Doppler broadening

An example of a Doppler broadened line profile. The solid line represents an un-broadened emission profile, and the dashed line represents a broadened emission profile.

In atomic physics, Doppler broadening is broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting (or absorbing) particles result in different Doppler shifts, the cumulative effect of which is the emission (absorption) line broadening. This resulting line profile is known as a Doppler profile.

A particular 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 (or absorbing) body being spectroscopically investigated.

Derivation (non-relativistic case)

When a particle moves (e.g., due to the thermal motion) towards the observer, the emitted radiation is shifted to a higher frequency. Likewise, when the emitter moves away, the frequency is lowered. In the non-relativistic limit, the Doppler shift is

$f=f_{0}\left(1+{\frac {v}{c}}\right),$

where $f$ is the observed frequency, $f_{0}$ is the frequency in the rest frame, $v$ 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 $\lambda$. Since $v/c\ll 1$, $\left|f/f_{0}-1\right|\ll 1$, and so ${\frac {\lambda -\lambda _{0}}{\lambda _{0}}}\approx -{\frac {f-f_{0}}{f_{0}}}$. Therefore,

$P_{\lambda }(\lambda )\,d\lambda ={\frac {c}{\lambda _{0}}}P_{v}\left[c\left(1-{\frac {\lambda }{\lambda _{0}}}\right)\right]\,d\lambda .$

Thermal Doppler broadening

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

$P_{f}(f)\,df={\frac {c}{f_{0}}}{\sqrt {\frac {m}{2\pi kT}}}\,\exp \left(-{\frac {m\left[c\left({\frac {f}{f_{0}}}-1\right)\right]^{2}}{2kT}}\right)\,df.$

We can simplify this expression as

$P_{f}(f)\,df={\sqrt {\frac {mc^{2}}{2\pi kTf_{0}^{2}}}}\,\exp \left(-{\frac {mc^{2}\left(f-f_{0}\right)^{2}}{2kTf_{0}^{2}}}\right)\,df,$

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)

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

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. Other causes of velocity distributions may exist, though, for example, 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 that can also broaden the lines. For example, a sufficiently high particle number density may lead to significant Stark broadening.

Doppler broadening can also be used to determine the velocity distribution of a gas given its absorption spectrum. In particular, this has been used to determine the velocity distribution of interstellar gas clouds.

Doppler broadening is used as a design feature in nuclear reactors. The nuclear fission reaction is sustained by neutrons causing fission in fissile nuclei. Doppler broadening results in a flatter neutron energy spectrum, which reduces the likelihood of a neutron causing fission. As reactor fuel heats up, Doppler broadening flattens the neutron spectrum and decreases the chance of fission or neutron absorption in the fuel. This creates a negative fuel temperature coefficient of reactivity that brings down the reaction rate as the reactor heats up. Fast-neutron reactors use Doppler broadening (alongside thermal expansion of the fuel) to ensure passive safety of the reactor. It also smooths small perturbations in reactivity, making the reactor easier to control.

Saturated absorption spectroscopy, also known as Doppler-free spectroscopy, can be used to find the true frequency of an atomic transition without cooling a sample down to temperatures at which the Doppler broadening is negligible.

See also

• Mössbauer effect
• Dicke effect