Relativistic states formalism

We will review here experimental tests of quantum electrodynamics (QED) and relativistic bound-state formalism in the positron-electron (e+,e ) system, positronium (Ps). Ps is an attractive atom for such tests because it is purely leptonic (i.e. without the complicating effects of nuclear structure as in normal atoms), and because the e and e+ are antiparticles, and thus the unique effects of annihilation (decay into photons) on the real and imaginary (related to decay) energy levels of Ps can be tested to high precision. In addition, positronium constitutes an equal-mass, two-body system in which recoil effects are very important. 

To summarize, the p + NR elastic scattering amplitude is conveniently calculated using the relativistic DWBA formalism, where the following dianges from the procedure desoibed in ref. [Ra 88b] are carried out (1) NR distorted waves and bound state wave functions are used for the upper components (2) the lower components are set to the NR or free partide limit [i.e., no potential terms included in (a k)/ E + m)] (3) ) corresponds to the dioice of t according to eq. (6.5) and may include density dependence (4) the pA recoil factor, EJ E + ), is included. 

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The first rigorous derivation of such a relativistic Hamiltonian for a two-fermion system that makes use of Feynman [13,14] formalism of QED was due to Bethe and Salpeter [30,31]. Recently, Broyles has extended it to many-eleetron atoms and molecules [32]. A detailed account of Broyle s derivation ean be found elsewhere [32,33] and will not be repeated here. Following Broyles, the stationary state many-fermion Hamiltonian based on QED ean be written as... 

The relativistic version (RQDO) of the quantum defect orbital formalism has been employed to obtain the wavefunctions required to calculate the radial transition integral. The relativistic quantum defect orbitals corresponding to a state characterized by its experimental energy are the analytical solutions of the quasirelativistic second-order Dirac-like equation [8]... 

Persson et al. states that the missing correlation effects in their two-electron QED calculations is estimated to be of the order of 0.1 eV for all elements. Formally this should only be only be applied to the range of elements 32 < Z < 92. The associated uncertainties for Z < 32 are unknown, but could be expected to increase in this regime. In the calculations of Drake, the uncertainty due to relativistic correlation effects in QED scales as a4Z4. The sources of the uncertainty are quite different in the calculations of Drake, and of Persson et al.. The lowest order Lamb shift is of order a3 ZA, and so the leading two-electron correction is of order a3Z3, i.e. smaller by a factor of 1 /Z. Higher order correlation... 

When describing arbitrary two-body systems fully relativistically, one would expect that the formalism produces explicitly CPT-invariant results. CVT-invariance requires symmetry of the terms under change of the sign of the system s total energy E + E% = E <— —E, also for bound states where the individual energies Ei and E% are not conserved. This means that the decisive equations should contain only even powers of E. 

We also mention that recently a density functional approach to excited states of relativistic systems has been formulated [45], using ensembles of unequally weighted states. This formalism is restricted to the electrostatic limit and the no-sea approximation (see Section 3.2). Moreover, it remains unclear how the spontaneous emission of photons, which is possible in QED in contrast to the standard nonrelativistic many-body theory, is handled for the excited states involved. 

The relativistic enhancement of the subshells with j = 112 is so large that in the elements 165 to 168 the 9s and 9pi/z states will be occupied instead of the 8pz/2 state. Hence the filling of the 8 3/2 electrons can occur only in elements 169 to 172. This surprising result makes it possible to give the formal continuation of the periodic table shown in Fig. 21, because there are six p electrons available from two different shells which are energetically very close, so that they will nicely form a normal p shell. Therefore, the 9 th period will be quite analogous to the 2nd and 3rd periods in the periodic system. This continuation and the differences from the normal expected continuation are discussed below. 

Thus, the main relativistic effects are (1) the radical contraction and energetic stabilization of the s and p orbitals which in turn induce the radial expansion and energetic destabilization of the outer d and f orbitals, and (2) the well-known spin-orbit splitting. These effects will be pronounced upon going from As to Sb to Bi. Associated with effect (1), it is interesting to note that the Bi atom has a tendency to form compounds in which Bi is trivalent with the 6s 6p valence configuration. For this tendency of the 6s electron pair to remain formally unoxidized in bismuth compounds (i.e. core-like nature of the 6s electrons), the term inert pair effect or nonhybridization effect has been often used for a reasonable explanation. In this context, the relatively inert 4s pair of the As atom (compared with the 5s pair of Sb) may be ascribed to the stabilization due to the d-block contraction , rather than effect (1) . On the other hand, effect (2) plays an important role in the electronic and spectroscopic properties of atoms and molecules especially in the open-shell states. It not only splits the electronic states but also mixes the states which would not mix in the absence of spin-orbit interaction. As an example, it was calculated that even the ground state ( 2 " ) of Bij is 25% contaminated by Hg. In the Pauli Hamiltonian approximation there is one more relativistic effect called the Dawin term. This will tend to counteract partially the mass-velocity effect. 

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