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For spin-degenerate electrons at atomic spacing $a$, alternate the hopping amplitudes $-(t+\delta )$ and $-(t-\delta )$, with $t>0$ and $|\delta |<t$. The doubled primitive unit cell gives an off-diagonal Bloch Hamiltonian entry $-(t+\delta )-(t-\delta )e^{-2ika}$. Its two eigenvalues are the displayed energy bands, for $|k|\le \pi /(2a)$. The band gap at the edge of the Brillouin zone is $4|\delta |$. At half filling there are two electrons per doubled primitive unit cell, so the lower band is full and the upper empty. This explicit model realizes the period-doubling form of a Peierls distortion.

An ortho-dioxygen aromatic framework that can bind through both oxygen atoms in several redox states. The semiquinonate monoanion has a ligand-centred unpaired electron, whereas the catecholate dianion has a filled-shell configuration. non-innocent ligand behaviour means metal oxidation state cannot be inferred from an overall complex charge without specifying the ligand redox state.

At either displayed wave vector, the three complex hopping amplitudes add to zero: $1+2\cos (2\pi /3)=0$. Therefore the energy bands touch. There are two inequivalent electronic valleys; other equivalent corners are related by reciprocal-lattice translations. Setting $\mathbf{k}={\mathbf{K}}_{\tau }+\mathbf{q}$ and retaining first-order terms giveswhere ${\sigma }_x,{\sigma }_y$ are Pauli matrices. For example, at ${\mathbf{K}}_{+}$, $f\simeq -3b(q_x-i{q}_y)/2$. The matrix has the form of a massless two-dimensional Dirac Hamiltonian. Here $v_F$ is a material-model velocity, rather than the vacuum speed of light; this description does not change an electron's vacuum rest mass. It describes the dispersion relation of electronic excitations near a band crossing.

direct electron detection records incident electrons in a sensor without first converting them through a scintillator into an optical image. Fast readout permits counting and movies that support motion correction.

In On the Constitution of Atoms and Molecules, Part I (1913), pp. 8–9, Niels Bohr interprets existing hydrogen discharge and astronomical spectra. Electrical discharge in a vacuum tube removes electrons; recombination supplies spectral lines. This is a theoretical interpretation, without a reported spectrograph of his own.^[ref]

An actual optical obstacle appears in Theodore Lyman's measurements: hydrogen discharge light entered an evacuated spectrograph directly through a slit, avoiding a transmitting window. A concave reflection diffraction grating dispersed the light onto photographic film. Pumping preserved the vacuum; a second slit supplied a known spectral displacement for wavelength calibration. Gas contamination and vacuum leaks limited the measurement. This apparatus resolved ultraviolet spectral lines; it did not image orbits.^[ref]

For a separate original calibration exercise, suppose an idealized spectrograph returns synthetic vacuum line positions $x_n={\lambda }_{n\to 2}+b$ for $n=3,4,5,6$, with chosen additive offset $b=0.200\mathrm{nm}$. Assume its dispersion scale is already known and only this offset is uncertain. Compare the finite-mass prediction with the fixed-nucleus prediction ${\lambda }_n^{(\infty )}=(\mu /m_e){\lambda }_{n\to 2}$. Calibrate each model using only the $3\to 2$ line: $\hat{b}=x_3-{\lambda }_3^{\mathrm{model}}$. Then hold that offset fixed and test the other lines with residualsFor the finite-mass model every $d_n=0$. The fixed-nucleus model has $\hat{b}=0.55733\mathrm{nm}$ and residuals $d_4=-0.09264\mathrm{nm}$, $d_5=-0.12100\mathrm{nm}$, $d_6=-0.13400\mathrm{nm}$. Algebraically $d_n=(1-\mu /m_e)({\lambda }_n-{\lambda }_3)$: one additive calibration cannot absorb a multiplicative change of the wavelengths. Allowing an unknown multiplicative dispersion factor as well would make the two models degenerate in this exercise, so independent scale calibration matters.

A four-vertex model proves the essential gauge invariance without solving an electron scattering problem. Use two directed paths $S\to U\to D$ and $S\to L\to D$. Assign an edge phase $a_{ij}$ and a complex unit link $U_{ij}=e^{i{a}_{ij}}$ to each directed edge, with $U_{ji}=U_{ij}^{-1}$. These links represent discretized magnetic line integrals multiplied by $q/\hslash$. Under a gauge transformation with arbitrary vertex phases ${\chi }_i$,Both path products acquire the same factor $e^{i({\chi }_D-{\chi }_S)}$. Their gauge-invariant ratio is the closed-loop holonomyEvery intermediate vertex phase cancels. For any fixed nonnegative real amplitudes $a,b$, the interference quantity $|a{P}_1+b{P}_2{|}^2$ is unchanged too. A gauge-dependent open-path phase consequently cannot be interpreted as an observable by itself.

For an exact example, measure phases in units of $\pi$ and order edges as $(SU,UD,SL,LD)$. Choose edge phases $(0,0,0,1/2)$ and vertex phases $({\chi }_S,{\chi }_U,{\chi }_L,{\chi }_D)=(1/3,-1/4,2/5,1/7)\pi$. The transformed edge phases areThe upper and lower sums become $-4\pi /21$ and $13\pi /42$; their difference remains $-\pi /2$. With amplitudes $a=1$, $b=1/2$, the intensity remains $5/4$. Sweeping one physical loop phase through $2\pi$ instead produces extrema $9/4$ and $1/4$, agreeing with the interferometric visibility example above.

The integrated electron number density along the line of sight, conventionally quoted in $\mathrm{pc}\mathrm{c}{\mathrm{m}}^{-3}$. A dispersion measure is a column, not a unique distance: converting it into distance requires a model of the interstellar medium. A clumpy nearby column and a longer dilute column can give the same pulsar dispersion delay.

A normalized product of two identical bonding orbitals built from separated orthogonal centers has finite weight in configurations with both electrons on one center. Repulsion penalizes that ionic weight even when hopping disappears. Mixing configurations can instead approach one electron per center while preserving the required spin and exchange symmetries.

A disulfide bond is the single covalent bond between the two adjacent sulfur chemical atoms in a disulfide, conventionally written $\mathrm{R}-\mathrm{S}-\mathrm{S}-{\mathrm{R}}^{\prime }$. Here $R$ and $R^{\prime }$ stand for the attached organic groups; they may belong to the same molecule or to initially separate molecules. A disulfide bond is a chemical bridge, rather than a hydrogen bond or a mere contact between nearby groups. Two thiols, $\mathrm{R}-\mathrm{SH}$ and ${\mathrm{R}}^{\prime }-\mathrm{SH}$, can supply its two sulfur endpoints. ^[ref]

The stoichiometry of forming a disulfide bond is especially revealing. Written as an oxidation half-reaction, it isThe reverse reduction half-reaction consumes two electrons and two hydrogen ions to restore two thiols. Both sides contain two sulfur chemical atoms; the two hydrogen chemical atoms have moved into the two hydrogen ions, and the net electric charge remains zero because $+2-2=0$. Thus one disulfide bond corresponds to two thiol groups and two transferred electrons. These half-reactions describe redox bookkeeping, not a complete standalone chemical reaction: an accompanying reactant must accept or supply the electrons, and pH affects the actual protonation of the groups. Mercaptoethanol can promote reduction through exchange of disulfide bonds, but adequate excess and appropriate conditions are needed; adding it does not by itself prove complete reduction. ^[ref]

In a protein disulfide bond, the endpoints are normally the amino acid side chains of two cysteine amino acid residues. An intrachain protein disulfide bond closes an intramolecular bridge within one polypeptide chain; an interchain protein disulfide bond connects different polypeptide chains, and can arise by intermolecular crosslinking of initially separate molecules. Forming or reducing these bridges changes covalent bond connectivity and can change protein structure, while preserving the order of amino acid residues and the peptide bonds in the protein backbone. A complete chemical account of protein primary structure includes crosslinks as well as sequence, so preservation of sequence does not imply preservation of every covalent bond. Mature insulin has two interchain protein disulfide bonds and one intrachain protein disulfide bond. ^[ref]

A bridge is useful because it constrains which parts of a polypeptide chain can move apart without breaking a covalent bond. Its placement therefore matters as well as its presence. The ribonuclease A refolding experiment provides the existing experimental example: breaking and reforming four protein disulfide bonds can accompany loss and recovery of function. Counting four bridges alone cannot tell whether their endpoints have the native pairing. This is an attractive connection between simple chemical counting and the richer problem of protein folding.

How can an experiment reveal an initially hidden disulfide bond? Ellman's reagent, also called DTNB, measures free thiols by producing a yellow absorbing species. Its reactive disulfide bond undergoes exchange with the conjugate base $\mathrm{R}-{\mathrm{S}}^{-}$ of a thiol, producing a mixed disulfide attached to $R$ and one released 2-nitro-5-thiobenzoate species, abbreviated TNB. At the assay pH, the strongly colored form is the dianion $\mathrm{TN}{\mathrm{B}}^{2-}$: one negative charge is on sulfur and the other is on the carboxylic acid-derived $\mathrm{CO}{\mathrm{O}}^{-}$ group. With excess DTNB and complete reaction, one released TNB reports one reacted thiol. George L. Ellman's 1959 paper introduced the tissue assay; it was subsequently reexamined quantitatively. ^{[ref] [ref]}

The modern apparatus can be simple: tubes, pipettes, a buffer solution, cysteine standards, DTNB, and an instrument that measures absorbance through a cuvette at wavelength $412\mathrm{nm}$. The Pierce protocol uses $50\mu \mathrm{L}$ of a $4\mathrm{mg}\mathrm{m}{\mathrm{L}}^{-1}$ DTNB stock, $2.50\mathrm{mL}$ of buffer solution, and $250\mu \mathrm{L}$ of sample. Its buffer solution is $0.1\mathrm{mol}{\mathrm{L}}^{-1}$ sodium phosphate at pH 8.0 with a metal-binding additive. After 15 minutes at room temperature, measure absorbance against a buffer solution plus DTNB blank; run known cysteine concentrations in parallel. The observable is increased yellow color and increased blank-corrected absorbance as more accessible thiols react, rather than direct imaging of disulfide bonds. These are published protocol conditions, not measurements performed for this article. ^[ref]

For a before/after test, split a purified protein sample into matched aliquots. Measure accessible free thiols in one aliquot; subject the other to complete reduction of its disulfide bonds, remove the reducing reagent, then measure the resulting thiols with fresh DTNB. Process the unreduced aliquot through a matching cleanup. Normalize to recovered protein and known dilution, since cleanup can change both. To estimate all reducible protein disulfide bonds, expose buried groups using matching protein denaturation conditions in both aliquots and calibrate in that same chemical solution; a native-accessibility measurement answers a different question. A larger corrected reading after reduction supports conversion of previously unreactive groups into detectable thiols. Attribution to disulfide bonds requires excluding other reducible sulfur modifications. This is a proposed comparison based on the detection principles, not a reconstruction of Ellman's original tissue measurements. ^[ref]

Under the Beer-Lambert law, the blank-corrected absorbance is $A=\epsilon lc$, where $l$ is the cuvette path length, $c$ is the released TNB concentration, and $\epsilon$ is its molar absorption coefficient. Riddles, Blakeley and Zerner measured ${\epsilon }_{412}=14150\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}\mathrm{c}{\mathrm{m}}^{-1}$ in dilute aqueous salt chemical solutions, confirming the value independently using cysteine. It must not be applied indiscriminately to other solvents or protein denaturation mixtures. ^[ref] For aliquots compared on the same final volume and protein concentration basis, the ideal estimate isThe factor of two follows from the reduction stoichiometry; ${\bar{n}}_{\mathrm{S}-\mathrm{S}}$ is the number of disulfide bonds per protein molecule, or an average for a mixed population. If the aliquots have different dilutions or recoveries, convert each reading to a common sample basis before subtracting.

As a hypothetical calculation, choose $c_{\mathrm{protein}}=10\mu \mathrm{mol}{\mathrm{L}}^{-1}$ in each final cuvette, two free thiols and two disulfide bonds per protein molecule, and $l=1\mathrm{cm}$. Before reduction, $c_{\mathrm{SH}}=20\mu \mathrm{mol}{\mathrm{L}}^{-1}$ gives $A=0.283$; complete reduction adds four thiols per protein molecule, so $c_{\mathrm{SH}}=60\mu \mathrm{mol}{\mathrm{L}}^{-1}$ gives $A=0.849$. The difference $0.566$ gives $c_{\mathrm{S}-\mathrm{S}}=20\mu \mathrm{mol}{\mathrm{L}}^{-1}$ and ${\bar{n}}_{\mathrm{S}-\mathrm{S}}=2$. The published tube volumes imply a dilution factor $2.80/0.250=11.2$; reversing that dilution multiplies both thiol and protein concentrations by 11.2 and leaves the bridge count per protein molecule unchanged.

Controls determine whether that arithmetic has chemical meaning. Use fresh cysteine standards and a processed blank without protein to detect color generated by residual reducing reagent. Even a reducing reagent containing no thiols can interfere: Han and Han showed that tris(2-carboxyethyl)phosphine, abbreviated TCEP, reduces DTNB to two TNB equivalents per TCEP. ^[ref] In the hypothetical model, $5\mu \mathrm{mol}{\mathrm{L}}^{-1}$ residual TCEP in only the reduced cuvette would add $A=0.1415$ and falsely add $0.5$ bridge per protein molecule. A stable endpoint and agreement after a further reduction treatment help check completion. A sample without DTNB checks its own color; cloudy samples can also attenuate light without producing TNB. Match pH, temperature, accessibility and calibration chemical solution, and minimize delay that permits reoxidation. If two already-free but buried thiols become accessible only in the reduced aliquot, they mimic one new disulfide bond per protein molecule. Finally, bulk absorbance supplies a count under these assumptions: it does not locate the cysteine endpoints, distinguish intrachain from interchain bridges, or establish the native pairing of a protein disulfide bond.

A double heterostructure laser electrically pumps its central layer for stimulated emission. Band offsets confine charge carriers; refractive index contrast confines the optical normal mode.^[ref]

For an original bookkeeping model, let $R$ be the injected pair rate per unit area, $d$ the active-layer thickness, $N$ its excess pair number density, $\tau$ a recombination lifetime, and ${\tau }_e$ an escape time. Before appreciable stimulated emission, supposeChoose two otherwise identical devices with ${\tau }_e=\tau$ and ${\tau }_e=9\tau$. The resulting steady state populations are $R\tau /(2d)$ and $9R\tau /(10d)$: improved confinement raises the population by a factor $1.8$ at fixed injection. This linear model assumes equal excess electron and electron hole densities, fixed lifetimes, and no laser gain clamping.

Optical confinement enters separately. For a chosen optical normal mode, define $0<\Gamma \le 1$ as its gain-overlap factor with the active layer. With material optical gain $g$, internal loss ${\alpha }_i$, and mirror loss ${\alpha }_m$, a simple laser threshold condition isFor stipulated losses ${\alpha }_i+{\alpha }_m=3$ in inverse-length units, $\Gamma =1/2$ requires $g=6$, whereas $\Gamma =3/4$ requires $g=4$. This calculation varies optical overlap only; the preceding population calculation varies carrier escape only. Their combination explains the usefulness of controlling both the energy band profile and the optical normal mode.

Herbert Kroemer proposed the double heterostructure laser in 1963; he did not participate in its 1970 experimental realization.^[ref] Zhores I. Alferov describes his group's independent 1963 proposal and later experimental program.^[ref]

Two electrons of opposite spin occupying the same spatial orbital. The mean occupation product measures its probability at a specified site. In the two-electron dimer, the sum over sites is also the probability of any doubly occupied site, since two sites cannot both be doubly occupied.

The ground-state probability of both electrons occupying one site is $P_{\mathrm{double}}=\frac{1}{2}[1-U/\sqrt{U^2+16t^2}]$. It equals the derivative of the relative ground energy with respect to $U$. It tends to zero for $|t|/U\to 0$ and to $1/2$ for $|t|/U\to \infty$.

A drift chamber uses the drift times of ionization electrons in a gas to infer positions along charged particle tracks. In a known magnetic field, track reconstruction can estimate momentum and the sign of electric charge.

The drift velocity is a charge carrier population's mean velocity relative to its material's rest frame. It describes the net transport superposed on microscopic motion. For a single species, current density is $\mathbf{J}=nq{\mathbf{v}}_d$; for electrons, $q=-e$ reverses the direction. It is distinct from the propagation speed of an electrical disturbance. ^[ref].

A motivating laboratory realization is The Hall Effect and Conductivity in Germanium by Rex Tayloe. The handout specifies a germanium sample holder with temperature and magnetic field control, a low-voltage supply, digital meters, and a current-reversing switch. Measure the sample dimensions, map the contacts, and establish the directions of electric current, magnetic field, and transverse voltage. At fixed temperature and zero magnetic field, sweep positive and negative electric currents and record longitudinal electric potential differences. A straight voltage-versus-electric current relation tests Ohm's law; sample geometry converts its slope to electrical resistivity. For the Hall effect, minimize the zero-field transverse offset and measure transverse voltage at all four combinations of $\pm I$ and $\pm B$, repeating for several electric currents. The readouts test longitudinal proportionality, transverse polarity, and transverse proportionality to electric current. These are protocol instructions, not reported measurement outcomes.^[ref]

For interpreting such measurements, an ideal uniform bar of wire cross-sectional area $A$ with longitudinal sensing contacts separated by $L$ gives $\sigma =LI/(A{V}_L)$. Four-terminal sensing suppresses voltage drops in current contacts when the voltage meter draws negligible electric current; it does not fix uncertain dimensions or a nonuniform current density. Reverse electric current and use $[V_L(+I)-V_L(-I)]/2$ to reject a constant voltage offset. Keep temperature stable, allow the current density to settle, and reduce electric current to test for Joule heating. Those are controls for interpreting the material coefficient, rather than reasons to assume every measured curve is linear.

For the transverse measurement define $\mathbf{J}=J_x\hat{\mathbf{x}}$, $\mathbf{B}=B_z\hat{\mathbf{z}}$, width $w$ along $y$, and thickness $d$ along $z$. Define $V_H=V(y=-w/2)-V(y=+w/2)=E_{y}w$, so $J_x=I/(wd)$ givesExtract the part odd in both electric current and magnetic field usingThis combination rejects a constant offset and longitudinal pickup even in magnetic field. It cannot reject another signal with the same two reversal symmetries. Check field polarity and magnitude independently; a sign error in the contact map changes the inferred Hall coefficient.

For one Drude model species in transverse open circuit, the steady Lorentz force balance is $0=q(\mathbf{E}+{\mathbf{v}}_d\times \mathbf{B})-m{\mathbf{v}}_d/\tau$ with $v_{d,y}=0$. Its $y$ component gives $E_y=v_{d,x}{B}_z$. ConsequentlyLongitudinal electrical conductivity and the Hall coefficient therefore separate number density from charge carrier mobility in this single-species model. Inferring a momentum relaxation time additionally requires an appropriate mass. A constant-field longitudinal measurement alone cannot supply all three.

A Hall coefficient is not universally the reciprocal of signed number density. For example, adding independent electron and electron hole contributions with densities $n,p$ and positive charge carrier mobilities ${\mu }_e,{\mu }_h$ in the weak-magnetic field limit givesThe Hall coefficient can vanish even while electrical conductivity is positive, and its sign weights charge carrier mobility as well as number density. The germanium handout explicitly asks students to track possible transverse sign changes with temperature; it does not establish a universal single-carrier interpretation. Real electronic band structure and unequal scattering also limit that interpretation.^[ref]

Switch on a constant electric field ${\mathbf{E}}_0$ at $t=0$. For prescribed initial current density $\mathbf{J}(0)$, the exact Drude model solution is the displayed formula. For electrons, the corresponding drift velocity isStarting from zero drift velocity, the current density reaches $1-e^{-1}\approx 0.632$ of its limiting value after one momentum relaxation time. Switching the electric field off at $T$ instead gives $\mathbf{J}(t)=\mathbf{J}(T)e^{-(t-T)/\tau }$ for $t\ge T$. A finite step of electric field changes the acceleration immediately but leaves drift velocity and current density continuous.

More generally, multiplying the current density equation by an integrating factor givesThis is the causal response of a first-order relaxation equation: only earlier electric fields enter the integral. It also shows why instantaneous Ohm's law is an approximation when the electric field changes appreciably during a momentum relaxation time.

Use physical electric fields $\mathbf{E}(t)=\mathrm{Re}[\hat{\mathbf{E}}{e}^{-i\omega t}]$ and current densities $\mathbf{J}(t)=\mathrm{Re}[\hat{\mathbf{J}}{e}^{-i\omega t}]$, with peak complex amplitudes. After the initial transient has decayed, substitution into the Drude model equation gives the complex conductivityHere the mobile species is electrons. With the alternative $e^{+i\omega t}$ convention, the denominator is $1+i\omega \tau$ and the imaginary part changes sign; the physical response is identical.^[ref]

For $E_x(t)=E_0\cos \omega t$ with $\omega >0$, the real-time current density isThus a positive argument of complex conductivity in the $e^{-i\omega t}$ convention means that current density lags the electric field. At $\omega \tau =1$, its real part and imaginary part are both ${\sigma }_0/2$, its modulus is ${\sigma }_0/\sqrt{2}$, and the lag is $\pi /4$.

For $\omega \tau \ll 1$, complex conductivity tends to the positive constant ${\sigma }_0$: current density nearly follows the electric field. For $\omega \tau \gg 1$ within this model,The current density amplitude decreases and its response becomes predominantly inertial. Average power absorbed per volume is $\overline{p}=\frac{1}{2}\mathrm{Re}\sigma |\hat{\mathbf{E}}{|}^2\ge 0$. These limits describe the material response to a specified local electric field, not the complete electromagnetic response of a finite specimen with contacts. The Drude response has its pole at $\omega =-i/\tau$, consistent with the decaying causal response of a first-order relaxation equation.

The simple Drude model does not predict the microscopic momentum relaxation time. It neglects electronic band structure and Fermi-Dirac statistics; a useful semiclassical extension replaces $m$ by an appropriate band effective mass, but neither a single scalar mass nor one constant momentum relaxation time describes every material. Several charge carriers, energy-dependent scattering, and transitions between electronic bands can spoil the simple Drude frequency response. A successful low-angular frequency fit does not justify extrapolation to arbitrarily large angular frequencies.

The Drude model explains finite electrical conductivity by balancing charge carrier acceleration in an electric field against loss of directed momentum through collisions. Its useful surprise is that continuous microscopic motion can produce a constant macroscopic current density: the applied electric field continually restores the directed momentum lost to the material. The simplest version has one mobile species, constant number density $n$, electric charge $q$, mass $m>0$, and momentum relaxation time $\tau >0$, in a stationary, homogeneous, isotropic material. At zero magnetic field, its mean drift velocity and current density obeyFor electrons, $q=-e$ with positive elementary charge $e$, so ${\sigma }_0=n{e}^2\tau /m$. The electron drift velocity points against a constant electric field, whereas conventional current points along it. Random microscopic velocities can have zero mean without being zero individually; drift velocity is this population mean, rather than each particle's speed.^[ref]

The same first-order linear differential equation predicts both Drude current relaxation after switching an electric field and the Drude frequency response to a sinusoidal electric field. Together they let a measured electrical conductivity be interpreted in terms of a density, a mass, and a time scale; a single constant-field measurement determines only their combination $n{q}^2\tau /m$.

The original 1900 paper Zur Elektronentheorie der Metalle by Paul Drude sought a common account of electrical transport and heat transfer using mobile charged particles and ideas from the kinetic theory of gases. Its introduction, especially printed pp. 568–569, explains that comparing different transport phenomena could constrain the populations of free and bound charged particles. The modern momentum relaxation time formulation above is a useful presentation of the model, not a verbatim translation of the original notation or its historical assumptions about mass.^[ref]

The correlated instantaneous avoidance of electrons beyond a single Slater determinant. It can remain important even when one configuration dominates, as for an isolated closed-shell helium atom; it is distinct from the near-degeneracy problem of static electron correlation.

Ernst Ruska and Max Knoll developed electron optics while improving the writing spot of a cathode-ray oscilloscope.^[ref] A structured object tests image formation more strongly than a single bright spot: each feature must retain its own image position.

On 7 April 1931, a cold cathode source and two short magnetic coils imaged aperture grids at $50\mathrm{kV}$. A platinum grid preceded the first coil; a bronze grid preceded the second. The fluorescent screen showed one-stage grid images and the first grid reimaged through both stages. Comparing these patterns tested successive image formation, not merely beam concentration. Heating could destroy even metal grids. This prototype did not establish resolution beyond an optical microscope.^[ref]

In 1933, Ruska measured $3\mathrm{mm}$ focal lengths at $75\mathrm{kV}$ with a pole-piece lens, then built an instrument giving $12000$-fold magnification.^[ref] TU Berlin dates images surpassing conventional optical microscope resolution to 1933.^[ref]

A fluorescent screen converts the spatial electron distribution into visible light.^[ref] A collector reporting only total electric current would discard that spatial pattern. In a two-stage image relay, the first lens makes an intermediate real image; the second treats that image as its object. The diagram below illustrates this geometry with invented distances and an ideal beam.