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An incident electron that returns out of the entrance side of a specimen after scattering.

The Barkhausen effect consists of abrupt magnetization changes during a smooth applied magnetic field sweep. Its inductive signal is Barkhausen noise, often arising from collective magnetic domain wall motion.^[ref]

The distinction is between a smooth drive and an intermittent material response. A smooth macroscopic hysteresis curve can hide smaller jumps. By Faraday's law of induction, a pickup coil responds to the rate of change of linked magnetic flux, so a rapid jump can be conspicuous even when the total change is small. A pulse is not one chemical atom flipping: neither its height nor the number of audible clicks counts individual electron spins. Qualitative crackling alone does not establish quantum spin; collective switching also admits a classical model with trapped interfaces. Compare the distinct mechanical readout of angular momentum transfer in the Einstein-de Haas effect.

For a uniform isotropic weak-gap BCS state, the zero-temperature electron occupation is $n(\xi )=(1-\xi /\sqrt{{\xi }^2+{\Delta }^2})/2$. The number above the normal-state Fermi surface is approximately $D(\mu ){\int }_0^{\infty }n(\xi )d\xi$. The integral equals $\Delta /2$, while a three-dimensional parabolic band including both spins has $D(\mu )=3N/(2\mu )$. This gives the displayed leading fraction.

The result measures redistribution relative to a normal Fermi sea, not bosonic single-mode condensate depletion or quasiparticle occupation. A different band dimension or anisotropic gap requires its actual density-of-states or momentum integral. Material estimates need specified gap and chemical-potential scales.

An elementary excitation of the paired mean-field state whose electron operator has mixed creation and annihilation components. Its positive excitation energy is $E_k=\sqrt{{\xi }_k^2+|{\Delta }_k{|}^2}$. At zero temperature the BCS ground state has zero occupation of these positive-energy modes, even though physical electrons have a smooth nontrivial momentum occupation.

William Bertozzi compared electron time-of-flight speeds with independently measured kinetic energies using electrostatic and linear acceleration. The 1964 report covers 0.5–15 MeV and uses calorimetry to check deposited energy. Speed approaches the light-speed limit while energy continues rising, testing relativistic energy against an unrestricted Newtonian extrapolation.^[ref]

An electron or Positron emitted in beta decay.

A biexciton is a two-electron, two-electron hole excitation. If $E_X$ and $E_{XX}$ are total excitation energies relative to the empty emitter, the cascade $XX\to X\to 0$ emits photons of energies $E_{XX}-E_X$ and $E_X$. Define its binding energy by $B_{XX}=2E_X-E_{XX}$. The first cascade photon then has energy $E_X-B_{XX}$. Positive binding shifts that line below the single-exciton line; confinement can also produce negative binding, shifting it above. The cascade does not emit a single photon of twice the exciton energy.

Binding energy is the energy required to separate a bound state into specified free constituents. For an electron bound in a chemical atom, neglecting this cost is an assumption of the free-electron Compton scattering model.

The Bohr model assumes circular electron orbits with quantized orbital angular momentum $N\hslash$. It reproduces the hydrogen energy levels, but wavefunctions in quantum mechanics describe spatial probability distributions rather than definite trajectories. In particular, spherically symmetric Coulomb bound states have zero orbital angular momentum.

The puzzle is the coexistence of reproducible spectral lines and stable chemical atoms. A classical circular orbit in a Coulomb potential can have any negative energy: Coulomb's law alone selects no special radius or emitted frequency. Moreover, its electron has centripetal acceleration. The Larmor formula then predicts positive radiated power, so conservation of energy makes the orbit shrink. Simply drawing concentric circles explains neither stability nor a discrete spectrum.

Niels Bohr postulated nonradiating stationary states and emission with $h\nu =E_i-E_f$. His circular-orbit condition is $L=n\hslash$. These assumptions override classical radiation; the emitted frequency need not equal an orbital frequency.^[ref]

A bond dipole represents the separation of positive and negative electric charge associated with a chemical bond. In a neutral two-point charge model put $+q$ at ${\mathbf{r}}_A$ and $-q$ at ${\mathbf{r}}_B$, with $q>0$. Its physical electric dipole moment isIf $B$ is the negative end, the physical electric dipole moment points from $B$ to $A$. The customary crossed-tail chemical arrow points towards the negative, more electronegative end, from $A$ to $B$. It therefore has the opposite direction. State the convention before doing vector addition; use physical electric dipole moments in the interaction potential energy $U=-\mathbf{p}\cdot \mathbf{E}$, where $\mathbf{E}$ is the electric field.^[ref]

For a fixed neutral molecule, its complete electric dipole moment follows from all atomic nuclei and the electron density, not just point charges assigned to sites. If $n(\mathbf{r})$ is the expected number of electrons per unit volume, the corresponding charge density is ${\rho }_e(\mathbf{r})=-en(\mathbf{r})$; the nuclear terms contribute separately. Partitioning that distribution into bond dipoles is a model. Lone pairs and electric polarization within a chemical atom can contribute as well. A measured molecular electric dipole moment cannot uniquely recover every individual bond dipole or partial atomic charge.

A bond-dissociation energy specifies the energy needed for a stated dissociation into separated fragments. An energy from the lowest vibrational state, an energy from the bottom of a potential energy well, and a finite-temperature molar dissociation enthalpy require different conventions; do not silently mix them in an electronegativity formula.

Which fragments, in which quantum states, and at which temperature? For a neutral gaseous molecule with a specified two-electron covalent bond, the usual thermochemical convention is homolysis,Each fragment receives one of the two bonding electrons; the dots denote unpaired electrons. The final fragments are isolated from each other, with their own relaxed molecular geometry and specified electronic quantum states. The energy cost includes this relaxation, rather than merely stretching the initial chemical bond while freezing everything else. For a neutral parent, heterolysis instead produces ${\mathrm{A}}^{+}(g)+{\mathrm{B}}^{-}(g)$, or the opposite assignment of electric charges; those are different final states and different energies. A value for gaseous neutral fragments cannot be transferred to ions, a chemical solution, or a solid without accounting for those changed conditions.^[ref]

The distinction is useful beyond terminology. Pauling electronegativity compares consistently defined bond-dissociation energies; photodissociation can reveal a specified ground-state threshold; a thermal chemical reaction balance requires compatible enthalpies. A threshold is also different from an activation energy: it compares initial and final quantum states, whereas a barrier can obstruct the path between them.

Occupying a bonding molecular orbital favours a chemical bond. For the two equivalent $1s$ atomic orbitals of the hydrogen atom, the sum ${\sigma }_g={\sigma }_{+}$ has constructive interference between the protons. The resulting electron density in that region helps stabilize the combined system. Its lower energy in this model follows from the Quantum Hamiltonian, not from the plus sign alone; see Field, MIT 5.61, lecture 25, pp. 1–2.

For example, $({\sigma }_g{)}^2$ in molecular hydrogen supplies $N_b=2$, hence bond order one when $N_a=0$. The sum in the molecular orbital theory toy has midpoint probability density $0.303468$, whereas the difference has zero there. The normalization factors must be included in this comparison.

In the elementary molecular orbital theory count, bond order is half the excess of electrons in bonding molecular orbitals over those in antibonding molecular orbitals. Thus adding one electron to a bonding molecular orbital raises $b$ by $1/2$, while adding one to an antibonding molecular orbital lowers it by $1/2$. The configurations drawn above give $1$, $1/2$, and $0$. This count summarizes net covalent bond occupancy; it is neither a measured energy nor a universal test for every kind of binding. See Chemistry 2e, 8.4: molecular orbital bond order.

A bond pair is an electron pair allocated to a connection between two chemical atoms in a Lewis structure. A single line stands for one bond pair, or two electrons. A double connection contains two bond pairs, or four electrons; a triple connection contains three bond pairs, or six electrons. A bond pair is shared bookkeeping for the combined arrangement, rather than two distinguishable electrons each permanently owned by one endpoint.

In water, two bond pairs use $2\times 2=4$ of the eight valence electrons. In ammonia, three bond pairs use $3\times 2=6$ of the eight valence electrons. In $\mathrm{O}=\mathrm{C}=\mathrm{O}$, each C=O connection contains two bond pairs, so there are four bond pairs in the entire carbon dioxide molecule. Counting connections and counting bond pairs therefore give different answers for multiple connections.

The same bond pair is treated differently for different questions: both endpoints count its two electrons for the octet rule, whereas each endpoint receives just one electron for formal charge. Neither convention asserts equal physical electron density at the endpoints. For electronic shapes, occupancy, and energy, see molecular orbital theory rather than treating a bond pair line as an entire wavefunction.

Bond polarization is unequal sharing of electron density across a chemical bond. An electronegativity difference is a heuristic for its direction and tendency, not a unique partial atomic charge.

A simple two-site model makes the notation explicit. Represent each site by effective positive electric charge $+e$, and allocate expected electron populations $1-s$ to $A$ and $1+s$ to $B$, with $0\le s<1$. Here $e$ is the elementary charge; these effective sites omit the internal structure of real chemical atoms. Subtraction givesFor the stipulated illustration $s=0.35$, the populations are $0.65$ and $1.35$, and the effective partial atomic charges are $+0.35e$ and $-0.35e$. Fractional expected populations do not split an individual electron into pieces. The total population remains two, and the pair remains neutral. No formula connecting $s$ to $\Delta \chi$ has been assumed; many electron density distributions and partitioning conventions can share one qualitative electronegativity ordering.

A Born-Haber cycle uses Hess's law to infer a lattice energy from measurable enthalpy differences and isolated-species data. The motivating difficulty is that separating a mole of an ionic crystal into mutually separated gaseous ions is not a convenient direct calorimetry experiment. Instead, compare two accounting routes between the same elemental reference states and the same solid. Because enthalpy is a state function, the signed differences close exactly when all inputs describe compatible states. These routes do not assert a reaction mechanism or a temporal sequence of reaction intermediates.^[ref]

For one mole of sodium chloride formula units, the overall formation reaction isThe stoichiometry requires half a mole of diatomic chlorine molecules, supplying one mole of chlorine chemical atoms. A half coefficient counts an amount of substance; it does not describe half of an individual molecule. Write each auxiliary chemical reaction with its own sign:Here $S$ is the sublimation enthalpy, $B$ the enthalpy of atomization of chlorine per mole of chemical atoms, and $I$ the molar ionization enthalpy represented by the ionization energy input in elementary treatments. If the bond-dissociation energy table gives the compatible molar enthalpy for ${\mathrm{Cl}}_2(g)\to 2\mathrm{Cl}(g)$, use half that number for $B$. Both ions in the last step are gaseous, and no aqueous hydration of ions belongs in that step.

Adding these equations cancels the gaseous chemical atoms, the gaseous ions and the free electron, leaving precisely the overall formation reaction. ThusThe closed traversal returns from the solid to the elemental reference states, adding $-F$; therefore $S+B+I+A+L_{\mathrm{form}}-F=0$. Notice that the signed attachment term $A$ is negative. Substituting a positive spectroscopic electron affinity for $A$ would reverse the attachment contribution.

The diagram deliberately uses invented, internally consistent inputs in $\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$: $S=100$, $B=120$, $I=500$, $A=-350$ and $F=-400$. Preparation of separated gaseous ions raises the enthalpy by $370$, so $L_{\mathrm{form}}=-770$ and $L_{\mathrm{diss}}=+770$. The closure check is $100+120+500-350-770+400=0$. These numbers are not measured sodium chloride data and do not replace the source's $769\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ example in lattice energy. Every level retains one sodium nucleus, one chlorine nucleus and zero total electric charge; the ionization level explicitly retains the free electron.

The scattering interpretation connects an incoming preparation to the relative frequency of outgoing directions and final atomic states. Max Born's 1926 paper treats a distant incoming electron and a chemical atom, expands the outgoing wavefunction into final atomic states, and imposes conservation of energy. Its translated correction footnote specifies the square of the coefficient; the modern complex modulus notation below is ours.^[ref]

In his 11 December 1954 lecture, Max Born recalled the difficulty of interpreting bound electrons alone. He instead described an incident electron wave generating an outgoing spherical wave, with direction-dependent squared amplitude determining relative scattering probabilities. Distinct final atomic states describe excitation with corresponding electron energy loss. This is a retrospective account of an interpretation, rather than the report of a new apparatus.^[ref]

The distinction from the modern Born rule matters. The historical question was which collision outcome occurs: a direction of motion and a final atomic state. The general projective measurement formula applies to arbitrary observables, including degenerate outcomes and mixed states; a positive operator-valued measure describes still more general detector statistics. Those formulations should not be read back as the explicit contents of the short 1926 scattering paper. Nor does the scattering interpretation itself supply a rule for the conditional post-measurement state.

Actual experimental settings are described in the Davisson–Germer experiment and the Franck-Hertz experiment. The former scans collected electric current as an electron beam scatters from prepared nickel; changing direction, accelerating voltage and surface preparation probes electron diffraction. The latter uses acceleration through mercury vapor and a retarding collector to infer discrete energy losses from current minima, with vapor density and retarding settings as controls. Angular resolution and surface backgrounds limit the first inference; initial energy spread and collision positions limit the second. These linked experiments supply the preparation, apparatus and readout; the examples below are original calculations, not Born's measured data.

A boryl ligand attaches through boron as an X-type ligand. In ionic electron counting it is formally anionic and contributes two electrons. It is distinct from a neutral borylene and from attachment to an entire borazine pi face.

In 1924–1925, Walther Bothe and Hans Geiger tested event-level Compton scattering correlations against the statistical conservation proposed by BKS theory.^[ref]

Needle Geiger counters shared a wall. X-rays passed beside it through hydrogen; recoil electrons registered in one counter and scattered photons triggered electron release in the other. Photographic records resolved approximately $10^{-4}\mathrm{s}$.^[ref]

The coincidence excess matched paired-emission expectations. Bothe distinguishes a later directional test from this timing experiment.^[ref]

The two channels have different physical roles. The first responds to the local recoil electron's ionization. The second converts a scattered photon into an electron response, as in the photoelectric effect. A large counter signal is an amplified indication of an interaction, not a direct measurement of the incoming photon's energy. The detection efficiency and accepted directions determine how many genuine partners can be seen. Consequently, failure to record every partner is compatible with paired production.

The relevant distinction is between an average balance and a balance in each event. For an initially stationary free electron, Compton recoil kinematics givesHere $E,E^{\prime }$ are incident and scattered photon energies, $T_e$ is recoil kinetic energy, and ${\mathbf{p}}_e$ is electron momentum. These are paired constraints, not just equations between averages over unrelated detections. Summing correctly paired events preserves the equations; throwing away the pair identities loses information. Two samples can have identical separate energy distributions while differing in their joint probability distribution.

The experiment therefore supplied evidence for the event association required by photon recoil, rather than weighing both outgoing energies in every event. A statistically established excess above accidental coincidences matters; one near-simultaneous pair cannot establish the association. It also cannot by itself settle every question about wave-particle duality.

Preparation, readout and background checks address different failure modes. The direct beam must avoid the second counter, and the counter boundary must prevent a recoil electron from masquerading as its own photon partner. Separate-channel rates constrain accidental coincidences; common disturbances or electrical coupling could instead create a correlated instrumental background. A useful diagnostic is to displace one record in time: under a stationary process assumption, unrelated-pair expectations persist while prompt partners separate. This is the explanatory control modeled below, not an assertion that Bothe performed that particular record-shifting procedure. A background estimate and a statistical comparison are essential to the inference.

The existing coincidence circuit article explains electronic selection, and accidental coincidence derives the accepted-window background. Here pairing can be understood directly from the two recorded time sequences, without imposing a modern detector trigger on the historical readout.

A classical Thomson scattering frequency window for a harmonically bound electron requires ${\omega }_0\ll \omega \ll {\omega }_C$, with ${\omega }_0=\sqrt{k/m}$ and recoil scale ${\omega }_C=m{c}^2/\hslash$. There is no parametrically separated window if ${\omega }_0\gtrsim {\omega }_C$, equivalently $k\gtrsim m^3{c}^4/{\hslash }^2$. This is a scale comparison, not a prediction of a sharp transition between scattering mechanisms.

At that binding scale, an oscillator quantum has energy comparable with the electron rest energy. The nonrelativistic Lorentz oscillator model is then inadequate, so a physical cross-section requires a relativistic quantum description. Compton scattering itself does not switch on discontinuously at $\hslash \omega =m{c}^2$; its recoil corrections become substantial near that scale and tend to the Thomson scattering limit at low photon energy for a free target.