Codex @codex  0

An electron filling an electron hole removes both excitations relative to the filled-band background. Their electric charges cancel, consistent with charge conservation; the remaining electrons are still present. The released energy can leave through photon emission or other channels, including phonons. Formation of an exciton instead retains the excited electron and electron hole as a bound pair; recombination removes the pair. A valence band electron filling a nearby vacancy merely moves the electron hole and does not remove the total vacancy count.

Electron holography uses electron object and reference waves to encode their relative wave phase in a hologram.^[ref]

Separate geometry from electronic response in an original one-dimensional toy tunnel junction. Write the geometric surface height as $h(x)$, tip height as $z(x)$, and positive gap as $s(x)=z(x)-h(x)$. Assume low fixed tip-sample electric potential difference: the sampled energy window is narrow enough that the tip density of states is featureless and the sample local density of states varies little across that window. Neglect thermal broadening, changes of tip shape and lateral averaging. Let $\rho (x)>0$ be the effective sample local density of states divided by a positive reference value, so it is dimensionless. Choose a spatially constant effective barrier energy $\Phi >0$, a positive coefficient $C$ with units of electric current, andHere $m_e$ is the electron mass, $\hslash$ the reduced Planck constant, and $\kappa$ the tunnelling decay constant. This is an illustrative low-bias model, not a universal calibration formula. At fixed $I=I_0>0$, taking a logarithm givesThe logarithms have dimensionless arguments. Only constant $\rho$ makes the recorded height differ from geometric height by a constant. Multiplying $\rho$ by $r>0$ raises the apparent height by $\ln r/(2\kappa )$ without moving the geometric surface.

For a concrete ambiguity, choose $\kappa =10{\mathrm{nm}}^{-1}$, $a=0.020\mathrm{nm}$, lateral period $L=1\mathrm{nm}$, $s_0=0.50\mathrm{nm}$, and $q(x)=\cos (2\pi x/L)$. CompareBoth produce exactly $z(x)=s_0+aq(x)$ at $I_0=C{e}^{-2\kappa s_0}=C{e}^{-10}$. Substitution checks this without identifying $\rho$ with geometric height: case A has constant gap $s_0$, while case B has gap $s_0+aq(x)$ and a compensating electronic factor. The chosen ${\rho }_B$ ranges from $e^{-0.4}$ to $e^{0.4}$, approximately $0.6703$ to $1.4918$; every gap remains positive. The chosen $\kappa$ corresponds to $\Phi \simeq 3.81$ electronvolts, a positive effective barrier.

At fixed tip height $z_f=s_0$, both cases instead giveTheir maximum-to-minimum electric current ratio is $e^{0.8}\simeq 2.226$. Thus even adding this fixed-height scan at the same electric potential difference cannot distinguish these deliberately chosen surfaces within the toy model. Independent structural information or additional electronic measurements are required. Changing electric potential difference can supply such information, but also changes the assumptions used to define $C$ and $\rho$.

Transfer of a system to an excited electronic state. Energy deposited into electrons need not become a permanent atomic displacement.

Electronic heat capacity is the contribution from electronic excitations to heat capacity. A simple normal metal with a smooth density of states near its Fermi energy has $C_{\mathrm{m}}^{\mathrm{el}}\simeq \gamma T$ well below its electron Fermi temperature. Only a narrow energy interval near the Fermi energy changes occupation on heating, explaining the small fraction of active electrons. The Debye T-cubed test separates this contribution from lattice phonons. Interactions, a superconductivity transition or other electronic structure can invalidate a constant $\gamma$ description.

For a free target electron at rest, $e^{-}{e}^{-}\to e^{-}{e}^{-}{e}^{-}{e}^{+}$ has threshold invariant mass $4m_e$. In natural units, $s=2m_e^2+2m_e{E}_{\mathrm{incident}}$ gives incident total energy $7m_e$, or incident kinetic energy $6m_e$. Atomic binding and recoil of the rest of an atom are excluded by this free-target model. This threshold differs from photon-induced pair production.

An electron lens uses an electric field or magnetic field to make trajectories from an object point converge towards an image point. It has no glass for the electrons to pass through. In a paraxial approximation, an effective focal length permits the same first-order image geometry used for a thin lens; finite beam angles, energy spread and misalignment limit this approximation.

The electron magnetic moment couples an electron's angular momentum to a magnetic field. Its intrinsic spin contribution isHere ${\mu }_B$ is the Bohr magneton, and $g_s$ is the positive electron spin g-factor under this sign convention. The minus sign makes magnetic dipole moment antiparallel to spin angular momentum. An electron in a chemical atom can also contribute an orbital magnetic dipole moment; the measured response of a whole atomic level combines these contributions.

Measuring a tiny departure of $g_s$ from $2$ tests how well a theory predicts an electron's interaction with its surroundings. The Dirac equation gives $g_s=2$ before radiative corrections; the anomalous magnetic moment makes a small correction experimentally accessible.^[ref] A useful strategy is to compare two atomic energy levels whose orbital and spin contributions have different weights, rather than trying to calibrate an absolute magnetic field more accurately.

Electron magnetohydrodynamics neglects ion motion and electron inertia while retaining transport of the magnetic field by the electron current. At constant electron number density, ${\mathbf{u}}_e=-\mathbf{J}/(en)=-c\nabla \times \mathbf{B}/(4\pi en)$. Flux freezing in this velocity givesFor a uniform guide field, the solenoidal linear equation implies ${\omega }^2=(c{B}_0/4\pi en{)}^2{k}_{\parallel }^2{k}^2$. With $v_A=B_0/\sqrt{4\pi n{m}_i}$ and $d_i=c\sqrt{m_i/(4\pi e^{2}n)}$, the coefficient is $c{B}_0/(4\pi en)=v_A{d}_i$.

The rest mass of an electron.

An electron microscope forms a spatially enlarged representation of a specimen using electrons, electron lenses, and a detector. Its appeal is access to structure too fine for an ordinary optical microscope, but a short de Broglie wavelength alone does not guarantee fine electron microscope resolution. Increasing magnification can enlarge an existing blur without revealing any additional structure.

In transmission electron microscopy, an electron gun illuminates a thin specimen; magnetic electron lenses form and enlarge its transmitted image on a fluorescent screen or detector. A vacuum reduces collisions with gas. Lens electric currents control focusing and magnification.^[ref]

Electron microscope resolution concerns distinguishable specimen detail under a specified contrast criterion. It depends on diffraction, lens defects, electron energy spread, specimen motion and detection. A smaller de Broglie wavelength permits finer detail in principle, but neither wavelength nor magnification is the instrument's measured resolution.

For an electron accelerated from negligible initial kinetic energy through a positive electric potential difference $U$, combine $K=eU$, the relativistic energy-momentum relation and the de Broglie wavelength:Here $h$ is the Planck constant, $m_e$ the electron mass, $e$ the elementary charge, and $c$ the speed of light. This includes the relativistic correction omitted by the low-voltage electron accelerating-voltage wavelength approximation. With $m_e\simeq 9.1093837\times 10^{-31}\mathrm{kg}$ and SI defining values of $h,e,c$, calculated wavelengths are approximately $6.979$, $3.701$ and $1.969\mathrm{pm}$ at $30$, $100$ and $300\mathrm{kV}$ respectively. The rounded mass agrees with the recommended value.^[ref]

For a circular aperture admitting a small semi-angle $\alpha$, the Rayleigh criterion suggests a diffraction-limited resolution scale $d_D=0.61\lambda /\alpha$. At $100\mathrm{kV}$ and $\alpha =10\mathrm{mrad}$, this gives $0.226\mathrm{nm}$, already much larger than the $0.00370\mathrm{nm}$ wavelength. This is an ideal aperture estimate. Transmission electron microscopy with coherent illumination and a particular specimen requires its own image-contrast analysis; this scale alone is not a universal two-feature resolution test.

An original illustrative blur model adds competing effects:Define $C_s{\alpha }^3$ as a characteristic transverse blur from the electron-lens analogue of spherical aberration, and $C_c\alpha \Delta K/K$ as one from energy-dependent focusing, the analogue of chromatic aberration. The coefficients $C_s,C_c$ have units of length, $\Delta K$ is a chosen characteristic kinetic energy spread, and $b_0$ is a fixed residual blur. Adding their squares is a declared heuristic, not a measured point spread function or a rigorous resolution theorem. The model omits contrast transfer, specimen scattering and signal statistics, so its minimum is only an estimate within its assumptions.

Choose $C_s=C_c=1\mathrm{mm}$, $\Delta K=0.7\mathrm{eV}$ and $b_0=0.10\mathrm{nm}$ at $K=100\mathrm{keV}$. At $5\mathrm{mrad}$, the four contributions are about $0.452$, $0.125$, $0.035$ and $0.100\mathrm{nm}$, giving $b\simeq 0.480\mathrm{nm}$. At $10\mathrm{mrad}$, the diffraction contribution falls to $0.226\mathrm{nm}$ but the assumed spherical aberration contribution grows to $1.00\mathrm{nm}$. Thus opening the aperture indefinitely worsens this model's image. All these values are calculations with chosen parameters, not observations of any microscope.

Electron microscopy is the practice of preparing specimens, producing images with an electron microscope, and interpreting their contrast. A dark region can reflect electron deflection out of the admitted beam rather than complete absorption. A useful image therefore needs a stated preparation, detection signal and length calibration; brightness by itself does not identify a material. Transmission electron microscopy probes thin specimens, whereas a scanning electron microscope can examine the exposed surface of a bulk specimen.^[ref]

The positive electron mobility relates the drift velocity of conduction electrons to an electric field: ${\mathbf{v}}_d=-{\mu }_e\mathbf{E}$. In the isotropic Drude model, ${\mu }_e=e\tau /m^{\ast }$, where $\tau$ is the momentum relaxation time and $m^{\ast }$ the band effective mass. The minus sign describes negative charge; the electron mobility itself is positive.

The electron neutrino is the neutrino flavour associated with the electron in the weak charged current.

Electron optics studies how electric fields and magnetic fields guide electrons and form images. The Lorentz force supplies a trajectory description, while the de Broglie wavelength supplies the wave scale needed for diffraction. Both descriptions matter: steering a narrow beam and resolving two nearby specimen features are different achievements.

For a magnetic electron lens,Here $e>0$ is the elementary charge, $\mathbf{v}$ the electron velocity, and $\mathbf{B}$ the magnetic field. The second equality follows because a cross product is perpendicular to each factor: a static magnetic field bends the trajectory without increasing the kinetic energy. Acceleration instead requires work by an electric field. This distinction lets an electron microscope adjust imaging with magnetic fields while an accelerating voltage sets the electron momentum.

An electron pair consists of two electrons with opposite spins sharing the same spatial part of their quantum state. The Lewis structure distinction between a bond pair and a lone pair records whether that pair is assigned to a connection between chemical atoms or to one chemical atom.

The electron pairing energy represents the energetic penalty for putting electrons into already occupied orbitals rather than suitable separate orbitals. A simple comparison with octahedral splitting explains high-spin and low-spin alternatives.

electron paramagnetic resonance measures resonant transitions involving unpaired electron magnetic moments. A spin label attached to a protein or lipid can report local motion or distances under appropriately designed pulse experiments.

The electron-phonon interaction couples electronic states to the atomic displacements represented by phonons. A lattice vibration changes the potential seen by an electron; its quantized displacement permits phonon absorption and emission. This interaction affects electrical resistance, electronic thermal conductivity, lattice dynamics and pairing in conventional superconductors.