Doppler broadening

Doppler broadening is the spread of a spectral line caused by different line-of-sight emitter or absorber velocities. To first order in $v/c$, a rest-frame line at frequency ${\nu }_0$ is shifted by a magnitude ${\nu }_0|v_{\parallel }|/c$, with sign fixed by approach or recession.

For a nonrelativistic thermal gas of particles of mass $m$, a Maxwell velocity distribution gives a Gaussian line contribution with standard deviation ${\sigma }_{\nu }=({\nu }_0/c)\sqrt{k_{B}T/m}$. Its full width at half maximum is $2\sqrt{2\ln 2}{\sigma }_{\nu }$. The width can therefore constrain temperature when instrumental width, collisions and other motions are modeled. Doppler broadening is an ensemble distribution of frequency shifts, distinct from the finite-lifetime width of each emitter.

A Maxwellian emitter has Gaussian Doppler frequency FWHM $\Delta {\nu }_D/\nu =\sqrt{8k_{B}T\ln 2/(m{c}^2)}$. Equating this width to the minimum adjacent weak-field sodium component separation $2{\mu }_{B}B/(3h)$ gives the title resolution scale. A Gaussian blend has no unique resolution threshold: disappearance of individual components and disappearance of the whole broadened multiplet are different criteria. Instrumental resolution, pressure broadening, selected polarization and line strengths must be specified for a sharper prediction.

If nonrelativistic emitters of mass $m$ have rest-frame angular frequency ${\omega }_0$, then the one-dimensional Maxwell-Boltzmann velocity distribution produces a Gaussian line with standard deviationMeasuring the line width therefore determines the gas temperature when nonthermal broadening is negligible or separately modeled.

A purely thermal hydrogen absorption line has velocity-space optical depth proportional to $\exp [-(v-v_0{)}^2/b^2]$ with the displayed width. Additional independent Gaussian broadening contributes positive terms to $b^2$. Fitting the lower envelope of width against neutral-hydrogen column density, with numerical calibration for density and pressure smoothing, estimates the temperature-density relation of photoionized intergalactic gas. A single broad line only bounds the temperature from above when its nonthermal width is unknown.