The Hydrogen Spectrum and its Fine Structure

Atoms in a gas all come with their element-specific fingerprints. Add energy, for example by heating or by bumping other particles onto them, and they will emit light at very specific wavelengths. This means that if the light is dispersed according to wavelength in a glass prism, a characteristic pattern of bright lines will appear, a so-called line spectrum. Here is part of the line spectrum of atomic hydrogen:


[Source: Wikipedia / NASA's Imagine the Universe.]
Three lines are clearly visible on top of a continuous background, called H-α (H-alpha, the red line), H-β (the blue line), and H-γ (the violet line). The light of the H-α line has a wavelength of about 656.3 nanometer (nm, or 10^-9 meter), or 0.0006563 mm. These three lines are part of a series of lines that follow an intriguing numerical pattern, the so called Balmer series. At closer inspection, the lines themselves exhibit a structure. Here is a detailed view of the H-α line:


Source: Arthur L. Schawlow's Nobel Lecture Spectroscopy in a New Light, The Nobel Foundation, 1981.

The upper curve shows the profile of the H-α line as it was known in the middle of last century. Intensity of light is plotted here as a function not of wavelength λ but of frequency ν - the relation is quite simple, ν = c/λ, where c is the speed of light. At the wavelength of the H-α line, a difference in frequency of 10 Gigahertz (GHz) - the unit used in the figure - corresponds to a tiny shift in wavelength of only 0.015 nm. The wide profile in the upper curve is caused not by the limitations of the spectrometer in use, but by the Doppler motion of the atoms under analysis: Due to temperature, the hydrogen atoms are always in motion, and the Doppler effect blurs lines that would be sharp if the atom would be at rest.

Around 1970, a clever method called saturation spectroscopy had been developed, where by an arrangement of laser beams the blurring effects of the Doppler shift could be strongly diminished. As a result, the H-α line shows a fine structure with a series of lines, as shown in the lower plot.

Hydrogen consists of an electron bound to a proton. As a result of the strange rules of nature called quantum mechanics, the energy of the electron can only have discrete values, and if the electron changes between different energy levels, light can be emitted with a frequency >ν corresponding to the energy difference by the rule ΔE = hν, where h is Planck's constant. Hence, understanding the spectrum of hydrogen amounts to understanding how the different energy levels of the electron come about. Here is a schematic representation of the energy levels of the electron in the hydrogen atom



Energy is plotted along the vertical axis. The series of lines in the left-hand column represents the energy levels calculated by Bohr in his famous model of the hydrogen model. Energy levels come in steps proportional to 1/n², which explains Balmer's curious formula for the lines of the hydrogen spectrum. The H-α line corresponds to the transition between the levels with n = 2 and n = 3 - theses transitions are indicated by the arrow. If an energy of about 13.6 eV is applied to the hydrogen atom, the electron can overcome the Coulomb force of the proton and get unbound - that's the continuum limit of ionisation.

Nature is a bit more complex than the Bohr model. The electron has a spin, and the interaction of spin and orbital angular momentum (which, again, comes in discrete steps which are labelled by the letters S, P, D, ...) results in small corrections to the Bohr formula. This so-called fine structure of the hydrogen spectrum comes out naturally in the Dirac theory of the relativistic electron. But even Dirac's sophisticated theory is not the end of the story - as a consequence of the quantum nature of the electromagnetic field, there are shifts to the energy levels which are known as Lamb shift. And finally, the proton also has a magnetic moment, and the interaction of the spin of the electron with the proton magnetic moment results in a further tiny splitting of the energy levels, called the hyperfine structure.

All these details of the hydrogen spectrum can be measured with different spectroscopic techniques, and calculated by theory - so far, there are no known discrepancies. But there is a lot of very interesting, and diverse, physics hidden behind the pattern of lines of the hydrogen spectrum.

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A very nice little book about atomic hydrogen and the history and many facets of the analysis of its spectrum is Hydrogen: The Essential Element by John S. Rigden.

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This post is part of our 2007 advent calendar A Plottl A Day.