0 electrodynamics corrections

There exist a number of methods to account for correlation [17, 45, 48] and relativistic effects as corrections or in relativistic approximation [18]. There have been numerous attempts to account for leading radiative (quantum-electrodynamical) corrections, as well [49, 50]. However, as a rule, the methods developed are applicable only for light atoms with closed electronic shells plus or minus one electron, therefore, they are not sufficiently general. 

Account has been taken here of the fact that the Lande factor gi which is connected with the electronic orbital momentum precisely equals unity gi = —1, whilst the spin-connected one equals gs = —2.0023. The recommended gs and hb values can be found in Table 4.1. The discrepancy of the gs value from two is due to quantum electrodynamical correction. It is important to mention that, in agreement with the definition of Cl = A+2 as a positive value (see Section 1.2) and gi, gs as negative (see Section 4.1), we have from (4.54) that if 2 < A, but <7S2 > <7/A, hq can take a positive value because Hu and Cl possess the same direction (positive Lande factor go). In other cases g,Q is negative, (J-Q being directed opposite to Cl as shown in Fig. 4.24 (negative go). 

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. 

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

The correction B(i,j) to the Coulomb potential is treated as a perturbation of the zero-order Hamiltonian, and may include relaxation effects, correlations, quantum electrodynamic corrections and the relativistic retardation of the two-electron potential. 

Up to now, we did not discuss any expansions in Z a series for the diagrams including vacuum polarisation contributions. For the lsi/2-state of hydrogenlike lead and uranium we present in Table 1 a direct comparison between these values and our direct numerical calculations [7]. From this presentation it is evident that a Za expansion also for the quantum electrodynamical corrections of order is completely... 

These agree very satisfactorily with the theoretical value 2 0337 X 105 Mc/s (equation 12.6) which includes the quantum electrodynamic corrections to order J a3. The uncorrected value, 2 044 x 106 Mc/s, is clearly inadequate. The importance of the annihilation term is demonstrated beyond doubt. 

Highly-ionized atoms DHF calculations on isoelectronic sequences of few-electron ions serve as the starting point of fundamental studies of physical phenomena, though many-body corrections are now applied routinely using relativistic many-body theory. Relativistic self-consistent field studies are used as the basis of investigations of systematic trends in ionization energies [137-144], radiative transition probabilities [145-148], and quantum electrodynamic corrections [149-151] in few-electron systems. Increased experimental precision in these areas has driven the development of many-body methods to model the electron correlation effects, and the inclusion of Breit interaction in the evaluation of both one-body and many-body corrections. 

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-... 

In addition, quantum electrodynamic corrections to the Coulomb interaction modify the potential at short range we refer here to the sum of all these corrections as the Lamb shift of a level, following Johnson and Soff and Mohr ] and note that this definition differs from that of Ericksont ]. In particular I shall be concerned with the... 

The quantum electrodynamical corrections to these Dirac-level results are small. The leading one is the correction factor of 1.001 159 652 193(4) to the free-electron -factor of —2. 

To explain these differences the quantum electrodynamic corrections have to be implemented. The velocity of light is finite and this means retardation of the interparticle interactions. This means that the Dirac-Coulomb Hamiltonian has to be corrected by further expressions. 

AE, all-electron results HF, Haitree-Fock DHF, Dirac-Hartree-Fock (spin-otbit averaged values) +QED, DHF including the Breit interaction and quantum electrodynamic corrections WB, Wood-Boring. 

C. Thierfelder, P. Schwerdtfeger. Quantum electrodynamic corrections for the valence shell in heavy many-electron atoms. Phys. Rev. A, 82 (2010) 062503. 

M. Douglas, N. M. Kroll. Quantum Electrodynamical Corrections to the Fine Structure of Helium. Ann. Phys., 82 (1974) 89-155. 

---