In a semiconductor, a conduction electron fills a vacant valence state and emits a photon. In a simple nondegenerate model the recombination rate is $Bnp$, while the equilibrium reverse generation rate is $B{n}_i^2$; their difference gives the displayed net pair-loss rate. This subtraction enforces detailed balance at thermal equilibrium. Photon momentum is small on the scale of a Brillouin zone, so a direct band gap permits a direct transition while an indirect band gap generally requires a phonon or another source of momentum.

In a weak-injection regime where the majority population is nearly fixed, a small excess minority charge carrier population often relaxes exponentially with time constant $\tau$. This lifetime describes carrier removal by recombination. It differs from the momentum relaxation time governing charge carrier mobility: many momentum-scattering events can occur before a carrier recombines. Traps and spatial transport can make a single exponential inadequate.

In a half-space with no pressure gradient, Stokes balance and Poisson electrostatics give $\eta v\prime \prime =\epsilon E\psi \prime \prime$. No slip at the wall and zero bulk shear imply $v=\epsilon E(\psi -\zeta )/\eta$. The bulk velocity is $U=-\epsilon \zeta E/\eta$; it is not the fluid speed at the no-slip wall.

With $\psi (0)=\zeta$, $\psi \prime (d)=0$ and no slip at either wall, linear screening gives $\psi /\zeta =\cosh [\kappa (d-x)]/\cosh (\kappa d)$. The velocity is $v/U=1-(1-\mathrm{sech}\kappa d)x/d-\cosh [\kappa (d-x)]/\cosh (\kappa d)$. This electrically neutral-wall boundary assumes no external normal field; it differs from a grounded wall.

A homogeneous charged channel can add a constant electro-osmotic drift to the ideal electrophoretic velocity at fixed field. It changes the coefficient of a linear length-versus-entry-time law without generating a new length exponent. Charge-dependent adsorption, heterogeneous trapping or cancellation of drift followed by diffusion lies outside that simple claim; wall charge cannot be excluded as an indirect cause in every real experiment.

In the $e^{i\omega t}$ phasor convention, electrical admittance is $Y=G+iB$, with conductance $G$ and electrical susceptance $B$. An ideal capacitor has $B=\omega C$ and an ideal inductor has $B=-1/(\omega L)$. electrical susceptances add for parallel branches; their cancellation gives a purely resistive input admittance.

An applied potential difference that sets a device operating condition or shifts reservoir electrochemical potentials. In a multi-terminal device the grounded lead, biased lead and gate reference must be stated; they determine the relation between measured slopes and capacitances.

The linear response relating an induced electric dipole moment to an applied electric field. In ordinary nonresonant Raman spectroscopy, changes in its symmetric components during a normal mode determine activity. These components transform like $x^2,y^2,z^2,xy,xz,yz$.

A bound charge in a harmonic restoring potential has induced electric dipole moment $d=(q^2/k)E$. Completing the square proves this for every static field within the strictly harmonic model. First-order nondegenerate perturbation theory gives the same coefficient because the position operator connects the unperturbed ground state only to the first excited state. This exact response coefficient does not make its two-state wavefunction approximation exact.

The matrix element of the electric dipole moment operator between two quantum states. Electric-dipole transition strength is proportional to its squared magnitude; molecular symmetry can force every Cartesian component to vanish.

A coherent transition involving two atom-field interactions. two-photon absorption uses a sum of photon energies; stimulated absorption-emission uses their difference. A second-order intermediate contribution need not represent real population in a stationary atomic state.

For an inversion-symmetric system, electric-dipole transitions require opposite parities. Transitions between d-shell terms of the same gerade parity can acquire weak vibronic or symmetry-breaking intensity.

For $0<x<w$, normalized states ${\psi }_n=\sqrt{2/w}\sin [(n+1)\pi x/w]$ have centre-reflection eigenvalue $(-1{)}^n$. Since $x-w/2$ is odd, off-diagonal electric-dipole matrix elements vanish between equal-parity states. Ground state n=0 cannot reach n=2 at first order; an initial n=1 can reach even labels including n=0. This does not impose an all-orders prohibition: two dipole interactions can restore the original parity. The general nonzero elements follow from position matrix elements in a symmetric infinite square well after shifting the origin.

A critically refracted wave associated with a faster lower layer that travels along an interface and radiates back into the slower upper layer. Its travel-time branch has the lower-layer horizontal slowness and a positive intercept time.

At a material interface, incident P or SV energy can scatter into both P and SV modes, subject to conserved horizontal slowness and traction/displacement conditions. SH is decoupled in a planar isotropic model; some conversions vanish at symmetry directions such as normal incidence.

An elastic weak interaction of a neutrino with an electron. For a stationary electron of mass $m$ and an effectively massless incident neutrino of energy $E$, the maximum recoil kinetic energy in natural units is $T_{\max }=2E^2/(m+2E)$. This kinematic limit does not specify the interaction probability.

A particle of kinetic energy $\alpha m{c}^2$ hitting an equal mass at rest, with equal and opposite outgoing angles $\theta$, obeys $\tan \theta =\sqrt{2/(\alpha +2)}$. Conservation of total energy gives equal final energies, and longitudinal momentum conservation fixes the angle.

A massive piston at velocity $-u\hat{\mathbf{x}}$ reverses a particle's normal velocity in its own frame. Transforming back gives $v_x\mapsto -v_x-2u$, with transverse components unchanged. The particle energy gain is $2mu{v}_x+2m{u}^2$.

Let one equal-mass disc arrive with velocity $\mathbf{v}$ while the other is stationary, and let $\mathbf{n}$ be the unit line of centres at contact. A smooth central elastic impulse exchanges their normal velocity components and leaves tangential components unchanged: ${\mathbf{v}}_A^{\prime }=\mathbf{v}-(\mathbf{v}\cdot \mathbf{n})\mathbf{n}$ and ${\mathbf{v}}_B^{\prime }=(\mathbf{v}\cdot \mathbf{n})\mathbf{n}$. The final translational velocities are orthogonal when both are nonzero, and no rotation about a disc centre is generated by the normal impulse. This result concerns nonrelativistic smooth discs; roughness introduces tangential contact dynamics.

For normalized Eisenstein series $E_2,E_4,E_6$ and $D=q\frac{d}{dq}$, the relations are $D{E}_2=(E_2^2-E_4)/12$, $D{E}_4=(E_2{E}_4-E_6)/3$, and $D{E}_6=(E_2{E}_6-E_4^2)/2$. They turn derivatives of these series into polynomial expressions in the same series.