Breit retardation term

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

This potential is referred to in electromagnetism texts as the retarded potential. It gives a clue as to why a complete relativistic treatment of the many-body problem has never been given. A theory due to Darwin and Breit suggests that the Hamiltonian can indeed be written as a sum of nuclear-nuclear repulsions, electron-nuclear attractions and electron-electron repulsions. But these terms are only the leading terms in an infinite expansion. 

Note that the subscript on the a matrices refers to the particle, and a here includes all of the tlx, tty and components in eq. (8.4). The first correction term in the square brackets is called the Gaunt interaction, and the whole term in the square brackets is the Breit interaction. The Dirac matiices appear since they represent the velocity operators in a relativistic description. The Gaunt term is a magnetic interaction (spin) while the other term represents a retardation effect. Eq. (8.27) is more often written in the form... 

The terms H5+H6 are often written in the form of the sum of magnetic (Hm) and retarding (Hr) interactions, sometimes also called the relativistic Breit operator... 

The first interaction term is, of course, the Coulomb interaction the second interaction term has the same form as the classical expression for the retarded interaction of two particles, derived in section 3.8. However, the Breit Hamiltonian suffers from the defect that the interaction terms are not Lorentz invariant. Detailed investigations using... 

The fully self-consistent handling is compared with a perturbative evaluation of only the beyond-Breit terms and a perturbative treatment of the complete Ej. Even for the heaviest atoms the perturbative evaluation of the retardation corrections to the Breit term seems to be sufficient. On the other hand, use of first-order perturbation theory for the complete Ej leads to errors of the order of 1 eV for heavy atoms. An accurate description of inner shell transitions in these systems requires the inclusion of second-order Breit corrections. 

A semirelativistic expression for the Breit interaction with the neglect of retardation can be obtained if we expand cos Ea — Eb)ti2) in Eq(160) in powers of aZ. In the nonrelativistic limit for the atomic electrons we have Ea — Eb m aZ) in r.u. The characteristic value for r,2 for the bound electron in atom is ri2 Oq/Z maZ), where Oq is the Bohr radius. Then (Ea — EB)rn aZ and we can expand the cosine in Eq(160). Since the a-matrices also introduce a smallness of order aZ (see Eqs(24)-(26)) we have to retain only the first term in cosine expansion when it is multiplied by Sia2 but to retain the third term when it is multiplied by 1. The second term vanishes due to the orthogonality of the wave functions. 

The first-order correction is known as the Breit term, and ai and velocity operators. Physically, the first term in the Breit correction corresponds to magnetic interaction between the two electrons, while the second term describes a retardation effect, since the interaction between distant particles is delayed relative to interactions between close particles, owing to the finite value of c (in atomic units, c -137). 

As fits to the crystal spectra became more detailed, a lack of balance in the theory appeared. The Coulomb interactions within the 4f shell and the effects of configuration interaction to second order can be taken into account by means of the four Slater integrals F (4f, 4f), the three Trees parameters a, J5, y, and the six three-electron parameters T . In contrast to these 13 electrostatic parameters, the spin-orbit interaction, until 1968, was represented by the single parameter This scheme overlooks the terms that arise from the Breit interaction, which was developed on relativistic grounds to account for the fine structures of the multiplets of Hel Isnp (see Bethe and Salpeter 1957). In the non-relativisitc limit parts of the Breit interaction, such as the retardation of the Coulomb interaction and the magnetic interactions that exist between the electrons in virtue of their orbital motions, can be represented by adjustments to the electrostatic parameters. Two terms cannot be absorbed in that way the spin-spin interaction and the spin-other-orbit interaction Marvin (1947) showed that, for the configurations I", ... 

The current density only occurs implicitly as we have obtained a new interaction operator, the Coulomb-Breit operator, for the charge density in an essentially stationary picture. Hence, we describe the interaction energy by the stationary states only and have all retardation- and current-density-specific terms hidden in the operator. This picture will be generalized for many-electron systems later in section 8.8. 

Because of the importance of Darwin s expression for the classical electromagnetic interaction of two moving charges (section 3.5), we are particularly interested in the frequency-independent radial form of the Breit operator. This represents the consistent interaction term to approximately include the retarded electromagnetic interaction of the electrons in our semi-classic formalism that describes only the elementary particles (electrons) quantum mechanically. In this long-wavelength limit, m —> 0, the radial operator Vv l,2) in Eq. (9.16) becomes D (l, 2) — already known from the Coulomb case in Eq. (9.9)... 

If one neglects the transverse contribution, one arrives at what is termed the Dirac-Coulomb approximation (a standard in quantum chemistry). Inclusion of the transverse term, which describes retardation and magnetic effects, in perturbation theory (weakly relativistic hmit) leads to the Dirac-Coulomb-Breit Hamiltonian. 

The retarded electron-electron interaction presented above arises from the first of the one-photon Feynman diagrams in figure 5.1. In terms of an expansion of the relativistic interactions in powers of 1 /c, this interaction contains the lowest-order terms. As pointed out above, the Breit interaction contains all terms of order c . After the Breit interaction, the lowest-order interactions come from the other two one-photon diagrams, the vacuum polarization and self-energy terms, which are 0 c ). The energy contribution from these two terms is called the Lamb shift, after its discoverer W. E. Lamb Jr. (1952), and its calculation has been an important testing ground for QED theories. 

---