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For a length $L$ and parabolic free-electron dispersion relation $E={\hslash }^2{k}^2/(2m)$, the displayed total density of states includes both signs of $k$ and both spin states. Per spin it is half as large. Its $E^{-1/2}$ dependence follows by transforming the constant signed-$k$ density $L/(2\pi )$ to energy and including both momenta. The continuum singularity at the band bottom is integrable; a finite specimen still has discrete levels.

For filled valence and empty conduction energy bands, the joint density of states counts pairs of states at a prescribed interband energy difference. In a direct optical transition the photon momentum is usually negligible compared with a Brillouin zone dimension, so initial and final states have nearly the same crystal momentum. The absorption spectrum weights this count by transition matrix elements; its onset and structure can constrain the band gap and electronic band structure. Indirect transitions additionally involve phonons to supply crystal momentum.

A many-to-one change of energy variable requires summing every inverse branch. For $E(\xi )=\sqrt{{\xi }^2+{\Delta }^2}$ and constant normal density of states $g_F$ on both sides of zero, there are two roots for $E>\Delta$, giving $D_{+}(E)=2g_{F}E/\sqrt{E^2-{\Delta }^2}$. A single branch instead gives $g_{F}E/\sqrt{E^2-{\Delta }^2}$. The distinction persists when $\Delta \to 0$: folding $\xi$ into $|\xi |$ doubles a two-sided constant density on the positive axis. Spin degeneracy is a separate normalization choice; it must be applied consistently to both normal and transformed densities. The BCS electronic density of states has the single-factor expression because it is spectrally weighted rather than an unweighted branch count.

For an ellipsoidal sample in a uniform applied magnetic field intensity $H_0$ along a principal axis, its uniform demagnetizing field is $-N_{d}M$. The SI demagnetizing factor obeys $0\le N_d\le 1$. Linear magnetic susceptibility $\chi ={\mu }_r-1$ gives $M=\chi (H_0-N_{d}M)$, hence $M=\chi H_0/(1+N_d\chi )$. An axially slender prolate limit has $N_d\to 0$, a sphere has $N_d=1/3$, and a thin plate with field normal to it has $N_d\to 1$. Thus $M=({\mu }_r-1)H_0/{\mu }_r$ belongs to the plate limit, not the slender axial-rod limit. A finite cylinder generally has nonuniform magnetization; an effective ellipsoidal factor is an approximation rather than an exact universal correction.

For a single constant delay $\tau >0$, a retarded delay differential equation has the displayed form: delayed values of the unknown function enter, but delayed first derivatives do not. The unknown can also be a vector.^[ref]

Here $t$ denotes time, and the rule applies for $t>0$, with a right derivative at $0$. On $0\le t\le \tau$, the argument $t-\tau$ lies in the prescribed past. Once this interval is known, it supplies the delayed values for $\tau \le t\le 2\tau$. This is the method of steps for a delay differential equation: solve successive ordinary differential equations with a known delayed input, keeping each new endpoint as the next initial condition. For a continuous function $f$ locally Lipschitz continuous in its two state arguments, locally uniformly in $t$, this gives local existence and uniqueness for each continuous history function of a delay differential equation, while the solution of a differential equation remains in the domain of $f$. A nonlinear rule can still produce finite-time blowup.

For an unstable particle undergoing exponential decay in its rest frame, the decay waiting time has an exponential distribution with mean $\tau$. Its probability of surviving to time $t\ge 0$ is $e^{-t/\tau }$, and its total decay width is $\Gamma =\hslash /\tau$. In units $\hslash =1$, $\tau =1/\Gamma$.

For a uniformly charged sphere of radius $a$, total electric charge $Q$, and an exterior dilute electrolyte of permittivity $\epsilon$, the Debye–Hückel approximation gives $(r^2{\varphi }^{\prime }{)}^{\prime }={\kappa }^2{r}^2\varphi$. Decay at infinity and Gauss's law, $-\epsilon {\varphi }^{\prime }(a)=Q/(4\pi a^2)$, select the displayed electric potential. Substituting $u=r\varphi$ reduces the proof to $u^{\prime \prime }={\kappa }^{2}u$. The screening parameter is ${\kappa }^2={\sum }_i{z}_i^2{e}^2{n}_{i,\infty }/(\epsilon k_{B}T)$, and the ion profiles are $n_i\simeq n_{i,\infty }(1-z_{i}e\varphi /(k_{B}T))$. The validity condition is $|z_{i}e\varphi (a)|\ll k_{B}T$ for every relevant species. Integrating the diffuse charge density using the radial equation gives total screening charge $-Q$.

For a plane bounding a symmetric electrolyte in $x>0$, assume no displacement field on the other side, bulk potential zero, and $|ze\psi |\ll k_{B}T$. The Debye–Hückel approximation gives ${\psi }^{\prime \prime }={\kappa }^2\psi$, whose decaying solution is ${\psi }_0{e}^{-\kappa x}$. The electrostatic interface boundary conditions give $\sigma =-\epsilon {\psi }^{\prime }(0^{+})=\epsilon \kappa {\psi }_0$. Equivalently, its diffuse charge density is ${\rho }_e=-\epsilon {\kappa }^2\psi$ and its integral is $-\sigma$. Here $\epsilon ={\epsilon }_0{\epsilon }_r$ in SI units and ${\kappa }^2=2z^2{e}^2{n}_{\infty }/(\epsilon k_{B}T)$ when $n_{\infty }$ counts each ionic species. An isolated sheet with identical electrolyte on both sides has two diffuse layers, so the total sheet charge is $2\epsilon \kappa {\psi }_0$ instead. A surface geometry and the field on its other side must therefore be specified.

A charmed strange meson. Its ground-state charged particle has valence content $c\bar{s}$ and spin-parity $0^{-}$. Its antiparticle has the conjugate valence content. Nonleptonic decays are governed by weak charged currents and hadronic transition matrix elements.

A cell plate is an internal partition assembled during plant cytokinesis by fusion of transport vesicles. It grows outward to join the parental cell membrane, providing daughter boundaries and material for a new plant cell wall.^[ref]

A cleavage furrow is the inward indentation of the cell membrane during cytokinesis. It is the visible deformation; the actomyosin ring is machinery beneath that boundary.^[ref]

An actomyosin ring is a contractile assembly containing actin filaments and myosin II. In typical animal cytokinesis, it forms in the cell cortex beneath the cell membrane. Its contraction draws the boundary inward to produce a cleavage furrow; an inward groove is not yet complete separation of the daughters.^[ref]

Define $\Delta V={\Phi }_{\mathrm{entry}}-{\Phi }_{\mathrm{exit}}$ as the electric potential drop along the particle's passage. If the electric field changes negligibly during that short crossing, the work givesThe sign matters: a positive ion gains kinetic energy while going to lower electric potential; a negative ion needs the opposite drop. A fixed-sign field would accelerate on one crossing and decelerate on the next. The alternating field must reverse before the particle comes back.

For a fixed electric charge sign and circulation direction, absorb this alternating crossing direction into the definition of the arrival phase ${\varphi }_n$. If $V_0$ is the peak electric potential difference across the whole gap, rather than the amplitude of either electrode relative to ground, a thin-gap model isArrival at ${\varphi }_n=0$ gives the largest accelerating kick; $|{\varphi }_n|<\pi /2$ gives a positive kick. Arrival at ${\varphi }_n=\pi$ gives deceleration. For an ideal classical particle initially on crest, the cyclotron resonance condition keeps successive arrival phases equal. It does not automatically bring an initially mistimed particle onto crest.

Cyclotron resonance is resonant absorption by charged particles in a magnetic field when an applied oscillating electric field drives their cyclotron motion. For a parabolic band, the resonance frequency is ${\omega }_c=|q|B/m^{\ast }$, so measuring it determines the cyclotron effective mass. In the quantum description, transverse electric dipole transitions connect adjacent Landau levels, whose separation is $\hslash {\omega }_c$. For an electron and a field along $z$, only the circular polarization rotating in the electron's orbital sense drives the positive-frequency resonance. Scattering broadens the absorption peak; resolving it requires an orbital period short compared with the relaxation time.

The Euler spiral is the planar curve parametrized by the Fresnel integrals. In the optical convention,The tangent has unit length and argument $\pi u^2/2$, so its signed curvature is $\pi u$, proportional to signed arc length. In Fresnel diffraction, an interval integral is the directed chord joining its endpoint parameters on the Cornu spiral. Chord length gives the interval factor's modulus, while chord direction gives its complex argument.

An interstitial defect is an atom occupying a position between the regular atomic positions in a crystal structure. Small solute atoms or an extra host atom can occupy such positions. Different interstitial orientations can induce anisotropic local strain; an applied stress changes their relative energies. Thermally activated hopping among those positions can then produce delayed elastic strain and the Snoek effect.

A missing atom at a lattice site that would be occupied in the reference crystal. A vacancy defect is a point defect and can contribute configurational entropy because many choices of empty sites can have the same macroscopic energy. Formation energy and pressure-volume work oppose this entropy gain. Vacancies can be present in thermal equilibrium; they need not all be preparation mistakes. Their allowed number and creation mechanism depend on whether the lattice size, atom number and surface reservoirs are fixed.