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Anderson localization is the confinement of electron wave functions, or other wave modes, by interference between repeated wave scattering paths in static disorder. A localized eigenstate occupies a limited region even in a much larger sample. Disorder can therefore suppress diffusion without absorbing the wave. In 1958, Philip Warren Anderson studied random site energies and intersite coupling, finding a regime without transport. His paper develops a theoretical model; it does not report an Anderson laboratory measurement.^[ref]

The distinction from ordinary multiple scattering is the retention of wave phases: alternative paths contribute probability amplitudes before their squared modulus is taken. Adding path probabilities alone loses the cross terms responsible for interference. This makes Anderson localization useful as a warning about transport models: repeated deflections need not produce unlimited diffusion. It also connects electron transport with wave propagation in a random medium, although the detailed equations and experimental diagnostics differ between electrons, light and acoustic waves.

Philip Warren Anderson's 1977 lecture distinguishes localized magnetic moments from disorder-localized states. It credits Nevill Francis Mott with recognizing the significance of the mobility edge. A Mott insulator instead requires electron interactions and can exist without disorder.^[ref]

John Hasbrouck Van Vleck developed quantum mechanics explanations of magnetic susceptibility, including its temperature dependence; this is a distinct contribution from Anderson localization.^[ref]

In 1986, Johannes Georg Bednorz and Karl Alexander Müller precipitated mixed oxalates from aqueous barium, lanthanum and copper nitrates using oxalic acid. Heating at $900{}^{\circ }\mathrm{C}$ for five hours preceded pressing at $4\mathrm{kbar}$ and reheating for sintering. X-ray diffraction resolved three phases. Rectangular pieces carried gold electrodes and indium wires; four-terminal sensing in a continuous-flow cryostat measured electrical resistance versus temperature. Annealing atmosphere and measuring electric current density were varied; larger electric current densities suppressed the drop. An onset near $35\mathrm{K}$ differed from the much lower temperature where the drop approached three orders of magnitude.^[ref]

This was evidence for possible granular superconductivity, with phase identification unfinished. The paper deferred Meissner effect and heat capacity tests; its proof note corrected bulk lanthanum/barium ratios. A resistive onset did not establish a uniform bulk superconducting transition temperature.^[ref]

Their 1987 authored lecture describes later magnetometer checks on lead, followed by magnetic susceptibility measurements: samples with a resistive drop developed diamagnetism at somewhat lower temperatures. Combining magnetic, transport and X-ray diffraction results identified barium-substituted $\mathrm{L}{\mathrm{a}}_2\mathrm{Cu}{\mathrm{O}}_4$ as the superconducting phase. These later results strengthened the case beyond the initial mixed-phase transport evidence.^[ref]

The coefficient in the high-temperature Curie Magnetic susceptibility.

For independent moments of angular momentum $J$ at small ${\mu }_B|B|/(k_{B}T)$, the zero-field variance is $⟨m^2⟩=J(J+1)/3$. Differentiating the canonical mean gives the magnetic susceptibilityThus susceptibility is inversely proportional to temperature. The small-field or high-temperature condition distinguishes this law from the exponentially vanishing fixed-field susceptibility of a saturated paramagnet at low temperature.

The Curie–Weiss law approximates the magnetic susceptibility of interacting localized moments by $\chi =C/(T-\Theta )$ in a suitable paramagnetic temperature range. A positive Weiss temperature often reflects predominantly ferromagnetic interactions, while a negative value often reflects antiferromagnetic interactions. It is not a universal exact law through an ordering transition, and crystal-field effects or itinerant carriers can require another description.

Diamagnetism is a magnetic response in which the induced magnetization opposes the applied magnetic field intensity. In a linear isotropic description, $\mathbf{M}=\chi \mathbf{H}$ with negative magnetic susceptibility $\chi$.

For a smooth single-spin density of states $g$ near a positive Fermi energy and fixed total particle number, the Sommerfeld expansion gives $\mu ={ϵ}_F-({\pi }^2/6)(g_F^{\prime }/g_F)(k_{B}T{)}^2+O(T^4)$. The spin contribution to the total-moment magnetic susceptibility, defined by differentiating with respect to $B$, is $2{\mu }_B^2\int g(E)[-n^{\prime }(E)]dE$. Expanding about $\mu$ and including the shift of chemical potential gives the displayed formula. The $g_F^{\prime \prime }$ term is thermal broadening; the $-g_F^{\prime 2}/g_F$ term enforces fixed particle number. A band edge or singular density of states within $k_{B}T$ of ${ϵ}_F$ requires a separate analysis. Orbital magnetic response is excluded.

For constant scalar response $\mathbf{m}={\alpha }_B{\mathbf{B}}_0$ and a body small compared with the applied field's spatial variation scale, the gradient of the induced magnetic dipole energy gives $\mathbf{F}={\alpha }_B\nabla B_0^2/2$. Along an axis this is $F_z=md{B}_0/dz$. For an ellipsoidal body, put ${\chi }_{\mathrm{eff}}=\chi /(1+N_d\chi )$ to obtain $F_z={\mu }_{0}V{\chi }_{\mathrm{eff}}{H}_0{H}_0^{\prime }$. Positive effective magnetic susceptibility attracts the body towards stronger field; a negative response repels it. Reversing the source current leaves the force unchanged because both the induced moment and the field gradient reverse. Shape and orientation enter through the demagnetizing factor, not merely through volume.

William Francis Giauque's 1949 lecture, Some Consequences of Low Temperature Research in Chemical Thermodynamics, credits Peter Debye with an independent proposal. It describes gadolinium sulfate octahydrate in a jacket immersed in liquid helium, with an external solenoid. Exchange gas conducts away heat during magnetization; evacuation isolates the sample before field reduction. A liquid-nitrogen shield reduces radiative heat transfer. Its photograph identifies the 1933 apparatus operated with Duncan Peck MacDougall. An inductance bridge reads the sample's magnetic susceptibility; measuring coils remain outside the evacuated jacket in liquid helium. Increasing magnetic susceptibility indicates cooling, but converting it to temperature using Curie's law becomes approximate as that law fails.^[ref]

The measurement logic deserves a separate check. A changed inductance alone is not an absolute temperature calibration: the sample geometry, applied measuring magnetic field, and constitutive response enter the inference. A modern control would compare the susceptibility-based thermometer with an independently calibrated thermometer over their common range, check recovery after repeated sweeps, and measure subsequent warming to bound incoming heat. These are proposed controls derived from the readout principle, rather than additional procedures attributed to the 1933 experiment. The historical working salt has a richer magnetic level structure than the illustrative two-level paramagnetic cooling model.

V. Cannella and J. A. Mydosh measured low-field magnetic susceptibility of gold-iron alloys using a mutual inductance bridge. Moving samples into and out of the pickup arrangement separated their signal from the background. Cooling revealed sharp susceptibility cusps; changing concentration, sample shape and applied magnetic field tested their interpretation. Larger fields rounded the peaks. Their 1972 paper interpreted low-concentration ordering in antiferromagnetic terms: the cusp alone did not demonstrate Parisi's later theory.^[ref]

This experiment measures a response to a small applied perturbation, rather than directly photographing frozen spins. Comparing field strengths is essential because the probing magnetic field itself changes the response.^[ref]

A Gouy balance detects the force exerted by a nonuniform magnetic field on a sample of uniform cross-section extending between high- and low-field regions. TeachSpin's implementation reverses the usual arrangement: a rare-earth permanent-magnet assembly sits on a one-pan electronic balance, while a separate holder brings the sample into the field. The magnet's apparent mass change measures the opposite of the sample's vertical force, by Newton's third law. The described pole faces have diameter $25\mathrm{mm}$ and separation $15\mathrm{mm}$; samples occupy cuvettes of cross-section $(10\mathrm{mm}{)}^2$ and height about $32\mathrm{mm}$. ^[ref]

The same balance calibrates magnetic flux density by measuring the reaction force on the magnet when a wire of known horizontal length $L$ carries known electric current $i$ perpendicular to the field: $|F_z|=iL{B}_0$. The manufacturer describes water being repelled by the high-field region, identifying negative magnetic susceptibility. This observed sign supplies a qualitative test, not a measured Curie's law curve. ^[ref]

For weak reversible linear response, a uniform sample displacing a weakly magnetic surrounding medium has, to leading order, downward forceThis follows by integrating the induced-dipole force density $({\chi }_v-{\chi }_{\mathrm{medium}}){\partial }_z(B_0^2)/(2{\mu }_0)$ along a downward coordinate. $A$ is the cross-sectional area, and the two $B_0$ values are applied magnetic flux densities. The force depends on the endpoint fields; the entire field-gradient profile need not be measured under these assumptions. In the reversed apparatus, $F_{\downarrow }=-g\Delta m_{\mathrm{magnet}}$ after subtracting the holder background, where $g$ is gravitational acceleration. This sign must be reversed before inferring volume magnetic susceptibility.

Useful controls are an empty-holder reading at the same position, a distant-sample baseline, repeated wire-current reversals during field calibration, and stable temperature. They separate holder response, balance drift, force offsets, and thermal changes; these are proposed checks for the stated inference. Finite endpoint fields, nonuniform cross-section, uneven powder packing, stray forces and a demagnetizing field limit the simple formula. If magnetization is nonlinear, the weak-response force involves $A{\int }_{B_{0,\mathrm{top}}}^{B_{0,\mathrm{bottom}}}M(B_0)d{B}_0$ instead of a constant-${\chi }_v$ endpoint square. A balance reading then measures an integrated response, not directly the local differential magnetic susceptibility.

David Morris Lee, Douglas Dean Osheroff and Robert Coleman Richardson compressed coexisting liquid and solid helium-3 in a cell precooled by a dilution refrigerator. Helium-4 pressure drove coupled bellows and a piston; a capacitance diaphragm gauge read pressure and platinum nuclear magnetic resonance supplied thermometry. Uniform compression revealed a slope break A and a small pressure drop B; decompression and varied solid fraction checked reproducibility. Their April 1972 paper, Evidence for a New Phase of Solid He3, interpreted a new solid phase, without establishing liquid superfluidity. ^[ref]

The October follow-up replaced the thermometer assembly with a sample nuclear magnetic resonance coil. An applied magnetic field gradient selected different sample layers, separating strong solid signals from the liquid background. At A the liquid resonance shifted; at B its shift disappeared and its magnetic susceptibility fell. Removing the gradient resolved the frequency shift directly. Independence from solid fraction and distinct relaxation times supported the liquid assignment. Thus the reinterpretation rested on phase-selective evidence, rather than relabeling the first paper's title. These magnetic observations identify changed liquid phases; they do not directly measure frictionless flow. ^[ref]

Douglas Dean Osheroff recalls the subsequent p-wave interpretation. ^[ref]

The inference has a useful limit. A pressure anomaly in two coexisting phases cannot, by itself, identify which phase changed: both contribute to the common phase coexistence curve. Thermometer lag, a temperature gradient or slow equilibration can also separate indicated temperature from the sample state. Correlating another response with the anomaly provides information that repeating the same pressure trace cannot supply. Conversely, a shifted magnetic resonance alone does not measure a frictionless flow rate.

An iron oxide, $\alpha \text{-}\mathrm{F}{\mathrm{e}}_2{\mathrm{O}}_3$. It typically has weaker induced magnetic susceptibility than magnetite but can preserve stable remanent magnetization.

Define the instantaneous spin magnetization $m=\frac{1}{4}{\sum }_i{\sigma }_i$. Global reversal sends $m$ to $-m$ without changing the zero-field energy. Pairing every microstate with its reversed partner proves $⟨m⟩=0$ at every positive temperature, even when almost all probability lies on the two fully aligned patterns. A vanishing mean does not imply weak alignment: the limiting probability distribution is half at $m=+1$ and half at $m=-1$.

The Ising ring magnetization fluctuations measure this distinction through variance. In the $d=0$ class, both patterns have $m^2=1$. Among the $d=2$ patterns, eight have $m^2=1/4$ and four have $m^2=0$; the two $d=4$ patterns also have $m^2=0$. ConsequentlyIt equals $6/17$ at $K=(\ln 2)/4$, approaches $1/4$ for four independent fair Ising spins at $K\to 0$, and approaches $1$ as $K\to \infty$. Unlike the energy variance, the spin magnetization variance remains large in the low-temperature symmetric mixture.

The same thermal fluctuation controls the field response. Allow $h$ to vary before setting it to zero; differentiating the Boltzmann distribution inserts $\beta {\sum }_i{\sigma }_i=4\beta m$, givingThis is a spin-normalized magnetic susceptibility with respect to the energy-valued parameter $h$; conversion to physical magnetization and magnetic field requires the magnetic moment and sample volume factors.

For finite $N$, $Z$ is a finite positive sum of exponential functions. Its free energy $-k_{B}T\log Z$ is a real analytic function for $T>0$, so the smooth heat capacity peak here is not a critical phase transition. Spontaneous magnetization concerns the order of limits: first a thermodynamic limit, then removal of a symmetry-breaking field. The four-spin Ising ring alone establishes neither that limit nor any two-dimensional Ising model result.^[ref]

For identical isotropic closed-shell atoms of number density $n$, with $Z$ electrons and mean squared radius $⟨r^2⟩$ per electron, the quadratic magnetic field energy is $Z{e}^2{B}^2⟨r^2⟩/(12m_e)$. It follows from the $e^2{\mathbf{A}}^2/(2m_e)$ term with $\mathbf{A}=\mathbf{B}\times \mathbf{r}/2$. Differentiating the energy with respect to field and using $B\simeq {\mu }_{0}H$ yields the negative SI volume magnetic susceptibility. For helium, $Z=2$; combining it with the independent spin-one-half nuclear response ${\mu }_{0}n{\mu }_a^2/(k_{B}T)$ gives a density-independent sign-change scale $T_0=3m_e{\mu }_a^2/(e^2{k}_B⟨r^2⟩)$.

The observable physical and compositional character of rock, including its constituent minerals, texture and grain size. Different lithologies can have distinct spectral reflectance, density and magnetic susceptibility, making geological mapping possible through several independent measurements.

A rock relatively rich in magnesium- and iron-bearing silicate minerals. Basalt and gabbro are common examples; their magnetic susceptibility depends particularly on magnetic-mineral abundance.

Magnetic susceptibility describes how readily a magnetic field changes a material's magnetization. The field variable and normalization matter: volume magnetic susceptibility in the International System of Units is dimensionless, whereas a derivative of total magnetic moment with respect to $B$ has units of $\mathrm{J}{\mathrm{T}}^{-2}$. A slope measured against the applied magnetic field intensity also depends on the sample's demagnetizing field.

For an isotropic, reversible linear response through the origin, writeHere $\mathbf{M}$ is magnetization, or magnetic moment per volume; ${\mathbf{H}}_{\mathrm{int}}$ is the internal magnetic field intensity; ${\mathbf{B}}_{\mathrm{int}}$ is the internal magnetic flux density; and ${\mu }_0$ is the vacuum permeability. Both $M$ and $H$ have units $\mathrm{A}{\mathrm{m}}^{-1}$. The magnetic permeability is ${\mu }_0(1+{\chi }_v)$. ^[ref]

The independent-moment magnetic response below shows why one number cannot describe every field: magnetization approaches a saturation magnetization, while its differential magnetic susceptibility approaches zero. Anisotropy requires a magnetic susceptibility tensor; hysteresis requires specifying the preparation and sweep branch as well as the bias magnetic field, temperature, and timescale. The Gouy balance turns a small response into a measurable force without presuming that its readout is already the intrinsic volume magnetic susceptibility.

The magnetic-susceptibility critical exponent is defined by the zero-field magnetic susceptibility $\chi \sim |t{|}^{-\gamma }$. Separate amplitudes, and occasionally separate exponents, may occur on the two sides of the transition.

The difference $\Delta \mathbf{M}={\mathbf{M}}_{\mathrm{target}}-{\mathbf{M}}_{\mathrm{reference}}$ between a target region and the reference material it replaces. A weakly magnetic crater infill inside magnetic basalt can have a negative induced contrast, even though both materials have positive magnetic susceptibility. The sign and direction of this contrast, including remanent magnetization, determine a total-field magnetic anomaly together with geometry and the ambient field. Geophysical density contrast does not determine magnetic contrast.