Doppler Broadening

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\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \section*{Doppler Broadening}

Doppler Broadening is one cause of the width $\Delta\nu$ in a line profile function $\phi(\nu)$. This type of spectral line broadening arises from the thermal motions of atoms. There will be a velocity distribution for these atoms which causes redshifted and blueshifted photons. Therefore, the Doppler effect leads to variation in the absorbed frequency $\nu$. \\

The Doppler shift is given by:

$$\nu = \nu_{0}\bigg(1+{\frac{v}{c}}\bigg)$$

where $v$ is the line-of-sight velocity of the absorbing atom. The amplitude of the frequency shift is therefore given by:

$$\Delta \nu = \nu_{0}{\frac{v}{c}}$$

Recall that the distribution of particle speeds in local thermodynamic equilibrium is Maxwellian:

$${\frac{1}{2}}mv^{2} \sim kT $$ $$v \sim \sqrt{{\frac{2kT}{m}}} $$

Plugging this velocity into the Doppler width gives:

$$\Delta \nu = {\frac{\nu_{0}}{c}}\sqrt{{\frac{2kT}{m}}} $$

The line profile function resulting from Doppler Broadening is a Gaussian Profile.

$$\phi(\nu) = {\frac{1}{\Delta\nu\sqrt{\pi}}}e^{-{\frac{(\nu-\nu_{0})^{2}}{\Delta\nu^{2}}}} $$ \\


The peak value of the profile occurs when $\nu = \nu_{0}$.

$$\phi_{peak} = {\frac{1}{\Delta\nu\sqrt{\pi}}}$$ \\

This can be used to calculate the FWHM by setting ${\frac{\phi_{peak}}{2}} = \phi(\nu)$ and solving for $2\times (\nu-\nu_{0})$. The result is:

$$\Delta\nu_{FWHM} = 1.665\Delta\nu$$ \\

One thing to note is that as temperature increases, there is a greater spread in velocities. However, the number of atoms remains the same, so the peak in the above plot would drop $\bigg($since $\int_{0}^{\infty} \phi(\nu) d\nu = 1\bigg)$. Therefore, it is possible to go from an optically thick medium to optically thin. Up until now the thermal velocity distribution has been used, but turbulence can also cause a spread in velocities. The total Doppler Broadening can be found by adding the two components in quadrature.

$$v_{total}^{2} = v_{thermal}^{2} + v_{turb}^{2}$$\\

Doppler Broadening as discussed above refers to the differential movement of atoms. There can also be a bulk motion of the atoms in a particular direction, which is a shift in $\nu_{0}$.