Rydberg constant mass-corrected

Use the values for fundamental constants (with m equal to the mass of an electron) in eqn 4.16 to calculate a value for the Rydberg constant. What correction is necessary to give the value for hydrogen ... 

The aim of this section is to extract from the measurements the values of the Rydberg constant and Lamb shifts. This analysis is detailed in the references [50,61], More details on the theory of atomic hydrogen can be found in several review articles [62,63,34], It is convenient to express the energy levels in hydrogen as the sum of three terms the first is the well known hyperfine interaction. The second, given by the Dirac equation for a particle with the reduced mass and by the first relativistic correction due to the recoil of the proton, is known exactly, apart from the uncertainties in the physical constants involved (mainly the Rydberg constant R0c). The third term is the Lamb shift, which contains all the other corrections, i.e. the QED corrections, the other relativistic corrections due to the proton recoil and the effect of the proton charge distribution. Consequently, to extract i oo from the accurate measurements one needs to know the Lamb shifts. For this analysis, the theoretical values of the Lamb shifts are sufficiently precise, except for those of the 15 and 2S levels. 

The Is-2s transition energy is primarily due to the (electron) Rydberg constant, where the antiproton mass contributes via the reduced mass only of the order of 10 . For the theoretical calculations an uncertainty exists at the level of 5 X 10 [17] (finite size corrections) due to the experimental error in the determination of the proton radius, even if only the more rehable Mainz value of = 0.862 0.012 fm [18] is used (for a detailed discussion of the pro-... 

At present, the dominant contribution to the uncertainty of the predicted 1S-2S interval (2.5 MHz) is the uncertainty of the Rydberg constant (10" ), and a measurement of the 1S-2S frequency can improve the Rydberg constant about tenfold, until we approach the uncertainties due to the electron mass (5x10" 70 kHz), the charge radius of the proton (10" 130 kHz), and approximations in the computation of QED corrections (60 kHz). The fine structure constant (10" ) contributes an uncertainty of only 4 kHz. 

Above we have proposed to measure the the small difference frequency df = f(1S-2S) - 3 f(2S-nS). This frequency depends critically on the Lamb shifts of the participating levels, and can provide a stringent test of QED. For n=100, the theoretical uncertainty of df is dominated by the nuclear size effect ( 70 kHz) and by approximations in the computation of electron structure corrections and uncalculated higher order QED corrections (65 kHz). The contribution of the Rydberg constant (1 kHz), the electron mass (0.05 kHz), and the fine structure constant (4 kHz) are negligible by comparison. The QED computations can be improved, and if theory is correct, a precision measurement of df can provide accurate new values for the charge radii of the proton and deuteron. 

We can see that the Bohr s factor of constants should agree with the experimental value of the Rydberg. We have already noted there needs to be a small correction to the value of m since a tiny center-of-mass correction should be applied (the electron and proton actually rotate about then-mutual center of mass which is very close to the position of the more massive proton). 

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