Helium Bethe logarithm

Table 6. 1/n expansion coefficients bi for the Bethe logarithms of helium. The coefficients di and da give the finite mass correction due to mass polarization effects on the wave function. See Eqs. (16) and (17)... 

This paper reviews progress in the application of atomic isotope shift measurements, together with high precision atomic theory, to the determination of nuclear radii from the nuclear volume effect. The theory involves obtaining essentially exact solutions to the nonrelativistic three- and four-body problems for helium and lithium by variational methods. The calculation of relativistic and quantum electrodynamic corrections by perturbation theory is discussed, and in particular, methods for the accurate calculation of the Bethe logarithm part of the electron self energy are presented. The results are applied to the calculation of isotope shifts for the short-... 

The basis sets for helium and lithium are more complicated in detail but the principles are the same. In each case the Bethe logarithm comes almost entirely from virtual excitations of the inner Is electron to p-states lying high in the photoionization continuum, and so the basis set must be extended to very short distances for this particle. The outer electrons are to a good approximation just spectators to these virtual excitations. 

Results for the low-lying states of helium and the He-like ions are listed in Table 4.4 (see also Korobov [39]). In order to make the connection with the hy-drogenic Bethe logarithm more obvious, the quantity tabulated is ln(A o/Z Ry). The effect of dividing by a facfor of is to reduce all the Bethe logarithms to approximately the same number y3(ls) = 2.984128556 for the ground state of hydrogen. It is convenient to express the results in the form /3(lsnL) = /3(ls) -F A/3(nL)/n, where A/3(nL) is a small number that tends to a constant at the series limit. 

The principle message of this paper is that few-body atomic systems such as helium and lithium can be solved essentially exactly for all practical purposes in the nonrelativistic limit, and there is a systematic procedure for calculating the relativistic and other higher-order QED corrections as perturbations. The solution of the problem of calculating Bethe logarithms means that the theoretical energy levels are complete up to and including terms of order Ry. 

The principal uncertainty in evaluating E 2 for the low-lying states of helium-like ions is the value of the Bethe logarithm In defined in analogy with (8). Calculations of reasonable... 

Table 2. Comparison of Bethe Logarithms for the Ground State of Helium-like ionso...

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