Hydrogenlike ion

The theory of high order corrections to the Lamb shift described above for H and D may also be applied to other light hydrogenlike ions. The simplest such ion is He+. Originally the classic Lamb shift in 17e+ was measured in [50] by the quenching-anisotropy method with the result L 2Si — 2Pi,17e+) = 14 042.52 (16) MHz. Later the authors of [50] discovered a previously unsuspected source of systematic error in their experiment. Their new measurement of the classic Lamb shift in 17e+ by the anisotropy method resulted in the value L 2Si — 2Pi,He ) = 14 041.13 (17) MHz [51]. Besides the experimental data this result depends also on the theoretical value of the hne structure interval. In [51] the value AE 2Pz —2Pi) = 175 593.50 (2) MHz was used. We recalculated this interval using tire latest theoretical results discussed above and obtained AE 2P3 — 2Pi) = 175 593.33 (1) MHz. Then the value of the... 

Recently, a new setup was developed [22] and tested [23] which allows the storage of a single hydrogenlike ion in a trap and the performance of gj measurements with a precision up to a few ppb [24], Up to now, experiments have been carried out on 12C5+ but are planned to be extended to systems up to U91+. The most recent experimental value is presented in detail elsewhere in this volume [25]. The experimental success demands a close look on the theoretical contributions to the g factor of an electron bound in a hydrogenlike system. In the following we will present the current status of these effects for the case of carbon. 

This PWE was used in [18] to obtain the numerical results. For the numerical implementation the B-spline approximation [21] was chosen that represents actually the refined version of the space discretization approach. In Table 1 the convergence of the PWE approach with the multicommutator expansion is presented for the lowest-order SE correction for the ground state of hydrogenlike ions with Z = 10. The minimal set of parameters for the numerical spline calcuations was chosen to be the number of grid points N = 20, the number of splines k = 9. This minimal set allowed to keep a controlled inaccuracy below 10%. What is most important for the further generalization of the PWE approach to the second-order SESE calculation is that with Zmax = 3 the inaccuracy is already below 10% (see Table 1). The same picture holds with even higher accuracy for larger Z values. The direct renormalization approach is not necessarily connected with the PWE. In [19] this approach in the form of the multicommutator expansion (Eq. (16)) was employed in combination with the Taylor expansion in powers of (Ea — En>)r 12 The numerical procedure with the use of B-splines and 3 terms of Taylor series yielded an accuracy comparable with the PWE-expansion with Zmax = 3. 

M. Diederich, H. Haffner, N. Hermanspahn, M. Immel, H. J. Kluge, R. Ley, R. Mann, S. Stahl, W. Quint, J. Verdu, and G. Werth The g-faktor of hydrogenlike ions . In Trapped charged particles and fundamental physics, AIP conference proceedings ed. by D. H. E. Dubin and D. Schneider (American Institute of Physics, Woodbury, New York 1999) pp. 43—51... 

The Bohr model can be readily extended to hydrogenlike ions, systems in which a single electron orbits a nucleus of arbitrary atomic number Z. Thus, Z = 1 for hydrogen, Z = 2 for He+, Z = 3 for Li++, and so on. The Coulomb potential (7.5) generalizes to... 

The first ionization energy, Ii, the minimum energy required to remove the first electron from helium, is experimentally 24.59 eV. The second ionization energy, h, is 54.42 eV, which can be calculated exactly since He+ is a hydrogenlike ion. We have... 

In reality, Bohr s theory accounted for the observed emission spectra of He" and Li " ions, as well as that of hydrogen. However, all three systems have one feature in common— each contains a single electron. Thus the Bohr model worked successfully only for the hydrogen atom and for hydrogenlike ions. 

The He ion contains only one electron and is therefore a hydrogenlike ion. Calculate the wavelengths, in increasing order, of the first four transitions in the Balmer series of the He" ion. Compare these wavelengths with the same transitions in a H atom. Comment on the differences. (The Rydberg constant for He-" is 8.72 X 10 J.)... 

The operation of the selection rule for l for hydrogen and hydrogenlike ions can be seen by the study of the fine structure of the lines. The phenomena are complicated, however, by the influence of electron spin.1 In alkali atoms the levels with given n and varying l are widely separated, and the selection rule for l plays an important part in determining the nature of their spectra. Theoretical calculations have also been made of the intensities of lines in these spectra with the use of wave functions such as those described in Chapter IX, leading to results in approximate agreement with experiment. 

In the previous sections we have collected all corrections which contribute to the Lamb shift in hydrogenlike ions. For the radiative corrections from QED it was... 

Fig. 20. Comparision of experimental measured values for the Lamb shift in hydrogenlike ions and the theoretical predicition. The energy shift is presented in the dimensionless unit F Za) similar to Fig. 10.

From this second point of view the many-body problem is simply a nuisance, and in fact hydrogen or hydrogenlike ions are generally considered to be the best places to search for new physics. The new physics that will be treated in this chapter is that of the weak interactions, which lead to parity nonconserving (PNC) transitions in atoms. While this effect has... 

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. 

Why is it much harder to explain the line spectra of polyelec-tronic atoms and ions than it is to explain the line spectra of hydrogen and hydrogenlike ions ... 

We consider the electron in a hydrogen atom or hydrogenlike ion (He, Li +, ) orbiting around a nucleus of atomic number Z. The attractive Coulomb potential in atomic units e /ATt Q = 1) can be written as... 

Equation (2-15) permits the calculation of the frequency of the photon emitted for any electronic transition of hydrogenlike ions consequently, the wavelength of the transition can be calculated from the expression A = c/v. The calculated wavelengths and experimental wavelengths agree closely in all cases for all the series identified for hydrogen. 

A hydrogenfike ion is an ion containing only one electron. The energies of the electron in a hydrogenlike ion are given by... 

As a result of the spherical symmetry of the potential energy, the energy of the atomic orbitals for hydrogen and hydrogenlike ions depends only upon the value of the principal quantum number (n), and is given by... 

To summarize, the quantum state of an electron in hydrogen (or a hydrogenlike ion) is completely specified by the four quantum numbers n, /, mi, and m,. The energy of this electron is determined only by the value of the principal quantum number n (according to Equation 1.37). The angular momentum quantum number I determines the tl etfc, n r icmantum number mi determines its... 

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