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Doppler-free saturation spectroscopy pairs a strong pump with an opposed weak probe. Shared populations produce a narrow Lamb dip within the Doppler broadening of a spectral line.

Laser spectroscopy uses a tunable laser to interrogate energy levels through photon absorption, stimulated emission or fluorescence. A narrow optical source is useful only if the sample and readout preserve the information: Doppler broadening can hide closely spaced spectral lines even when the source itself is narrow.

Absorption of a gamma ray into an excited nuclear level followed by re-emission. The absorber-frame photon energy must overlap the absorption line; nuclear recoil, Doppler broadening and the natural decay width affect resonance. This nuclear process differs from atomic resonance fluorescence used in quantum optics.

The closest components above differ by $\Delta {\nu }_{\min }={\mu }_{B}B/(2h)\simeq 7.0\mathrm{GHz}{\mathrm{T}}^{-1}B$. At $B=0.1\mathrm{T}$, their separation is about $0.70\mathrm{GHz}$. A visible spectral line near $500\mathrm{nm}$ would consequently require spectral resolving power of order $\nu /\Delta {\nu }_{\min }\simeq 9\times 10^5$. An ordinary low-resolution spectrometer would merge the components.

A suitable arrangement places an excited low-pressure strontium discharge or a transversely observed atomic beam between wide electromagnet pole pieces. Collect the emission perpendicular to the magnetic field, select the desired spectral line with a prefilter or monochromator, and send it through a high-finesse scanning Fabry–Pérot interferometer to a detector. A rotatable linear polarizer separates the two $\pi$ components from the four $\sigma$ components. Calibrate both the interferometer's free spectral range and the local magnetic field; take zero-field and several field-strength scans, and fit component positions against $B$.

The magnetic field should be uniform across the viewed volume, with its associated Zeeman splitting larger than the combined natural linewidth, Doppler broadening, pressure broadening and instrumental width, but much smaller than the separations of the relevant atomic fine-structure levels. A collimated transverse atomic beam, low collision rate and restricted collection angle reduce unwanted spectral line broadening. Avoid appreciable power broadening, resolve or model isotope and hyperfine structure, and check for self-absorption. A zero-field scan and the predicted linear scaling help distinguish actual Zeeman splitting components from neighbouring spectral lines.

Natural broadening is about $8\times 10^2$ times narrower than the thermal Doppler width and is negligible for this comparison. The supplied Doppler broadening formula is a half width at half maximum. Using the integer molecular mass $m=28m_{\mathrm{u}}$,For an excited-state population lifetime $\tau$, the radiating amplitude decays as $e^{-t/(2\tau )}$. Its Lorentzian line shape has natural linewidthEquivalently, compare full widths $2.11\mathrm{GHz}$ and $2.65\mathrm{MHz}$. Using the same linewidth convention in both expressions matters more than the small isotope-mass correction. This estimate assumes that the lower state's lifetime is long and leaves any pressure broadening as a separate contribution.

Using mass $7\mathrm{u}$ for the dominant lithium isotope gives a transition wavenumber ${\tilde{\nu }}_0=14925\mathrm{c}{\mathrm{m}}^{-1}$. Inserting the thermal speed into the stated Doppler broadening expression givesThe supplied formula is the Gaussian half width at half maximum; the corresponding full width is about $0.256\mathrm{c}{\mathrm{m}}^{-1}$. Meanwhile exponential population decay with lifetime $\tau$ produces a natural linewidth expressed as a full widthHere $c$ must be in $\mathrm{cm}{\mathrm{s}}^{-1}$ in the second expression. Equivalently the fine-structure separation is about $9.0$ GHz, the Doppler half width about $3.83$ GHz, and the natural linewidth expressed as a full width about $5.89$ MHz. The natural linewidth is therefore negligible compared with thermal broadening; low density also makes collision broadening less important.

The doublet is resolvable in principle, but only marginally against the Doppler width. The separation exceeds one Doppler full width, so two components can survive, although their profiles overlap strongly. The $J=3/2$ and $J=1/2$ upper levels have different statistical weights, so the weaker member need not make an equally conspicuous maximum. A spectral resolving power $R\gtrsim {\tilde{\nu }}_0/0.3\simeq 5\times 10^4$ is the basic instrumental requirement; a width well below $0.1\mathrm{c}{\mathrm{m}}^{-1}$ and adequate signal would make the test more convincing. An arbitrarily fine spectrometer cannot remove the physical Doppler overlap. This conclusion uses the stipulated thermal, optically thin picture; additional photospheric velocity fields or optical-depth effects would require more information.

The Rosseland mean opacity is the transport opacity appropriate to an optically thick region close to local thermodynamic equilibrium. If ${\kappa }_{\nu }$ is monochromatic transport opacity and $B_{\nu }(T)$ the Planck function, thenIt is a weighted harmonic mean: low-opacity frequency windows carry much of the diffusive flux. One must combine the monochromatic processes before averaging, rather than adding their separately averaged opacities.

In hot fully ionized material, electron-scattering opacity supplies an almost density- and temperature-independent contribution at fixed composition. In the nonrelativistic Thomson scattering limit, ${\kappa }_{\mathrm{es}}={\sigma }_T{n}_e/\rho \simeq 0.2(1+X)\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$. Free-free opacity, the absorption counterpart of thermal bremsstrahlung, comes from photon absorption during encounters between free electrons and ions. Its approximate mean obeys Kramers' opacity law, ${\kappa }_{\mathrm{ff}}\propto \rho T^{-7/2}$, with composition and quantum correction factors.

When ions retain bound electrons, bound-free absorption is photoionization; edges, occupations and ionization fractions strongly affect its frequency dependence. Bound-bound absorption produces absorption lines. Numerous metal lines can close low-opacity windows and produce important opacity features in partially ionized layers. Line populations depend on excitation as well as ionization, while Doppler broadening, pressure broadening and overlapping lines change the transport windows. Stimulated inverse processes must be included consistently with detailed balance and the local thermal radiation field.

In cooler hydrogen-rich atmospheres, negative hydrogen ion opacity from both bound-free and free-free processes is often important. Its strong temperature dependence helps determine convective surface relations. In still cooler layers, molecular bands, and at sufficiently low temperatures dust absorption/scattering, may be important. These processes cannot be represented by one Kramers power law over the whole density-temperature plane.

An opacity calculation therefore needs the chemical abundances, the equation of state, electron density and bound-state populations. Saha ionization equation and thermal excitation populations give useful dilute LTE estimates; high-density pressure ionization, nonideal effects and altered level populations require corrections. Composition matters even when one writes only $\kappa (\rho ,T)$: hydrogen, helium and the metal abundance pattern are additional fixed inputs.

In dense degenerate cores, thermal conduction by electrons can dominate energy transport. An equivalent conductive opacity may be introduced so that $1/{\kappa }_{\mathrm{eff}}=1/{\kappa }_R+1/{\kappa }_{\mathrm{cond}}$, because the two diffusive fluxes add; conduction is not itself a photon absorption process. Very hot regimes also require departures from the simple nonrelativistic scattering approximation. Near the optically thin photosphere, a Rosseland diffusion description alone is inadequate, and frequency-dependent radiative transfer with appropriate boundary conditions is required. These regime changes are why realistic stellar opacity tables, rather than one global power law, enter stellar structure.

Typical weak Lyman-alpha forest absorbers have neutral hydrogen column densities roughly $10^{12}$–$10^{16}{\mathrm{cm}}^{-2}$, with stronger forest absorption extending towards $10^{17}{\mathrm{cm}}^{-2}$. Common Doppler broadening parameters are a few tens of ${\mathrm{km}\mathrm{s}}^{-1}$, consistent with gas at approximately $10^4$ K together with bulk velocity gradients. The thermal contribution isA line width is consequently a temperature upper bound if all other broadening is neglected, not a unique thermometer. The gas commonly lies in diffuse sheets and filaments at modest density contrast; only the strongest systems need substantial self-shielding or a galactic environment.

For optically thin gas illuminated by the cosmic ionizing background, photoionization equilibrium requireswhere the last approximation uses almost complete hydrogen ionization. For example, $n_H=10^{-5}{\mathrm{cm}}^{-3}$, $\alpha \simeq 4\times 10^{-13}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$ and ${\Gamma }_{\mathrm{HI}}=10^{-12}{\mathrm{s}}^{-1}$ give $x_{\mathrm{HI}}\simeq 4\times 10^{-6}$. A $100$ kpc path then has $N_{\mathrm{HI}}\sim 10^{13}{\mathrm{cm}}^{-2}$ despite its tiny neutral fraction. This illustrates the Lyman-alpha forest photoionization scaling: low density makes the two-body radiative recombination rate weak relative to the one-body photoionization rate, but a long path and a strong resonance still produce measurable absorption.

Independent information is important because a column density alone is not a volume density. Similar absorption in close pairs of quasars indicates substantial transverse coherence, while an ionizing optical depth $N_{\mathrm{HI}}{\sigma }_{912}\ll 1$ shows that these weak systems do not shield themselves from the background. The continuum-shielding threshold is $1/{\sigma }_{912}\simeq 1.6\times 10^{17}{\mathrm{cm}}^{-2}$. Together with the observed line widths and ionization balance, these facts favor extended, low-density, highly ionized gas over small cold neutral clouds. The physical interpretation and characteristic scales are consistent with Rauch's forest analysis and the measured column and line-width distributions.

A spectral line has finite width when the emitted or absorbed field loses coherence, or when different emitters have different transition frequencies. It is useful to distinguish homogeneous spectral broadening of each emitter from inhomogeneous spectral broadening across the ensemble.

Finite lifetime and dephasing. For an upper-state population lifetime $\tau$ and a stable lower state, the radiating amplitude decays as $e^{-t/(2\tau )}$. Its Fourier spectrum is Lorentzian, with full width at half maximumMore generally, an optical coherence time $T_2$ gives $\Delta {\omega }_{\mathrm{hom}}=2/T_2$. Both level lifetimes and pure dephasing can contribute to $1/T_2$. Distinguishing population lifetime from amplitude coherence time prevents the usual factor-of-two error.

Thermal motion. To first order in velocity, $\delta \nu ={\nu }_0{v}_{\parallel }/c$. A Maxwell-Boltzmann velocity distribution therefore produces a Gaussian profile, withThis is inhomogeneous doppler broadening: different velocity classes have different resonance frequencies. A distribution of bulk velocities can add further broadening, while a common bulk velocity shifts the line.

Interactions with the surroundings. Collisions can interrupt the dipole wave phase and perturb the energy levels, producing pressure broadening and shifts. In the impact approximation, approximately exponential wave phase relaxation produces a Lorentzian contribution; its rate increases with perturber density times an appropriate velocity-averaged collision cross section. Slowly varying electric microfields can instead produce non-Lorentzian Stark profiles. Neutral perturbers, charged perturbers and resonance interactions need not produce the same line shape. These mechanisms and the Gaussian thermal limit are distinguished in NIST's atomic-spectroscopy compendium.^[ref]

Finite observation time limits coherence; transit-time broadening also occurs for particles crossing a narrow optical beam. Strong resonant excitation causes power broadening: for an ideal lifetime-limited two-level transition, $\Delta \omega =\Gamma \sqrt{1+s}$ with population decay rate $\Gamma$ and saturation parameter $s$. Unresolved fine structure, hyperfine structure, isotope components, or spatially varying Zeeman and Stark shifts may look like a single broadened feature even though several frequencies are present.

The observed spectrum is additionally convolved with the instrument response. A Gaussian Doppler contribution convolved with Lorentzian homogeneous broadening gives a Voigt profile; their full widths do not generally add in quadrature. An informative experiment varies temperature, perturber pressure and excitation intensity separately, calibrates the spectrometer response with a narrower reference, and checks optical thickness or self-absorption. Extrapolation to weak excitation and low pressure helps isolate the lifetime contribution, while a Doppler-free technique is needed if velocity spread still dominates.

In the weak-field Zeeman effect, spin-orbit coupling keeps $L,S,J$ good and a level shifts by ${\mu }_B{g}_{J}B{m}_J$. At fields large compared with fine-structure coupling but still within the linear electronic Zeeman approximation, $L$ and $S$ decouple: the Paschen-Back effect gives shifts ${\mu }_{B}B(m_L+2m_S)$. Electric-dipole transitions obey $\Delta m_L=0,\pm 1$, $\Delta m_S=0$ in that limit, recovering an orbital triplet pattern. Intermediate fields mix the coupled states.

For the weak-field sodium D-line Zeeman pattern, the given Landé g-factor formula givesAllowed photon shifts are $\delta E={\mu }_{B}B(g_u{m}_u-2m_l)$ with $m_u-m_l=0,\pm 1$. ThusThese are the four- and six-component patterns. Both $\pi$ and $\sigma$ components are represented, so an observation strictly along the field, which removes $\pi$ emission, would not show this complete pattern. Hyperfine structure is neglected.

To determine the magnetic field from a Zeeman diagram, the unsplit separation provides its energy scale:The smallest adjacent split spacing $s_E=2{\mu }_B{B}_1/3$ is approximately $0.070$ of that separation in the sketch. ThereforeThis is a graphical estimate; the drawing does not warrant high precision. Its splitting energy is small compared with $\Delta E_D$, consistent with the weak-field approximation.

For Doppler resolution of a Zeeman pattern, thermal velocities give Gaussian Doppler broadening with widthAt $350\mathrm{K}$, the photon-energy FWHM near $589\mathrm{nm}$ is $5.88\times 10^{-6}\mathrm{eV}$. Equating the smallest adjacent Zeeman spacing to this width gives $B\sim 0.15\mathrm{T}$ for loss of individually resolved components. There is no sharp Gaussian resolution threshold; using whole-pattern width instead gives a smaller field of order $0.03$–$0.04\mathrm{T}$ for complete disappearance into a single envelope. Instrumental and collisional broadening would also affect the estimate.

Pressure broadening is proportional to collision frequency and therefore becomes small in the upper atmosphere. Doppler broadening remains finite at finite temperature, with a characteristic width of order ${\nu }_0\sqrt{k_{B}T/(m{c}^2)}$ up to the stated width convention. Where that exceeds the collisional width, the line core is Doppler dominated. A Voigt profile combines the two mechanisms; natural broadening can contribute at still lower pressures. Thus a Lorentz pressure-broadened profile cannot hold all the way to $p=0$.

At line centre, replacing the fictitious $1/p$ opacity by a finite Doppler-core coefficient removes the divergent column optical depth. The emitting pressure is then finite rather than arbitrarily close to zero. If that level still lies in the isothermal stratosphere and the core is opaque, the central brightness temperature stays approximately $T_{\mathrm{strat}}$, so the flux floor changes little. If it lies below the stratosphere, or the Doppler-core column becomes thin, warmer layers or the surface contribute and the core brightens. The core-to-wing transition changes shape and width, but Lorentz wings formed mainly at higher pressures remain approximately as in part (c). The size of the correction needs the missing spectroscopic widths, but no unique numerical correction is requested.

In the two-level saturation population model, $p_e$ and the population-difference reduction $q$ reach half their resonant values at $|\Delta |=(\Gamma /2)\sqrt{1+s}$. Their half width at half maximum is therefore $(\Gamma /2)\sqrt{1+s}$ and their full width at half maximum is $W_{\omega }=\Gamma \sqrt{1+s}$ in angular frequency, or $W_{\nu }=W_{\omega }/(2\pi )$ in hertz. This is power broadening: stronger excitation addresses a wider range of detunings even though the lifetime has not changed.

Increasing $s$ improves the local population-difference reduction $q(0)=s/(1+s)$ but broadens the response. From $s=1$ to $s=9$, this reduction increases from $0.5$ to $0.9$, while the width increases from $\sqrt{2}\Gamma$ to $\sqrt{10}\Gamma$. The contrast approaches a limit and the width keeps growing. This single-atom width is not automatically the width of a velocity-averaged pump–probe feature.

For example, if the distribution of velocities is approximately constant across the narrow resonant classes, the population-only overlap integral contains two Lorentzian line shapes: a probe of half width $\gamma =\Gamma /2$ and a pump hole of half width $a=\gamma \sqrt{1+s}$. Their centres in $kv$ are separated by $2\delta$. The convolution of these profiles has half width $a+\gamma$ in that separation, so its full width at half maximum as a function of $\delta$ is $a+\gamma =(\Gamma /2)(1+\sqrt{1+s})$. Even this simplified pump–probe width differs from the single-atom power broadening formula. Finite Doppler broadening and coherent or multilevel effects require further analysis.

Collisions and fluctuating neighboring-particle electric fields alter atomic energy levels and coherence times, broadening a spectral line. In a simple impact treatment the coherence-decay rate gives a Lorentzian width of order $\Gamma /(2\pi )$; the precise definition of full or half width must be stated. Unlike Doppler broadening, this depends primarily on collision rates and perturber density. In stellar opacity calculations it alters line overlap and the low-opacity windows entering the Rosseland mean opacity.

Raman scattering is inelastic scattering of light in which the outgoing photon has a different frequency because matter gains or loses internal energy. Molecular vibration is a particularly useful recipient: a visible photon can reveal an internal energy spacing normally associated with infrared frequencies. Rayleigh scattering supplies the much stronger, approximately unchanged-frequency reference. The distinction concerns the transferred energy, rather than merely a change of direction or the colour of the illuminating beam.

The elementary bookkeeping is conservation of energy:Here $h$ is the Planck constant, ${\nu }_0$ and ${\nu }_s$ are the incident and scattered photon frequencies, and $E_i,E_f$ are initial and final internal molecular energies. This expression neglects translational recoil and Doppler broadening; those can matter in sufficiently precise gas measurements. A positive internal energy change gives Stokes Raman scattering, and a negative one gives Anti-Stokes Raman scattering. The energy difference determines a possible line position, but it does not by itself establish that a transition is optically allowed or predict its strength.

The spontaneous molecular process developed here has a connection to the Raman transition used in atomic measurement methods: two photon interactions transfer an internal energy difference. Coherently driving that process with two applied light fields is a different experimental regime from collecting weak spontaneous Raman scattering beside an illuminating beam.

A Voigt profile is the convolution of a Gaussian distribution and a Cauchy distribution. For an absorption line, thermal Doppler broadening contributes the Gaussian core while finite excited-state lifetime contributes the Lorentzian wings. Instrumental broadening and unresolved motions must also be modeled when inferring column density and temperature.