Rydberg constant

A constant which relates the wavelengths of light emitted (or absorbed) by an atom. The Rydberg constant differs somewhat for different elements, and the value usually given is that calculated for an atom of infinite nuclear mass. The best current value is 10 973 731.568 527 ± 0.000 073 per meter (2006 CODATA value)¹. Symbol, R.

As an example of the constant's role, the various wavenumbers (reciprocal of wavelengths) of light emitted by the hydrogen atom can be predicted from the equation:

where
λ is wavelength,
ν is frequency,
c is the velocity of light,
R is the Rydberg constant for hydrogen,
n is an integer greater than x.

X is an integer, the choice of which leads to different series of lines.
x = 1, produces the Lyman series of lines, in the far ultraviolet.
x = 2, produces the Balmer series, in the near ultraviolet and the visible spectrum.
x = 3, produces the Paschen series, in the near infrared.
x = 4, produces the Brackett series, in the far infrared.
x = 5, produces the Pfund series, in the far infrared
x = 6, produces the Humphreys series, in the far infrared.

Each choice of x and n leads to a different wavelength. So, for example, (wavelengths in nanometers, not wavenumbers), the Lyman series (x = 1) includes Lyman alpha, 121.567 nm; Lyman beta, 102.573 nm; Lyman gamma, 97.2537 nm; Lyman delta,  94.9743 nm; Lyman epsilon, 93.7803 nm, etc.

In general, for heavier atoms there is no such simple way of calculating wavenumbers.

The Rydberg constant was at first determined experimentally by the man for whom it is named, Johannes (Janne) Robert Rydberg, a Swedish physicist.  He introduced the concept of wavenumber, replacing wavelengths, and a kind of curve-fitting led him to discover a mathematical relationship between various wavenumbers. He published in 1890.

Bohr showed that the Rydberg constant could be expressed in terms of other atomic constants.

where
α is the fine-structure constant
me is the mass of the electron
c is the velocity of light
h is the Planck constant

Investigations attempting to measure the constant experimentally continue, because it can serve to check the validity of various theories. We can't improve on NIST's excellent press release (29 April 2008) describing a proposed experiment:

High-Flying Electrons May Provide New Test of Quantum Theory

Illustration courtesy NIST

(a) In a Rydberg atom, an electron (black dot) is far away from the atomic nucleus (red and grey core).
(b) Probability map for an electron in a Rydberg atom shows that it has virtually no probability of being near the nucleus in the center.
(c) An optical frequency comb for producing ultraprecise colors of light can trigger quantum energy jumps useful for accurately measuring the Rydberg constant.

Researchers at the National Institute of Standards and Technology (NIST) and Max Planck Institute for Physics in Germany believe they can achieve a significant increase in the accuracy of one of the fundamental constants of nature by boosting an electron to an orbit as far as possible from the atomic nucleus that binds it. The experiment, outlined in a new paper,² would not only mean more accurate identifications of elements in everything from stars to environmental pollutants but also could put the modern theory of the atom to the most stringent tests yet.

The physicists’ quarry is the Rydberg constant, the quantity that specifies the precise color of light that is emitted when an electron jumps from one energy level to another in an atom. The current value of the Rydberg constant comes from comparing theory and experiment for 23 different kinds of energy jumps in hydrogen and deuterium atoms. Researchers have experimentally measured the frequencies of light emitted by these atomic transitions (energy jumps) to an accuracy of as high as 14 parts per quadrillion (one followed by 15 zeros), but the value of the Rydberg constant is known only to about 6.6 parts in a trillion—500 times less accurate. The main hurdle to a more accurate value comes from uncertainties in the size of the atom’s nucleus, which can alter the electron’s energy levels and therefore modify the frequency of light it emits. Another source of uncertainty comes from the fact that electrons sometimes emit and reabsorb short-lived “virtual photons,” a process that also can slightly change the electron’s energy level.

To beat these problems, NIST physicist Peter Mohr and his colleagues propose engineering so-called hydrogen-like Rydberg atoms—atomic nuclei stripped of all but a single electron in a high-lying energy level far away from the nucleus. In such atoms, the electron is so far away from the nucleus that the latter’s size is negligible, and the electron would accelerate less in its high-flung orbit, reducing the effects of “virtual photons” it emits. These simplifications allow theoretical uncertainties to be as small as tens of parts in a quintillion (one followed by 18 zeros).

NIST researchers Joseph Tan and colleagues hope to implement this approach experimentally in their Electron Beam Ion Trap Facility. The idea would be to strip an atom of all its electrons, cool it and inject a single electron in a high-flying orbit. Then the researchers would use a sensitive measurement device known as a frequency comb to measure the light absorbed by this Rydberg atom. The result could be an ultraprecise frequency measurement that would yield an improved value for the Rydberg constant. Such a measurement would be so sensitive that it could reveal anomalies in quantum electrodynamics, the modern theory of the atom.

1. The best current value of the constant is always to be found at NIST: http://physics.nist.gov/cgi-bin/cuu/Value?ryd|search_for=Rydberg

J. R. Rydberg.
Recherches sur la constitution des spectres d'emission des elements chimique.
Den Kungliga Svenska Vetenskapsakademiens Handlingar, vol. 23, no. 11 (1889).

Niels Bohr.
Philosophical Magazine vol 26. page 1 (1913).

2. U. D. Jentschura, P. J. Mohr, J. N. Tan and B. J. Wundt.
Fundamental constants and tests of theory in Rydberg states of hydrogen-like ions.
Physical Review Letters, vol. 100, page 160404, 2008.

B. de Beauvoir, C. Schob, O. Acef, L. Josefowski, L. Hilico, F. Nez, L. Julien, A. Clairon, F. Biraben.
Metrology of the hydrogen and deuterium atoms: determination of the Rydberg constant and Lamb shifts.
European Physics Journal, volume 12, pages 61-93 (1997).

E. R. Cohen.
The Rydberg Constant and the Atomic Mass of the Electron.
Physical Review, vol. 88, no. 2), pages 353-360 (1952).

G. W. Series.
The Rydberg Constant.
Contemporary Physics, vol. 14, no. 1, pages 49-68 (1974).

D. H. McIntyre and T. W. Hänsch.
Precision Measurements of the Rydberg Constant.
Metrologia, vol. 25, pages 61-66 (1988).

P. Zhao, W. Lichten, Z.-X. Zhou, H. P. Layer, and J. C. Bergquist.
Rydberg constant and fundamental atomic physics.
Physical Review A 39, no. 6, pages 2888-2898 (1989)

T. Hänsch and D. McIntyre
The Ryberg constant. In
Units and Fundamental Constants in Physics and Chemistry: Subvolume b: Fundamental Constants in Physics and Chemistry.
J. Bortfeldt and B. Kramer, eds., subvolume b.
New York: Springer-Verlag, ch. 3.2.10, pp. 132-139 (1992).

P. J. Mohr and B. N. Taylor.
Fundamental Constants and the Hydrogen Atom. In
The Hydrogen Atom: Precision Physics of Simple Atomic Systems,
S. G. Karshenboim, F. S. Pavone, G. F. Bassani, M. Inguscio, and T. W. Hänsch, eds.,
Berlin: Springer, pp. 145-156 (2001).

T. W. Hänsch.
Nobel Lecture: Passion for precision.
Rev. Mod. Phys. 78(4), 1297-1309 (2006).