Proton radius

After the discovery that atoms have masses which are integral multiples of the mass of the hydrogen atom, it was realised that, if a suitable mechanism could be found, all atoms could be created from the fusion of hydrogen. The problem is that the electrostatic force implies a strong repulsion between atomic nuclei, which all carry positive electric charge. Typically, the e-s potential energy for 2 protons, separated by 2 proton radii ( 10 15 m) is... 

An alternative treatment of the correction of order Z a) Za) m/M)m was given in [4]. The idea of this work was to modify the standard definition of the proton charge radius, and include the first order quantum electrodynamic radiative correction into the proton radius determined by the strong interactions. Prom the practical point of view for the nS levels in hydrogen the recipe of [4] reduces to elimination of the constant 11/72 in (5.6) and omission of the Pauli correction in (5.7). Numerically such a modification reduces the contribution to the lA energy level in hydrogen by 0.14 kHz in comparison with the naive result in (5.6), and increases it by 0.03 kHz in comparison with the result in (5.8). Hence, for all practical needs at the current level of experimental precision there are no contradictions between our result above in (5.8), and the result in [4]. 

Fig. 6.1. Proton radius contribution to the Lamb shift. Bold dot corresponds to the form factor slope...

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). 

Using the dipole form factor one can connect the third Zemach moment with the proton rms radius, and include the nuclear size correction of order Za) m in (7.62) on par with other contributions in (7.58) and (7.74) depending on the proton radius. Then the total dependence of the Lamb shift on Vp acquires the form [3, 25]... 

This expression also may be used for determination of the proton rms radius from the experimental data. Numerically it makes almost no difference because contributions in (7.63) and (7.64) with high accuracy coincide. However, the coefficient before r in (7.76) is modef dependent, so it is conceptuaffy advantageous to use the experimentaf vafue of the third Zemach moment obtained in [56] for cafculation of the nucfear size correction of order Za) m, and use the expression in (7.75) for determination of the proton radius from experimentaf data. 

Model dependence of the Zemach correction, as well as its dependence on the proton radius is theoretically unsatisfactory. A much better approach is suggested in [8], where the values of the proton and deuteron first Zemach moments were determined in a model independent way from the analysis of the world data on the elastic electron-proton and electron-deuteron scattering. The respective moments turned out to be [8]... 

The last term in the braces is ultraviolet divergent, but it exactly cancels in the sum with the point proton contribution in (11.12). The sum of contributions in (11.12) and (11.13) is the total proton size correction, including the Zemach correction. According to the numerical calculation in [6] this is equal to AE = —33.50 (55) x lO Ep. As was discussed above, the Zemach correction included in this result strongly depends on the precise value of the proton radius, while numerically the much smaller recoil correction is less sensitive to the small momenta behavior of the proton form factor and has smaller uncertainty. For further numerical estimates we will use the estimate AE = 5.22 (1) X 10 Ep of the recoil correction obtained in [6]. 

Both the theoretical and experimental data for the classic 2S i/2 — 2Pi/2 Lamb shift are collected in Table 12.2. Theoretical results for the energy shifts in this Table contain errors in the parenthesis where the first error is determined by the yet uncalculated contributions to the Lamb shift, discussed above, and the second reflects the experimental uncertainty in the measurement of the proton rms charge radius. We see that the uncertainty of the proton rms radius is the largest source of error in the theoretical prediction of the classical Lamb shift. An immediate conclusion from the data in Table 12.2 is that the value of the proton radius [27] recently derived form the analysis of the world data on the electron-proton scattering seems compatible with the experimental data on the Lamb shift, while the values of the rms proton radius popular earlier [28, 29] are clearly too small to accommodate the experimental data on the Lamb shift. Unfortunately, these experimental results are rather widely scattered and have rather large experimental errors. Their internal consistency leaves much to be desired. 

Since the main contribution to the uncertainty of the theoretical value of the IS Lamb shift comes from the uncertainty of the proton charge radius we can invert the problem and calculate the proton radius using the average of the self-consistent Lamb shifts in Table 12.3 L IS) = 8 172 847 (14) as input. Then we obtain the optical value of the proton charge radius... 

Individual uncertainties of the proton and deuteron charge radii introduce by far the largest contributions in the uncertainty of the theoretical value of the isotope shift. Uncertainty of the charge radii are much larger than the experimental error of the isotope shift measurement or the uncertainties of other theoretical contributions. It is sufficient to recall that uncertainty of the 15 Lamb shift due to the experimental error of the proton charge radius is as large as 50 kHz (see (12.11)), even if we ignore all problems connected with the proton radius contribution (see discussion in Subsects. 12.1.5, 12.1.6). 

After calculation of these corrections, the uncertainty in the sum of all theoretical contributions except those which are directly proportional to the proton radius squared will be determined by the uncertainty of the proton polarizability contribution of order (Za). This uncertainty of the proton polarizability contribution is currently about 0.002 meV, and it will be difficult to reduce it in the near future. If the experimental error of measurement... 

P — 2S Lamb shift in hydrogen will be reduced to a comparable level, it would be possible to determine the proton radius with relative error smaller that 3 X 10 or with absolute error about 2 x 10 fm, to be compared with the current accuracy of the proton radius measurements producing the results with error on the scale of 0.01 fm. 

The very negative value of the estimated enthalpy of hydration of the proton is consistent with a Born radius of 63 pm (from equation 2.43). Since that equation overestimates the ionic radius by —67 pm it follows that a bare proton (radius —0 pm) would be expected to have such a very negative enthalpy of hydration. Studies of the proton and water in the gas phase have shown that the stepwise additions of water molecules in the reaction ... 

In Table II, energy levels for two distances r = 6.36 x 10 16 m and rfs = 1.417 x 10 16 m which are under the limit hcp are also presented. This table covers six energy levels altogether. In Table II and positions of Xce and Xcp, the proton radius rp, and the proton-neutron distance in the deuteron, are also presented. 

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. 

We are setting up an experiment to measure the energy difference AE 2 3 S i /2 — 2 3/2) in a laser resonance experiment to a precision of 30 ppm, which corresponds to 10% of the natural linewidth, to deduce the proton radius with 10-3 relative accuracy, 20 times more precise than presently known. An important pre-... 

Experiments in optical spectroscopy of electronic hydrogen have come to the point where a precise value of the proton radius is urgently needed for the interpretation of the results. Laser technology and X-ray detection schemes have... 

A precision of 30 ppm for the Lamb shift in muonic hydrogen, i.e. an uncertainty on the proton radius of 10-3, can therefore be achieved within a reasonably short measuring time. Better accuracy of the proton radius will then be limited by effects of proton polarizability [36,13]. 

Theoretical values of 6 were calculated by Mohr (5 = 1057.864(14) /4/ and by Erickson (5 = 1057.910(10)) /5/. Experimental values of the proton radius were obta-... 

Hand, Miller and Wilson (1/2 = 0.805(11) fm) /7/. Comparison of experimental values of the proton radius with theoretical values of S were performed by Bhatt and Groteh /8/ and by Kinoshita and Sapirstein /9/. A set of new theoretical corrections introduced by these authors results in... 

The experiment and theory are in excellent agreement if the old value for the proton radius is used the agreement is poor if the new value of the proton radius is used. Because of the discrepant values for the proton radius, one cannot say how well experiment and theory agree. There are also uncalculated theoretical contributions which could be as large as 10 kHz.[3]... 

Reference to the hydrogen ion, H" ", is also common. However, in the presence of solvent, the extremely small size of the proton (radius approximately 1.5 X 10" pm) requires that it be associated with solvent molecules or other dissolved species. In aqueous solution, a more correct description is H30 (a ), although larger species such as H9O4 are also likely. Another important characteristic of H" " that is a consequence of its small size is its ability to form hydrogen bonds. 

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-... 

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