Lecture #2-- Atomic Spectroscopy



Spectroscopy is born from quantum mechanics, which is in turn born from the recognition that all particle motions are governed by the rules of waves rather than the rules of continuous trajectories. When motion is ruled by wave mechanics, the particle can go anywhere that it experiences mostly constructive interference, and nowhere that it experiences only destructive interference. This gives rise to the classical wave equation, but something similar gives rise to the quantum mechanical Schroedinger equation.

The Schroedinger equation tells us that, in a nutshell, wave functions evolve via sinusoidal variations in their amplitudes projected onto each eigenstate of the energy operator, and the rate of variation of the amplitudes is proportional to the energy eigenvalue. Further, the energy operator depends on the potential function, and on the second spatial derivative of the wave function. This means that a wave function can minimize its energy either by being concentrated into regions of low potential, or by spreading out to minimize the spatial second derivative. Those two ways are contradictory, so the tension between them means that the ground state wavefunction will always be somewhat localized to regions of lowest potential, but not to smaller regions than necessary. So the wave function will be localized until its second spatial derivative competes at order unity with the potential contribution, and that sets the scale of the Bohr radius.

The existence of such a minimum energy is the first step toward establishing a discrete set of energy eigenvalues. The next step is the mathematical fact that a compact (i.e., confined) wave function can be expanded on a basis with discrete eigenvalues, similar to how Fourier transforms in a non-infinite domain can be expanded using a discrete k-spectrum. Physically, this is because of the importance of constructive interference and the association with the energy of a wavelength for the particle wave function. Any time you have a concept of wavelength and a concept of compact size, you will have a concept of a discrete set of basis functions, with discrete eigenvalues, because there are "jumps" between the situations where you can get constructive interference. This is entirely analogous to the spectrum of possible standing waves in a resonant cavity, or notes in a musical instrument. In quantum mechanics, these jumps are associated with "quantum numbers" (music calls them "higher harmonics"), which form the basis of atomic spectroscopy. The jumps are also associated with topological differences, namely, the discrete number of extrema (or "bumps") in the eigenfunction, which in turn connects to the second derivative because n bumps requires a second derivative that scales like n squared, all else being equal. Spin and angular momentum quantum numbers add additional structure to the allowed energies that lead to constructive interference.

Astrophysically, the quantum numbers tell us something different about each atom and about the environment of that atom, so studying how they interact with the observable frequencies of emitted and absorbed light, we learn a great deal about the conditions we are looking at. Without a quantized atom, this would be virtually impossible, as there would be a terrible problem with redundancies in what kinds of atoms can make what kinds of spectrums.

At high densities, the quantum systems are so perturbed by their environments that the redundancies return, and can even be handy (as with a "blackbody spectrum"), because they allow you to infer something (temperature) without knowing further details about the atoms. But when the "much else" is actually what you want to know about the gas, the convenience becomes an obstacle. Quantum mechanics limits the possibilities dramatically, just as destructive interference limits what can happen, and those limitations allow us to create a much more one-to-one connection between what we see and what is causing it, when the densities are low and the degree of perturbation is minimal. Hence the diagnostic importance of atomic spectroscopy of low-density gases. (Fortunately, stars are kind enough to provide us with low-density outer layers, and the ISM is low density almost everywhere.)

Atomic spectroscopy is not just of diagnostic importance, but also of physical importance, because not only does the widespread destructive interference drastically limit the allowed behavior, the selected constructive interference induces resonances that greatly enhance the ability of the atom to interact with its environment. In particular, the atomic cross section for absorbing light is vastly increased at the resonant frequencies of these allowed transitions, much like the response of a whistle is vastly increased by the presence of resonances in the cavity vibrations. This is why atoms are so important to the absorption and emission of light by the ISM and by all other astrophysical sources at appropriate wavelengths.

A Little History and the Usefulness of Lines

If you use a spectrograph to break down the spectrum of a star, you find many dark lines. Spectral lines contain a lot of useful information, in part because they are distance-independent, and contains a host of additional information about the structure and content of the observed atmosphere.

The dark lines in stellar spectra are called Fraunhoefer lines, after Joseph Fraunhoefer, the glassmaker who made them famous in 1814. He was merely using sunlight to calibrate his glassware. William Wollaston had already discovered these lines in 1802, but missed out on having them named after him when he incorrectly mistook them as natural breaks between the colors, rather than attributes of the Sun. Fraunhoefer noticed the lines did not appear in spectrum of Venus, so must carry information about the source. William Herschel realized in 1823 that the lines served to label the composition of the emitting gas. Even he probably did not realize how much additional information was carried in the line shapes, which can be extracted using radiative transfer modeling.

The first step toward deciphering this information was taken around 1860 by Kirchhoff and Bunsen, who realized that lines that appear bright in the lab when a given gaseous element is heated should appear dark if seen against the bright continuum of a star. This led to the enumeration of what are called Kirchhoff's laws, which simply state that if you get enough hot gas together, it will emit a continuum of light, and the lines that appear in emission if there is not much gas (i.e., if the gas is optically thin in the continuum) will shift to dark absorption lines if the gas is thick enough and the continuum is bright enough.

In the ISM, we generally find emission lines when we look at astrophysical gas against the blackness of deep space (or the CMB). However, we can also see absorption by ISM lines when we look directly at stars, which can be easier to do since stars are much brighter. When we look at entire galaxies, we see broad ISM emission lines in regions where they are not drowned out by the galactic continuum or unresolved bright stars.