Codex @codex  0

With charge density $\rho$, current density $\mathbf{J}$, and speed of light $c=({ϵ}_0{\mu }_0{)}^{-1/2}$, the four Maxwell equations in vacuum areThe electric potential and magnetic vector potential determine the electric field and magnetic field byThese definitions satisfy Gauss's law for magnetism and Faraday's law identically. The Lorenz gauge condition isSubstituting into Gauss's law and using the Lorenz gauge condition givesFor the Ampère-Maxwell equation, the double-curl identity givesThe two gradient terms cancel by the Lorenz gauge condition. Defining the d'Alembert operator with the sign convention $\square ={\nabla }^2-c^{-2}{\partial }_t^2$, the remaining equations areConversely, if the electric potential and magnetic vector potential satisfy these wave equations and the Lorenz gauge condition, the same substitutions in reverse give Gauss's law and the Ampère-Maxwell equation. The other two Maxwell equations follow identically from the definitions, proving equivalence within the Lorenz gauge.

Use the sign $s=\pm 1$ to label a point charge $sq$ at $s\mathbf{p}/(2q)$, with velocity $s{\mathbf{p}}^{\prime }/(2q)$. All quantities on the right below are evaluated at $t_0=t-r/c$. To first order,The denominator in each Liénard–Wiechert potential therefore expands assoIn the sum of the two electric potentials, the $q/r$ terms cancel and the finite terms add. This gives the retarded potentials of a point electric dipole:For the magnetic vector potential, the signed electric charge and signed velocity multiply to give the same sign for both contributions:Since ${\mu }_0{ϵ}_0{c}^2=1$,The ${\mathbf{p}}^{\prime }$ term in the electric potential arises from the moving-source denominator and remains finite in the point-electric dipole limit; discarding it would fail to give the full retarded electric field.

The leading electric dipole radiation fields areThe radiation electric field is transverse, $\mathbf{n}\cdot {\mathbf{E}}_{\mathrm{rad}}=0$. The vector triple product therefore gives the leading Poynting vectorIf $\theta$ is the angle between ${\mathbf{p}}^{\prime \prime }(t-r/c)$ and $\mathbf{n}$, thenUsing ${\mu }_0{ϵ}_0{c}^2=1$, the final result isThe leading Poynting flux is outward, vanishes along the instantaneous retarded ${\mathbf{p}}^{\prime \prime }$ axis, and is maximal in its perpendicular plane. If ${\mathbf{p}}^{\prime \prime }=0$, the radiation term is zero and no angle needs to be assigned. The large-$r$ expansion assumes bounded retarded electric dipole moment and time derivatives; it may also be read at fixed retarded time.

Set $K=1/(4\pi {ϵ}_0)$, $t_0=t-r/c$ and $\mathbf{n}=\mathbf{r}/r$, and evaluate $\mathbf{p}$, ${\mathbf{p}}^{\prime }$, ${\mathbf{p}}^{\prime \prime }$ at $t_0$. The chain rule givesIn particular, for $\mathbf{u}(t_0)$ and any positive integer $m$,Apply this gradient identity to the electric potential $\varphi =K[(\mathbf{n}\cdot \mathbf{p})/r^2+(\mathbf{n}\cdot {\mathbf{p}}^{\prime })/(cr)]$:The magnetic vector potential is $\mathbf{A}=K{\mathbf{p}}^{\prime }/(c^{2}r)$, with time derivative ${\partial }_t\mathbf{A}=K{\mathbf{p}}^{\prime \prime }/(c^{2}r)$. Thus the electric field $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$ isFor the magnetic field, use the curl product rule and the chain rule:Consequently the fields of a time-dependent electric dipole includeThese formulas hold for $r>0$. The $r^{-3}$ electric field is the retarded electric dipole field, the $r^{-2}$ terms are induction contributions, and the $r^{-1}$ terms describe electric dipole radiation. Retaining the dependence of $t_0$ on position in every spatial derivative is what produces both the induction and radiation contributions.

In photoemission electron microscopy, illuminate a surface with ultraviolet light or X-rays and form an electron-optical image of the emitted electrons. An extraction field accelerates them away from the sample; an immersion objective and projection lenses magnify their spatial distribution onto an electron detector. Unlike a scanned probe, this is a full-field image. Surface preparation and vacuum matter because the emitted low-energy electrons are strongly affected by contamination and local electric fields.

Contrast can come from work-function differences, composition, occupied electronic states, illumination and topography. Near threshold, a small work-function change can produce a large emission-yield change. Therefore brightness is not simply geometrical height. An energy filter can select an emitted-electron energy interval; scanning the selection yields spatially resolved spectra. Sweeping the photon energy across an absorption edge can add element and chemical-state sensitivity. Appropriate polarization-dependent measurements can probe magnetic domains.^[ref]

Resolution depends on electron-lens aberrations, the accepted angular and energy spread, extraction geometry, sample flatness, space charge, mechanical stability and detector sampling. Rejecting electrons improves some aberrations at a cost in signal. An unfiltered photoelectron image and a chemically selected image therefore answer different questions even when they use the same microscope.

A scanning tunneling microscope can modify an adsorbate's potential landscape as well as image it. First locate the atom with a weakly perturbing tunnel-current setpoint. Bringing the tip closer strengthens tip–atom coupling. In lateral atomic manipulation, moving the tip parallel to the surface can pull or push the atom between adsorption sites; reducing the coupling afterwards leaves it at the chosen location. A change in tunneling current during the motion can signal a jump, and a subsequent low-current image checks the final arrangement.

In vertical atomic manipulation, a suitable tip interaction or voltage pulse transfers an atom between substrate and tip. The tip is repositioned and the atom is deposited elsewhere by changing that interaction again. Electric fields and inelastic tunneling excitation can also change adsorption, induce hopping or modify bonds. The useful mechanism depends on the atom, substrate, tip and bias: there is no universal pulse amplitude that works for every system. Low temperature suppresses thermal diffusion and ultrahigh vacuum helps preserve a clean, stable surface.

A motivating demonstration is IBM in atoms. Don Eigler and Erhard Schweizer used a low-temperature STM to position individual xenon atoms on a nickel surface, constructing a chosen atomic pattern. The 1990 paper reported operation at about $4\mathrm{K}$ with atomic positioning precision. This demonstrates controlled rearrangement of surface adsorbates, rather than making the image alone prove that every STM feature is an isolated atom.^[ref]

In near-edge X-ray absorption fine structure, a photon promotes a core electron into an unoccupied molecular or solid-state level. Near an oxygen K-edge this is principally an O $1s$ excitation. Bound or resonant final states, orbital symmetry, local bonding and multiple scattering shape the edge. A monochromatic tunable X-ray beam can measure transmission through a thin specimen, or fluorescence or electron yield as an absorption proxy. In scanning transmission X-ray microscopy, a focused spot is scanned while transmitted intensity is recorded; energy stacks across an edge distinguish regions with different chemistry even when their thicknesses are similar. A zone plate or another suitable X-ray optic sets spatial resolution, while energy resolution and reference spectra control chemical discrimination.

The dipole transition amplitude contains $\mathbf{e}\cdot \mathbf{r}$, so the incident electric-field direction selects final-orbital orientations. For an approximately free diatomic molecule with axis $\mathbf{n}$, the axial ${\sigma }^{\ast }$ transition has strength proportional to $|\mathbf{e}\cdot \mathbf{n}{|}^2$. Summing the two perpendicular ${\pi }^{\ast }$ directions gives strength proportional to $1-|\mathbf{e}\cdot \mathbf{n}{|}^2$, with separate radial matrix-element factors for the two resonance families. This NEXAFS orientation dependence of a diatomic molecule explains the complementary polarization behavior; the molecular axis is not itself the direction of a ${\pi }^{\ast }$ orbital.

In the supplied geometry, normal incidence has electric field parallel to the surface, whereas near-grazing incidence with the illustrated polarization has a large surface-normal field component. The single-layer curves show a stronger ${\pi }^{\ast }$ resonance and much weaker ${\sigma }^{\ast }$ structure in the latter geometry. This is consistent with oxygen axes lying largely parallel to the surface: a normal electric field is perpendicular to those axes and excites ${\pi }^{\ast }$ efficiently, while suppressing the axial ${\sigma }^{\ast }$ transition. For flat molecules with random in-plane azimuths, the ideal axial factor is $1/2$ for a tangential field and zero for a normal field.

The multilayer curves instead have a strong ${\pi }^{\ast }$ peak at normal incidence and substantially stronger relative ${\sigma }^{\ast }$ response at grazing incidence. Their axes therefore have a substantial surface-normal component, consistent with the proposed tilted, more upright packing. The distinction supports a different orientation distribution in the multilayer and monolayer, rather than one isotropically oriented molecular population in both.

For illustration, a molecular polar tilt $\beta$ and random azimuth give $⟨|\mathbf{e}\cdot \mathbf{n}{|}^2⟩={\cos }^2\beta {\cos }^2\alpha +\frac{1}{2}{\sin }^2\beta {\sin }^2\alpha$, where $\alpha$ is the electric-field angle to the surface normal. The original diagram plots these ideal factors for two chosen tilts; it is not a reconstruction or fit of the supplied spectra. Quantitative tilt extraction would additionally need intensity normalization, polarization purity, azimuthal distribution and substrate-induced changes of the resonances. The spectra alone do not determine unique bond lengths or every adsorption site.

Set $\kappa =|k|$ and first work with a representative positive $k=\kappa$. For $\kappa {\lambda }_{\mathrm{De}}\ll 1$, the thermal core is strongly screened. However, the causal dielectric function has weakly damped Langmuir wave roots ${\omega }_s+i{\gamma }_s$, with $s=\pm 1$, ${\omega }_s\simeq s{\omega }_{pe}$ and ${\gamma }_s<0$. Their phase velocities $v_s={\omega }_s/\kappa$ are in the tail. Both roots must be retained for each spatial Fourier series harmonic.

Write $ϵ(-i\omega +0^{+},\kappa )={ϵ}_R(\omega )+i{ϵ}_I(\omega )$. The supplied approximation givesExpanding $ϵ({\omega }_s+i{\gamma }_s)=0$ to first order gives the Landau damping rateIn particular, stable damping means ${\omega }_s{f}_0^{\prime }(v_s)<0$. Near either root,The narrow Lorentzian resonance approximation then gives the leading resonant contributionThis is a formula under the integral over velocity; it is not a pointwise approximation away from the resonances. Insert it into the heating formula in c. For one spatial harmonic, the contribution from one root isbecauseThus the tiny tail slope cancels the tiny Landau damping rate: weak absorption is compensated by a large noise-driven wave intensity. The two temporal roots supply two equal halves of the answer. Negative $k$ gives the same result by the conjugate response of a real electric field, without altering the original convention that the spatial sum includes both signs. ThereforeAs in c, equality means the leading result in the stated asymptotic limit. This calculation assumes nonzero damping, observation times $t\gg {\max }_s|{\gamma }_s{|}^{-1}$, and forcing weak enough that $f_0$ changes negligibly during that relaxation. Taking zero damping before the long-time limit invalidates the stationary calculation.

The long-wavelength limit in d is resonant heating. Most electrons screen the slowly varying spatial field. The noise nevertheless excites collective Langmuir waves, whose sharp temporal spectrum lies near $\omega =\pm {\omega }_{pe}$. Electrons with $v\simeq \omega /k$ remain in phase with a wave and exchange energy with it. A stable tail absorbs net wave energy through Landau damping. The heating is initially deposited in narrow tail regions of velocity space, even though the volume-integrated rate equals that in c.

The short-wavelength limit in c is stochastic heating in the terminology intended here. With $ϵ\simeq 1$, the imposed electric field is essentially unscreened. Its temporally uncorrelated kicks cause a random walk of electron velocity, so $⟨(\Delta v{)}^2⟩=2D_0\Delta t$. Electrons throughout the thermal population contribute to the resulting kinetic energy increase; no narrow collective wave peak selects a small tail population.

Both mechanisms are driven by white noise, and the formal condition $\omega =kv$ enters the stationary quasilinear diffusion calculation in both cases. The physical distinction is the sharp collective resonance in d versus the broad direct random forcing in c, rather than the mere presence of randomness. In collisionless theory this heating is an increase of the averaged kinetic energy; subsequent collisions can establish thermal equilibrium.

Put $b=\mathcal{E}{e}^{i(kr-\omega t)}$, allowing $\mathcal{E}$ to be complex. The Hermitian electric field is $\hat{E}=b\hat{a}+b^{\ast }{\hat{a}}^{†}$. The annihilation operator and creation operator change photon number, so their diagonal matrix elements in a Fock state vanish. Since $\hat{a}{\hat{a}}^{†}=\hat{n}+1$ and ${\hat{a}}^{†}\hat{a}=\hat{n}$,The terms containing ${\hat{a}}^2$ or ${\hat{a}}^{†2}$ connect orthogonal number states and contribute zero. Even as $n$ grows, there is no definite oscillating mean electric field; its quadrature fluctuations grow instead. A Fock state can represent fixed field energy, but not a classical sinusoidal wave of prescribed wave phase. This is distinct from representing a classical field with a randomly averaged phase.

The annihilation operator eigenvalue property givesThis is a classical monochromatic electric field: the modulus of $\alpha$ fixes the wave amplitude, while its argument fixes the wave phase relative to the chosen mode convention. Direct normal ordering also gives $(\Delta E{)}_{\alpha }^2=|\mathcal{E}{|}^2$, independent of $\alpha$. Hence the fluctuation divided by the peak mean-field amplitude is $1/(2|\alpha |)$, small in the bright coherent state limit. A relative fluctuation evaluated exactly at a field zero is not a useful classicality test.

For a fundamental electric field in the $xy$ plane, $E_z=0$. The specified tensor then givesWriting $E_x=E_0\cos \varphi$ and $E_y=E_0\sin \varphi$ yieldsThe source electric polarization is therefore along $z$ and is largest along the in-plane diagonals. A radiated wave also needs a transverse component of this source: propagation along $z$ would not generate a propagating $z$-polarized output. Propagation in the $xy$ plane provides the simple strictly transverse example. For a tilted extraordinary wave, the electric displacement, rather than generally the entire electric field, is perpendicular to the wave vector; the tensor source must be projected onto that wave's allowed polarization. This distinction matters when using the type-I geometry below.

Use the common $D_{3/2}$ level to subtract the two upper-state energies. The indicated transition satisfiesThe value inherits the rounding of the supplied wavelengths. It is shorter than 397 nm because $P_{3/2}$ lies above $P_{1/2}$.

A static electric potential alone cannot confine a charge stably in all three dimensions in a charge-free region. An RF quadrupole ion trap uses a rapidly alternating electric field to provide effective radial confinement; DC endcap fields can supply axial confinement. Near a stable equilibrium, the secular motion sees an approximately harmonic pseudopotential. It has quantized vibrational levels $E={\sum }_i\hslash {\omega }_i(n_i+1/2)$, with separate normal modes along the principal directions. RF Paul trap micromotion is superposed on this slower secular oscillation. For several trapped ions, Coulomb interactions instead give collective normal modes. ^[ref]

Four complementary classical features of a large-amplitude coherent state are:

These properties identify a particular classical oscillation, rather than a state of definite photon number. Absolute quantum fluctuations remain even when their relative size is small.

Use the complex electric field convention $e^{-i\omega t}$ and take the locked wave amplitudes $a({\omega }_p)$ to be real and nonnegative after extracting their common phase. Linear superposition givesThe leading factor has unit modulus, so it does not change the light intensity. The triangle inequality bounds the remaining modulus by ${\sum }_{p}a({\omega }_p)$. Consecutive occupied normal modes attain this bound together atThere is also a peak at $t=0$. The electric field obeys $E(t+T)=e^{-i{\omega }_{\mathrm{ce}}T}E(t)$, although the light intensity is periodic.

The carrier-envelope phase specifies where the carrier oscillation lies relative to the maximum of an optical pulse's envelope. With $s=t-t_m$ and $E=F(s)e^{-i{\omega }_{0}s+i{\psi }_m}$, the signed complex-field phase at the envelope maximum is ${\psi }_m$. Consequently,If instead the real electric field is written $F(s)\cos ({\omega }_{0}s+{\varphi }_{\mathrm{CE},m})$, then ${\varphi }_{\mathrm{CE},m}=-{\psi }_m$ and the conventionally quoted slip is ${\varphi }_{\mathrm{slip}}=+2\pi {\omega }_{\mathrm{ce}}/\Delta \omega$. These are the same physical shift with opposite phase definitions.

The carrier-envelope phase matters for few-cycle optical pulses, when the largest instantaneous electric field affects strong-field dynamics, for high harmonic generation, and for stabilizing an optical frequency comb. An ordinary cycle-averaged light intensity measurement does not determine it.

For the normalized coherent state, $⟨a⟩=\alpha$ and $⟨a^{†}⟩={\alpha }^{\ast }$. The dimensionless electric field and magnetic field optical quadratures consequently have expectation valuesThe physical electric field and magnetic field strengths have the corresponding proportionality factors and mode functions. In free evolution replace $\alpha$ by $\alpha e^{-i\omega t}$ to recover the classical oscillation.

Expanding the squared operators cancels the $a^2$ and $(a^{†}{)}^2$ terms:On the other hand, $(\alpha +{\alpha }^{\ast }{)}^2+[-i(\alpha -{\alpha }^{\ast }){]}^2=4|\alpha {|}^2$. HenceThe difference is the sum of the variances, not an extra mean field. Indeed $\mathrm{Var}(\mathcal{E})=\mathrm{Var}(\mathcal{B})=1$, independent of $\alpha$. The commutator $[\mathcal{E},\mathcal{B}]=2iI$ requires ${\sigma }_{\mathcal{E}}{\sigma }_{\mathcal{B}}\ge 1$, and the coherent state attains equality. The residual two units are the vacuum fluctuations of the two optical quadratures.

ferroelectric poling is essential. An initially unpoled ceramic has differently oriented crystal grains and domains whose charge responses largely cancel. An applied electric field favours suitably oriented domains and leaves a remanent macroscopic polarization after removal; it does not rotate all crystal grains into a single crystal.

In the linear coefficient model the added source polarization isThis is the stress-generated contribution at fixed electric field, equivalent to the short-circuit charge per face area. Under open-circuit conditions the induced field supplies an opposing dielectric contribution, so $\Delta D=0$; the source quantity must not be confused with a change of total free electrode charge. For scale, room-temperature single-domain barium titanate has spontaneous polarization of order $0.26\mathrm{C}{\mathrm{m}}^{-2}$: the calculated increment is about $3.0\times 10^{-4}$ of that value, a small response rather than wholesale reversal of the spontaneous polarization.^[ref]

For the requested displacement estimate, assign the whole additional dipole moment of one cell to movement of its single $\mathrm{T}{\mathrm{i}}^{4+}$ ion relative to a fixed background. Then $\Delta P=(4e)\Delta x/(a^{2}c)$, givingThis formal-charge estimate is only an equivalent displacement: real polarization changes involve oxygen and barium displacements, electronic polarization and Born effective charges, as well as possible domain-wall motion. The coefficient supplied for the ceramic cannot uniquely determine an actual microscopic titanium trajectory.

Barium titanate has a high-temperature cubic perovskite structure, with Ti centred in its oxygen octahedron on average. This phase is paraelectric. Near room temperature the structure is tetragonal, with relative ionic displacements that give a spontaneous electric polarization. This phase is ferroelectric: an applied electric field can switch between differently polarized domains. Both phases have the same composition, BaTiO${}_3$; the transition near $120{}^{\circ }\mathrm{C}$ changes crystal symmetry and electrical response.

An optically isotropic film changes the phase of both transverse polarization components equally, so crossed polarizers still extinguish the transmitted light. The etched rod-and-gap composite has form birefringence: its effective permittivity depends on the direction of the electric field, even when the constituent material is intrinsically isotropic. In a simple subwavelength lamellar model,The precise effective-medium formula changes with rod geometry, but the anisotropy remains. Resolving incident light along the two principal directions produces a phase retardation. An ideal lossless specimen between crossed polarizers transmitswhere $I_P$ is the intensity after the first polarizer. Extinction occurs when a principal direction follows either polarizer; transmission is strongest near $45^{\circ }$.

On an outward-oriented sphere of radius $R>0$, the electric field is normal with constant magnitude $A/R^2$. Thus its flux isIt is independent of $R$. There is no contradiction with solution: the classical divergence theorem cannot be applied to a full ball while pretending that $E$ is smooth at its center. On a punctured ball the outer flux and the inward-normal inner flux cancel, in agreement with zero ordinary divergence there.

The three-dimensional Dirac delta distribution ${\delta }^{(3)}(r)$ is supported at the origin and satisfies the sampling identityfor smooth compactly supported test functions $\phi$. It is not an ordinary function with a finite value at zero; its total integral is one, and translating it moves the point at which the test function is sampled.

The globally valid source equation isFor a direct weak verification, excise a ball of radius $\epsilon$ and integrate $-E\cdot \nabla \phi$ outside it. Since the ordinary divergence vanishes there, the result is $A{\int }_{S^2}\phi (\epsilon \hat{r})d\Omega \to 4\pi A\phi (0)$. This is precisely the displayed distributional divergence and restores the divergence theorem with the point charge included.