Atomic Spectroscopy

I’m doing an ‘experiments in modern physics’ lab this semester. The structure of the course is simple: we do 4 labs chosen from a tantalizing list of many, complete with a report and an APS-style talk. I decided to do my first experiment on atomic spectroscopy: measuring energy level splittings using optical resonators, and I figured I’d write a post on it (well, paraphrase my report) because it may be interesting to a non-scientific audience. So, this post only has a few equations and is mainly an overview of the basic concepts involved.

(For those interested in the details, here’s the executive summary: the experiment was basically to use Fabry-Perot and Michelson interferometers to characterize the spectral lines of atomic sodium and hydrogen/deuterium, and measure the spectral splitting of the sodium 3p_3/2→3s_1/2 / 3p_1/2→3s_1/2 D line and the hydrogen/deuterium 3s→3p H-α transition due to spin-orbit coupling and isotope mass differences, respectively. The H/D splitting measurement also allows one to deduce the proton/deuteron mass ratio by playing around with the numbers a bit. Note that the picture on the left is a photo of slightly split sodium fringes from a Fabry-Perot interferometer, from [1] - click Read on… and scroll to the bottom for the references.)

In this post I’ll briefly review the historical context of this experiment, go over the basic physics responsible for the spectral splittings and the physics behind the experimental tools, and outline the experiment itself.

1. Introduction and Theoretical Details
Developed in 1913, the Bohr model of hydrogenic atoms was a key step in explaining a number of outstanding questions regarding their spectral emission lines – among other things, it provided a theoretical foundation for the Rydberg formula. For example, considering a negatively charged electron orbiting a positively charged atomic nucleus purely electrostatically, one would expect the electron to spiral into the core due its being accelerated, leading to radiation of energy and an unstable system.

However, it was known before this time that atoms not only were stable, but had characteristic spectra – that is, they emitted light at characteristic wavelengths. For example, the Rydberg formula for the wavelength (lambda) of the spectral lines of hydrogen is:

where the prefactor on the right-hand side is the Rydberg constant for hydrogen. Various electronic transitions (corresponding to various spectral lines) occur and correspond to different values of the integers n₁ and n₂. These include the Lyman, Balmer, or Paschen series, the second of which is in the visible part of the light spectrum.

Figure 1. Schematic describing various transitions (and an image with the corresponding visible spectral lines) of atomic hydrogen. From [1].

The most intense line in the visible hydrogen spectrum is that of the 3s→2p (also known as the H-α) transition, and this is the line I looked at in my experiments. (For completeness I want to note that this line is actually split by non-isotopic spin-orbit effects, which I’ll discuss further on, but this splitting is too small to be relevant to this experiment.)

Drawing on the equation above and previous theoretical work by Planck and Einstein regarding quanta in the context of blackbody radiation or the photoelectric effect, Bohr proposed that these ‘valence’ electrons were confined to specific orbits around the atomic nucleus – that is, the angular momentum of the electron was considered to be quantized. Transitions between such orbits could only occur if exactly hv (h is Planck’s constant, while v is the frequency characterizing the transition) of energy – in the form of photons – was emitted or absorbed. The fact that many such allowed orbits exist led to the notion of energy ‘levels’ and the spectrum of such atoms, which had previously been experimentally determined. Using these principles, Bohr constructed a model for hydrogenic atoms, and found that the energy associated with each spectral line (i.e. each transition) was given by:

The terms in the first parentheses (m_e is the electron mass, e is the electron charge, epsilon nought is the permittivity of free space, and c is the speed of light) constitute the Rydberg constant for hydrogen in MKS units. This constant was calculated from the various other constants making it up and compared to spectroscopic data, with the two being in good agreement. Later experiments measuring the spectrum of ionized helium forced Bohr to slightly revise his postulate: the total angular momentum of the system was quantized, not just that of the electron. One has to consider the reduced mass of the two-body system, not just the mass of the electron, and the above equation is replaced by:

where the term in the first parentheses is the reduced mass (m_n is the nuclear mass). This led Harold Urey to discover deuterium by measuring a shift in its spectral lines, for which he won the Nobel Prize in 1934. Deuterium is atomic hydrogen with a neutron in the nucleus in addition to the proton, and since the neutron mass is the same as the proton mass (to within less than 0.2%), it is often referred to as ‘heavy hydrogen’ (since it is like hydrogen with a doubly massive nucleus).

Anyway, the point is that measuring the difference between these two energies (one corresponding to hydrogen, the other corresponding to deuterium) allows one to determine the reduced mass, and since the proton-to-electron mass is known to be around 1836.15 [2], the proton-to-deuteron mass ratio can be found from this measurement. Among other things, this is what I did in my experiment.

This isn’t the whole story, though: things get complicated when considering other ‘hydrogenic’ atoms (that is, atoms with one valence electrons, but one or more ‘inner’ electrons which we’ll ignore for now) - like sodium. One has to go through some slightly more advanced quantum mechanical calculations and considerations of details such as spin-orbit coupling (we did perturbation theory in my quantum mechanics class a few weeks ago, which is when I slogged through the details - but I’ll spare them here).

The physical intuition is relatively simple, and draws on some considerations of relativity: think of the electron as a magnetic dipole (kind of like a tiny bar magnet) orbiting around the nucleus. In the electron reference frame, the nuclear proton appears to be moving around the electron. This orbital motion (of angular momentum L) can be thought of as a current loop, which sets up a magnetic field that exerts a torque on the electron – which has spin angular momentum S (hence the term spin-orbit or L-S coupling). I’m not going to present the details here [3-4], I’ll simply state that this has the effect of ‘perturbing’ the system in such a way as to split the lines in the sodium spectrum in an experimentally measurable manner (a similar effect occurs in hydrogen, but it is much harder to see for reasons I won’t go into).

Experimentally, the most intense spectral line of atomic sodium is the ‘D’ line (with a wavelength of approximately 589 nm). This is a doublet (due to the splitting noted above) and corresponds to the 3p_3/2→3s_1/2 / 3p_1/2→3s_1/2 electronic transitions (where 3p3/2, for example, corresponds to the 3p state with j = 3/2, where j is the total angular momentum quantum number). The transitions corresponding to this doublet are indicated in the figure to the left, which is from [1].

2. Optical Interferometry and Experimental Details
The shifts and splittings of the spectral lines of atoms can be measured and quantified using a variety of experimental techniques. Perhaps one of the simplest and most powerful is using optical resonators – in this experiment, the instruments used were Fabry-Perot and Michelson interferometers.

In general, a resonator (be it a driven string attached at both ends or electromagnetic modes in a reflective cavity) picks out certain frequencies through constructive interference. The apparatus used in this experiment are optical resonators, made using two partially reflective mirrors to confine light in a ‘cavity’, as the figure below illustrates. The apparatus on top is a Fabry-Perot interferometer, with the two parallel rectangles at the end representing two carefully-aligned partially-silvered mirrors defining a ‘cavity’ within which the light bounces around (and interferes with itself) before it exits, whereas the bottom is a Michelson interferometer that works by a very similar principle - the only difference is that the ‘cavity’ within which light bounces isn’t just the space between two parallel mirrors, but is defined in two directions, as shown. (S/PS refer to fully/partially-silvered mirrors. The end lens in both cases is the lens of the observer’s eye. The glass is used as a compensator. )

In both of these experiments, a focused incident light beam (produced by optical transitions from the source, which is either a hydrogen/deuterium or sodium discharge lamp) is made to bounce between a ‘cavity’ defined by two partially silvered mirrors.

Consider an incident light beam that is not parallel (image adapted from [5]):

(Note that the very first image in this post - from [1] - is an example of what sodium fringes look like when they’re slightly split.) Anyway, the point is that some elementary geometry allows one to calculate the path length difference between two adjacent paths indicated in the figure (let’s call it delta l), and to have constructive interference - that is, interference resulting in fringes that you can see, as in the figure - this path length difference must equal an integer multiple (n) of the light wavelength:

I won’t go into more details, save to note that the point of this is that, thinking of n as being kind of like the number of standing wave modes in the cavity, it is useful to index the fringes seen by this integer. It turns out that fringes of a higher index (resulting from constructive interference of the light in the cavity) lie closer to the center, and what the equation above tells us is that changing d - that is, as the spacing between the two mirrors is increased, the fringes move outward from the center and new fringes are produced. Experimentally, the light source, lenses, and vibration-isolated Fabry-Perot cavity all lie on a linear track, and d is changed using a micrometer with a coarse knob (for changing d on the order of millimeters) and a fine knob (for submicron changes of d). And so, by counting the number of fringes produced and noting the corresponding changes in d, it’s possibly to use the above formula (in a slightly modified form) to measure the wavelength of the light you’re seeing (averaging over any splittings).

But the punchline is yet to come. Since the light is actually of two slightly different wavelengths (due to the splittings that we’re trying to measure in the first place), the fringe pattern you see isn’t just one fringe pattern - it’s the combination of two fringe patterns, resulting in a visible ’splitting’ of the fringes (an example of which can be seen in the photo of the yellow sodium fringes, at the very beginning of this post). The cool thing is that since the wavelength of light characterizing each individual fringe pattern is different, the rate at which the fringes move outward is different for the two fringe patterns, and for certain values of d they coincide, while for certain values of d, they are evenly spaced. A few more elementary calculations show that using the wavelength we measured before, and by measuring the d’s at which the fringes perfectly coincide (or are perfectly split) and looking at the difference between these, it is possible to calculate the difference in wavelength corresponding to the two superimposed fringe patterns - and hence, the splitting!

So, that’s basically it - the experiment and physics involved in a nutshell.
Here are the main points:
- When valence electrons transition between different energy levels of an atom, they emit photons (or particles of light) characterized by a certain frequency. This is the light that you see, and different colors are characterized by transitions of differing energies. Because of this, atoms of a certain element emit light of specific frequencies - this is known as the atomic spectrum of that element, and the study of spectra is known as spectroscopy.
- Atomic spectroscopy has been a very important experimental technique in the history of quantum physics. A key example of how it has been useful is the observation of ’splittings’ in the spectral lines of certain elements in a variety of circumstances. Understanding these splittings has led to increased understanding in quantum physics.
- Two examples of spectral splittings are that present in the spectrum of hydrogen and deuterium, which results from their isotopic difference in nuclear mass, and that present in the spectrum of sodium, which results from what is known as spin-orbit coupling.
- Resonators - characterized by certain frequencies - are ubiquitous in nature, and are very important experimental tools for picking out specific frequencies of an excitation.
- Optical resonators like Fabry-Perot or Michelson interferometers are particularly useful in atomic spectroscopy, and can be used to measure both the wavelength of a spectral line, as well as certain splittings that may be present. Interesting physical quantities can be deduced from these measurements.

Voilà!

References
[1] From HyperPhysics, Georgia State University:
http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html. This is a very good resource for concepts in elementary physics.
[2] A more accurate value is 1836.152 666 5(40), from a very interesting experiment done using a Penning trap – see D. L. Farnham et al., Phys. Rev. Lett. 75, 3598 (1995).
[3] Relativistic effects are very important, but some of these have classical interpretations, such as Thomas precession, which explains a factor of 1/2 that comes about in the equations. See, for example, Classical Electrodynamics 3rd ed., by J. D. Jackson.
[4] An excellent derivation is in Modern Quantum Mechanics 2nd ed., by J. J. Sakurai. A more elementary presentation can be found in Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, by R. Eisberg and R. Resnick.
[5] Adapted from University of Sydney pre-lab notes on Fabry-Perot Interferometry:
www.physics.usyd.edu.au/comphys2/cp2optics/pdf_files/cp2opset3.pdf.

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