Blackbody Radiation

Example

For the n = 2 to n = 4 transition of hydrogen, E = 109677.583 cm^-1*(1/2²-1/4²) = 109677.583 cm^-1 * 3/16 = 20564.547 cm^-1. Naively computing λ = 1 / ν, λ = 486.2738 nm. The dispersion relation gives n = 1.0002793 and λ_air= 486.2738 nm/1.0002793 = 486.1380 nm. This shift of 0.1358 nm may seem small, but with the resolution of typical spectrometers, it is 10 to 1000 times the measurement resolution.

For atoms with more electrons than hydrogen, there are at least two ways to look at the spectrum (the lines emitted or absorbed by the atoms). One is to describe the various energy levels with quantum numbers including shell number (n), orbital angular momentum (L), projection of orbital angular momentum on a reference axis (m_L), and spin (S) and to use the tools of perturbation theory, electrostatics, and quantum electrodynamics to evaluate the energies. The second, less rigorous but entirely adequate for quantitative analysis, is to simply observe the emission lines, trust the many spectroscopists of the last century or so who classified those lines, and look up the energies, wavelengths, and "term symbols" (summaries of quantum numbers) that they have determined. We take this approach.

Where can one find lists of emission lines for the elements? The most general portal on the web is maintained by the Harvard-Smithsonian Center for Astrophysics. The data page is here. One can find lists of hundreds of thousands of transitions for the various ionization stages (neutral or I, singly charged or II, doubly charged or III, ...) of all the elements, ranging in energy from gamma rays to microwaves. A listing that is reasonably complete (at least for the strongest lines) yet not overwhelming is maintained by the National Institute for Standards and Technology (see the NIST Tables). At this writing, the output from the site can give wavelengths, energy levels, and VERY unreliable "intensities" (intensities that are not comparable from one atom to another, were collected using photographic plates from DC arcs, and haven't been revised since the 1940's). New, graphical capabilities that will include computation of quantitative intensities and line-widths, appear to be under development.

What about intensity? Statistical mechanics describes how excited states are populated, while transition probabilities tell how a given distribution of excited states will generate emitted light or absorb incident light. While there are many excited state distributions one might consider, the most general, approximately accurate distribution is Local Thermal Equilibrium (LTE). For a system in LTE, the population of vibrational, rotational, and electronic excited states are in equilibrium as described by a temperature. In addition, the distribution of free electron velocities is described by the same temperature. The only aspect of the system not at equilibrium is light -- all of it is presumed to escape the system. Another way to describe this situation is that the system is optically thin. Weak exceptions to such transparency sometimes matter, and we discuss this later.

LTE contrasts with a system in which the light is in equilibrium with the rest of the system. Fully equilibrated light and matter is possible only in a closed system, where light can't escape. This is blackbody radiation, which you can learn about if you aren't already familiar with it. Energy flows from hot to cold objects, so one can only measure absorption when the absorbing atoms or molecules are cooler than the light source illuminating those atoms/molecules. Similarly, atomic emission can only be seen when the potentially-emitting atoms are hotter than their surroundings.