Non-relativistic operators of electronic transitions

Let us consider the non-relativistic limit of the relativistic operators describing radiation. Expressing the small components of the four-component wave functions (bispinors) in terms of the large ones and expanding the spherical Bessel functions in a power series in cor/c, we obtain, in the non-relativistic limit, the following two alternative expressions for the probability of electric multipole radiation  

By evaluating the commutator of Q q with Hnr the generalization of the acceleration form of the transition operator to cover the case of electric multipole transitions of any multipolarity was obtained [77]. Unfortunately, the resultant operator has a much more complex and cumbersome form than Q q and Q q, since it contains both one-electron and two-electron parts. 

Now we can establish the correspondence between relativistic expressions (4.3), (4.4) and the non-relativistic ones obtained above, as well as the particular cases when specifying gauge condition K. The relationship between the values of K = 0, —y/(k + 1 )/k and the forms of the electron 

The transformation of the relativistic expression for the operator of magnetic multipole radiation (4.8) may be done similarly to the case of electric transitions. As has already been mentioned, in this case the corresponding potential of electromagnetic field does not depend on the gauge condition, therefore, there is only the following expression for the non-relativistic operator of Mk-transitions (in a.u.)  

It differs by multiplier i from the normally used one, but this multiplier does not affect the transition quantities. 

There exists another more consistent way of obtaining the electron transition operators. We can start with the quantum-electrodynamical description of the interaction of the electromagnetic field with an atom. In this case we find the relativistic operators of electronic transitions with respect to the relativistic wave functions. After that they may be transformed to the well-known non-relativistic ones, accounting for the relativistic effects, if necessary, as corrections to the usual non-relativistic operators. Here we shall consider the latter in more detail. It gives us a closed system of universal expressions for the operators of electronic transitions, suitable to describe practically the radiation in any atom or ion, including very highly ionized atoms as well as the transitions of any multipolarity and any type of radiation (electric or magnetic). 

It is convenient to treat the interaction of atomic electrons with an electromagnetic field in the framework of perturbation theory. 

This is due to the comparative weakness of the electromagnetic interaction, the theory of which contains a small dimensionless parameter (fine structure constant), by the powers of which the corresponding quantities can be expanded. The electron transition probability of the radiation of one photon, characterized by a definite value of angular momentum, in the first order of quantum-electrodynamical perturbation theory may be described as follows [53] (a.u.)  

Here O, A denotes the components of the four-dimensional vector-potential of the electromagnetic field corresponding to a definite state 

Let us underline that the matrix elements denoted ( ) will always 

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As was shown in Chapter 26, the pecularities of relativistic operators of electronic transitions are confined in their one-electron submatrix elements. Therefore formulas (26.7), (26.10), (26.12), (26.13), (26.15), (26.17)-(26.19), (26.21) and (26.23) are equally applicable for both the relativistic Ek- and Mk-transitions between complex electronic configurations. This is denoted by the subscripts e, m at the operator 0 k The same holds for sum rules (26.8), (26.9), (26.24)-(26.28). Therefore, we have only to present the appropriate expressions for the submatrix elements of non-relativistic operator (4.16) of Mk-transition in general cases of complex electronic configurations. For Mk-transitions between the levels of a shell of equivalent electrons the following formula is valid ... 

As we have seen, the line and oscillator strengths as well as the probabilities of electronic transitions between configurations of any type are expressed in terms of the submatrix elements of the appropriate operators. These submatrix elements may be found in a similar way as for the energy operator (see Part 5). Therefore, further on we shall consider only submatrix elements and present only final results. It is convenient to write the submatrix element of the non-relativistic operators of E/c-radiation (4.12) and (4.13) for transitions of the type (25.2), namely Zf Z2 — Zf Z3, as follows ... 

The general definition of the electron transition probability is given by (4.1). More concrete expressions for the probabilities of electric and magnetic multipole transitions with regard to non-relativistic operators and wave functions are presented by formulas (4.10), (4.11) and (4.15). Their relativistic counterparts are defined by (4.3), (4.4) and (4.8). They all are expressed in terms of the squared matrix elements of the respective electron transition operators. There are also presented in Chapter 4 the expressions for electric dipole transition probabilities, when the corresponding operator accounts for the relativistic corrections of order a2. If the wave functions are characterized by the quantum numbers LJ, L J, then the right sides of the formulas for transition probabilities must be divided by the multiplier 2J + 1. 

Relativistic corrections of order v2/c2 to the non-relativistic transition operators may be found either by expanding the relativistic expression of the electron multipole radiation probability in powers of v/c, or semiclas-sically, by replacing p in the Dirac-Breit Hamiltonian by p — (l/c)A (here A is the vector-potential of the radiation field) and retaining the terms linear in A. Calculations show that in the general case the corresponding corrections have very complicated expressions, therefore we shall restrict ourselves to the particular case of electric dipole radiation and to the main corrections to the length and velocity forms of this operator. 

Here submatrix elements of the operator Tk are defined in accordance with (7.5). As in the non-relativistic case (25.23), only transitions described by one term in (26.10) take place, depending on which subshell an electron is jumping in. In this case the other summand must be considered as being equal to zero. 

The dependence of photoabsorption cross-section on many-electron quantum numbers (sets of quantum numbers of a chain of electronic shells) is mainly determined by the submatrix element of the transition operator. Their non-relativistic and relativistic expressions for the most widely considered configurations are presented in Part 6. When exciting an atom by X-rays the main type of transitions are as follows ... 

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