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1,2-Bis(dimethylphosphino)ethane has the connectivity $\mathrm{M}{\mathrm{e}}_2\mathrm{PC}{\mathrm{H}}_2\mathrm{C}{\mathrm{H}}_2\mathrm{PM}{\mathrm{e}}_2$. Chelation through both phosphorus donors contributes four electrons in ionic electron counting.

The 18-electron rule is a useful closed-shell guideline for many low-valent transition-metal complexes, filling nine valence orbitals. It is not universal: many early-metal $d^0$ complexes have fewer electrons, while stable square-planar $d^8$ complexes are often 16-electron.

The experiment involving Ferenc Krausz combined a subfemtosecond soft-X-ray pulse with a few-cycle visible electric field. Its delay-dependent krypton photoelectron spectra tested the field-controlled response. The reported pulse duration was $650\pm 150$ attoseconds. Changing delay compared the emitted electron distributions at different reference phases; the spectral-width trace supplied a further duration-sensitive observable.^[ref]

The distinction from RABBITT is the measurement regime and reconstruction, rather than merely calling one trace a clock. The attosecond streaking momentum map is a limiting model; a finite-duration wave packet requires a quantum transition calculation. Inferring an emission delay also requires separating the illuminating pulse's spectral phase from the target response. The measurement supports time-dependent photoionization in a specified model, with no unique classical electron trajectory inside an atomic orbital implied.

DPPH is a neutral organic radical used as a spin reference in electron paramagnetic resonance (EPR), also called electron spin resonance (ESR). Its unpaired electron remains bound within the chemical species: “free radical” does not mean a freely moving electron. The DPPH electron spin resonance experiment makes its magnetic moment observable.

A 2,4-Diazadiboretidine contains an alternating four-membered B–N–B–N ring. nitrogen lone pairs provide sigma-donor sites, while the ring can bind an electron-rich metal through a pi interaction involving both B and N.

Electron microscopy with optical correctors compensating selected lens aberrations. Multipole correctors can reduce spherical-aberration blur and permit finer probes or improved image transfer. Resolution still depends on chromatic effects, source properties, stability, specimen scattering and damage.

A hot, comparatively optically thin plasma region associated with an accretion flow. Thermal or nonthermal electrons can transfer energy to softer seed photons through inverse Compton scattering, producing a harder radiation component. Its geometry, heating and coupling to the cooler disc are model dependent; a hard radiation spectrum alone does not uniquely determine these properties.

An increase in oxidation state records oxidation; a decrease records reduction. Thus ${\mathrm{Fe}}^{2+}\to {\mathrm{Fe}}^{3+}$ raises the iron oxidation state by one, while ${\mathrm{MnO}}_4^{-}\to {\mathrm{Mn}}^{2+}$ lowers the manganese oxidation state from $+7$ to $+2$. Five iron ions supply the five-electron inventory required per permanganate ion.

In an acidic chemical solution, the complete half-reactions areFirst balance oxygen with four water molecules, then hydrogen with eight protons, and finally electric charge with five electrons. Multiply the iron half-reaction by five and add; the exchanged electrons cancel:Both sides contain one manganese, four oxygen, eight hydrogen and five iron chemical atoms, with net electric charge $+17e$. The ${\mathrm{H}}^{+}$ notation represents the acidic aqueous proton inventory; it does not imply bare protons moving independently of water. These half-reactions and their sum are the acidic balance used in the documented permanganate-iron titration.^[ref]

The coefficients predict amounts consumed, not a collision of five ions or an elementary reaction. Bulk chemical solution also contains compensating counterions, which are omitted from this net chemical equation. The hydrogen and oxygen oxidation states are unchanged; balancing their inventories remains essential even though they do not account for the exchanged electrons.

A. D. Challinor coauthored, with A. N. Lasenby, analytic corrections to the thermal Sunyaev-Zeldovich effect in hot galaxy clusters. The calculation uses an expansion of the Kompaneets equation to include relativistic effects in scattering by electrons.

The resulting change in spectral shape matters when combining X-ray and cosmic microwave background observations to infer cluster properties or a Hubble constant. A small correction to the physical signal can produce a systematic shift in a derived parameter.^[ref]

Slow widening of a constriction approximately conserves the transverse mode action, reducing transverse momentum as the width $W$ increases. A lowering band bottom also increases longitudinal kinetic energy. Together they can narrow the angular distribution of an emitted electron beam. An abrupt narrow aperture alone can instead diffract strongly.

Flavor conversion in a slowly varying medium when an occupied instantaneous propagation eigenstate follows its eigenvalue branch without transitions to the other branch. For positive electron-neutrino matter potential and normal two-state ordering, a high-density electron neutrino follows the upper branch into the vacuum mass eigenstate ${\nu }_2$. Detection in the electron flavor then has probability ${\sin }^2\theta$, rather than the phase-averaged vacuum value $1-\frac{1}{2}{\sin }^{2}2\theta$. High production density alone does not guarantee adiabatic evolution.

Aerobic respiration uses molecular oxygen as the final electron acceptor. For complete oxidation of glucose, the net material balance isThis records the overall conversion of glucose, molecular oxygen, carbon dioxide and water; it is not one enzyme-catalyzed step, and it omits the separate ATP accounting. For an original stoichiometry check, each side contains six carbon, twelve hydrogen and eighteen oxygen chemical atoms. If exactly $1\mathrm{mmol}$ of glucose undergoes this complete conversion, the equation requires $6\mathrm{mmol}$ of molecular oxygen and yields $6\mathrm{mmol}$ each of carbon dioxide and net water. This is an equation-based calculation, not an observed seed measurement.

The molecular oxygen is reduced to water, rather than being the source of the carbon dioxide's carbon. A formal redox electron balance, using acidic half-reaction notation, makes the roles explicit:Adding and cancelling gives the net equation above. Both half-reactions conserve chemical atoms and electric charge. The $24$ electrons balance a net transfer; they are carried through cellular intermediates, rather than released as a pool of free electrons. The final respiratory reduction can be written ${\mathrm{O}}_2+4{\mathrm{H}}^{+}+4e^{-}⟶2{\mathrm{H}}_2\mathrm{O}$. These equations are bookkeeping, not a tracing experiment that assigns every product oxygen chemical atom to its original molecule.^[ref]

The Aharonov-Bohm effect is the phase $q\Phi /\hslash$ acquired when a charge $q$ encircles magnetic flux $\Phi$, even when the magnetic field vanishes along the charge's path.

In the magnetic Aharonov-Bohm effect, a coherent electron wave takes two routes around inaccessible magnetic flux. Recombination converts their relative wave phase into displaced interference fringes. In 1959, Yakir Aharonov and David Bohm proposed a shielded solenoid geometry and identified the obstruction to removing the magnetic vector potential throughout the accessible region.^[ref]

Use International System of Units, electric charge $q$, and magnetic minimal coupling $H=(-i\hslash \nabla -q\mathbf{A}{)}^2/(2m)$. Let ${\gamma }_1,{\gamma }_2$ run from the same source $S$ to the same detector point $D$. The oriented closed curve $C$ follows ${\gamma }_1$ and then ${\gamma }_2$ backwards. For a static magnetic vector potential, the magnetic contribution to the relative wave phase isHere ${\Phi }_C={\int }_{\Sigma }\mathbf{B}\cdot d\mathbf{S}$ is the oriented magnetic flux through a spanning surface. Equal or calibrated nonmagnetic propagation phases are assumed. For an electron, $q=-e$, where $e>0$ is the elementary charge; reversing $C$ reverses the sign. The observable is the gauge-invariant holonomy $e^{i\delta }$, rather than a unique local value of the magnetic vector potential.

The accessible region is a multiply connected region: $C$ cannot contract to a point without entering the excluded flux region. Stokes theorem does not set the line integral to zero, because a spanning surface must cross that excluded region, where the magnetic field need not vanish. There is real magnetic flux in the apparatus; only the accessible paths are idealized as free of the magnetic field. The exterior of a finite straight solenoid also permits routes around its ends, so its exact topology and leakage differ from those of an infinite solenoid or an excluded solid torus.

An explicit model makes the obstruction visible. Outside an infinite flux tube of radius $a$, choose cylindrical coordinates $(r,\theta ,z)$ andIts curl vanishes there, but a loop with winding number $n$ has ${\oint }_C\mathbf{A}\cdot d\mathbf{l}=n\Phi$. Locally $\mathbf{A}=\nabla (\Phi \theta /(2\pi ))$; the angular coordinate changes by $2\pi$ on a circuit. For generic $q\Phi /\hslash$, that candidate gauge removal changes the wave function's single-valued boundary condition. On a simply connected region, a smooth static magnetic vector potential with zero curl instead has a globally defined scalar primitive and is a pure gauge potential. Thus local zero magnetic field and globally removable magnetic vector potential are different conditions.

For an original interference example, take mutually coherent scalar amplitudes with constant individual intensities $I_1,I_2$ and an imposed carrier phase $ky$, as in off-axis holography. ThenThe interference fringe spacing is $d=2\pi /k$. An electron enclosing $\Phi =h/(4e)$ has $\delta =-\pi /2$, shifting maxima by $d/4$ for this orientation. Increasing the magnetic flux by $h/e$ changes $\delta$ by $-2\pi$ and restores the same pattern: the electron Aharonov-Bohm effect has magnetic flux quantum $h/e$. This uses $h=2\pi \hslash$, the Planck constant. It does not require the source's magnetic flux to vary continuously during a single exposure.

For $I_1=1$, $I_2=1/4$, the extrema are $I_{\max }=9/4$, $I_{\min }=1/4$, and the interferometric visibility is $4/5$. A fixed Aharonov-Bohm phase translates these fringes without changing their contrast. Averaging unresolved fluctuating phases can reduce the contrast; a darkened pattern alone cannot identify enclosed magnetic flux. These formulas assume coherence, indistinguishable alternatives and stable beam envelopes.

The appeal of the Aharonov-Bohm effect is that a local Lorentz force calculation along either ideal path misses information carried by their global arrangement. It gives a physical example of holonomy and supplies the encircling phase used in Chern--Simons flux attachment. This alone does not establish intrinsic topological order. The static comparison also matters: changing magnetic flux can induce an electric field, so a time-dependent experiment requires the scalar-potential contribution as well as the displayed spatial line integral.

Akira Tonomura used electron holography to test the Aharonov-Bohm effect with magnetic flux confined away from the electron paths. A coated magnetic toroid let electron waves pass through its hole and outside it while a superconducting shield excluded the magnetic field from those paths. The measured relative phase still depended on the enclosed flux. The experiment isolates a quantum phase effect that cannot be inferred from the local classical magnetic force along either path.^[ref]

A physicist who developed electron holography and investigated the Aharonov-Bohm effect.^[ref]

A physicist whose work connects special relativity, general relativity, Brownian motion and the photoelectric effect. He worked at the Institute for Advanced Study from 1933 until his death in 1955.^[ref]

In 1905, Albert Einstein used the photoelectric effect to explore quantized light, related Brownian motion to molecular agitation, formulated special relativity, and connected changes in mass and energy. These problems link visible measurements to otherwise inaccessible structure: emitted electron energies, suspended-particle displacements and comparisons of clocks.^[ref]

The Millikan photoelectric experiment follows one of those predictions into the laboratory. Its stopping-voltage measurements test a quantitative relation; the experiment's apparatus, surface preparation and electrical corrections matter to that inference. The Einstein field equations and general relativity lead to a different set of tests involving gravity and spacetime.

The Nobel Prize in Physics in 1921 recognized his theoretical work, particularly the law of the photoelectric effect; the award was presented in 1922.^[ref]

Albert Fert coauthored the discovery of giant magnetoresistance in alternating iron and chromium layers. In the 1988 measurements, a magnetic field of 2 tesla nearly halved the electrical resistivity of one thin-chromium sample at 4.2 kelvin. Spin-dependent electron transmission through the chromium layers explained why changing the magnetic arrangement changed the electrical resistance so strongly. The experiment made nanoscale layer structure and electron spin directly relevant to an easily measured electrical signal.^[ref]

A physicist.^[ref]

An experimental physicist who studied electron diffraction by sending cathode rays through thin celluloid films. His contemporary scientific obituary identifies this research and the definitive paper that followed the preliminary report. A transmitted electron beam developing an angular diffraction pattern tests the matter wave description of electrons, connecting a particle beam produced in a vacuum apparatus with phenomena familiar from wave optics.^[ref]

An alkenyl group is a monovalent carbon–hydrogen substituent derived from an alkene by removing one hydrogen atom. The attachment can be at a double-bonded carbon or elsewhere in the carbon skeleton. For an acyclic parent with one double bond, the group composition is ${\mathrm{C}}_{\mathrm{n}}{\mathrm{H}}_{2n-1}$.^[ref]

For example, a vinyl group attaches as $\mathrm{R}-\mathrm{CH}=\mathrm{C}{\mathrm{H}}_2$ and contributes ${\mathrm{C}}_2{\mathrm{H}}_3$, whereas an allyl group attaches as $\mathrm{R}-\mathrm{C}{\mathrm{H}}_2-\mathrm{CH}=\mathrm{C}{\mathrm{H}}_2$ and contributes ${\mathrm{C}}_3{\mathrm{H}}_5$. The attachment atom in the first example belongs to the double bond; in the second it is adjacent to the double bond. A prenyl group is another alkenyl group, with five carbon atoms.^[ref]

The missing hydrogen identifies an attachment site in a larger molecule; it does not mean that the completed attached group must contain an unpaired electron. Formulae alone also do not specify connectivity. For $n=3$, the composition ${\mathrm{C}}_3{\mathrm{H}}_5$ does not by itself tell where the attachment or double bond lies. A structural formula supplies information that the atom count omits.

An alkyl group is a monovalent group obtained formally by removing one hydrogen atom from an acyclic saturated hydrocarbon. The general composition is ${\mathrm{C}}_{\mathrm{n}}{\mathrm{H}}_{2n+1}$: examples include $\mathrm{C}{\mathrm{H}}_3-$ and $\mathrm{C}{\mathrm{H}}_3\mathrm{C}{\mathrm{H}}_2-$. The dash denotes an attachment chemical bond, rather than necessarily an unpaired electron.^[ref]

“Alkyl” specifies a structural fragment. The corresponding free radical, carbocation and carbanion have different electronic states and reactivities. Branched alkyl groups can also share an elemental formula while giving different connectivity and steric environments around the attachment site.

With a momentum-independent allowed nuclear matrix element, negligible nuclear recoil and negligible neutrino mass, the electron spectrum in natural units is $d\Gamma /dp=CF(Z,E)p^2(E_0-E{)}^2$, where $E=\sqrt{p^2+m_e^2}$. Transforming to an energy spectrum requires the Jacobian determinant $dp/dE=E/p$.