The Braun Formula

Before returning to the recoil corrections of order (ZoLf we will digress to the Braun formula. We will not give a detailed derivation of this formula, referring the reader instead to the original derivations [16, 17, 18, 14]. We will however present below some physically transparent semiquantative arguments which make the existence and even the exact appearance of the Braun formula very natural. 

Another useful perspective on the Braun formula is provided by the idea, first suggested in the original work [16], and later used as a tool to rederive (4.13) in [18, 14], that the recoil corrections linear in the small mass ratio m/M are associated with the matrix element of the nonrelativistic proton 

There is a clear one-to one correspondence between the terms in this non-relativistic Hamiltonian and the respective terms in (4.13). The latter could be obtained as matrix elements of the operators which enter the Hamiltonian in (4.16) [18, 14]. 

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Lower Order Recoil Corrections and the Braun Formula... 

Being exact in the parameter Za and an expansion in the mass ratio m/M the Braun formula in (4.13) should reproduce with linear accuracy in the small mass ratio all purely recoil corrections of orders (Za) (m/M)m, (Za)4(m/M)m, Zaf m/M)m in (3.5) which were discussed above. 

The recoil correction in (4.19) is the leading order (Za) relativistic contribution to the energy levels generated by the Braun formula. All other contributions to the energy levels produced by the remaining terms in the Braun formula start at least with the term of order (Za) [17]. The expression in (4.19) exactly reproduces all contributions linear in the mass ratio in (3.5). This is just what should be expected since it is exactly Coulomb and Breit potentials which were taken in account in the construction of the effective Dirac equation which produced (3.5). The exact mass dependence of the terms of order Za) m/M)m and Za) m/M)m is contained in (3.5), and, hence,... 

Calculation of the recoil contribution of order (Za) (m/M)m to the nS states generated by the Braun formula was first performed in [14]. Separation of the high- and low-frequency contributions was made with the help of the e-method... 

The agreement on the magnitude of Za) m/M)m contribution for the 15 and 25 states obtained in the diagrammatic approach and in the framework of the Braun formula achieved in [14] seemed to put an end to all problems connected with this correction. However, it was claimed in a later work... 

Recoil corrections of order Za) m/M)m to the energy levels with nonvanishing orbital angular momentum may also be calculated with the help of the Braun formula [27]. We would prefer to discuss briefly another approach, which was used in the first calculation of the recoil corrections of order Za) m/M)m to the P levels [30]. The idea of this approach (see, e.g.. 

Further progress was also achieved in numerical calculations of higher order recoil corrections without expansion over Za in the framework of the approach based on (4.13). To describe these results we following [33, 34] (compare (3.98) and (3.102)) write all recoil corrections of order (Za) and higher in the form (note absence of the characteristic factor (m /m) which cannot be reproduced in the Braun formula framework)... 

Over the years different methods were applied for calculation of the radiative-recoil correction of order a Za). It was first considered in the diagrammatic approach [1, 3, 2]. Later it was reconsidered on the basis of the Braun formula [4]. The Braun formula depends on the total electron Green function... 

Calculation of the radiative-recoil correction generated by the one-loop polarization insertions in the exchanged photon lines in Fig. 5.2 follows the same path as calculation of the correction induced by the insertions in the electron line. The respective correction was independently calculated analytically both in the skeleton integral approach [8] and with the help of the Braun formula... 

Calculation of the same contribution with the help of the Braun formula was made in [4]. In the Braun formula approach one also makes the substitution in (3.35) in the propagators of the exchange photons, factorizes external wave functions as was explained above (see Subsect. 5.1.1), subtracts the infrared divergent part of the integral corresponding to the correction of previous order in Za, and then calculates the integral. The result of this calculation [4] nicely coincides with the one in (5.5). ... 

Braun i has shown that alcohols of the type of phenyl-ethyl alcohol, containing an aliphatic and an aromatic radicle, can be prepared by the reduction of nitriles of the general formula X. CN with the corresponding... 

A number of people have compiled data on the range and average composition of glauconite (Hendricks and Ross, 1941 Smulikowski, 1954 Borchert and Braun,1963). For the present review 69 analyses and 82 structural formulas from the literature were selected. 

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