Energy Dirac

Let us emphasize once more that hyperfine structure, radiative, recoil, radiative-recoil, and nonelectromagnetic corrections are all missing in the Dirac energy spectrum. Discussion of their calculations is our main topic below. 

We will first discuss corrections to the basic Dirac energy levels which arise in the external field approximation. These are leading relativistic corrections with exact mass dependence and radiative corrections. 

This equation has the same contributions of order (Za)" as in (3.4), but formally this expression also contains nonrecoil and recoil corrections of order Zaf" and higher. The nonrecoil part of these contributions is definitely correct since the Dirac energy spectrum is the proper limit of the spectrum of a two-particle system in the nonrecoil limit m/M = 0. As we will discuss later the first-order mass ratio contributions in (3.5) correctly reproduce recoil corrections of higher orders in Za generated by the Coulomb and Breit exchange photons. Additional first order mass ratio recoil contributions of order (Za) ... 

However, there still remain QED corrections to the electron-electron interaction and purely relativistic corrections of order a4Z6 corresponding to the one-electron Dirac energies of this order and their two-electron corrections. In a recent series of papers, Pachucki [52,53,54] has made important progress in the derivation of effective operators for the relativistic corrections for triplet states. The results can be expressed in the form... 

The energy levels generated by this formula are those you are all entirely familiar with. They are the Dirac energy levels. I need hardly say that a is the fine structure constant, now written as e2/ c[/i0c2/4x], that you will recognise k as (j +i) and discover that (k+w) has the same values as our present integer n, which is Bohr s n. 

Such comparisons promise interesting tests of QED. Unfortunately, however, the theory of hydrogen is no longer simple, once we try to predict its energy levels with adequate precision [36]. The quantum electrodynamic corrections to the Dirac energy of the IS state, for instance, have an uncertainty of about 35 kHz, caused by numerical approximations in the calculation of the one-photon self-energy of a bound electron, and 50 kHz due to uncalculated higher order QED corrections. 

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

The deviation of all corrections from this Dirac energy eigenvalue is widely known as Lamb shift of a state nlj. This is an analogy to the classical Lamb shift, which is the energy difference between the 2sj/2 and the 2pi/2 states. However, by convention [12] the so-called nonrelativistic reduced mass correction ... 

The two most dominant corrections to the Dirac energy eigenvalue are the self energy and the vacuum polarization. The leading diagrams are shown in Fig. 3. They contain only one photon line and are thus of order a. Therefore they yield the significant contributions of both effects. The diagrams of order which contain two photon lines will be discussed later on. 

In order to investigate the performance of the various orders of the DK method in greater detail, it is convenient to examine the dependence of the results on the nuclear charge Zina systematic manner and compare it to the well-known dependence of the exact Dirac eigenvalues on Z. The exact Dirac energy for lsi/2 levels is given by... 

In equation (58), e is the electron charge, c is the speed of light,. / A- (r) is the vector potential describing the magnetic interactions, and denotes the strength parameter of the vector potential. Differentiation of the Dirac energy... 

The equations derived in Sections 2, 3, and 4 have been implemented as described in the previous Section and the methods have been applied on one-electron atoms. The energies for the lowest l.v states obtained using the infinite-order regular approximation (lORA) Hamiltonian [52] are given in Table I. A few Dirac energies are for comparison listed in Table 2. 

Figure 2. The difference between the Dirac and the lORA energies as a function of nuclear charge. The energy difference between the ERA and the Dirac energies is also shown.

The difference between the energies obtained at the lORA level and the Dirac energies (AE) are plotted in Figure 2 as a function of the nuclear charge (Z). The same graph for the ERA model is also shown in the figure. It can clearly be seen that for smaller Z values both the ERA and the lORA models scale with the same power of aZ, whereas for the heavier elements the slope of the ERA curve is significantly smaller than Z. ... 

Figure 5. The difference between the ERA energy and the Dirac energy for the the 2.v, 2 state of as a function of the scaling factor y.

Dirac energy when the optimal y value of 1.24 is used than with y = 1. 

A similar study for the first excited s state shows that at the ERA level, the optimal y value of 1.7 provides a quasi-relativistic energy that is about a factor of 4 closer to the Dirac energy than when the default y value of 1.0 is used. The optimal y value obtained in the ERA+PT calculations on the 2.Vj 2... 

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