Recoil correction

Dirac theory (incl. binding) Finite-size correction Recoil... 

The density here refers to the spatial coordinate, i.e. the concentration of the reaction product, and is not to be confused with the D(vx,vy,vz) in previous sections which refers to the center-of-mass velocity space. Laser spectroscopic detection methods in general measure the number of product particles within the detection volume rather than a flux, which is proportional to the reaction rate, emerging from it. Thus, products recoiling at low laboratory velocities will be detected more efficiently than those with higher velocities. The correction for this laboratory velocity-dependent detection efficiency is called a density-to-flux transformation.40 It is a 3D space- and time-resolved problem and is usually treated by a Monte Carlo simulation.41,42... 

Once we have the appropriate nuclide, we must separate the radiation of interest from all other radiation present. A typical gamma spectrum is shown in Figure 3 for cobalt-57 in palladium. The radiations which can be identified include the 6-k.e.v. x-ray, the 14-k.e.v. y-ray of interest, and a sum peak and palladium x-ray peak, both lying at about 21 k.e.v. If one now sets the single-channel analyzer window correctly, one observes essentially only the 14-k.e.v. peak, but all of this is not recoil-free radiation it includes other radiation which falls into the window from various gamma quantum de-excitation processes. 

As mentioned, most calculations we have done so far have concerned molecular systems. However, prior to development of the non-BO method for the diatomic systems, we performed some very accurate non-BO calculations of the electron affinities of H, D, and T [43]. The difference in the electron affinities of the three systems is a purely nonadiabatic effect resulting from different reduce masses of the pseudoelectron. The pseudoelectrons are the heaviest in the T/T system and the lightest in the H/H system. The calculated results and their comparison with the experimental results of Lineberger and coworkers [44] are shown in Table 1. The calculated results include the relativistic, relativistic recoil. Lamb shift, and finite nuclear size corrections labeled AEcorr calculated by Drake [45]. The agreement with the experiment for H and D is excellent. The 3.7-cm increase of the electron affinity in going from H to D is very well reproduced by the calculations. No experimental EA value is available for T. 

VP vacuum polarization SE self-energy part of the Lamb shift LS = VP + SEE Lamb shift RC nucleus recoil correction, polarization Relativistic PT accounts for the main relativistic and correlation effects HOPT higher-order PT contributions. Data are from refs [1-10]. 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

Contributions to the energy which depend only on the small parameters a. and Za. are called radiative corrections. Powers of a arise only from the quantum electrodynamics loops, and all associated corrections have a quantum field theory nature. Radiative corrections do not depend on the recoil factor m/M and thus may be calculated in the framework of QED for a bound electron in an external field. In respective calculations one deals only with the complications connected with the presence of quantized fields, but the two-particle nature of the bound state and all problems connected with the description of the bound states in relativistic quantum field theory still may be ignored. 

Corrections which depend on the mass ratio m/M of the light and heavy particles reflect a deviation from the theory with an infinitely heavy nucleus. Corrections to the energy levels which depend on m/M and Za are called recoil corrections. They describe contributions to the energy levels which cannot be taken into account with the help of the reduced mass factor. The presence of these corrections signals that we are dealing with a truly two-body problem, rather than with a one-body problem. 

Leading recoil corrections in Za (of order (Za) (m/M)") still may be taken into account with the help of the effective Dirac equation in the external field since these corrections are induced by the one-photon exchange. This is impossible for the higher order recoil terms which reflect the truly relativistic two-body nature of the bound state problem. Technically, respective contributions are induced by the Bethe-Salpeter kernels with at least two-photon exchanges and the whole machinery of relativistic QFT is necessary for their calculation. Calculation of the recoil corrections is simplified by the absence of ultraviolet divergences, connected with the purely radiative loops. 

Radiative-Recoil corrections are the expansion terms in the expressions for the energy levels which depend simultaneously on the parameters a, m/M and Za. Their calculation requires application of all the heavy artillery of QED, since we have to account both for the purely radiative loops and for the relativistic two-body nature of the bound states. 

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. 

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) ... 

Recoil corrections depending on odd powers of Za are also missing in (3.5), since as was explained above all corrections generated by the one-photon exchange necessarily depend on the even powers of Za. Hence, to calculate recoil corrections of order Za) one has to consider the nontrivial contribution of the box diagram. We postpone discussion of these corrections until Sect. 4.1. 

The external field approximation is clearly inadequate for calculation of the recoil corrections and, in principle, one needs the machinery of the relativistic two-particle equations to deal with such contributions to the energy levels. The first nontrivial recoil corrections are generated by kernels with two-photon exchanges. Naively one might expect that all corrections of order Za) m/M)m are generated only by the two-photon exchanges in Fig. 4.1. However, the situation is more complicated. More detailed consideration shows that the two-photon kernels are not sufficient and irreducible kernels in Fig. 4.2 with arbitrary number of the exchanged Coulomb pho-... 

Complete formal analysis of the recoil corrections in the framework of the relativistic two-particle equations, with derivation of all relevant kernels, perturbation theory contributions, and necessary subtraction terms may be performed along the same lines as was done for hyperfine splitting in [3]. However, these results may also be understood without a cumbersome formalism by starting with the simple scattering approximation. We will discuss recoil corrections below using this less rigorous but more physically transparent approach. 

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