Main Proton Size Contribution

The nucleus is a bound system of strongly interacting particles. Unfortunately, modern QCD does not provide us with the tools to calculate the bound state properties of the proton (or other nuclei) from first principles, since the QCD perturbation expansion does not work at large (from the QCD point of view) distances which are characteristic for the proton structure, and the nonper-turbative methods are not mature enough to produce good results. 

x Eides et al. Theory of Light Hydrogenic Bound States, STMP 222, 109-130 (2007) 

Fortunately, the characteristic scales of the strong and electromagnetic interactions are vastly different, and at the large distances which are relevant for the atomic problem the influence of the proton (or nuclear) structure may be taken into account with the help of a few experimentally measurable proton properties. The largest and by far the most important correction to the atomic energy levels connected with the proton structure is induced by its finite size. 

The leading nuclear structure contribution to the energy shift is completely determined by the slope of the nuclear form factor in Fig. 6.1 (compare (2.1)) 

The respective perturbation potential is given by the form factor slope insertion in the external Coulomb potential (see (2.3)) 

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Nuclear size and structure corrections for the electronic hydrogen were considered in Chap. 6 and are collected in Table 7.1. Below we will consider what happens with these corrections in muonic hydrogen. The form of the main proton size contribution of order (Za) m (r ) from (6.3) does not change... 

The leading nuclear size correction of order Za) m r )EF may easily be calculated in the framework of nonrelativistic perturbation theory if one takes as one of the perturbation potentials the potential corresponding to the main proton size contribution to the Lamb shift in (6.3). The other perturbation potential is the potential in (9.28) responsible for the main Fermi contribution... 

Theoretically, light muonic atoms have two main special features as compared with the ordinary electronic hydrogenlike atoms, both of which are connected with the fact that the muon is about 200 times heavier than the electron First, the role of the radiative corrections generated by the closed electron loops is greatly enhanced, and second, the leading proton size contribution becomes the second largest individual contribution to the energy shifts after the polarization correction. 

The Zemach correction is essentially a nontrivial weighted integral of the product of electric and magnetic densities, normalized to unity. It cannot be measured directly, like the rms proton charge radius which determines the main proton size correction to the Lamb shift (compare the case of the proton size correction to the Lamb shift of order Za) in (6.13) which depends on the third Zemach moment). This means that the correction in (11.4) may only conditionally be called the proton size contribution. 

To overcome this limitation will require the measurement of the Lamb shift (the 2S-2P energy difference) in muonic hydrogen. Here the main QED contribution is vacuum polarization, for which calculations are now available at a precision level of 10-6 [11,12,13,14]. Because the effect of the finite proton size contributes as much as 2% to the pp Lamb shift, a precise measurement of the shift will provide an accurate value of the proton radius. The knowledge of the proton radius has intrinsic interest as a fundamental quantity, and is important in other measurements. A measurement of rp at 0.1% precision will permit QED calculations of bound systems to be compared with the ep experiments at a precision level of fewxlO-7 gaining an order of magnitude over the present limits. 

The main part of the nuclear size (Za) contribution which is proportional to the nuclear charge radius squared may also be easily obtained in a simpler way, which clearly demonstrates the source of the logarithmic enhancement of this contribution. We will first discuss in some detail this simple-minded approach, which essentially coincides with the arguments used above to obtain the main contribution to the Lamb shift in (2.4), and the leading proton radius contribution in (6.3). 

A neutron is characterized by having no electrical charge but has one unit of atomic mass, the same as that of a proton (Figure 46.2). Neutrons, like protons, reside in the atomic nucleus and contribute to the mass of the atom. The chemistry of an atom, like its size, is determined by the electrons in the atom. The mass of the atom is characterized mainly by the total number of neutrons and protons in the nucleus (atomic binding energies are ignored in this discussion). For mass spectrometric purposes of measurement, it is the mass that is important in establishing m/z values. 

All yet uncalculated nonrecoil corrections to the Lamb shift almost cancel in the formula for the isotope shift, which is thus much more accurate than the theoretical expressions for the Lamb shifts. Theoretical uncertainty of the isotope shift is mainly determined by the unknown single logarithmic and nonlogarithmic contributions of order ZaY m/M) and a(Za) (m/M) (see Sects. 4.3 and 5.2), and also by the uncertainties of the deuteron size and structure contributions discussed in Chap. 6. Overall theoretical uncertainty of all contributions to the isotope shift, besides the leading proton and deuteron size corrections does not exceed 0.8 kHz. 

The water reorganization energy is mainly determined by the size of the water clusters between which the proton is transferred (H30+, H9C>4+) and the effective distance over which the proton is shifted during the process. The contribution of the medium reorganization outside the reactant water clusters (outer-sphere contribution) is calculated in Ref. 43, 44 as a function of reactant cluster sizes, transfer distance, pore size, and dielectric properties of the contacting media (water, polymer). 

For proton-driven spin diffusion, the most important difference to abundant high-y spin systems is that the main contribution to the zero-quantum line-width is no longer the homonuclear dipolar coupling, but rather, the hetero-nuclear dipolar coupling to protons or other abundant high-y spins in the sample. Because the zero-quantum lineshape is largely independent of the size of the homonuclear dipolar couplings and, therefore, independent of the... 

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