2S-nS and 2S-nD transitions

From the 2S-8S/D and 2S-12D measurements in hydrogen, we deduced in 2000 the mean value of the Rydberg constant :
\(R_{\infty} = 109 737.315 685 5 (11) \,\text{cm}^{-1}\).
This result is the most precise one if we exclude theoretical assumptions concerning the 1S and 2S Lamb shifts \(L_{1S}\) et \(L_{2S}\). The \(10^{-11}\) uncertainty is mainly due to the one of the 2S Lamb shift. One can get rid of this limitation by comparing the frequencies of several optical transitions involving the 1S or 2S levels. Indeed, the coulombian energy term and the QED corrections do not have the same dependence with the principal quantum number \(n\) (\(1/n^{2}\) et \(1/n^{3}\) respectively). Comparing the various transition frequencies gives then more precise values of the Rydberg constant and QED corrections. In fact, theory [*] gives a precise calculation of the Lamb shift deviation from the \(1/n^{3}\) law. One can then go further using the 1S-2S frequencies measured in Garching by the group of T.W. Hänsch [**] in hydrogen and deuterium. Combining all the available optical frequency measurements (1S-2S, 2S-8S/D and 2S-12D transitions), one obtains the following result for the Rydberg constant :
\(R_{\infty} = 109 737.315 685 4 (12)\,\text{cm}^{-1}\)
This value is independent from the RF measurements of the 2S Lamb shift. The method also gives the 1S and 2S Lamb shifts :
\(L_{1S} = 8172.837 (26)\,\text{MHz}\)
\(L_{2S-2P} = 1057.8447 (34)\,\text{MHz}\)
This last value is in perfect agreement and more precise than the best RF determination.

Finally, one can perform a least squares adjustment of all available experimental data in hydrogen and deuterium, to get the Rydberg constant :
\(R_{\infty} = 109 737.315 685 50(84)\,\text{cm}^{-1}\) with an uncertainty of \(7.7 \times 10^{-12}\).

A complete analysis of the results obtained in hydrogen and deuterium is done in the review paper [de Beauvoir et al. 2000].

The CODATA adjustment of fundamental constants performed in 2010 [***] gives the following value for the Rydberg constant :
\(R_{\infty} = 109 737.315 685 39(55)\,\text{cm}^{-1}\) with an uncertainty of \(5.0 \times 10^{-12}\).

Experiments on various hydrogen transitions are presently ongoing in order to further improve this uncertainty.

[*] S.G. Karshenboim, J. Phys. B 29, L29 (1996) ; Z. Phys. D 39, 109 (1997)[**] M. Fischer, N. Kolachevsky, M. Zimmermann, R. Holzwarth, Th. Udem, T.W. Hänsch, M. Abgrall, J. Grünert, I. Maksimovic, S. Bize, H. Marion, F. Pereira Dos Santos, P. Lemonde, G. Santarelli, P. Laurent, A. Clairon, C. Salomon, M. Haas, U.D. Jentschura, and C.H. Keitel, Phys. Rev. Lett. 92 (2004), 230802[***] P.J. Mohr, B.N. Taylor and D.B. Newell, arXiv:1203.5425 (2012)