### Reference Material

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

Collisional, or Pressure Broadening, is one cause of the width $\Delta\nu$ in a line profile function $\phi(\nu)$. This type of spectral line broadening arises from collisions that interfere with natural emission processes. Collisions amongst atoms in a high pressure gas can trigger the release of photons by reducing the effective lifetime of energy states, and therefore results in a steeper decay rate and the broadening of absorption lines. \\

The collision timescale is given by:

$$t_{collision} \sim t_{spontaneous\ decay} \sim {\frac{1}{A_{10}}}$$

And the collision rate is dependent on number density, cross section, and velocity:

$$A_{10} \sim n\sigma v$$ \\

In order for Collisional Broadening to dominate, a very dense gas and high velocities are needed. This is related to pressure, which is why Collisional Broadening is sometimes called Pressure Broadening. And this is also why white dwarfs produce broader spectral lines than giants of the same spectral types. \\

The cross section is also important in dictating the timescale for interaction. For fixed number density and velocity:

$$t \propto {\frac{1}{\sigma}}$$ \\

Atoms can interact in a variety of ways. There are different collisional cross sections due to different $\vec{E}$ field drop-off rates, such as $1/r^{2}$ for the Coulomb interaction between ions or charged particles, and $1/r^{3}$ for neutral atoms (dipole field), etc. Because these fields fall off at different rates, there are different cross sections for interaction. Therefore, the time scales for Collisional Broadening depend on which type of interaction dominates, which in turn affect the shape of the line profile function. \\

The line profile function due to Collisional Broadening is a Lorentzian Profile, similar to that caused by Natural Broadening. If all 3 types of broadening mechanisms are present, then Collisional Broadening will add width to the Voigt Profile. Different mechanisms can be responsible for broadening at different distances from the line center.