In spectroscopy, the Doppler profile is a spectral line profile which results from the thermal motion of the emitting atom or molecule. 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:

$\backslash nu\; =\; \backslash nu\_0\backslash left(\backslash frac\{c\}\{c+v\}\backslash right)$

where $\backslash nu$ is the observed frequency, $\backslash nu\_0$ 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 gas, the net effect will be to broaden the observed line. The distribution of speeds towards and away from an observer is given by the Maxwell distribution. If $P(v)dv$ is the fraction of particles with velocity component $v$ to $v+dv$ along a line of sight, then:

$P(v)dv\; =\; \backslash sqrt\{\backslash frac\{m\}\{2\backslash pi\; kT\}\}\backslash ,\backslash exp\backslash left(-\backslash frac\{mv^2\}\{2kT\}\backslash right)dv$

where $m$ is the mass of the emitting particle, $T$ is the temperature and $k$ is the Boltzmann constant. The Doppler shift equation can be used to express velocity in terms of the frequency. Using the relationship $\backslash nu\backslash lambda=c$ relating the wavelength $\backslash lambda$ to the frequency, we can substitute into the above equation to obtain the fraction of particles emitting at wavelength $\backslash lambda$ to $\backslash lambda+d\backslash lambda$ according to the observer as:

$P(\backslash lambda)d\backslash lambda\; =\; \backslash sqrt\{\backslash frac\{mc^2\}$

{2\pi kT\lambda_0^2}}\,\exp\left(-\frac{mc^2(\lambda-\lambda_0)^2}{2kT\lambda_0^2}\right)d\lambda

This is seen to be just a normal distribution with standard deviation

$\backslash Delta\backslash lambda=\backslash lambda\_0\backslash sqrt\{\backslash frac\{kT\}\{mc^2\}\}.$

For widths that are small with respect to the central wavelength, we can make the approximation

$\backslash frac\{\backslash lambda-\backslash lambda\_0\}\{\backslash lambda\_0\}\backslash approx\backslash frac\{\backslash nu\_0-\backslash nu\}\{\backslash nu\_0\}$

and the Doppler profile will now be a normal distribution in frequency with standard deviation:

$\backslash Delta\; \backslash nu=\backslash nu\_0\backslash sqrt\{\backslash frac\{kT\}\{mc^2\}\}$

The width of the Doppler profile is sometimes given in terms of its full width at half maximum (FWHM). The FWHM is related to the variance $\backslash sigma$ by:

$FWHM=2\backslash sigma\backslash sqrt\{2\backslash ln(2)\}$

which allows the calculation of the FWHM for either wavelength or frequency by substituting the respective variance in the above equation.

Spectroscopy | Doppler effects