Metrology of the hydrogen and deuterium atoms: Determination of the Rydberg constant and Lamb shifts

We present a detailed description of several experiments which have been previously reported in several letters: the determination of the 1S Lamb shift in hydrogen by a comparison of the frequencies of the 1S-3S and 2S-6S or 2S-6D two-photon transitions, and the measurement of the 2S-8S/D and 2S-12D optical frequencies. Following a complete study of the lineshape of the two-photon transitions, we provide the updated values of these frequencies which have been used in the 1998 adjustment of the fundamental constants. From an analysis taking into account these results and several other precise measurements by other authors, we show that the optical frequency measurements have superseded the microwave determination of the 2S Lamb shift and we deduce optimized values for the Rydberg constant, $ R_{\infty }$ = 109 737.315 685 50(84) cm-1 (relative uncertainty of $ 7.7\times 10^{-12}$) and for the 1S and 2S Lamb shifts L(1S) = 8 172.840(22) MHz and L(2S-2P) = 1 057.8450(29) MHz (respectively, 8 183.970(22) MHz and 1 059.2341(29) MHz for deuterium). These are now the most accurate values available