On the Dipole Approximation in the Theory of the Atom

In the nonrelativistic many-electron approximation of the theory of the atom in the formalism of secondary quantization and the theory of irreducible tensor operators, analytical structures for the quadrupole transition operator and its matrix element are established. To obtain these structures, the integral Loudoun representation for the nonrelativistic operator of the interaction of an electric field with an atom and the decomposition of plane waves into infinite alternating power series are used. A new criterion for the applicability of the dipole approximation is formulated as the ratio of the cross sections of single photoionization of an atom via the channels of dipole and quadrupole transitions. On the example of calculating the cross-sections of a single ionization of a deep shell of a neon-like ion of an iron atom by an X-ray photon, it is shown that the new applicability criterion reveals the leading role of the dipole transition operator even outside the long-wave radiation regime. The results obtained are predictive in nature.