Voigt Profile

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\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \section*{Voigt Profile}

The Voigt Profile is a convolution of the Lorentzian Profile from Natural Broadening and the Gaussian Profile from Doppler Broadening. It is dominated by the Lorentzian wings and thermal Doppler Broadening in its center. It is a normalized function.

The Voigt profile shown here is for the Lyman-alpha transition, where $A_{10} = 5 \times 10^{8}$ $s^{-1}$ and $\nu_{0}$ is the transition frequency for the Ly-$\alpha$ transition ($1216$ angstroms). The Doppler width used here corresponds to a temperature of $100$ K.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\

The absorption features due to each type of broadening is also plotted. The optical depth is given by: \\ \\ $$\tau = n \sigma s$$ $$ = N\sigma$$ \\

where $N$ is the column density. An absorption feature reduces the intensity of light by $e^{-\tau} = e^{-N\sigma}$ when there is no emission ($I_{\nu} = I_{\nu,0}e^{-\tau}$). The line profile function is incorporated into the cross-section $\sigma$ by the following:

$$\sigma = {\frac{\pi e^{2}}{m_{e}c}}f\phi(\nu)$$ \\

Therefore, different cross-sections can be calculated for each line profile function (using a Ly-$\alpha$ oscillator strength of $f=0.416$. The factor $e^{-N\sigma}$ by which a continuum is reduced creates the absorption feature. Again, the Voigt feature is dominated by the Gaussian at its center and by the Lorentzian wings.