# Relationship between oscillator strength and cross-section

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.

This may be very late but I went through the book you mentioned and I believe a factor of $$4\pi\epsilon_{0}$$ is missing from equations (6.19) and (6.20) of the book. In particular, to the best of my knowledge, the oscillator strength is a dimensionless quantity and a cross section is in units of area. Multiply the definition in (6.19) by $$4\pi\epsilon_{0}$$ and the rest falls into place. That is: $$\begin{equation} f_{ij}=4\pi\epsilon_{0}\frac{m_{\rm e}c}{\pi e^{2}}\int \sigma_{ij}(\nu)d\nu \end{equation}$$ Also note that the value 0.667 [cm$$^{2}$$s$$^{-1}$$] in equation (6.20) is equivalent to $$\begin{equation} \frac{1}{4\pi\epsilon_{0}}\frac{8\pi^{2}e^{2}}{m_{\rm e}c} \end{equation}$$ The oscillator strength is related to the radiative decay rate (i.e., transition probability or Einstein spontaneous emission coefficient) of an excited species, and therefore is used more in the context of the lifetime of excited species. Cross section on the other hand, I think is more a quantitative value for measuring the rate of absorption, emission, excitation of a particle (photons for instance which in turn relates it to energy).
• As is standard in astronomy, on page 3 the book states that it uses cgs units (including for electromagnetism). More formally, and particularly in the context of EM, this is known as Gaussian units: en.wikipedia.org/wiki/Gaussian_units. Among other things, it provides a different definition for charge than does SI that causes the $4\pi\epsilon_0$ factor to not appear. – NeutronStar Aug 8 '19 at 15:27
• @NeutronStar Of course. I was trying to figure out how to make the oscillator strength dimensionless and the cross section in units of area. Although, the exact value of the the proportionality factor (i.e., $4\pi\epsilon_{0}$) stems from calculations of the density of states, transition dipole moment, etc. Plus, I added some further explanations with regards to cross sections. Hope it helps! – Reza Janalizadeh Aug 8 '19 at 15:46