Lowest order results

We use four potentials here. We have already mentioned the Coulomb and core-Hartree potentials. We will also use the modified core-Hartree potential (MCH) and the Kohn-Sham (KS) potentials. The former is obtained from the CH potential by replacing the factor 2r by r in Eq. 14, and the latter by using 

It is important to stress that none of these potentials is in any sense exact. They simply represent starting points for QED perturbation theory, which theory should build in the actual physics arising from inter-electron interactions perturbatively. 

It is a simple matter to solve the Dirac equation in all four potentials, and the results for the 2 3/2 — 2si/2 transition in lithiumlike bismuth are collected in the first row of Table 1. We also give the 2si/2 and 2p i2 energies separately in Tables 2 and 3 respectively. We note that we will always drop any term contributing to the energy that affects only the core states. These cancel out of the transition we are studying, and also do not affect the valence removal energies. 

Cumulative lowe.st order, one-photon, and two-photon contributions to the 2pa/2 — 2si/2 transition energy of lithiumlike bismuth, 2788.139(37) eV, in different potentials units eV. 

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The systematie development of the equations needed to determine the Ek and the /k is presented in Appendix D. Here, we simply quote the few lowest-order results. 

We now proceed to carry out a consistent implementation of MBPT based on parity-mixed single-particle states. The above result (88) is the lowest-order result in such a perturbation theory, corresponding to the parity-mixed HF . To evaluate the second-order corrections, we linearize the second-order amplitude from Eq. (69) in the weak interaction. As discussed in the previous section, we use RPA amplitudes, rather than lowest-order amplitudes on the RHS of Eq. (69). Evaluating the resulting expression, one obtains the correction... 

One can sensibly ask why we combine the lowest-order results for the SR and LR terms, which are of 0(y") for fixed r and of O(y ) for fixed yr, respectively, rather than to take the result to order 0(y ) for fixed r together with which is of order O(y ) for fixed yr. The answer lies in the fact that in general the thermodynamic contribution of J d( ) d(2)p(l)p(2)c(l 2) to a given order of y involves the SR functions for fixed r to that order, but the LR functions to one order higher in y for fixed yr. (See LSB for further details.) We feel this question deserves further consideration, however. 

The first term on the right-hand side of (62) is of order and the second and third of order y. There is one further term of order y (not shown) and a remainder of order y. Thus the lowest-order result for F (l 2) is... 

To go beyond (66) we shall proceed to second order in F ordering, supplemented by some further approximations to simplify the form of our final results. The second-order F-ordered approximation, which we shall call SOGA, requires knowledge of F (1 2), and the logic of F ordering demands that the F (l 2) used in the expression F(1 2) = F (1 2)+F (l 2), like 2) itself, be exact to one higher order in y than the lowest-order result. Thus we seek an F (l 2) that is exact through y for fixed ti2 to accompany an... 

Despite this essential difference, the perturbative analyses of the individual cases have much in common. Further it will be shown that the lowest order results for 77 = 2 are those obtained for the model Hamiltonian W [Eq. (8a)j. Thus Sec. 2 contains most of the tools necessary to analyze the 77 = 2 case. For this reason, in this section, to the extent possible, we treat the most general, 77 = 5 case (the Af odd, spin-orbit included, no symmetry case), first and then show how the simpler 77 = 3 case (the odd electron, spin-orbit included, Cg or greater symmetry, case) and 77 = 2 (the N even, spin-orbit included case and the non-relativistic case) can be recovered as limits. On the other hand, the 77 = 5 and 3 results are quite new compared to those for 77 = 2. Thus, as the subsequent presentation attests, the formalism for 77 = 2 is further advanced and more substantative numerical studies have been carried out. 

Note the peculiar feature that 2.3 times the lowest order result (LO) is remarkably close to the higher order one (LO-I-NLO). This so-called RT-factor also shows up in the Drell-Yan process (see Section 17.4). 

Note that the lowest-order result (3.4.10a) is recovered when (4.1.20) is substituted into (4.1.17) with v = 1. 

When combined with boundary layer analysis, potential flow theory provides estimates for the velocity of a rising bubble. The lowest-order result for the terminal velocity of a freely rising bubble is... 

The lowest-order result for the total-energy flux approach is solely based on the use of the expansion vector and can be written as... 

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