Problem: What does the phrase “spectral line shape” mean?
Solution: A spectrum is typically a plot of the intensity \(I(\omega)\sim|E(\omega)|^2\) of some underlying time domain signal \(E(t)\).
Problem: What are the \(2\) most prominent kinds of spectra one encounters?
Solution: There are both emission spectra and absorption spectra (and also, it seems there are only broadening mechanisms, no such thing as narrowing mechanisms).
Problem: Distinguish between homogeneous broadening mechanisms and inhomogeneous broadening mechanisms.
Solution: Homogeneous broadening mechanisms affect all atoms indistinguishably, giving rise to a Lorentzian spectral line shape. By contrast, inhomogeneous broadening mechanisms affect atoms distinguishably; this leads to non-Lorentzian spectral broadening.
Problem: In the presence of Doppler broadening alone, what is the spectral line shape?
Solution: Along \(1\)-dimension, the Maxwell-Boltzmann distribution of velocities is normally distributed (no factor of \(4\pi v^2\)):
\[\rho(v)=\sqrt{\frac{\beta m}{2\pi}}e^{-\beta mv^2/2}\]
The non-relativistic Doppler shift gives the angular frequency random variable \(\omega\) as an affine transformation of the velocity random variable \(v\):
\[\omega=\omega_0+k_0v\]
so:
\[\rho(\omega)=\frac{dv}{d\omega}\rho(v)=\sqrt{\frac{\beta m}{2\pi k_0^2}}e^{-\beta m(\omega-\omega_0)^2/2k_0^2}\]
In particular it is also a Gaussian (hence Doppler broadening is inhomogeneous) but with expectation \(\omega_0\) and standard deviation (aka line width) \(\Delta\omega\):
\[\Delta\omega=\omega_0\sqrt{\frac{k_BT}{mc^2}}\propto\sqrt{T}\]
(slogan: variance proportional to temperature) so the quality factor \(Q:=\omega_0/\Delta\omega_D\) of depends on the ratio of rest mass energy \(mc^2\) of each particle to their thermal energy \(k_BT\).
Problem: In the presence of natural/lifetime/radiative broadening alone, what is the spectral line shape?
Solution: For a \(2\)-level system with ground state \(|0\rangle\) and excited state \(|1\rangle\) associated to a lifetime \(\tau\) of spontaneous emission, then the natural line width is:
\[\Gamma\sim\omega\sim\frac{1}{\tau}\]
and the spectral line shape is a Lorentzian \(\sim\frac{1}{(\omega-\omega_0)^2+\Gamma^2}\) (is there a clear way to understand why? From optical Bloch equations for instance…). So natural broadening is a homogeneous broadening mechanism.
Problem: In the presence of pressure/collisional broadening alone, what is the spectral line shape?
Solution: Also a Lorentzian as with natural broadening (so also homogeneous), but the line width is determined from the relaxation time \(\tau=\ell/\langle v_{\text{rel}}\rangle\) between collisions. Using the fact that the mean free path is \(\ell=1/n\sigma\), this yields the line width due to pressure broadening:
\[\Delta\omega=n\sigma\langle v\rangle\]
Problem: What does the overall spectral line shape look like when multiple broadening mechanisms are at play?
Solution:
sum of independent random variables yields convolution.
Power Broadening
This is also homogeneous.
Doppler Broadening
Need to use that formula for converting a probability distribution of one random variable into another function of it with the derivative…(this is how you can experimentally verify the Maxwell distribution btw).