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Physical Review A

Theory of forbidden transitions in light atoms

Krzysztof Pachucki

Published Jan 22, 2003• 19 citations
Publisher

Abstract

Forbidden transitions such as intercombination ones, appear due to relativistic effects. In light atoms these effects are usually incorporated through the use of Breit-Pauli Hamiltonian. We show that for some transitions such as $E2$ or $M1$ this treatment is incomplete. There exist additional corrections to the transition amplitude which can be equally significant. We analyze also the role of the anomalous magnetic moment of the electron in some forbidden transitions.

DOI
10.1103/physreva.67.012504

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Forbidden transitions such as intercombination ones, appear due to relativistic effects. In light atoms these effects are usually incorporated through the use of Breit-Pauli Hamiltonian. We show that for some transitions such as E2E2E2 or M1M1M1 this treatment is incomplete. There exist additional corrections to the transition amplitude which can be equally significant. We analyze also the role of the anomalous magnetic moment of the electron in some forbidden transitions.
A series of magic wavelengths for the forbidden transition of $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}$ in helium are determined by performing large-scale full relativistic configuration interaction calculations based on Dirac-Coulomb-Breit Hamiltonian with the mass shift operator included. QED corrections to magic wavelengths are taken into account by using the perturbation theory in nonrelativistic configuration interaction calculations. The 1335.55(2)-nm magic wavelength for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}(M=\ifmmode\pm\else\textpm\fi{}1)\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}(M=0)$ transition is found to be sensitive to the finite nuclear mass, relativistic, and QED effects, and its measurement might provide a test of atomic structure theory.
Basic electromagnetism and relativistic quantum theory are used to evaluate the magnetic field due to the electron in a one-electron atom. The field patterns are displayed and discussed for several eigenstates with low values of the quantum numbers. The effect of increasing the number of protons in the nucleus is shown explicitly. Comparison with the non-relativistic result illustrates the effects of relativity in atoms. There is a brief discussion of the meaning of the terms spin and orbital magnetic moment of an atom in the light of these results.
Abstract The law of Biot and Savart, derived by observation of the magnetic field produced by electric current flowing in a macroscopic conductor, is shown to yield the correct expression for the nuclear hyperfine interaction, when applied to the electron circulation current derived for hydrogenic atoms from the theories of Pauli and Dirac, as further developed independently by Darwin, Gordon and Hartree. The law thus applies also on the microscopic scale to the magnetic field generated by the electron probability flux of the various atomic eigenstates, and may clearly be generalised formally for application to many-electron atoms, as outlined by Hartree. The present, fully relativistic, treatment is advantageous in avoiding the usual artificial splitting of the hyperfine interaction into a magnetic dipole–dipole contribution and a separate contribution arising from electron orbital motion. The formula obtained forms the basis for the inclusion of higher order (quantum electrodynamic and nuclear recoil) corrections. Interest in this subject is enhanced by observations of ‘strongly forbidden’ (magnetic dipole) atomic transitions arising from the nuclear magnetic interaction, notably the 21 cm line of hydrogen, which has played an important role in astrophysics, in both measurements of galactic rotation and studies of the early Universe.
Introduction The selection rules discussed in Sec. 3.5 forbid many kinds of transitions that are frequently observed or exploited. It must be remembered that these “rules” are based on the approximations of Sec. 3.2, but of course, those can be extended. Such expansions to higher orders of approximation may seem to be little more than a tedious exercise, but this chapter will show that the results are extremely important, because they cause qualitatively different kinds of transitions. Since they involve both the electric and magnetic components of the light, their atomic dipole moments will be identified with the superscript E and B for the remainder of this chapter. The magnetic dipole transitions, called M1, discussed in Sec. 4.2.1 that arise from extension of the electric dipole approximation encompass all of magnetic resonance, including atomic clocks, forced evaporative cooling in Bose–Einstein Condensation (BEC), MRI, and astrophysical processes such as generation of the all-important 21 cm radio frequency line. The electric quadrupole transitions, called E2, discussed in Sec. 4.2.2 describe myriad stellar spectral features as well as the vast majority of nuclear γ-ray transitions. Even with these extensions there are many cases where the relaxation of the electric dipole (E1) selection rules to include M1 and E2 transitions as in Sec. 4.2 still do not account for observed processes. Thus other approximations have to be extended so that their limitations can be relaxed as well. Extension of the perturbation approximation (see Sec. 4.3.1) provides an analytical description of most of non-linear optics, including second harmonic generation, four wave mixing, Raman spectroscopy, and the fabled decay of the hydrogen 2S state. A semi-classical description of the interaction between radiation and atoms can begin with a multipole expansion of the Hamiltonian. The lowest-order term is just from the point charges α located at, and vanishes for neutral atoms. The next-order term is the electric dipole interaction that arises from the non-isotropic distribution of charges induced by the electric field of the light. This term produces the electric dipole transitions (E1) that have been described in some detail in Chap. 3.