The Rydberg Constant A Fundamental Constant

Johannes Robert Rydberg, 1890 • Theodor W. Hansch, 1992

The Rydberg constant is one of the most important constants of atomic physics because of its connection with the fundamental atomic constants (e, h, me, c) and because of the high accuracy with which it can be determined.

The Rydberg constant first appeared in the literature of physics in 1890. Today, more than a century later, this constant still challenges physicists as they carefully design experiments with state-of-the-art instruments to measure the Rydberg constant with ever-increasing precision. There are good reasons for the interest in this constant, but before we consider these reasons, three background questions assert themselves: first, what makes a constant "fundamental"? Second, where do fundamental constants come from? And third, why are fundamental constants important?

Constants deemed fundamental are those that emerge from the core of the overarching theories of physics; they are constants whose values determine the magnitudes of the basic interactions of nature; and finally, they are the constants whose values are linked to and help establish the values of other significant physical constants. The melting point of water stands alone as an important property of water, but its import does not extend beyond water. The speed of sound is different for every medium in which the mechanical disturbance called sound propagates. Neither the melting point of water nor the speed of sound is a fundamental constant. By contrast, the fine-structure constant appears in many different physical contexts and links important domains of physics. The fine-structure constant is a fundamental constant.

The fundamental constants originate from both nature itself and physical theories. Every atom in the universe consists of electrons and protons. The charge carried by these fundamental particles as well as their masses are in any list of fundamental constants. A basic property of both the electron and the proton is their magnetic moments. These basic particle properties are fundamental constants that come from nature. The great theories of physics such as gravitation, electromagnetism, relativity, quantum mechanics, and quantum electrodynamics (QED) also give rise to the fundamental constants such as the gravitational constant, G, the speed of light, c, Planck's constant, h, and the fine-structure constant, a.

The importance of the fundamental constants is manifold. The fundamental constants often come into play when experimental results are compared with theoretical predictions. When these constants are known with high precision, they can expose both the strengths and the weaknesses in the theories that physicists employ to explain the physical processes of nature. It is through the interplay between measured result and predicted result that physical theories are put to the test: questioned, refined, or discarded. The lure of measuring the fundamental constants to ever-increasing precision has stimulated new experimental techniques that have paid dividends throughout science. The fundamental constants not only link experiment and theory, they also link different theories. Their appearance in diverse theoretical contexts speaks for their significance.

As another indicator of their basic nature and hence their importance, the fundamental constants often appear grouped together in clusters, so that they provide a natural means of cross checking one against the other. For example, the ratio of two fundamental constants, e/h, consisting of the charge of the electron, e, and Planck's constant, h, has significance in condensed matter physics, QED, high-energy physics, atomic physics, and X-ray physics. Still another reason underlying the importance of fundamental constants is this: The system of weights and measures has historically been based on artifacts such as the platinum-iridium bar, safeguarded by the International Bureau of Weights and Measures in Paris, which is the standard for length, the meter, defined as the distance between two scratches on the bar. The fundamental constants provide a potential for defining standards on a basis that draws directly from the workings of nature itself and is thereby independent of arbitrary scratches on a metal bar.

The Rydberg constant is a constant that meets all the criteria established for being deemed fundamental. In addition, the Ryd-berg constant connects all theoretical calculations and experimental measurements of the energy states of any atom.

Fourteen years after Balmer published his work that showed how the wavelengths of the hydrogen spectrum could be represented by a simple mathematical formula, Johannes Robert Ryd-berg came across Balmer's paper. Rydberg was a mathematician and physicist from Sweden who was fascinated by the periodic system of the chemical elements. Rydberg believed that the spectra of the elements held the key to a deeper understanding of the elements themselves as well as the reasons for the periodicity the elements exhibited in the Periodic Table. He was particularly interested in the observation that the spectra of many chemical elements appear to be members of a series of wavelengths, hinting at relationships among the wavelengths. Rydberg succeeded in developing a simple mathematical formula that accurately reproduced the observations associated with these spectral series.

Just about the time Rydberg composed his simple mathematical formula, he also discovered Balmer's paper on the spectrum of the hydrogen atom. Rydberg realized immediately that he could recast the Balmer formula for the hydrogen spectrum into the more general form of his just-discovered mathematical formula. In Rydberg's recast form, the Balmer formula looked like this:

where X is the wavelength of a spectral line, m is a running integer, and N is a constant, as determined by Rydberg himself, equal to 109,721.6 cm-1. Rydberg claimed that Nwas a "universal constant common to all the series and to all the elements examined."1 With this formula, Rydberg was able to give an accurate account of the spectra of many different chemical elements. The Rydberg constant, N, was later given the symbol R in honor of Rydberg. Still later, the constant was given the symbol Rx, which reflects the need to take into account the mass of the nucleus of the atom producing the spectral transitions.

On the basis of Rydberg's work alone, the constant did not qualify as a fundamental constant. In 1890, when Rydberg published his results, the constant was an empirical number, which means simply that it emerged from an analysis of experimental data. No deeper significance could be brought to this number other than that it was consistent with a good range of physical data and was therefore probably important.

The status of the constant changed dramatically when Niels Bohr crafted his model of the hydrogen atom. Specifically, Bohr's theory revealed that the Rydberg constant was not just a number, but was a combination of other fundamental constants. Here is the result that emerged from Bohr's work:

Bohr was able to express Rydberg's constant in terms of the electron's mass, me, the electron's charge, e, Planck's constant, h, and the speed of light, c. After Bohr finished with it, Rydberg's constant was no longer a simple empirical constant, but had a theoretical basis and a deep significance because it was directly linked to other fundamental constants.

The constants that make up the Rydberg constant reach into other major domains of physics. The constants me and e are key properties of the electron and each of these constants appears individually or in combination with other constants in many physical contexts. The constant c arises in electromagnetic phenomena and relativity theory, and the constant h is ubiquitous in quantum mechanics. Perhaps more significant, the constants that come together to define the Rydberg constant are, on an individual basis, difficult to measure with great precision. As we shall see, the Rydberg constant can be measured very precisely and thus it becomes one of the cornerstones for the determination of other basic constants. In addition, a highly accurate value of the Rydberg constant would provide a stringent test of QED.

Over the decades of the twentieth century, experimental physicists worked diligently to measure the Rydberg constant with ever greater precision. It has been the hydrogen atom that has provided the principal means for this quest. More specifically, it has been primarily the first and most prominent spectral line in the Balmer series, the bright red Ha line, that has been the focus of interest. For this reason among others, the Ha line "has been studied more intensively than any other line in experimental spec-troscopy."2

When one considers the myriad of measurements that are made in the conduct of science, the measurements of spectral wavelengths stand apart in that they can be measured with great precision. Thus, from the measurement of the wavelength of the Ha line, physicists over the years have determined the Rydberg constant. A few of the early results are given in Table 19.1.

The experimental results shown in the table cluster around the value 109,737.3 cm-1 with uncertainties precluding values

Table 19.1 Values of the Rydberg constant determined by optical spectroscopy



Value (cm 1)

J. R. Rydberg



W. V. Houston3




R. T. Birgeb




J. W. M. DuMond et al.c




B. P. Kibble et al.d




a. William V Houston, "A Spectroscopic Determination of e/m," Physical Review 30, 608-613 (1927).

b. R. T. Birge, "The Values of R and of e/m, from the Spectra of H, D and He+," Physical Review 60, 766-785 (1941).

c. Jesse W. M. DuMond and E. Richard Cohen, "Least-Squares Adjustment of the Atomic Constants," Reviews of Modern Physics 25, 691-708 (1953).

d. B. P. Kibble, W. R. C. Rowley, R. E. Shawyer, and G. W Series, "An Experimental Determination of the Rydberg Constant," Journal of Physics B 6, 1079-1089 (1973).

a. William V Houston, "A Spectroscopic Determination of e/m," Physical Review 30, 608-613 (1927).

b. R. T. Birge, "The Values of R and of e/m, from the Spectra of H, D and He+," Physical Review 60, 766-785 (1941).

c. Jesse W. M. DuMond and E. Richard Cohen, "Least-Squares Adjustment of the Atomic Constants," Reviews of Modern Physics 25, 691-708 (1953).

d. B. P. Kibble, W. R. C. Rowley, R. E. Shawyer, and G. W Series, "An Experimental Determination of the Rydberg Constant," Journal of Physics B 6, 1079-1089 (1973).

with greater precision. The situation changed around 1985. The breakthrough that allowed a significant improvement in the measured precision of the Rydberg constant came when physicists confronted an inherent limitation in the measurement of spectral wavelengths. The limitation arises because the Ha spectral transition, as well as all other spectral transitions, does not consist of a single wavelength but a cluster of wavelengths distributed symmetrically around the "true" wavelength. This clustering of wavelengths arises because the hydrogen atoms emitting the detected photons are in random motion. The photons emitted by atoms moving toward the detector are seen by the detector as shifted toward a shorter wavelength—the faster the atom is moving, the bigger the shift. Alternately, the photons emitted by atoms moving away from the detector are seen by the detector as shifted toward longer wavelengths—the faster the atom is moving, the bigger the shift. This shift in wavelength due to the relative motion between the emitting atom and the detector is called the Doppler effect—this same effect is well known in cosmology as the red shift of distant galaxies, which occurs as the universe expands. The challenge facing experimentalists in their quest to measure the Rydberg constant more accurately was to reduce the Doppler effect, to narrow the clustering of wavelengths, to locate the exact center of the cluster of the observed wavelengths more accurately, and thereby to reduce the uncertainties.

The means to accomplish these ends was one rather obvious and one not-so-obvious step. The obvious step was to find a way to lower the temperature of the hydrogen sample, which would decrease the speed of the randomly moving atoms. The lower temperature, however, produced relatively minor improvements. The not-so-obvious step was the ingenious application of lasers to achieve Doppler-free results.

It can be said with confidence that lasers revitalized, if not resuscitated, studies of the atom. Atomic physics was one of the most active areas of physical research following the creation of quantum mechanics. Already in the 1930s, however, nuclear physics was rising to prominence and after World War II, particle physics and a little later condensed matter physics became dominant areas of interest. The laser provided a new tool for precision studies of atoms, spurring a genuine renaissance in atomic physics about 1970. Precision measurements of the Rydberg constant were a part of this resurgence of physicists' interest in atoms.

Many outstanding physicists have struggled to extend the measured value of the Rydberg constant to the next level of accuracy. Theodor W. Hansch, however, is one of the most persistent and most successful in designing new experiments to probe the hydrogen atom and establish more accurate values of the Rydberg constant.

Hansch was born and raised in Heidelberg, Germany. He became interested in hydrogen in 1967 as a graduate student when he heard an inspiring talk by G. W. Series on the hydrogen atom.

Series was known for his extensive work on the hydrogen atom and for his 1957 book, The Spectrum of Atomic Hydrogen.3 From his first paper on hydrogen in 1972,4 Hansch's interest in hydrogen has been a staple of his physical research. After spending sixteen years on the faculty at Stanford University, Hansch returned to his native Germany where he has a dual appointment as professor of physics at the University of Munich and director of the MaxPlanck-Institut fur Quantenoptik.

Before 1974, all wavelength measurements of the hydrogen Ha line, from which the value of the Rydberg constant was deduced, suffered from Doppler broadening of the spectral line, thereby limiting the accuracy of the result. During the early 1970s, an extremely clever method was devised by Hansch to eliminate this broadening effect. First, Hansch developed a new type of dye laser that would be suitable for exciting hydrogen atoms and observing the Ha transition.5 With this laser, two beams were directed at the sample of hydrogen atoms—one strong beam passed through the sample of hydrogen atoms in one direction and the second, weak beam passed through from the opposite direction. Atoms moving either toward or away from the oncoming laser beams "see" different wavelengths because of the Doppler effect. Only when the wavelength of the two laser beams is equal to the actual wavelength of the hydrogen spectral transition, Ha, do both beams interact with the same group of atoms; namely, those atoms that are effectively standing still.

Here then is the crux of the experiment: in response to the strong beam, tuned to the transition wavelength, essentially all hydrogen atoms that are motionless relative to the strong laser beam absorb energy from the beam and make the Ha transition. Essentially, the strong beam clears a path for the weaker laser beam. The weaker beam, passing through the sample in the opposite direction, then passes through and, with no atoms to stimulate, no absorption occurs and the beam exits the sample with essentially the same energy it had on entering. It is the intensity of the weak laser beam that is monitored and the magic wavelength is identified as that particular wavelength that permits the weak probe beam to pass through the sample with its intensity unchanged. This experiment, called saturation spectroscopy, is beautifully conceived and generates excellent results.

This experimental measurement of the Ha wavelength produced a value of the Rydberg constant with an accuracy that was a tenfold improvement over previous experiments. The value of the Rydberg constant that Hansch and his collaborators obtained was

Two years later, in 1976, Hansch and a graduate student, Carl Wieman, devised another experimental method, called polarization spectroscopy, that improved still further the accuracy of the Rydberg constant. In this method polarized light is used. Two beams of laser light—one a strong beam that is circularly polarized, the other a weak beam that is linearly polarized—pass through a sample of hydrogen atoms in opposite directions. Once again, only when the wavelengths of the two beams are exactly equal to the wavelength of the Ha transition do the two beams interact with the same class of hydrogen atoms—those that are not moving relative to the light beam. The strong, circularly polarized beam interacts with a select population of hydrogen atoms and essentially removes them from the sample. The linearly polarized weak probe beam, encountering a sample of atoms with a select population missing, has its polarization axis rotated by the remaining atoms and this enables light from the probe beam to be detected. When light is detected, the wavelengths of the two laser beams exactly equals the Ha transition wavelength. From the wavelength, the value of the Rydberg constant can be determined. From this experiment, the determined value was

R» = 109,737.31476 ± 0.00032, which is about three times more accurate than the 1974 result.7

During the 1974 and 1976 experiments, Hansch's attention was shifting from the Ha line of hydrogen to a transition that was long recognized "as one of the most intriguing transitions to be studied by Doppler-free high-resolution laser spectroscopy."8 This intriguing transition is not a part of the Balmer series. The Ha transition is between the second energy state of hydrogen and the third energy state; that is, from n = 2 to n = 3. The intriguing transition, part of the Lyman series and called the Lyman-alpha transition, is between the first and second energy states, identified as the 1S-2S transition: n = n = 2. Its fascination arises because this transition is inherently narrow, which means that with Doppler-free techniques, even more accurate values of the Ryd-berg constant might be obtained.

The Balmer series has wavelengths that are visible; by contrast, the wavelength of the 1S-2S transition is invisible; it is in the ultraviolet region. Dye laser sources at such short wavelengths did not exist. Once again, an ingenious approach, called two-photon spectroscopy, was devised to open the door to a study of the 1 S-2S transition of the hydrogen atom. In two-photon spectroscopy, the laser beam has a frequency that is half the frequency (or twice the wavelength) of the desired transition. A beam from the laser is sent through the sample of hydrogen atoms, where it strikes a mirror and is reflected back through the sample. Thus, the hydrogen atoms are bathed in laser light, tuned at exactly twice the wavelength of the Lyman-alpha transition, going in opposite directions. Regardless of how a particular hydrogen atom is moving, the Doppler shifts cancel out when an atom absorbs a photon from each of the oppositely moving laser beams. The absorption of the two photons stimulates the Lyman-alpha transition.

Employing the two-photon method, the Rydberg constant was again measured. From this experiment its determined value was

Later, with refinements, the two-photon method yielded a more accurate value:

With this result, Hansch and his collaborators announced in 1992 that "our new value represents the most accurate measurement of any fundamental constant."11 However, Hansch's superlative claim had a rather short life.

By October 1997, Hansch and his co-workers had a new "most accurate" value. Still using the provocative 1S-2S transition and building on the two-photon method used to get the 1992 result, the 1997 result was

Table 19.2 provides a more complete account of the measured values of the Rydberg constant, including Hansch's 1997 result.

What is the payoff for this relentless push toward greater and greater accuracy? Why were Hansch's 1997 results so crucial? The first paragraph of Hansch and colleagues' 1997 paper explains:

For almost three decades, the 1S-2S two-photon transition in atomic hydrogen with its natural linewidth of only 1.3 Hz has inspired advances in high-resolution spectroscopy and optical frequency metrology. This resonance [the 1S-2S transition] has become a de facto optical frequency standard. More importantly, it is providing a cornerstone for the determination of fundamental constants and for stringent tests of quantum electrodynamic theory. In the future, it may unveil

Table 19.2 Measured values of the Rydberg constant





















































































* Other individuals were included as authors in the papers reporting the results. Note that in the earlier era it was more common for individuals to work alone.

* Other individuals were included as authors in the papers reporting the results. Note that in the earlier era it was more common for individuals to work alone.

conceivable slow changes of fundamental constants or even differences between matter and antimatter.13

Clearly, these experimental results are not going to impact the lives of today's world citizens. However, from the perspective of the physicist, there is an enormous payoff.

The experiments themselves—saturation spectroscopy, polarization spectroscopy, and two-photon spectroscopy—were magnificent in their design and execution. These experimental methods, developed for the explicit purpose of measuring features of the hydrogen spectrum, will have applications elsewhere. For example, the challenge of measuring the wavelength of the Ha transition with great accuracy motivated the advancement and refinement of lasers and laser techniques, which have wide-ranging applications.

For physics itself, however, the story is still unfolding. When Lamb discovered that the 2S1/2 and 2P1/2 states of hydrogen were slightly different, the result became a challenge for theorists, and QED emerged in a refined form as perhaps the most powerful theory of physics. In the experiments of Hansch, the Lamb shift for the 1S ground state of hydrogen was measured with great accuracy. This opens further challenges for physical theorists: the Lamb shift embraces such basic phenomena as the difference between the electron's self-energy in free and bound states, the effect of vacuum polarization on binding energy, and nuclear size effects. A comparison of spectra of the two hydrogens, hydrogen and deuterium, can provide stringent tests to QED, the proton-electron mass ratio (mp/me), and the charge radius of the proton. In fact, Hansch has already determined the difference between the mean square charge radii of the proton and the deuteron:

r,2 - rp2 = 0.0000000000000038212

± 0.0000000000000000015 m2.14

Hansch has also determined the deuteron structure radius to be rar = 0.00000000000000197535

± 0.00000000000000000085 m.15

Since the deuteron is the most important nucleus for understanding the inner workings of nuclei in general, this result will attract the attention of nuclear physicists.

Finally, the Rydberg constant is now known with sufficient accuracy that it may become the basis for a new definition of the second. In this capacity, the Rydberg constant will assume a more visible place in the hierarchy of fundamental constants. Whatever happens, physicists will be scratching their heads trying to devise a way to bring this constant to the next level of accuracy.

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    What is the significance of the Rydberg constant?
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