Lower Order Recoil Corrections and the Braun Formula

2 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. 

Corrections of lower orders in Za are generated by the simplified Coulomb-Coulomb and Coulomb-transverse entries in (4.13). The main part of the Coulomb-Coulomb contribution in Eq. (4.13) may be written in the form 

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 

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