Inhomogeneous Linewidth The inhomogeneous linewidth in Rare Earth Ion crystals are often fitted with a lorentzian, I do not see any reason for that, I would expect a gaussian due to the fact that, too many random parameter affecting the central oscillation frequency and the central limit theorem should take over and it must be a gaussian. How do I explain this..?
 A: The shape of the inhomogeneous linewidth depends of the type of defects responsible for it. In the case of rare-earth crystals, point defects (0D) are most dominant, like impurity atoms or vacancies. Each point defect causes local strain in the crystal, which causes the emitter's frequencies to shift. The strain falls off like $\sim \frac{1}{r^3}$ with the distance $r$ from the 0D defect. So, a few emitters are close to the defect and therefore shifted by a lot, a handful emitters are at intermediate distance and are shifted by less. The rest is either barely affected or already closer to yet another defect.
This is how a distribution of the shifts relative to the transition frequency in an ideal crystal can be modeled. If you rigorously go through the math, you will find that 0D defects will lead to a Lorentzian lineshape, where the width and the offset from the perfect crystal depend on the concentration of defects. Luckily, people have already calculated this. For example in A. M. Stoneham, Shapes of Inhomogeneously Broadened Resonance Lines in Solids, Reviews of Modern Physics 41(1), 82–108 (1969).
