with no temperature dependence because no level populations change
Paramagnetism and the Curie law
[Reserved: Curie's systematic measurements establishing \(\chi=C/T\) for paramagnets and the sharp distinction between the three classes of response [Curie:1895]; Langevin's classical statistical derivation, with the Langevin function \(L(x)=\coth x-1/x\) [Langevin:1905]; its quantum replacement, the Brillouin function for a moment of angular momentum \(J\) [Brillouin:1927], and the saturation curves of chrome, iron and gadolinium salts that confirmed the quantized rather than the classical form; the effective moment \(p_{\mathrm{eff}}\) extracted from a measured Curie constant.]
Substances containing permanent atomic moments are attracted into a field, with a susceptibility inversely proportional to the absolute temperature,
over a wide temperature range and to good accuracy — one of the three distinct classes of magnetic behaviour that Curie's systematic survey separated [Curie:1895]. At low temperature and high field the magnetization departs from Equation (129.1) and saturates, and the measured saturation curves of paramagnetic salts follow the stepped, quantized form appropriate to a moment of definite angular momentum [Brillouin:1927] rather than the classical curve for a freely orientable vector [Langevin:1905] [Blundell:2001]. The Curie constant \(C\) therefore measures the size of the atomic moment, and the shape of the saturation curve measures its angular momentum.
Derivation. Take the simplest case, \(n\) independent moments per unit volume, each of magnitude \(\mu\) and each with only two accessible orientations along the field, of energies \(\mp\mu B\). Their thermal average is
so that \(M=n\mu\tanh(\mu B/k_{B}T)\). Two limits follow at once. For \(\mu B\gg k_{B}T\) the hyperbolic tangent saturates at unity and \(M\to n\mu\): the magnetization has an upper bound, which is the saturation observed. For \(\mu B\ll k_{B}T\), expanding \(\tanh x\approx x\) gives \(M=n\mu^{2}B/k_{B}T\) and hence
which is Equation (129.1) with \(C=n\mu_{0}\mu^{2}/k_{B}\). The inverse temperature is the competition of alignment energy against thermal disorder, and nothing else; it is the same \(1/T\) that governs any two-level population. For a moment of general angular momentum \(J\) and Landé factor \(g\) the sum over \(2J+1\) orientations replaces Equation (129.2) by the Brillouin function [Brillouin:1927], whose small-argument limit gives Equation (129.1) with the effective moment \(\mu_{\mathrm{eff}}=g\sqrt{J(J+1)}\,\mu_{B}\) in place of \(\mu\); the saturation value is \(ngJ\mu_{B}\), so the two limits measure different combinations of \(g\) and \(J\) and between them determine both. Note that Equation (129.2) already assumes that the orientations are discrete, which no classical theory permits (Phenomenon 129.3).
∎The Bohr–van Leeuwen theorem
[Reserved: the theorem that in classical statistical mechanics (Statistical Mechanics) the partition function is independent of the vector potential, because the field shifts the momentum integration variable, so the equilibrium magnetization vanishes identically; proved independently by Bohr [Bohr:1911] and van Leeuwen [vanLeeuwen:1921]; the consequence that magnetism is evidence for quantum mechanics in bulk matter, and the specific ingredients (quantized levels, spin, the exclusion principle) that evade it.]
Matter in thermal equilibrium is measurably magnetic: diamagnetic (Phenomenon 129.1), paramagnetic (Phenomenon 129.2) or, in a few elements, spontaneously magnetized. Classical statistical mechanics, applied to any collection of charges in thermal equilibrium at any temperature and in any applied field, gives exactly zero magnetization [Bohr:1911] [vanLeeuwen:1921]. The disagreement is not one of magnitude but of existence, so every measurement in this chapter, including the most banal, is a measurement of quantum mechanics operating in bulk matter at room temperature.
Derivation. Consider \(N\) classical charges \(q_{i}\) of masses \(m_{i}\) in an arbitrary potential \(U(\vect{r}_{1},\dots,\vect{r}_{N})\) and an external magnetic field described by a vector potential \(\vect{A}\). The field enters the Hamiltonian only through the minimal coupling,
and the classical canonical partition function is the integral of \(\ee^{-\beta\Ham}\) over all positions and all momenta. Hold the positions fixed and change momentum variables to \(\vect{\pi}_{i}=\vect{p}_{i}-q_{i}\vect{A}(\vect{r}_{i})\). The shift is a translation of each momentum by a constant, so its Jacobian is unity, and — this is the whole of the argument — each momentum ranges over all of \(\R^{3}\) both before and after, so the domain of integration is unchanged. Hence
in which \(\vect{A}\) has disappeared entirely. The free energy \(F=-k_{B}T\ln Z\) is therefore independent of the applied field, and the equilibrium magnetization \(M=-V^{-1}\pp F/\pp B\) vanishes identically, for every temperature, every field and every interaction \(U\). The result is stronger than a null prediction for some model: it says no classical model can succeed. What evades it is the discreteness of the momentum-space integration in quantum mechanics — the levels are a countable set that the field genuinely rearranges rather than a continuum that a shift maps onto itself — together with spin, which is not of the form Equation (129.4) at all, and the exclusion principle of Identical Particles, which controls how the levels are filled.
∎Magnetism of the conduction electrons
[Reserved: Pauli paramagnetism — a temperature-independent susceptibility \(\chi=\mu_{0}\mu_{B}^{2}g(E_{F})\) because only electrons within \(k_{B}T\) of the Fermi surface can flip [Pauli:1927a], a direct measurement of the density of states of Electrons in Solids: Band Theory; Landau diamagnetism from the orbital quantization of the same electrons, exactly \(-1/3\) of the Pauli term for free electrons [Landau:1930]; why the two nearly cancel and what the residual measures; the de Haas–van Alphen oscillations as the same physics at high field.]
Where the moments come from
Local moments and the effective moment
[Reserved: partially filled shells, Hund's rules and the Landé \(g\)-factor as established in Atoms and Molecules; the measured effective moments of the rare-earth ions, in excellent agreement with the free-ion \(J\); the transition-metal ions, in agreement only with the spin-only value, and crystal-field quenching of the orbital moment as the explanation [VanVleck:1932] [Blundell:2001]; the Bohr magneton \(\mu_{B}=e\hbar/2m_{e}\) as the natural unit and its value from [Mohr:2025].]
The gyromagnetic ratio measured
[Reserved: the Einstein–de Haas experiment — a suspended iron cylinder magnetized along its axis acquires a measurable rotation, because angular momentum is conserved and the magnetization carries some [Einstein:1915c]; Barnett's converse, magnetization by rotation [Barnett:1915]; and the point that makes the pair worth recounting — Einstein and de Haas reported a gyromagnetic ratio agreeing with the orbital value \(g=1\) that their own theory expected, whereas the correct value is \(g\approx2\), as Barnett's measurement [Barnett:1915] and every later repetition found; the founding experiment of the subject therefore erred in the direction of its authors' prediction, which makes it a case study for the error analysis of Measurement, SI Units, and the Theory of Errors; the moment is spin, not orbital, with the modern precision value from Experiment: The Electron Anomalous Magnetic Moment and the formalism of Angular Momentum and Spin.]
An iron cylinder hung on a fine fibre and magnetized along its axis by a surrounding solenoid begins to rotate about that axis, and reverses its rotation when the magnetization is reversed [Einstein:1915c]. The converse also holds: an unmagnetized ferromagnetic rod spun about its axis acquires a magnetization proportional to its angular velocity [Barnett:1915]. Magnetization is therefore not merely accompanied by angular momentum, it is angular momentum, in a fixed ratio
with \(m_{\mathrm{tot}}\) and \(L_{\mathrm{e}}\) the total magnetic moment and the total electronic angular momentum. The measured value is \(g\approx2\), not the \(g=1\) that orbital motion would give: the moment of a ferromagnet is carried by spin. The measurement is also a standing lesson in experimental method. Einstein and de Haas reported a ratio consistent with the orbital value \(g=1\) that their own theory expected, whereas Barnett's independent measurement [Barnett:1915] and every later repetition give approximately \(2\); the founding experiment of the subject erred in the direction of its authors' prediction.
Derivation. The suspended cylinder is mechanically isolated about its axis, so the component of total angular momentum along that axis is conserved. Split it into the mechanical angular momentum \(L_{\mathrm{mech}}\) of the body as a rigid rotor and the internal electronic angular momentum \(L_{\mathrm{e}}\) that the aligned moments carry. Conservation gives \(\Delta L_{\mathrm{mech}}=-\Delta L_{\mathrm{e}}\), and Equation (129.6) converts the second into a change of magnetic moment:
Every quantity in Equation (129.7) except \(g\) is directly measurable: the change in magnetic moment from the fluxmeter or from the saturation magnetization, and the change in mechanical angular momentum from the amplitude of the torsional oscillation the magnetization reversal excites, given the known moment of inertia and fibre constant. The experiment therefore determines \(g\) with no adjustable parameter. Because the two candidate answers, \(1\) for orbital and \(2\) for spin, differ by a factor of two rather than by a percentage, the measurement discriminates decisively — provided it is not biased, which is exactly the point of the historical note above.
∎Ferromagnetism
Weiss's molecular field
[Reserved: Weiss's postulate of an internal field proportional to the magnetization, \(B_{\mathrm{mol}}=\lambda M\), and the self-consistent equation it yields [Weiss:1907]; spontaneous magnetization below \(T_{C}\), the Curie–Weiss law \(\chi=C/(T-\theta)\) above it, and the measured Curie temperatures — \(1043\,\mathrm{K}\) for iron, \(1394\,\mathrm{K}\) for cobalt, \(631\,\mathrm{K}\) for nickel [Blundell:2001]; the fatal quantitative problem, that a molecular field of the required size corresponds to \(10^{3}\,\mathrm{T}\) and cannot be magnetic in origin; mean-field theory's exponents and their disagreement with experiment, treated in Phase Transitions and Critical Phenomena.]
A few elements are magnetized in zero applied field. Below a sharp temperature \(T_{C}\) — \(1043\,\mathrm{K}\) for iron, \(1394\,\mathrm{K}\) for cobalt, \(631\,\mathrm{K}\) for nickel [Blundell:2001] — a specimen carries a magnetization that grows continuously from zero as the temperature falls, with no applied field to define its direction. Above \(T_{C}\) the spontaneous moment is gone and the susceptibility follows not the Curie law Equation (129.1) but the Curie–Weiss form
with \(\theta\) positive and close to \(T_{C}\) [Weiss:1907]. The ordering temperature is the informative number. It corresponds to an internal field of order \(10^{3}\,\mathrm{T}\), whereas the magnetic field one aligned atomic moment produces at its neighbour is of order \(0.1\,\mathrm{T}\): whatever aligns the moments, it is four orders of magnitude too strong to be their magnetic interaction.
Derivation. Weiss's hypothesis is that each moment experiences, in addition to any applied field \(B\), an internal field proportional to the magnetization already present, \(B_{\mathrm{mol}}=\lambda M\) [Weiss:1907]. Substituting \(B\to B+\lambda M\) into the two-level result Equation (129.2) makes the magnetization implicit in itself,
and everything follows from that one equation. Put \(B=0\) and write \(m=M/n\mu\) and \(k_{B}T_{C}=n\mu^{2}\lambda\); then Equation (129.9) becomes
The right-hand side of Equation (129.10) is an odd, monotonically increasing, concave function of \(m>0\) with slope \(T_{C}/T\) at the origin and saturating at unity. It therefore meets the line \(m\) at a non-zero point if and only if that initial slope exceeds one, that is if and only if \(T<T_{C}\): below \(T_{C}\) a spontaneous magnetization exists, above it only \(m=0\) solves the equation. For \(T>T_{C}\) and small applied field, linearize Equation (129.9) in \(B\) and \(M\), \(M=n\mu^{2}(B+\lambda M)/k_{B}T\), and solve for the susceptibility:
which is Equation (129.8) with the same Curie constant as Equation (129.3): a paramagnet with a positive feedback term diverges at \(T_{C}\) instead of at zero. What the derivation does not supply is \(\lambda\). Setting \(k_{B}T_{C}=n\mu^{2}\lambda\) with the observed \(T_{C}\) gives \(\lambda M\) of order \(10^{3}\,\mathrm{T}\), which no arrangement of magnetic dipoles can produce; the molecular field is a bookkeeping device for an interaction that must be electrostatic in origin (Section 129.3.2). The mean-field treatment also predicts critical exponents that measurement contradicts, which is the subject of Phase Transitions and Critical Phenomena.
∎Exchange
[Reserved: the resolution — the antisymmetry of the many-electron wavefunction (Identical Particles) correlates spin orientation with spatial overlap, so the Coulomb energy itself depends on the relative spin; Dirac's formulation of the exchange energy in terms of a spin permutation operator [Dirac:1926]; Heisenberg's application to ferromagnetism [Heisenberg:1928]; the Heitler–London calculation for two atoms as the transparent worked case [Heitler:1927]; magnitudes of order \(\mathrm{eV}\), which is what makes room-temperature magnets possible.]
The Heisenberg model
[Reserved: the Hamiltonian \(H=-\sum_{ij}J_{ij}\vect{S}_{i}\cdot \vect{S}_{j}\) [Heisenberg:1928], the sign convention for ferromagnetic and antiferromagnetic \(J\), and its status as an effective model derived from the electronic problem rather than postulated; the Ising and \(XY\) limits and the role of anisotropy; the Mermin–Wagner theorem forbidding spontaneous breaking of a continuous spin symmetry in one and two dimensions at finite temperature [Mermin:1966]; exchange constants extracted from measured spin-wave dispersions in Section 129.6.2; the connection to the lattice models of Phase Transitions and Critical Phenomena.]
Itinerant ferromagnetism
[Reserved: the difficulty that iron, cobalt and nickel have non-integer moments per atom — \(2.22\), \(1.72\) and \(0.606\) Bohr magnetons — which no localized-moment model can produce; Stoner's band theory of collective-electron ferromagnetism, with exchange splitting the spin-up and spin-down bands and the criterion \(I g(E_{F})>1\) for spontaneous polarization [Stoner:1938]; the spin-split bands as measured by spin-resolved photoemission; the failure of Stoner theory near \(T_{C}\) and what replaces it; half-metals, band-insulating in one spin channel and metallic in the other [deGroot:1983], and their relevance to Section 129.9.2.]
Antiferromagnetism and ferrimagnetism
Néel's theory
[Reserved: Néel's two-sublattice molecular-field theory, predicting a state of zero net moment with a susceptibility that peaks at the ordering temperature and then falls [Neel:1936]; the Néel temperature and the Curie–Weiss law with negative \(\theta\) above it; ferrimagnetism as the unequal-sublattice case, which explains magnetite and the ferrites [Neel:1948] — the material class behind ferrite cores and the 1970 Nobel prize; the honest point that until 1949 this was a hypothesis about something no instrument could see.]
Neutron diffraction as the decisive evidence
[Reserved: the neutron's magnetic moment scattering off ordered electron spins, so that magnetic order produces extra Bragg peaks; Shull and Smart's observation of the superlattice reflections of manganese oxide, doubling the chemical unit cell and confirming Néel's two-sublattice picture directly [Shull:1949]; the systematic determination of magnetic structures that followed [Shull:1951]; the diffraction formalism of Phonons and Lattice Dynamics and the neutron sources of Nuclear Forces and Nuclear Structure; the 1994 Nobel prize.]
Some compounds order magnetically without becoming magnets. Manganese oxide has a susceptibility that rises on cooling, passes through a maximum at a definite temperature and then falls — the signature Néel predicted for two sublattices magnetized oppositely [Neel:1936] — and above that temperature obeys Equation (129.8) with \(\theta\) negative. No magnetometer can see the order itself, since the net moment is zero at every temperature. Neutron diffraction can: below the transition, manganese oxide shows extra Bragg reflections corresponding to a magnetic unit cell twice the chemical one, exactly as two alternating sublattices require, and those reflections disappear on warming through the transition [Shull:1949] [Shull:1951]. The related case of unequal sublattices, ferrimagnetism, accounts for magnetite and the ferrites [Neel:1948]. Until 1949 this was a hypothesis about something no instrument could detect; the neutron made it an observation.
The two-sublattice molecular-field theory, giving the Néel temperature, the susceptibility maximum at it and the negative Curie–Weiss intercept above it; and the magnetic neutron cross section, in which the neutron's own moment couples to the ordered electronic moments so that the scattering amplitude alternates between sublattices, producing reflections at the reciprocal-lattice vectors of the doubled magnetic cell that are absent from the nuclear pattern
Superexchange and RKKY coupling
[Reserved: how manganese ions separated by an intervening oxygen couple antiferromagnetically at all — Kramers' proposal that the coupling is mediated by virtual excitations of the intervening ion [Kramers:1934], made quantitative by Anderson [Anderson:1950], with the Goodenough–Kanamori rules relating the sign of the coupling to the geometry and occupancy of the bond [Goodenough:1955] [Kanamori:1959]; the RKKY interaction, oscillating in sign with distance as conduction electrons mediate it [Ruderman:1954], which is the mechanism behind the interlayer coupling of Section 129.9.1; double exchange in the mixed-valence manganites [Zener:1951].]
Domains and hysteresis
Domain structure and the Barkhausen effect
[Reserved: why a ferromagnet below \(T_{C}\) can have zero net moment — the dipolar field energy is lowered by breaking into domains of differing orientation, at the cost of the wall energy; Barkhausen's observation of discrete clicks in a coil wound on a slowly magnetized iron rod, the first direct evidence that magnetization proceeds in irreversible jumps [Barkhausen:1919]; jump-size statistics and their power law; the hysteresis loop, coercivity and remanence as the measured quantities, and hard against soft materials.]
Imaging domains
[Reserved: Bitter's technique of decorating a polished surface with a colloidal suspension of fine magnetic particles, which collect at the stray field of the walls and make the domain pattern visible under a microscope [Bitter:1931]; the magneto-optic Kerr effect, the rotation of the polarization of light reflected from a magnetized surface [Kerr:1877], together with Lorentz microscopy and magnetic force microscopy, as the modern successors [Blundell:2001]; what each measures and its resolution; the agreement between imaged wall widths and the theory of Section 129.5.3.]
Domain walls and anisotropy
[Reserved: the Bloch wall as the compromise between exchange, which wants a wide wall, and magnetocrystalline anisotropy, which wants a narrow one, giving widths of tens of \(\mathrm{nm}\) and the scaling \(\delta\propto\sqrt{A/K}\) [Landau:1935]; the Landau–Lifshitz equation of motion for the magnetization [Landau:1935]; magnetostriction and shape anisotropy [Blundell:2001]; wall pinning by defects as the microscopic origin of coercivity and of the Barkhausen jumps of Section 129.5.1.]
Spin waves
Magnons and the three-halves law
[Reserved: the low-lying excitations of an ordered ferromagnet are not single flipped spins but collective precessional waves, with \(\hbar\omega=Dk^{2}\) at long wavelength, found by Bloch [Bloch:1930b]; their quantization as magnons, bosons whose thermal population gives the observed \(M(0)-M(T)\propto T^{3/2}\) law — qualitatively different from the exponential a gapped excitation would give, and thus a test of the picture; the Holstein–Primakoff representation that makes the spin operators bosonic [Holstein:1940]; the antiferromagnetic case with its linear dispersion.]
The spontaneous magnetization of a ferromagnet falls below its zero-temperature value, at temperatures far below \(T_{C}\), as a power of the temperature and not exponentially:
the law Bloch obtained by treating the excitations as collective spin waves rather than as individually flipped spins [Bloch:1930b]. The distinction is testable, because a flipped spin costs a finite energy and would give an exponentially small reduction. The excitations themselves are seen directly: inelastic neutron scattering from magnetite resolves a magnon branch and measures its dispersion [Brockhouse:1957], which is quadratic, \(\hbar\omega=Dk^{2}\), at long wavelength — and it is the vanishing of \(\omega\) as \(k\to0\), guaranteed by the rotational symmetry the ordered state breaks, that supplies the low-energy excitations the exponent requires.
Derivation. Take the measured dispersion \(\hbar\omega=Dk^{2}\) as given, and take the magnons to be independent bosons, which is the leading approximation at low temperature since their density is then small. Each magnon reduces the total spin projection by one unit, so the magnetization deficit is proportional to the magnon number density,
with no chemical potential, magnon number not being conserved. Substitute \(u=Dk^{2}/k_{B}T\), so that \(k=(k_{B}Tu/D)^{1/2}\) and \(k^{2}\dd k=\tfrac{1}{2}(k_{B}T/D)^{3/2}u^{1/2}\dd u\); the angular integral gives \(4\pi\) and
which is Equation (129.12). The exponent is entirely a statement about the low-energy end of the spectrum: it is \(3/2\) because three-dimensional phase space contributes \(k^{2}\dd k\) while the energy goes as \(k^{2}\), so the number of thermally accessible modes scales as \(T^{3/2}\). Had the excitation spectrum a gap \(\Delta\) — as a single flipped spin, held in place by its neighbours, would have — the Bose factor would supply \(\ee^{-\Delta/k_{B}T}\) instead, and the measured power law excludes that picture. The fitted stiffness \(D\) from the neutron dispersion and the fitted coefficient of \(T^{3/2}\) from the magnetization must agree, which closes the loop between two independent experiments.
∎Magnons measured by neutron scattering
[Reserved: inelastic neutron scattering as the direct measurement of \(\omega(\vect{k})\) for magnons, exactly as for the phonons of Phonons and Lattice Dynamics; Brockhouse's observation of spin-wave scattering in magnetite, the first such measurement [Brockhouse:1957]; the exchange constants \(J\) read off the fitted stiffness \(D\), closing the loop with Section 129.3.3; magnon gaps from anisotropy, and the scattering theory of Scattering Theory.]
Magnetic phase transitions
[Reserved: the ferromagnetic and antiferromagnetic transitions as worked examples of the general theory in Phase Transitions and Critical Phenomena — order parameter, spontaneous symmetry breaking, universality classes, and the measured critical exponents of nickel and of the model magnets, which disagree with the mean-field values implied by [Weiss:1907]; the divergence of the susceptibility and of the correlation length; the specific-heat anomaly; a cross-reference rather than a repetition, since the renormalization group belongs to The Renormalization Group.]
Magnetic resonance
Electron spin resonance
[Reserved: Zavoisky's observation of resonant microwave absorption by paramagnetic salts at \(\hbar\omega=g\mu_{B}B\), the first magnetic resonance experiment [Zavoisky:1945]; the measured \(g\)-tensor as a probe of the local crystal field and of orbital quenching; hyperfine structure and its use in identifying defects and radicals; ferromagnetic resonance and its relation to the Landau–Lifshitz dynamics [Landau:1935].]
Nuclear magnetic resonance
[Reserved: the independent 1946 detections of nuclear magnetic resonance in condensed matter, by absorption in paraffin [Purcell:1946a] and by induction in water [Bloch:1946]; the Bloch equations, \(T_{1}\) and \(T_{2}\); the chemical shift and the Knight shift, the latter measuring the conduction-electron spin susceptibility of Section 129.1.4; the nuclear moments themselves belonging to Nuclear Forces and Nuclear Structure; the 1952 Nobel prize, and magnetic resonance imaging as the applied consequence.]
Magnetotransport and spintronics
Giant magnetoresistance
[Reserved: the independent discovery, in iron–chromium multilayers, that the resistance changes by tens of percent when an applied field aligns the antiferromagnetically coupled ferromagnetic layers [Baibich:1988] [Binasch:1989] — an effect an order of magnitude larger than any previously known magnetoresistance, hence the name; spin-dependent scattering and the two-current model as its explanation; the RKKY oscillation [Ruderman:1954] that sets the interlayer coupling; the 2007 Nobel prize and the read heads that followed.]
Tunnelling magnetoresistance and spin transport
[Reserved: Jullière's magnetic tunnel junction, in which the tunnelling conductance between two ferromagnets depends on their relative magnetization through the spin polarizations of their densities of states [Julliere:1975], and the large room-temperature ratios later obtained with crystalline barriers; spin-transfer torque as the reciprocal effect, a spin-polarized current switching a magnet [Slonczewski:1996]; the honest boundary of this chapter — device physics is only included where it constitutes a measurement of the magnetism itself.]