Polarization of vacuum in a hydrogen-like relativistic atom: Hyperfine structure

An analysis is made of the contribution of the polarization of vacuum to the hyperfine splitting of the ground state of a hydrogen-like atom. An expression for the correction to the energy is obtained as an explicit function of the parameter Zα. The final expression derived in terms of generalized hypergeometric functions and their derivatives is a function of the particle mass ratio in orbit and in a vacuum loop, and is therefore valid for both ordinary and muonic atoms. Various asymptotic forms are also given.     

Energy spectrum of a relativistic two-dimensional hydrogen-like atom in a constant magnetic field of arbitrary strength

We compute, via a variational mixed-base method, the energy spectrum of a two-dimensional relativistic atom in the presence of a constant magnetic field of arbitrary strength. The results are compared to those obtained in the non-relativistic and spinless case. We find that the relativistic spectrum does not present s states.     

Estimate of the polarization fields of a relativistic atom in a solid

The magnetic and electric polarization fields of a relativistic hydrogen atom in a solid are analyzed. At atomic distances, these fields differ only slightly from the corresponding fields of an ionized atom.     

On emission from a hydrogen-like atom

A solution of the Dirac equation for an electron in the field of a point nucleus (Ze) has been obtained as an eigenfunction of the Schrödinger Hamiltonian and the spin projection operator Σ₃. With the use of this solution, the probability W^(ν) of the emission of a neutrino per unit time from a hydrogen-like atom, \((Ze)* \to (Ze) + \nu \bar \nu\), has been calculated for the first time in the first order of the parameter Ze ? 1. The probability W^(ν) appears to be rather small, and the corresponding lifetime τ^(ν) = [W^(ν)]^–1 is much larger than the age of the Universe; correspondingly, this process cannot affect the balance of low-energy neutrinos. The smallness of W^(ν) is due not only to the presence of the obvious “weak” factor (Gm_p²)²(m/mp)⁴ in the expression for W^(ν), but also primarily to the “electromagnetic” factor (Zα)¹², which can be revealed only in a particular calculation. It has been argued within quantum electrodynamics with the mentioned wavefunctions that photon emission, (Ze)* → (Ze) + γ, can be absent (analysis of photon emission requires the further development of the method), whereas axion emission, (Ze)* → (Ze) + a, can occur, although the last two effects have not been considered in detail.     

Evidence for a small component of circular polarization in relativistic electron bremsstrahlung

Using a transmission type polarimeter, an asymmetry has been found that can be ascribed to a circularly polarized component in the bremsstrahlung beam of unpolarized relativistic electrons from the Mainz electron linac, in the region of electron energiesE _el between 140 and 400 MeV. The experiment is described in detail. Systematical errors are discussed. For a comparison with electroweak theory, an exact calculation for small photon emission angles (of the order ofm c ²/E _el) is needed.     

Relativistic correction for the Zeeman effect in a hydrogen-like atom

Based on an algebraic system of equations whose eigenvalues in the semirelativistic approximation determine the magnetic splitting of energy levels of two coupled Dirac particles with charges ±e and different masses, the splitting in extremely weak and weak magnetic fields is determined. In addition, a system of equations for charged particles with anomalous magnetic moments is given. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 3–13, December, 2006.     

Two-quantum transitions between the fine-structure levels of a hydrogen-like atom

A study is made of the probability of two-quantum spontaneous transitions between fine-structure levels for fixed value of the principal quantum number (the radiation spectrum and the total transition probabilities are calculated) on the basis of the solution of the Dirac equation in a Coulomb field with arbitrary value of the charge of the nucleus.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 20–24, July, 1980.     

The uncertainties in radial position and radial momentum of an electron in the non-relativistic hydrogen-like atom

Similar to the case of a simple harmonic oscillator, an increase in azimuthal quantum number l will result in simultaneous decrease in both the uncertainty in radial position and the uncertainty in radial momentum for the same principal quantum number n in the non-relativistic hydrogen-like atom. Thus, in some cases of hydrogen-like atom and in the case of a simple harmonic oscillator, the more precisely the position is determined, the more precisely the momentum is known in that instant, and vice versa.     

Emission of axions by a hydrogen-like atom in an ultrastrong magnetic field

The resonance effect of emission of axions by a hydrogen-like atom in an ultrastrong magnetic field B ≫ B ₀ = m ²/e = 4.41 × 10¹³ Gs, which is induced by polarization of electron-positron vacuum, is considered. The emission probability and the radiation intensity are on the order of (B/B ₀) × 10⁻¹² of electromagnetic radiation characteristics, which exceeds the conventional ratio by many orders of magnitude. It is shown that, at the temperature of early Universe ≲(Zα)² m, the contribution from the resonance mechanism prevails. However, the relation between the concentrations of relic photons and axions cannot explain the origin of cold dark matter. The axion energy density in “our epoch” is 10⁻⁴(B/B ₀) eV/cm³.     

Renormalization of the two-photon vacuum polarization and the self energy vacuum polarization for a tightly bound electron

The renormalization method of Bogoljubov-Parasiuk-Hepp-Zimmermann (BPHZ) is used in order to derive the renormalized energy shift due to the gauge invariant K?llén-Sabry diagram of the two-photon vacuum polarization (VPVP) as well as the self energy vacuum polarization S(VP)E beyond the Uehling approximation. It is outlined, that no outer renormalization is required for the two-photon vacuum polarization and that only the inner renormalization has to be accomplished. It is shown that the so-called nongauge invariant spurious term is absent for a wide class of vacuum polarization (VP) diagrams if one applies the widely used spherical expansion of bound and free-electron propagator. This simplifies significantly calculations in bound state quantum electrodynamics. As one result of our paper the use of the BPHZ-approach in bound state QED is established. Received 28 September 2001     

Perturbation of the excited states of the hydrogen atom and hydrogen-like ions by a neutral atom

Owing to the degeneracy of the energy levels, the wavefunction of the electron in the excited states of the hydrogen atom and hydrogen-like ions perturbed by a neutral atom B is significantly different from the wavefunction of the unperturbed state. The perturbed function has a wide high maximum in the region of atom B, which is explained by multiple collisions of the electron with atom B, because the classical trajectories in the Coulomb field are closed and the size of atom B is much smaller than the size of the excited-state orbit. The radiative lifetimes of the excited states are much larger than those of unperturbed states. The orbital angular momentum L of the excited electron is strongly changed in collisions with atom B owing to the quantum interference or mixing of the temporal phases of adiabatic wavefunctions. The cross sections for such a change in the orbital angular momentum are several orders of magnitude larger than the cross sections found in early investigations in the approximation of the single collision of the electron with atom B.     

Stochastic ionization of a relativistic hydrogenlike atom

The stochastic ionization of a relativistic hydrogenlike atom in a monochromatic field is investigated. Using Chirikov’s criterion for stochasticity, an analytical formula is obtained for the critical value of the external field for which stochastic ionization of a relativistic atom occurs. Zh. Tekh. Fiz. 69, 127–129 (March 1999)     

Stochastic autoionization of a relativistic two-electron atom

On the basis of the analysis of absorption spectra of Er³⁺:PbMoO₄ crystals made for the transitions from the ground ⁴ I _15/2 state to excited states of Er³⁺ ions by the Judd-Ofelt method, the main spectroscopic characteristics of the crystals were obtained, including the transition probabilities and the radiative lifetimes.     

Ionization of a single hydrogen-like atom by laser pulse of near-atomic strength

The dynamics of high-harmonic generation and atom ionization by a strong and superstrong laser field are studied. In contrast to many earlier works, the present theory does not impose limitations on the laser field’s strength. We solve the nonrelativistic problem of a single hydrogen-like atom’s ionization from the ground state by a short laser pulse of subatomic, atomic, and superatomic field strength. Within the framework of the proposed method, we investigated the matrix elements of the ionization transition and revealed its substantially nonlinear dependence on the laser field strength. Both ionization and recombination processes are taken into account. The proposed method enables us to take into account the arbitrary order multiphoton ionization processes.     

Self-energy effect in a relativistic bound state problem

A modification of the Wick-Cutkosky equation for the relativistic bound state of two scalar particles interacting through the exchange of a massless scalar field within the ladder approximation has been considered by incorporating the self-energy diagrams in the integral kernel. An exact analytical solution of the equation is obtained at vanishing total energy and it is shown that the self-energy effects generally diminish the eigenvalues in agreement with the findings of Liet al, who, however solved the equation numerically for the case of massive scalar exchange. An additional feature of the modified equation is that it preserves the 0(5) symmetry at zero total energy as was first noted by Cutkosky for the scalar bound state equation without self-energy effects.