Reduced mass nuclear

The energy difference between the 2 i/2 and 2pi/2 levels in hydrogen and in hydrogenlike ions (the Lamb shift) includes contributions from radiative corrections, reduced mass, nuclear recoil, and finite nuclear size. These corrections are discussed here and in the following two subsections. 

Nuclides that lie below the belt of stability have low neutron-proton ratios and must reduce their nuclear charges to become stable. These nuclides can convert protons into neutrons by positron emission. Positrons (symbolized jS ) are particles with the same mass as electrons but with a charge of -Ft instead of-1. Here are three examples ... 

Properly speaking, me should be the reduced mass for the specific nuclear mass of the system. The values given in Table C.l for me and R,Xj pertain to the infinite-mass limit, which suffices for practical purposes (and can be easily corrected, if necessary). 

As mentioned, most calculations we have done so far have concerned molecular systems. However, prior to development of the non-BO method for the diatomic systems, we performed some very accurate non-BO calculations of the electron affinities of H, D, and T [43]. The difference in the electron affinities of the three systems is a purely nonadiabatic effect resulting from different reduce masses of the pseudoelectron. The pseudoelectrons are the heaviest in the T/T system and the lightest in the H/H system. The calculated results and their comparison with the experimental results of Lineberger and coworkers [44] are shown in Table 1. The calculated results include the relativistic, relativistic recoil. Lamb shift, and finite nuclear size corrections labeled AEcorr calculated by Drake [45]. The agreement with the experiment for H and D is excellent. The 3.7-cm increase of the electron affinity in going from H to D is very well reproduced by the calculations. No experimental EA value is available for T. 

The effective hamiltonian in formula 29 incorporates approximations that we here consider. Apart from a term V"(R) that originates in nonadiabatic effects [67] beyond those taken into account through the rotational and vibrational g factors, other contributions arise that become amalgamated into that term. Replacement of nuclear masses by atomic masses within factors in terms for kinetic energy for motion both along and perpendicular to the internuclear axis yields a term of this form for the atomic reduced mass. 

If internal nuclear motion is neglected, then the reduced mass p is approximated by the electron mass m, and a in these formulas is replaced by a0 (the Bohr radius), where... 

The contributions that have been considered in order to obtain precise theoretical expressions for hydrogenic energy levels are as follows the Dirac eigenvalue with reduced mass, relativistic recoil, nuclear polarization, self energy, vacuum polarization, two-photon corrections, three-photon corrections, finite nuclear size, nuclear size correction to self energy and vacuum polarization, radiative-recoil corrections, and nucleus self energy. 

Here and M are the masses of the muon and the nucleus, and p are their magnetic moments, and I is the nuclear spin (1/2 for hydrogen and 1 for deuterium). The reduced mass factor is... 

Another contribution due to nuclear properties is the so-called recoil contribution. It takes into account the finite mass of the nucleus even beyond the non-relativistic reduced mass approximation. Up to now there is a lack of calculations non-perturbatively in (Za) which exist only for the QED corrections of order (a/n) up to now but would definitely be required for heavy systems. 

In the non-relativistic quantum mechanics the nuclear recoil effect for a hydrogenlike atom is easily taken into account by using the reduced mass p = mM/(m + M) instead of the electron mass m (M is the nuclear mass). It means that to account for the nuclear recoil effect to first order in m/M we must simply replace the binding energy E by E(1 — m/M). 

The purely leptonic hydrogen atom, muonium, consists of a positive muon and an electron. It is the ideal atom, free of the nuclear structure effects of H, D and T and also of the difficult, reduced mass corrections of positronium. An American-Japanese group has observed the 1S-2S transition in muonium to a precision somewhat better than a part in 107. [10] Because there were very few atoms available, the statistical errors precluded an accurate measurement. The "ultimate" value of this system is very great, being limited by the natural width of the 1S-2S line of 72 kHz, set by the 2.2 nsec lifetime of the muon. 

Positronium (e+e ) is a purely leptonic system, free of nuclear structure effects, but suffers from reduced corrections in the worst possible case of equal masses. This makes the system difficult to treat, since quantum electrodynamical calculations start from an infinite nuclear mass and treat reduced mass effects as a perturbation. 

PROBLEM 3.1.2. Bohr s 1913 derivation of the energy of the hydrogen atom (nuclear charge = e, electron charge = e, reduced mass of the electron-nucleus couple = m, electron-nucleus distance = r, linear momentum = p) is based on the classical energy... 

PROBLEM 3.1.3. The energy of a one-electron atom (nuclear charge Z e, electron charge — e, reduced mass of the electron-nucleus couple p) is obtained by solving the Schrodinger equation for the one-electron atom ... 

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