Vacuum Polarisation

Fig. 1. Feynman diagrams representing various contributions to the Lamb shift. A solid line represents an electron, a wavy line a virtual photon and a cross denotes exchange of a Coulomb photon (a) Leading self-energy term (b) One-loop vacuum polarisation term. The loop represents a virtual electron-positron pair (c) Some diagrams contributing to the two-loop binding correction...

In the subsequent discussion we shall need the irreducible 2- and 3-point functions i.e. the electron self energy Sy(p), the vacuum polarisation /7k,pv( ) and the full vertex function rv, PuP2% as knowledge of I vip) and determines the corresponding propagators Gy(p) and Dk,mv( ) completely and fy. iPuPz) represents the perturbative corrections to the free vertex The connection between these quantities is established by the Dyson equations (see e.g. [26])... 

For the subsequent discussion we also note the normalisation condition for the vacuum polarisation kernel real electrons, i.e. to reproduce the Coulomb interaction for well separated electrons, (Oy g q ) must vanish for vanishing momentum,... 

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... 

FIGURE 5.1 Various individual contributions to the ground-state Lamb shift in H-like uranium together with the experimental accuracies achieved so far [13-16], SE and VP denote the self energy and the vacuum polarisation contributions, respectively. 

TABLE 5.1 Comparison of our experimental result for Z = 92 with the RMBPT calculations of Persson [25]. 2eSE and 2eVP denote two-electron self-energy and vacuum polarisation contributions, respectively. All values are in eV... 

The exchange of a muon for the electron in hydrogen leads to an atomic system having considerably enlarged QED effects. However, in the case of pp and pHe the major contribution to the Lamb shift comes now from the vacuum polarisation rather than the self-energy... 

The Lamb shift and fine structure (2s - 2pi/a,3/2) in the lightest muonic atoms corresponds to wavelengths in the vicinity of 6 ym for yH, 812 nm and 897 nm for y He and 822 nm and 1061 nm for y He. In experiments analogous to the rf measurements on n= 2 of hydrogen the muonic measurements lead to precise tests of the vacuum polarisation and to the determination of the nuclear charge radii. Good experimental precision can be attained using laser resonance techniques, and the natural linewidth of the Lamb shift transition in yH for example is only 300 ppm while for yHe it is 1000 ppm. 

To date the muonic Lamb shift has only been measured by Zavattini and co-workers in y He using the infrared transition at 897 nmC32] jf the He rms charge radius is taken from electron scattering experiments, the vacuum polarisation contribution can be verified to 1700 ppm which is at present one of the best tests of a specific QED effect. A similar experiment is currently underway at SIN. 

AE all-electron HF Hartree-Fock DHF DIrac-Hartree-Fock DC DIrac-Coulomb-Hamlltonlan +B Breit interaction in quasi-degenerate perturbation theory +QED quantum electrodynamic corrections (vacuum polarisation, self-energy) p.n. point nucleus f.n. finite (Fermi) nucleus exp. experimental data. 

This leaves option 3b to be scrutinised closely. When the present writer did this, he realised that his puzzlement had arisen because he like others, had fallen into the trap of which he had frequently warned his students and which he has emphasised in his writings it is a serious error to attempt to understand electrochemical phenomena by thinking of ions in isolation, because this puts them putatively into a vacuum. But the ions of concern to us do not exist in a vacuum. Ions would not leave their positions of low energy in a crystal lattice to go into solution or be formed from neutral molecules by the transfer of a charged fragment from one molecule to another if those processes were not made exo-energetic by the interaction of the ions with polar or polarisable species in their environment, most commonly the solvent. For that reason, one should always think, and indeed talk, about... 

For accurate measurements of conductivity under normal conditions it is usual to use platinised Pt electrodes because such electrodes are less affected by polarisation, but this type of electrode is not only difficult to reproduce when it is incorporated into a complex vacuum apparatus, it is difficult to clean, not very robust, and can produce degassing problems. Therefore, they are not recommended for conductivity measurements in vacuo. 

In order to suppress the formation of elliptically polarised light rather than plane polarised light in a TN-LCD, it is necessary to control the cell thickness, d, and the birefringence of the nematic medium. An, with respect to the wavelength of light in a vacuum, X, according to the equation ... 

As far as explicit approximations for the polarisation functions Tltidq) are concerned only very little is known, even in the static limit. The complete frequency dependence is available for the noninteracting limit ( ). i e. the relativistic generalisation of the Lindhard function [95, 114]. In addition to its vacuum part (A.26) one has... 

The magnetic coil shown in figs. 2.15 and 2.16 was used to orient the electron polarisation vector P parallel to the axis of the analysing target—Mott detector system. The deflection system is part of the differential pumping stage which is necessary for the maintenance of the required ultra-high vacuum in the source chamber. 

For transitions in the vacuum ultraviolet it is usually easier to measure angular rather than polarisation correlations, due to the difficulty in measuring polarisation in this wavelength region. 

---