Relationship between oscillator strength and cross-section

Question

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:

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

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

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 ($int phi_nu dnu = 1$), 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.

Reza Janalizadeh · Answer

This may be very late but I went through the book you mentioned and I believe a factor of $4piepsilon_{0}$ 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 $4piepsilon_{0}$ and the rest falls into place. That is:
begin{equation}
f_{ij}=4piepsilon_{0}frac{m_{rm e}c}{pi e^{2}}int sigma_{ij}(nu)dnu
end{equation}
Also note that the value 0.667 [cm$^{2}$s$^{-1}$] in equation (6.20) is equivalent to 
begin{equation}
frac{1}{4piepsilon_{0}}frac{8pi^{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).

Also, cross sections are indispensable for gas discharge phenomena (plasmas) where one has to calculate collision integrals in order to obtain particle (e.g., electron) energy (or velocity) distribution functions, and in turn calculate rate coefficients.

Relationship between oscillator strength and cross-section

One Answer

Add your own answers!

Ask a Question