# What is the relation between cross-section and spectral line intensity?

I was told by my supervisor that there is a relation between these two and he talked about a coefficient connecting the two of them. I am currently brushing up a lot of chemistry as I'm a physics student currently working on an internship mainly working with physical chemistry.

• Review the guide How to ask and Asking FAQs to prevent need of clarification, objections, down-voting or closure. Sep 4, 2023 at 8:51

A plane wave moving along $$z$$ has intensity $$I$$ that decreases as

$$dI=-\alpha Idz$$

which yields Beer's law

$$I=I_oe^{-\alpha z}$$

where $$\alpha$$ is the absorption coefficient in cm$$^{-1}$$, at frequency $$\omega$$, for a transition from state $$i$$ to $$f$$, and which depends on the population difference and on the optical cross section $$\sigma_{if}$$ (cm$$^2$$) as

$$\alpha_{i,f}=\sigma_{i,f}(N_i-N_f)$$

and, as is almost always the case, $$N_f\ll N_i$$ then $$\alpha_{i,f}=\sigma_{i,f}N_i$$. As $$N$$ is the number density it is common to replace this with concentration (mol/dm$$^3$$) and then Beer's law becomes

$$I=I_oe^{-\epsilon [C] z}$$

where $$\epsilon$$ is the extinction coefficient in dm$$^3$$/mol/cm at a given wavelength.