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A diffraction order is the integer $m$ labeling a principal constructive interference direction of a diffraction grating: adjacent elements differ in path length by $m$ wavelengths. In the normal-incidence transmission geometry described above,Positive and negative diffraction orders lie on opposite sides of the forward normal. The condition expresses the range of the sine of a real angle, so it also limits the propagating diffraction orders.

For $\lambda =600\mathrm{nm}$ and $d=2\mu \mathrm{m}$, $\sin {\theta }_m=0.3m$. Taking the inverse sine givesThese are seven allowed propagating diffraction orders. Order $m=4$ would require $\sin {\theta }_4=1.2$, and $m=-4$ would require $-1.2$, so neither has a propagating far-field direction. The allowed diffraction orders are equally spaced in $\sin \theta$, while their actual angles become farther apart toward grazing propagation.

Each principal peak has finite width. For the ideal uniform coherent array, the nearest zeros around order $m$ obeyThus the one-sided distance from the order direction to the first zero is $\Delta (\sin \theta )=\lambda /(Nd)$. Near the first diffraction order in our example, $N=10$ puts the zeros at $0.27$ and $0.33$, while $N=40$ puts them at $0.2925$ and $0.3075$. The respective one-sided widths are $0.03$ and $0.0075$; the full zero-to-zero widths are $0.06$ and $0.015$. These are calculated coherent peak widths, with the slit envelope omitted, and do not describe a measured instrument response. ^[ref].

Four times as many illuminated slits therefore give a fourfold narrowing in this coordinate without moving the diffraction order. That sharper directional selection helps distinguish nearby wavelengths in a spectrograph. Actual performance also depends on illumination, optical coherence, and source width; the separate slit-limited resolving power of a grating describes the contribution of the entrance slit.

Directional emissivity is the ratio of emitted radiance to blackbody radiance at the same temperature in a specified outgoing direction. It can be total over wavelength or additionally spectrally resolved. Normal-direction emissivity is not automatically total hemispherical emissivity: the latter integrates all directions with the projected area weighting. A diffuse thermal emitter removes this directional dependence of emitted radiance.

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 dispersion relation specifies which angular frequencies and wavevectors a linear wave model admits. Fix the plane wave conventionHere $a$ is a constant complex amplitude, $\mathbf{k}$ is the wavevector, and $\omega$ is the angular frequency. Substitution into a constant-coefficient linear partial differential equation replaces ${\partial }_t$ by $-i\omega$ and $\nabla$ by $i\mathbf{k}$, turning the wave problem into an algebraic condition. A system can have several roots; $\omega =\omega (\mathbf{k})$ means one chosen branch, rather than a promise that the full dispersion relation is single-valued. For real $\mathbf{k}$, $\mathrm{Im}\omega >0$ means temporal growth in this convention, while $\mathrm{Im}\omega <0$ means damping. The real-branch discussion of phase velocity and group velocity below assumes negligible loss.

For the one-dimensional wave equation $u_{tt}=c^2{u}_{xx}$, the dispersion relation is ${\omega }^2=c^2{k}^2$. The branch $\omega =ck$ carries right-moving travelling waves for $c>0$; both its phase velocity and group velocity are $c$. One may instead use the positive-angular frequency branch $\omega =c|k|$, with right-moving components at $k>0$ and left-moving components at $k<0$. A real wave pairs each $(k,\omega )$ with $(-k,-\omega )$; that conjugate pair represents the same travelling pattern. Branch and sign choices therefore matter when interpreting a dispersion diagram.

A particularly tangible dispersive example is a small-amplitude surface gravity wave on still water of uniform depth $h$. Neglect surface tension and viscosity, assume incompressible flow and irrotational flow, and impose a flat impermeable bottom. With $k>0$ and positive angular frequency,Here $g$ is gravitational acceleration and $\tanh$ is the hyperbolic tangent. Deep water means $kh\gg 1$: $\omega \simeq \sqrt{gk}$ and longer wavelengths have larger phase speed. Shallow water means $kh\ll 1$: $\omega \simeq \sqrt{gh}k$, an approximately nondispersive wave. Thus depth relative to wavelength, rather than depth alone, determines whether crests and wave packets travel together.^[ref]

The dispersion diagram below also includes the ideal deep-water capillary wave limit $\omega =\sqrt{\gamma /\rho }{k}^{3/2}$, where $\gamma$ is surface tension and $\rho$ is mass density. This limit requires the surface tension restoring term to dominate gravity; it reverses the trend of the surface gravity wave, giving larger phase speed at shorter wavelength. The combined gravity-capillary wave has ${\omega }^2=gk+(\gamma /\rho )k^3$.^[ref]

The curves use dimensionless normalizations chosen separately for each model; they are analytical diagrams, not experimental data. Their beauty is that a single dispersion relation encodes several observable behaviours: its value gives the angular frequency, its origin-to-point slope gives the one-dimensional phase velocity, its tangent slope gives the group velocity, and variation of that tangent slope predicts distortion of a wave packet. A straight line through the origin describes a nondispersive wave; an affine branch $\omega =ck+b$ with $b\ne 0$ has constant group velocity but phase velocity $c+b/k$, so those properties must be distinguished.

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 doped semiconductor at frequencies far below its bound-electron resonances can haveFor a weakly absorbing nonmagnetic half-space adjacent to vacuum, the normal-incidence optical reflectance is minimized near refractive-index matching, ${ϵ}_r\simeq 1$. When ${ϵ}_b>1$ and ${\gamma }_c$ is small this gives ${\omega }_{\min }\simeq {\omega }_{pc}/\sqrt{{ϵ}_b-1}$. The zero of the real part of ${ϵ}_r$, near ${\omega }_{pc}/\sqrt{{ϵ}_b}$, is a distinct frequency. Increasing the carrier density shifts the minimum to shorter wavelength if the carrier effective mass and background response remain fixed. With appreciable loss, the minimum must be found from the full complex Fresnel equations.

A laser whose gain medium contains dye molecules. Its tunable wavelength permits laser spectroscopy of different spectral lines.

Electromagnetic radiation is a propagating disturbance of the electric field and magnetic field that carries energy and momentum away from its source. It can be a sustained electromagnetic wave or a finite wave packet: periodic oscillation is not required. In vacuum the disturbance travels at the speed of light. A receiver responds to fields arriving now from an earlier source event, as expressed by retarded time. This is how electrical activity at one place can deliver energy and information elsewhere without a connecting wire.

Stored electromagnetic energy alone does not establish outgoing electromagnetic radiation. A stationary electric charge in electrostatics, or a steady localized electric current in magnetostatics, can store electromagnetic energy with zero Poynting vector. Static combinations can have circulating energy flux without net outward power. Near a changing source, reactive energy can return later. The near and far field distinction separates these contributions from outgoing radiation zone fields.^[ref]

A single traveling vacuum plane electromagnetic wave obeying the Maxwell equations has $\mathbf{B}=\hat{\mathbf{k}}\times \mathbf{E}/c$. Its Poynting vector gives the energy density and flux of a plane electromagnetic wave:Here $I$ is wave intensity; ${ϵ}_0$ and ${\mu }_0$ are vacuum permittivity and vacuum permeability. For a monochromatic wave with linear polarization and peak amplitude $E_0$, squared cosine averages to $1/2$. Pulses require the instantaneous Poynting vector; arbitrary incident-and-reflected superpositions need not obey this relation.^[ref]

The electromagnetic momentum density is $\mathbf{g}=\mathbf{S}/c^2$. Completely absorbing a unidirectional vacuum pulse of energy $U$ transfers momentum $U/c$; see radiation pressure on a perfectly absorbing surface. A calculated $1.0\mathrm{mJ}$ pulse transfers $3.34\times 10^{-12}\mathrm{kg}m/s$. Symmetric emission can lose energy without net recoil because opposite momenta cancel.^[ref]

The direction of emission matters as much as its total power. For an oscillating electric dipole moment along $z$, the angular distribution of electric dipole radiation is strongest across the axis and vanishes along it. Its origin is geometric: the electric dipole radiation field is the transverse projection of the second time derivative of the electric dipole moment, so its magnitude contains $\sin \theta$. Squaring this electric field to obtain wave intensity gives ${\sin }^2\theta$. The electric-dipole approximation for radiation requires a source much smaller than its wavelength and observation in the radiation zone. ^[ref]

For example, the power per solid angle at $\theta =30^{\circ }$ is one quarter of its equatorial value, while the electric field amplitude is one half. To normalize the angular distribution to total power, use spherical coordinates with $d\Omega =\sin \theta d\theta d\varphi$:The integral follows by substituting $x=\cos \theta$, giving ${\int }_{-1}^1(1-x^2)dx=4/3$. This is a probability-like normalization over direction, with units of inverse steradian; it does not make the dipole an isotropic radiator. The same angular integral underlies the Larmor formula for nonrelativistic accelerated electric charge.

Detection requires the incident electromagnetic field to do something observable. An antenna develops an electrical response; a thermopile converts absorbed energy into a thermal and then electrical signal; a photodetector can respond through photon absorption. Hertz spark transmission and detection experiment makes this conversion visible as a tiny spark. Detector orientation, sensitivity, and surrounding reflections all matter: failure to see a signal does not prove that no electromagnetic radiation reached the receiver.

Let an Electron start with negligible kinetic energy and move through an increase $U>0$ in electric potential. Its electric charge is $q=-e$, where $e>0$ is the elementary charge, so the decrease in potential energy is $eU$. In a field-free region after acceleration,This is an Electron formula, not a universal conversion from voltage to wavelength. It assumes negligible initial kinetic energy, negligible losses, and $eU\ll m_e{c}^2$, where $c$ is the speed of light. It describes the de Broglie wavelength outside the crystal; the potential energy inside a material can change the local kinetic momentum.

For $U=54\mathrm{V}$, take $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$, $e=1.602176634\times 10^{-19}\mathrm{C}$, and $m_e\simeq 9.1093837\times 10^{-31}\mathrm{kg}$. The rounded Electron mass is consistent with the 2022 recommended constants.^[ref]The electronvolt is an energy unit: $54\mathrm{eV}$ must be converted to joules before combining it with $m_e$ in SI units. Dimensional analysis checks both steps:The resulting velocity is about $4.3584\times 10^6\mathrm{m}{\mathrm{s}}^{-1}$, or $0.01454c$, so the nonrelativistic assumption is consistent. For a larger accelerating voltage, the relativistic energy-momentum relation gives $pc=\sqrt{K(K+2m_e{c}^2)}$, with $K=eU$, while $\lambda =h/p$ still holds. At $54\mathrm{V}$ the relativistic wavelength is shorter by about $0.00264\%$.

Explicitly, the exact special relativity formula and its nonrelativistic quantum mechanics limit areHere $K=eU$ is the gained kinetic energy; the total energy is $E=m_e{c}^2+K$. At $80\mathrm{kV}$, the nonrelativistic quantum mechanics expression gives $4.336\mathrm{pm}$, about $3.84\%$ larger than the exact $4.176\mathrm{pm}$. This difference matters when using electron diffraction ring positions to infer lattice plane spacings.

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.

Ellman's reagent is 5,5'-dithiobis(2-nitrobenzoic acid), abbreviated DTNB, a disulfide used to quantify accessible thiols. Reaction releases the colored TNB species measured by absorbance at wavelength $412\mathrm{nm}$. The disulfide bond experiment above explains why a thiol measurement can become an indirect bridge count only after controlled reduction. ^[ref]

Built by André Baranne, Michel Mayor, Didier Queloz and colleagues, the ELODIE fiber-fed spectrograph carried light from Haute-Provence's 1.93-metre telescope through optical fibers. Its echelle grating dispersed light in high diffraction orders; an optical prism and grism cross-disperser separated them perpendicularly on a charge-coupled device. This cross-dispersed echelle spectrograph achieved spectral resolving power $R=42000$. A simultaneous thorium reference tracked instrumental drift. Wavelength calibration mapped detector positions to wavelengths; spectral cross-correlation combined many spectral lines into a fitted profile. A barycentric velocity correction removed the observer's motion.^[ref]

Why can a spectrograph measure a Doppler shift much smaller than one spectral resolution element? Resolution concerns separating neighbouring features, whereas locating the centre of a known profile uses information across that profile and across many spectral lines. For a chosen example, at $\lambda =500\mathrm{nm}$ a recession velocity of $60\mathrm{m}{\mathrm{s}}^{-1}$ givesAt $R=42000$, one spectral resolution element corresponds to $c/R\simeq 7.14\mathrm{km}{\mathrm{s}}^{-1}$, about $119$ times larger. A well measured centre can move by a small fraction of that width. More photons and more useful spectral lines improve the estimate, but do not remove wavelength calibration errors or stellar activity.

A simple mathematical picture of spectral cross-correlation makes the readout concrete. Define $x=c\log (\lambda /{\lambda }_0)$, with fixed reference wavelength ${\lambda }_0$, and let $d(x)$ be continuum-normalized line absorption. For small velocities, a common Doppler shift approximately translates $d(x)$. A positive absorption mask $m(x)$ then givesIf $d(x)=m(x-v_r)$, the maximum occurs at $u=v_r$; a convention correlating the received observed radiative flux instead produces a dip. A fitted correlation centre is an effective collective line position, not a claim that each individual spectral line has been resolved into a narrower feature. The lamp tracks instrumental drift, while the barycentric velocity correction removes the observer's changing line-of-sight velocity; neither removes changes in the star itself.