Breit formula

The total number of photons counted in the experiment envisioned is given by the Breit formula,128 which in our notation is... 

The quantum-mechanical treatment of level-crossing spectroscopy [834,837] starts with the Breit formula... 

The complete theory of resonance scattering is given by Breit [7.27] and Franken [7.28]. The Breit formula from 1933 gives the intensity S of emitted photons with polarization vector e obtained after absorption of photons with polarization (e- and are unity vectors) in the interaction with an atomic system (Fig.7.14). 

If the excited state has magnetic sublevels with indices M and M and the ground state has sublevels with indices /x and /x the Breit formula can be written... 

The general expression for the intensity I of fluorescence with polarization vector I2 following excitation by absorption of light with frequency w and polarization vector is given by the Breit formula [2.21]. Let a = (a", J", M") be the initial state of the atomic system, b = (a , J, M ) the intermediate state and c = (a, J, M) the final state of the absorption-emission sequence a-> b-> c. We then obtain... 

The capture rate is dominated at thermal energies around 30 keV by 5-waves for which the Breit-Wigner formula gives... 

Because the cross section of a resonance process is expressed by the Breit-Wigner formula, the energy-integrated cross section is written as follows ... 

The two parts of this formula are derived from the same QED Feynman diagram for interaction of two electrons in the Coulomb gauge. The first term is the Coulomb potential and the second part, the Breit interaction, represents the mutual energy of the electron currents on the assumption that the virtual photon responsible for the interaction has a wavelength long compared with system dimensions. The DCB hamiltonian reduces to the complete standard Breit-Pauli Hamiltonian [9, 21.1], including all the relativistic and spin-dependent correction terms, when the electrons move nonrelativistically. 

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,... 

Again the situation is much simpler when only asymptotic states containing stable particles are considered. Then unstable particles enter neither into the completeness relation nor into the unitary relations of the theory.5 However, in the intermediate states unstable particles may appear. They manifest themselves as poles exactly as in Eq, (16). We may then describe such poles by various approximate formulas of the Breit-Wigner type. But again this approach is severely limited. By definition we have to exclude the production or destruction processes involving unstable particles. It is even not easily seen how this can be done in a consistent manner. 

Let us first consider the case of Y/D 1. This means that at certain values of the compound nucleus excitation energy, individual levels of the compound nucleus can be excited (i.e., when the excitation energy exactly equals the energy of a given CN level). When this happens, there will be a sharp rise, or resonance, in the reaction cross section akin to the absorption of infrared radiation by sodium chloride when the radiation frequency equals the natural crystal oscillation frequency. In this case, the formula for the cross section (the Breit-Wigner single-lev el formula) for the reaction a + A —> C b + B is... 

For reactions involving isolated single resonances or broad resonances, it is possible to derive additional formulas for a(E) [R + R] in the Breit-Wigner form, that is,... 

Thus, we have expressed the non-relativistic Hamiltonian of a many-electron atom with relativistic corrections of order a2 in the framework of the Breit operator (formulas (1.15), (1.18)—(1.22)) in terms of the irreducible tensorial operators (second term in (1.15), formulas (19.5)—(19.8), (19.10)— (19.14), (19.20), respectively). 

Thus, introducing parameters a, / and T we can account for the essential part of the correlation effects. However, it turned out that in the framework of the semi-empirical approach, all relativistic corrections of the second order of the Breit operator improving the relative positions of the terms, are also taken into consideration (operators H2, and H s, described by formulas (1.19), (1.20) and (1.22), respectively). Indeed, as we have seen in Chapter 19, the effect of accounting for corrections Hj and H s in a general case may be taken into consideration by modifications of the integrals of electrostatic interaction, i.e. by representing them in form... 

Equations (15) and (16) are Breit-Wigner s one-level formula for the phase shift. If the pole lies close to the real E axis, i.e., if T is very small, the part 5r of the phase shift increases very rapidly with E by tt/2 within the energy region of width T and centered at Er. It increases by nearly it within several times T. This is a resonance phenomenon. 

A number of closely lying resonances in multichannel scattering is a difficult problem to treat theoretically. Even the representation of the S matrix is very complex for these overlapping resonances as compared with the Breit-Wigner one-level formula. Various alternative proposals are found in the literature, as is reviewed by Belozerova and Henner [61]. This is mainly due to the formidable task of constructing an explicitly unitary and symmetric S matrix having more than one pole when analytically continued into the complex k plane. Thus, possible practical forms of the S matrix for overlapping resonances may be explicitly symmetric and implicitly unitary, or explicitly unitary and implicitly symmetric. 

As described above, time-delay analysis [389] of the energy derivative of the phase matrix 4> determines parametric functions that characterize the Breit-Wigner formula for the fixed-nuclei resonant / -matrix R[N(q e). The resonance energy eKS(q), the decay width y(q). and the channel-projection vector y(q) define R and its associated phase matrix such that tan = k(q)R , where... 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

Breit-Breit interaction is calculated within disregard of negative frequency states, interaction with crossed photons (Fig.lh) and retardation. Thus Breit-Breit interaction is expressed by Feynman diagram Fig.lg. The formulas for box dia-... 

Besides taking into account the two-electron diagrams Fig.lc, d, e, f, Coulomb-Breit interaction for three-electron atom represented by the three-electron diagrams Fig.2b, c (as it is mentioned above the contribution of the diagrams Fig.2b, c is considered as doubled contribution of the diagram Fig.2b). The formulas for irreducible part of the diagram Fig.2b is (see the Appendix)... 

Breit-Breit interaction for three-electron atom is represented by both the two-electron diagrams Fig.lg, h and the three-electron diagram Fig.2d. The formula... 

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