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In the context of absorption of photons by atoms, I have come across two seemingly very related quantities, cross section and oscillator strength. In the book Physics of the Interstellar and Intergalactic Medium cross section $\sigma$ and oscillator strength $f$ are defined as follows:

$$\sigma_{lu}(\nu) = \frac{g_u}{g_l}\frac{c^2}{8\pi \nu^2_{lu}}A_{ul}\phi_\nu,$$

$$f_{lu} = \frac{m_e c}{\pi e^2} \int \sigma_{lu}(\nu)d\nu,$$

where $l$ means lower state, $u$ means upper state, $lu$ designates that the transition is from a lower to an upper state, $g$ is the degeneracy of the energy level, $A_{ul}$ is the Einstein A coefficient from $u$ to $l$, $\phi_\nu$ is a normalized line profile ($\int \phi_\nu d\nu = 1$), and the units (true to astrophysics form) are CGS.

I've dealt quite a bit with cross sections before, but never with oscillator strengths. I'm used to cross sections being in units of [length]$^2$, but it appears that this definition of cross section is [length/time]$^2$. Also, the units of the oscillator strength seem to be [time].

The cross section and the oscillator strength are obviously related quantities, but I am confused about in which situations each tends to be useful. If we've defined both of them, despite being related quantities, I imagine there are situations where one is more useful than the other.

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