A spatially uniform time-dependent reaction coefficient can be removed from a linear diffusion equation by a scalar integrating factor. If $A^{\prime }(t)=a(t)$ and $v=e^{A(t)}u$, substitution gives $u_t=u_{xx}$. Initial data and boundary values must also be divided by $e^{A(t)}$. A nonzero boundary profile can then be removed by boundary lifting before applying separation of variables. This reduction distinguishes growth of the whole amplitude from decay of the spatial diffusion modes.

Diffusion of particle position, as distinguished from rotational diffusion of its orientation. Its diffusion coefficient has units of length squared per time.

For $N$ coherently illuminated grating lines used in diffraction order $m$, the Rayleigh criterion yields the displayed ideal resolving power of a diffraction grating. The angular maximum-to-first-zero separation is $\lambda /(ND\cos {\theta }_m)$, while nearby wavelengths are separated by $|m|\delta \lambda /(D\cos {\theta }_m)$. The order must exist, carry appreciable light intensity and remain free from dominant instrumental broadening. Line count alone therefore does not determine whether a particular instrument resolves a specified doublet.

A normal-incidence diffraction grating of pitch $D$ has order $m$ at $D\sin \theta =m\lambda$ and angular separation $\Delta \theta \simeq m\Delta \lambda /(D\cos \theta )$. A lens whose axis follows the selected order maps a small angular interval to $\Delta x\simeq f\Delta \theta$. Detector pixels substantially smaller than this separation are needed; two pixels per resolution interval correspond to $t\lesssim \Delta x/2$. If instead detector coordinates obey $x=f\tan \theta$ relative to the original axis, the scale is $\Delta x\simeq f{\sec }^2\theta \Delta \theta$. A numerical pixel limit therefore requires an optical layout and a sampling criterion as well as the resolving power of a diffraction grating.

Landau diamagnetism is the orbital response of mobile electrons to an applied magnetic field, arising from quantization of their transverse motion. For a noninteracting three-dimensional isotropic electron gas with quadratic dispersion, electron spin with $g=2$ and magnetic fields weak enough for the smooth-response limit, its susceptibility is ${\chi }_L=-{\chi }_P/3$, where ${\chi }_P$ is the Pauli paramagnetism susceptibility. Band effective masses, interactions, dimensionality and resolved magnetic quantum oscillations can change this simple relation.

For a zero-bias abrupt p-n junction, choose the p side at $x<0$ and the n side at $x>0$. Ionized acceptors and donors remain fixed while mobile electrons and electron holes diffuse and establish a built-in electric field. In the depletion approximation, mobile carriers are neglected on $-x_p<x<x_n$ and the net charge density isCharge neutrality gives $N_A{x}_p=N_D{x}_n$. The quasi-neutral bulk can have a large mobile carrier density while its net charge density is nearly zero, because fixed dopant charge compensates it. With complete ionization and nondegenerate carriers, equilibrium also gives $np=n_i^2$; real carrier profiles vary smoothly rather than forming exact step functions at depletion edges.

For a planar abrupt p-n junction with ionized densities $N_A,N_D$, p-side width $w_p$ and n-side width $w_n$, zero external field requires $N_A{w}_p=N_D{w}_n$. Integrating Poisson equation gives $V_{\mathrm{bi}}-V=e(N_A{w}_p^2+N_D{w}_n^2)/(2{\epsilon }_s)$, with positive $V$ denoting forward bias. Therefore $W=w_n+w_p$ has the displayed value, $w_n=N_{A}W/(N_A+N_D)$ and $w_p=N_{D}W/(N_A+N_D)$. The positive depletion capacitance per area is $C_A={\epsilon }_s/W$. It equals $d{Q}_p/dV=-d{Q}_n/dV$ when $Q_p=-e{N}_A{w}_p$ and $Q_n=e{N}_D{w}_n$ are signed space charges per area. Differentiating a positive charge magnitude with respect to forward voltage instead gives a negative derivative. The square-root law assumes partially depleted regions, negligible mobile charge and no substantial injected-carrier storage.

The local density of states resolves the density of states by position as well as energy. With a consistent counting of spin states, its integral over the specimen's volume gives the total density of states, $g(E)=\int \rho (\mathbf{r},E)d^3\mathbf{r}$. It describes the spatial distribution of available states, rather than their occupation.

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]