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The 1933 report Ein neuer Effekt bei Eintritt der Supraleitfähigkeit by W. Meissner and R. Ochsenfeld describes cooling tin and lead cylinders in a fixed transverse magnetic field. Two parallel cylinders were about $140\mathrm{mm}$ long and $3\mathrm{mm}$ thick, separated by $1.5\mathrm{mm}$. A roughly $10\mathrm{mm}$ rotating search coil between them drove a ballistic galvanometer. At about $5\mathrm{G}$, the reported below/above-transition induction ratios were $1.70$ for tin single crystals and $1.77$ for polycrystalline lead. The increased gap magnetic field indicates redistribution outside the cylinders, rather than an unchanged frozen field. Coil-winding geometry and noncircular tin specimens limited comparison with the ideal prediction. ^[ref]

Their second arrangement used a lead tube, about $130\mathrm{mm}$ long with $3\mathrm{mm}$ outer and $2\mathrm{mm}$ inner diameters. A rotating search coil could sit inside or alongside it. The bore retained magnetic flux after removing the applied magnetic field; the coil nearly filled the bore, preventing a test of interior field uniformity. A hole containing magnetic field is distinct from magnetic field penetrating the surrounding material exhibiting superconductivity. ^[ref]

The modern Field-cooled Meissner experiment provides a complementary readout. MIT's probe I places a sample in a brass holder, with an $810$-turn test coil of $7.1\mathrm{mm}$ inner diameter, $10.5\mathrm{mm}$ outer diameter and $12.3\mathrm{mm}$ length, inside a longer solenoid. A nearby silicon-diode thermometer tracks cooling by helium gas. With lead still normal, the guide applies about $200\mathrm{mA}$ to the solenoid and connects the test coil to a current-integrator box and oscilloscope. Cooling produces a transient; warming gives the opposite sign. Switching off the solenoid while the sample is normal checks the sign. ^[ref]

The inferred change is in linked magnetic flux, not a direct scan of $B(z)$. For a stationary test coil the pulse obeys $\int Vdt=-\Delta \Psi$, where $\Psi$ is its magnetic flux linkage. An integrator or AC-coupled oscilloscope needs its transfer response accounted for before extracting this pulse area. Keep the source electric current fixed during cooling, monitor temperature, and compare warming and cooling slowly enough to reduce sample–thermometer lag. An AC inductance change alone can reflect changed normal-state electrical conductivity; the fixed-field expulsion test distinguishes that alternative. ^[ref] Neither this coil readout nor the historical gap-field measurement resolves the microscopic London penetration depth by itself.

A change between a material behaving as an electrical conductor and as an electrical insulator. The distinction concerns the limiting electronic response as temperature tends to zero; finite thermally activated electrical conductivity alone does not establish a metallic state. A Peierls transition opens a band gap through a change in the crystal structure.

A mobility edge separates localized from extended energy eigenstates on the energy axis. Three-dimensional disordered models can have both. In the infinite one-dimensional tight-binding model with independent continuous random site energies, all states are localized for any nonzero disorder, with probability one. This statement excludes specially correlated disorder.^[ref]

A mobility edge is not a band gap: states can exist on both sides of $E_c$, but differ in spatial extent. For electrons, the location of the Fermi energy relative to a mobility edge matters for low-temperature electrical conductivity. A three-dimensional sample need not have an extended interval at every disorder strength. In particular, the finite one-dimensional calculations below neither locate a three-dimensional mobility edge nor establish a nonzero universal critical disorder.

When spin-flip scattering is weak, two spin populations have separately defined current responses and share a terminal electric field. Their electrical conductivities add: $\sigma ={\sigma }_{\uparrow }+{\sigma }_{\downarrow }$, so $\rho ={\rho }_{\uparrow }{\rho }_{\downarrow }/({\rho }_{\uparrow }+{\rho }_{\downarrow })$. Spin-dependent band structure and scattering set the channel parameters; being the majority population does not universally imply lower channel resistivity. The model is named for Nevill Francis Mott and is the independent-channel basis for many giant magnetoresistance models.

Yoichiro Nambu's superconductivity work led to the spontaneous symmetry breaking ideas recognized by his share of the 2008 Nobel Prize in Physics.^[ref] His 1960 treatment made the Meissner effect response consistent with gauge invariance by including collective pair contributions alongside the quasiparticles. The resulting Ward identity relates the interaction correction to the current and the self-energy; retaining the quasiparticles alone omits part of the response.^[ref]

The individual Nobel lecture explains the bridge to particle: the superconductivity gap suggested a dynamically generated fermion mass, and the accompanying collective excitation suggested a light pion. It also notes that the long-range Coulomb interaction turns the charged phase response into a plasmon. A neutral global phase mode and a charged material therefore need different low-momentum descriptions.^[ref]

In 1961, Yoichiro Nambu and Giovanni Jona-Lasinio supplied a four-fermion model with a self-consistent mass and a massless pseudoscalar bound state: an idealized pion. Their original constituents were nucleon fields. Their introduction explicitly distinguishes the zero-mass neutral collective excitation from its charged plasma counterpart and allows approximate chiral symmetry as the explanation of a nonzero pion mass. This is a model of dynamical symmetry breaking, rather than a microscopic derivation of Quantum chromodynamics.^[ref]

In the modern two-light-quark account, chiral symmetry breaking leaves isospin intact and produces three Goldstone bosons in the exact non-anomalous chiral limit. Small quark masses explicitly break that chiral symmetry, so the observed pions are pseudo-Goldstone bosons. The chiral protection of a fermion mass identifies the distinct obstruction to inserting an explicit fermion mass into a chirally symmetric equation. The useful lesson is that massive individual excitations and much lighter collective excitations can coexist because they describe different motions of an interacting state.

A concrete experimental motivation for treating the electromagnetic response carefully is the Field-cooled Meissner experiment. MIT's realization cools lead in a solenoid field using helium gas. A nearby thermometer follows the cooling; a test coil feeds an integrator and oscilloscope. Expulsion gives a pulse, warming reverses it, and switching off the field while the sample is normal checks its sign. A change in alternating-field response alone can reflect normal electrical conductivity. This readout tests magnetic flux expulsion, not the spectrum or existence of a neutral Goldstone boson.^[ref]

For a global symmetry in field theory, different orientations can be physically distinct states. A local gauge symmetry instead includes redundant descriptions of the same physical state. A gauge-dependent field expectation alone is consequently insufficient as a physical criterion for spontaneous symmetry breaking. The Higgs mechanism changes the physical mode content when the gauge field is dynamical; the global Goldstone theorem alone does not prove that mechanism. In particular, the massless direction of a gauge-fixed scalar potential need not appear as a separately observable massless particle.

A nonelectrolyte supplies neutral molecules rather than ions on dissolving in the specified solvent, as illustrated by sucrose in water.

For example, dissolving an idealized $1\mathrm{mmol}$ sucrose inventory gives $1\mathrm{mmol}$ dissolved sucrose molecules, rather than separate cations and anions. Dissolving the same amount of substance of a fully dissociated one-to-one strong electrolyte instead gives $1\mathrm{mmol}$ of each ion. Both can disappear visibly into the solvent; the resulting chemical species distinguish them.^[ref]

A nonelectrolyte is not necessarily insoluble: sucrose provides a soluble counterexample. Nor does the label assert exactly zero measured electrical conductivity. Self-ionization of water, impurities or residual electrolyte from a probe can supply background ions. Comparing with a solvent blank and rinsing between samples, as in the strong electrolyte experiment, tests whether the dissolved substance itself produces appreciable additional mobile ions. This separates visible dissolution from ion production without treating lamp darkness as an absolute absence of electric charge.

A material whose current density is proportional to its electric field, with proportionality given by its electrical conductivity.

Ohm's law states that an Ohmic conductor has electric current proportional to its applied electric potential difference, with constant electrical resistance under fixed material conditions:Here $V=\varphi (A)-\varphi (B)$ is the terminal electric potential difference, and positive $I$ enters at $A$ and exits at $B$. This sign convention makes $VI$ the electrical power absorbed. The straight-line relation through the origin lets one constant predict the response at many applied electric potential differences. Merely defining $R=V/I$ at one nonzero operating point does not establish Ohm's law: a component may have a ratio that varies with applied electric potential difference. Material and temperature must be specified and held fixed; many components have a nonlinear response. ^[ref].

For the ideal resistor model $R=2\Omega$ at electric potential difference $V=5\mathrm{V}$,This power is converted to internal energy by Joule heating; a watt is a joule per second. If the model holds for time $10\mathrm{s}$, the transferred energy is $125\mathrm{J}$. Doubling electric potential difference doubles electric current and quadruples power for the same fixed $R$. The model keeps material temperature fixed even while calculating Joule heating; appreciable changes in temperature may require a changing electrical resistance. ^[ref].

In 1827, Georg Ohm published his mathematical account of electrical transport in Die galvanische Kette, mathematisch bearbeitet. On printed page 36, his relation $S=A/L$ expresses electric current in terms of total driving tension and a reduced length incorporating electrical conductivity and geometry. This is historical notation: his $A$ is not the wire cross-sectional area used here, and his $L$ is not simply the wire's physical length. The striking idea is that material response and geometry can be combined into one quantity governing an electrical circuit. ^[ref].

For a uniform axial magnetic field, the anisotropic Shear Alfvén wave is predominantly horizontally polarized, whereas the pseudo-Alfvén wave is predominantly vertically polarized. Without rotation both have characteristic frequency ${\omega }_A\sim |k_{\parallel }|v_A$. Pressure projection makes the effective Coriolis acceleration couple these two transverse wave polarization states at the inertial-wave rateThus the useful comparison is not $\Omega$ alone, nor anisotropy alone, butIn this limit rotational coupling of shear and pseudo-Alfvén waves is negligible on an Alfvén propagation time, so the two familiar linear polarizations are decoupled to leading order. The factor $k_{\parallel }$ cancels: simply making both disturbances more anisotropic does not by itself remove their relative rotational coupling.

For a nonlinear reduction one must also make the rotational mixing slow compared with the nonlinear evolution being retained. Under Alfvénic critical balance, $|k_{\parallel }|v_A\sim k_{\perp }U$, the same inequality gives ${\omega }_I\ll k_{\perp }U$. Rapid rotation relative to the horizontal nonlinear rate can therefore coexist with leading Alfvénic decoupling in the windowHigh electrical conductivity makes magnetic diffusion slow, and the corresponding inviscid approximation is also assumed. If $2\Omega \gtrsim k_{\perp }{v}_A$, rotational and magnetic restoring effects are comparable and the modes are mixed magneto-Coriolis disturbances; in the rotation-dominated limit they separate into fast inertial-like and slow magnetostrophic branches, rather than into independently evolving shear and pseudo-Alfvén polarizations.

At finite amplitude, independence is generally one-way in the usual anisotropic ideal magnetohydrodynamics reduction: the Alfvénic subsystem has no leading feedback from pseudo-Alfvénic perturbations, but it can still advect them. The small rotational-coupling condition removes an additional two-way coupling, rather than all nonlinear advection.

This is an asymptotic finite-timescale statement. The two nonrotating Alfvén polarizations are degenerate, so even a weak nonzero rotational coupling can rotate their superposition over the much longer time ${\omega }_I^{-1}$. Exact decoupling for arbitrarily long times does not follow from a small ratio alone. The strictly $k_{\parallel }=0$ nonpropagating limit is separate.

Use the nonrelativistic moving-conductor Ohm law in the formThe Maxwell equations give Faraday's law of induction, ${\partial }_t\mathbf{B}=-\nabla \times \mathbf{E}$. For a nonmagnetic conductor with constant permeability ${\mu }_0$, the quasistatic Ampere-Maxwell law gives $\mathbf{j}\simeq (\nabla \times \mathbf{B})/{\mu }_0$. This step neglects displacement current, requiring it to be small compared with conduction current; nonrelativistic velocity alone is not the complete justification.

For spatially uniform scalar electrical conductivity $\sigma$,Apply the identity $\nabla \times (\nabla \times \mathbf{B})=\nabla (\nabla \cdot \mathbf{B})-{\nabla }^2\mathbf{B}$ and Gauss's law for magnetism, $\nabla \cdot \mathbf{B}=0$. The resistive induction equation isThe first term transports and deforms the magnetic field with the moving fluid; the second diffuses it. The magnetic diffusivity $\eta$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$, so both terms have units tesla per second. No assumption of incompressible flow was needed. If $\sigma$ varies spatially, the conductivity-gradient term in magnetic induction, $-\nabla \eta \times (\nabla \times \mathbf{B})$, must also be retained.

Substitutional arsenic has one more valence electron than germanium, so it acts as a donor impurity. Thermal ionization of its shallow bound state supplies mobile conduction-band electrons and leaves fixed positive donor charge. The resulting n-type semiconductor can have $n_e\gg n_i$ even when only a small fraction of host atoms are replaced. Its electrical conductivity then rises because the mobile carrier count increases; the opposite donor charge preserves macroscopic charge neutrality.

This argument assumes that the donors are substantially ionized at the measurement temperature and that the added impurity scattering does not overwhelm the carrier increase. The screened bound-donor picture and its departures from a simple hydrogenic model are described in institutional semiconductor spectroscopy notes.^[ref]

The equal supplied masses give equal effective density of states. At the specified temperature,Using the same energy units in the exponential,For equal electron and hole mobilities, both carriers contribute to the intrinsic electrical conductivity:The factor two is essential. A dimensional check givesusing $\mathrm{A}=\mathrm{C}{\mathrm{s}}^{-1}$.

In a nondegenerate semiconductor, the Fermi-Dirac distribution reduces near the conduction edge to $f(E)\simeq e^{-(E-E_F)/k_{\mathrm{B}}T}$. The corresponding dilute hole probability near the valence edge givesAn intrinsic, neutral sample has $n=p=n_i$. Multiplying the two expressions therefore yieldsEach excitation creates an electron-hole pair; charge neutrality is why the carrier concentration has half the gap in its activation exponent. The electrical conductivity isThus $\sigma \propto e^{-E_g/(2k_{\mathrm{B}}T)}$ to leading exponential accuracy. The prefactor contains the mobilities and the density-of-states temperature dependence; for fixed parabolic masses, $\sqrt{N_c{N}_v}\propto T^{3/2}$.

Measure conductivity in the intrinsic regime and plot $\ln \sigma$ against $1/T$. If the prefactor is constant or its temperature dependence has been corrected, a slope $m$ givesUse the intrinsic branch rather than a donor freeze-out or extrinsic branch. A temperature-dependent gap or mobility changes the apparent activation energy, so a precision extraction must account for those prefactors.

Heating promotes electrons across the band gap, creating mobile conduction electrons and electron holes. Their contributions add:where $u_e,u_h$ are the electron mobility and hole mobility, and $e$ is the positive elementary-charge magnitude. The increase in carrier density gives the rising electrical conductivity in the intrinsic activation regime. Mobility need not itself increase with temperature; the model attributes the dominant increase to carrier generation.

Potassium intercalates between graphite layers and donates electron density to the graphitic pi system. Occupation of formerly less populated electronic states raises the mobile-carrier density, increasing electrical conductivity under the stated comparison.

The gain is electronic doping of the carbon framework, rather than a claim that potassium strengthens every C-C bond or simply acts as a separate wire through the sample. Intercalation also changes layer separation and scattering, so conductivity depends on both carrier population and mobility.

The material is potassium graphite, a graphite intercalation compound.

Sulfur substituting for phosphorus acts as a donor impurity, giving n-type semiconductor behaviour and usually increasing electrical conductivity. A P site normally supplies five valence electrons; S supplies six. After forming the surrounding bonds, the extra electron can occupy a donor level below the conduction band and be thermally ionized into that band. The donor impurity then carries positive charge.

This assignment is supported experimentally: infrared transmission measurements on S-doped gallium phosphide resolved bound-donor transitions and determined a sulfur donor optical ionization energy of about $0.104\mathrm{eV}$, far smaller than the host band gap.^[ref]

The effect depends on substitution on P sites, donor ionization, compensation by acceptors and charge carrier mobility. A dilute active donor population increases the electron term in $\sigma =e(n{\mu }_n+p{\mu }_p)$; not every dopant must donate a mobile electron at room temperature. Sulfur's larger Allen electronegativity does not overturn this electron-counting argument or make a P-site sulfur atom an acceptor.

A $2.24\mathrm{eV}$ band gap is consistent with a semiconductor. In pure gallium phosphide, the valence band is essentially full and the conduction band essentially empty at low temperature; thermal or optical excitation can produce mobile electrons and electron holes.

At room temperature, $k_{B}T\approx 0.026\mathrm{eV}$ is much smaller than the band gap, so intrinsic excitation is strongly suppressed. Pure material consequently has a low intrinsic carrier density, while semiconductor doping can markedly increase its electrical conductivity.

With equal constant electron mobility and hole mobility $\mathcal{M}$, the sheet electrical conductivity is $\sigma =e\mathcal{M}(n+p)$. In the intrinsic regime, $n+p\simeq 2n_i\propto T{e}^{-E_g/(2k_{B}T)}$, soIt decreases steeply with increasing $1/T$ on the high-temperature side. There is a small logarithmic curvature rather than a perfectly straight activation line.

In the fully ionized extrinsic regime, charge neutrality and the semiconductor mass-action law give $n-p=n_d$, $np=n_i^2$, henceThis gives a nearly horizontal region passing through $\sigma (200\mathrm{K})=0.01{\Omega }^{-1}$. At lower temperatures semiconductor carrier freeze-out decreases the ionized donor density and bends the curve downward again.

To locate that bend realistically, include donor charge-state degeneracy. For a neutral donor with two spin states and an empty donor with one state, the ionized density isSolve $n-n_i^2/n=n_d^{+}$ for the mobile density. In the asymptotic freeze-out regime, where $p$ is negligible and $n\ll n_d$,Its activation scale $E_B/2\simeq 3.14\mathrm{meV}$ is much smaller than the intrinsic scale $E_g/2=180\mathrm{meV}$.

Within the specified interval the low-temperature downturn is modest: $N_c(30\mathrm{K})\simeq 2.81\times 10^{10}{\mathrm{cm}}^{-2}$, and the partial-ionization equation gives $n(30\mathrm{K})\simeq 0.65n_d$. The sample is not yet deep in the asymptotic freeze-out limit at 30 K. Calibrating the mobility to the stated 200 K value gives representative valuesThe donor statistical weights were not explicitly supplied, so they are a stated assumption for these illustrative curve details; a one-state donor approximation changes the low-temperature rounding but not the three physical regimes or their activation scales.

Take $d$ as the centre-to-centre separation and $ϵ={ϵ}_0{ϵ}_r$. Use the ideal lossless two-conductor transmission line model, with nonmagnetic surroundings and negligible internal-wire inductance. Opposite line charges $+{\lambda }_q$ and $-{\lambda }_q$ approximate the cylindrical conductors when $d\gg a$. Integrating their electric fields, or adding their logarithmic electric potentials, gives the potential differenceThis is the two-wire capacitance at large separation. For opposite electric currents $+I$ and $-I$, the magnetic fields between the wires add. The linked magnetic flux per unit length is approximatelyThus the external inductance of a thin two-wire line is $L^{\prime }\simeq {\mu }_0\ln (d/a)/\pi$. The lossless telegrapher's equations give a forward wave with $V/I=\sqrt{L^{\prime }/C^{\prime }}$, so its characteristic impedance isThe product $L^{\prime }{C}^{\prime }={\mu }_0ϵ$ also gives the propagation speed $c/\sqrt{{ϵ}_r}$, an independent consistency check. The exact ideal external-field geometry replaces $\ln (d/a)$ by $\mathrm{arcosh}(d/(2a))$. If the wire separation were specified between surfaces, its centre separation would be $d+2a$, with the same leading large-separation result. Finite electrical conductivity and internal-wire magnetic energy require a more detailed line model.

Write $\mu ={\mu }_0{\mu }_r$ and $ϵ={ϵ}_0{ϵ}_r$ for homogeneous scalar material constants, and use $\mathbf{J}=\sigma \mathbf{E}$. Taking the curl of Faraday's law of induction and substituting the Ampere-Maxwell law givesThe vector calculus identity for a double curl therefore yieldsThe second term represents electric current response, rather than displacement current. A dispersive $\sigma (\omega )$ is used in the frequency domain; its general time-domain counterpart is a temporal response convolution.

For the convention $e^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$, the plane wave equation isDecompose the electric field into components parallel and perpendicular to $\mathbf{k}$. With $S=\mu ϵ{\omega }^2+i\mu \sigma \omega$, the longitudinal eigenvalue is $S$ and each transverse eigenvalue is $S-k^2$; thus the determinant is $S(S-k^2{)}^2$. For a transverse electromagnetic wave, $\mathbf{k}\cdot {\mathbf{E}}_0=0$ andwhen ${\mu }_r=1$ and the background ${ϵ}_r\simeq 1$. A real dissipative electrical conductivity generally makes $k$ complex, with the decaying branch chosen for a passive medium.

For a cold, collisionless, unmagnetized plasma, treat ions as stationary and write $e>0$ for the magnitude of the elementary charge. The electron equation of motion is $-i\omega m_e\mathbf{v}=-e\mathbf{E}$, soThis reactive response includes no collision-induced Joule heating. Substitution into the transverse dispersion relation givesThe quantity ${\omega }_p$ is the electron plasma frequency. The refractive index is real above this cutoff; below it, the wave is evanescent. The electron response is represented here by $\sigma (\omega )$, so it must not also be counted a second time in the background permittivity.

For a nonzero longitudinal component, taking the dot product of the matrix equation with $\mathbf{k}$ instead gives $S(\mathbf{k}\cdot {\mathbf{E}}_0)=0$. The same cold plasma response makes $S=({\omega }^2-{\omega }_p^2)/c^2$, henceAt nonzero $k$, the remaining equation forces ${\mathbf{E}}_0$ to be parallel to $\mathbf{k}$, and Faraday's law of induction gives ${\mathbf{B}}_0=0$. These are electrostatic Langmuir waves, with no $k$ dependence of angular frequency in the zero-temperature model. Equilibrium neutrality does not forbid their perturbed charge density; Gauss's law requires such a density when $\mathbf{k}\cdot {\mathbf{E}}_0\ne 0$.