In atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting particles result in different (Doppler) shifts, the cumulative effect of which is the line broadening. The resulting line profile is known as a Doppler profile. A particular and perhaps the most important case is the thermal Doppler broadening due to the thermal motion of the particles. Then, the broadening depends only on the frequency of the spectral line, the mass of the emitting particles, and their temperature, and therefore can be used for inferring the temperature of an emitting body.

## Derivation

When thermal motion causes a particle to move towards the observer, the emitted radiation will be shifted to a higher frequency. Likewise, when the emitter moves away, the frequency will be lowered. For non-relativistic thermal velocities, the Doppler shift in frequency will be: $f = f_0\left(1+\frac{v}{c}\right)$

where $\ f$ is the observed frequency, $\ f_0$ is the rest frequency, $\ v$ is the velocity of the emitter towards the observer, and c is the speed of light.

Since there is a distribution of speeds both toward and away from the observer in any volume element of the radiating body, the net effect will be to broaden the observed line. If $\,P_v(v)dv$ is the fraction of particles with velocity component $\,v$ to $\,v+dv$ along a line of sight, then the corresponding distribution of the frequencies is $P_f(f)df = P_v(v_f)\frac{dv}{df}df$,

where $\,v_f = c\left(\frac{f}{f_0} - 1\right)$ is the velocity towards the observer corresponding to the shift of the rest frequency $\,f_0$ to $\,f$. Therefore, $P_f(f)df = \frac{c}{f_0}P_v\left(c\left(\frac{f}{f_0} - 1\right)\right)df$.

We can also express the broadening in terms of the wavelength $\,\lambda$. Recalling that in the non-relativistic limit $\frac{\lambda-\lambda_{0}}{\lambda_{0}} \approx -\frac{f-f_0}{f_0}$, we obtain $P_\lambda(\lambda)d\lambda = \frac{c}{\lambda_0}P_v\left(c\left(1 - \frac{\lambda}{\lambda_0}\right)\right)d\lambda$.

In the case of the thermal Doppler broadening, the velocity distribution is given by the Maxwell distribution $P_v(v)dv = \sqrt{\frac{m}{2\pi kT}}\,\exp\left(-\frac{mv^2}{2kT}\right)dv$,

where $\,m$ is the mass of the emitting particle, $\,T$ is the temperature and $\,k$ is the Boltzmann constant.

Then, $P_f(f)df=\left(\frac{c}{f_0}\right)\sqrt{\frac{m}{2\pi kT}}\,\exp\left(-\frac{m\left[c\left(\frac{f}{f_0}-1\right)\right]^2}{2kT}\right)df$.

We can simplify this expression as $P_f(f)df=\sqrt{\frac{mc^2}{2\pi kT {f_0}^2}}\, \exp\left(-\frac{mc^2\left(f-f_0\right)^2}{2kT {f_0}^2}\right)df$,

which we immediately recognize as a Gaussian profile with the standard deviation $\sigma_{f} = \sqrt{\frac{kT}{mc^2}}f_0$

and full width at half maximum (FWHM) $\Delta f_{\text{FWHM}} = \sqrt{\frac{8kT\ln 2}{mc^2}}f_{0}$.

Similarly, $P_\lambda(\lambda)d\lambda = \sqrt{\frac{mc^2} {2\pi kT\lambda_0^2}}\,\exp\left(-\frac{mc^2(\lambda-\lambda_0)^2}{2kT\lambda_0^2}\right)d\lambda$

with the standard deviation $\sigma_{\lambda} = \sqrt{\frac{kT}{mc^2}}\lambda_{0}$

and FWHM $\Delta \lambda_{\text{FWHM}} = \sqrt{\frac{8kT\ln 2}{mc^2}}\lambda_{0}$.

## Applications and caveats

In astronomy and plasma physics, the thermal Doppler broadening is one of the explanations for the broadening of spectral lines, and as such gives an indication for the temperature of observed material. It should be noted, though, that other causes of velocity distributions may exist, e.g., due to turbulent motion. For a fully developed turbulence, the resulting line profile is generally very difficult to distinguish from the thermal one. Another cause could be a large range of macroscopic velocities resulting, e.g., from the receding and approaching portions of a rapidly spinning accretion disk. Finally, there are many other factors which can also broaden the lines. For example, a sufficiently high particle number density may lead to significant Stark broadening.

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