Relativistic Electron Wave Functions Research Articles

Resonant phase-matching between a light wave and a free-electron wavefunction

As charged particles surpass the speed of light in an optical medium they produce radiation - analogously to the way jet planes surpass the speed of sound and produce a sonic boom. This radiation emission, known as the Cherenkov effect, is among the most fundamental processes in electrodynamics. As such, it is used in numerous applications of particle detectors, particle accelerators, light sources, and medical imaging. Surprisingly, all Cherenkov-based applications and experiments thus far were fully described by classical electrodynamics even though theoretical work predicts new Cherenkov phenomena coming from quantum electrodynamics. The quantum description could provide new possibilities for the design of highly controllable light sources and more efficient accelerators and detectors. Here, we provide a direct evidence of the quantum nature of the Cherenkov effect and reveal its intrinsic quantum features. By satisfying the Cherenkov condition for relativistic electron wavefunctions and maintaining it over hundreds of microns, each electron simultaneously accelerates and decelerates by absorbing and emitting hundreds of photons in a coherent manner. We observe this strong interaction in an ultrafast transmission electron microscope, achieving for the first time a phase-matching between a relativistic electron wavefunction and a propagating light wave. Consequently, the quantum wavefunction of each electron evolves into a coherent plateau, analogous to a frequency comb in ultrashort laser pulses, containing hundreds of quantized energy peaks. Our findings prove that the delocalized wave nature of electrons can become dominant in stimulated interactions. In addition to prospects for known applications of the Cherenkov effect, our work provides a platform for utilizing quantum electrodynamics for applications in electron microscopy and in free-electron pump-probe spectroscopy.

L-Shell Ionization during the Alpha Decay of Superheavy Nuclei from 117 294 Ts Tennessine Decay Chain and the Alpha Decay of the Polonium Isotope 84 210 Po

The probabilities for L-shell ionization following the alpha decay of 117 294 Ts, 113 286 Nh, 109 278 Mt, and 105 270 Db superheavy nuclei from the decay chain of tennessine, which was synthesized at the Joint Institute for Nuclear Research (JINR Dubna), and the alpha decay of the isotope 84 210 Po are calculated and analyzed. The respective calculations rely on a quantum-mechanical model that takes into account alpha-particle tunneling through the Coulomb barrier of an atom. The relativistic electron wave functions obtained by the Dirac-Fock method were employed. The probabilities for L-shell ionization that were calculated here for the isotope 84 210 Po are in better agreement with experimental data than the results of earlier calculations. In contrast to what we have for K-shell ionization, where monopole transitions make a dominant contribution to the ionization probability, a significant contribution to the probability for L-shell ionization also comes from dipole and quadrupole transitions. The results of the present study will be of use in interpreting superheavy-element spectra obtained by methods of combined alpha, gamma, and electron-conversion spectroscopy.

Bound-state double-beta decay

We study a new mode of the neutrinoless and two-neutrino double-beta decays in which a single electron is emitted from the atom. The other electron is assumed to occupy one of the vacant s1/2 or p1/2 subshells of the daughter ion. Such process could manifest itself through an additional background signal in the single-electron spectra, which will be accessible in the next-generation experiment SuperNEMO. We calculate the phase-space factors in terms of relativistic electron wave functions obtained as the solutions to the Dirac equation and evaluated at the nuclear radius, while taking into account the shielding effect of nuclear charge via the multiconfiguration Dirac–Hartree–Fock package Grasp2K. Half-lives are estimated for the most relevant double-beta-decay isotopes and experimental significance is discussed.

BetaShape: A new code for improved analytical calculations of beta spectra

The new code BetaShape has been developed in order to improve the nuclear data related to beta decays. An analytical model was considered, except for the relativistic electron wave functions, for ensuring fast calculations. Output quantities are mean energies, log ft values and beta and neutrino spectra for single and multiple transitions. The uncertainties from the input parameters, read from an ENSDF file, are propagated. A database of experimental shape factors is included. A comparison over the entire ENSDF database with the standard code currently used in nuclear data evaluations shows consistent results for the vast majority of the transitions and highlights the improvements that can be expected with the use of BetaShape.

Partial cross sections for radiative recombination of bare ions : effects of relativity and radiation multipoles

SynopsisWe derived an original expression for the partial cross sections for radiative recombination (RR) of highly charged bare ions in the nonrelativistic electric-dipole approach. Three sets of cross sections for RR to individual magnetic sublevels of the excited levels of various recombined H-like ions were computed: the first two in the electric-dipole approximation with the nonrelativistic and relativistic electron wavefunctions and the third one in the exact relativistic method including all multipole orders of the radiation field.

Polarization of the Lyman line emission following radiative recombination of bare ions: effects of relativity and radiation multipoles

We have calculated the degree of linear polarization of the Ly-α1 (2p3/2 → 1s1/2) and Ly-β1 (3p3/2 → 1s1/2) lines emitted after radiative recombination of spinless bare nuclei with unidirectional, unpolarized electrons. These calculations have been performed for various recombined H-like ions with atomic numbers in the range 10 ⩽ Z ⩽ 92. Several incident-electron kinetic energies ε0 ranging from 0.01 to 10 times the electron binding energy in the 1s1/2 ground state were considered and the radiative cascades following recombination into states with principal quantum number up to n = 6 were included. Three sets of cross sections for radiative recombination to individual magnetic sublevels of the excited levels of the recombined H-like ions were computed: the first two in the electric-dipole approximation with the nonrelativistic and relativistic electron wavefunctions and the third one in the exact relativistic method including all multipole orders of the radiation field. Results for the polarization calculated using these three sets of cross-section data were compared in order to stress the relative importance of relativistic and multipole effects as Z or ε0 increases.

Asymmetry and Energy Spectrum of Electrons in Bound-Muon Decay

Theoretical values for the muon decay rate and the emitted electron energy spectrum and several kinds of asymmetry parameters for the angular distribution of the electrons in bound-muon decay are given in numerical tables and figures for 16O, 27Al, 28Si, 40Ca, 56Fe, 90Zr, 96Mo, 118Sn, 208Pb, and 209Bi. The present results are based on the general formalism of the angular distribution of emitted electrons in bound-muon decay published in Prog. Theor. Phys. 78, 114 (1987), in terms of the multipole series of the relativistic electron wave functions. The effect of finite nuclear size on the wave functions is properly taken into account for both the bound muon and the emitted electron.

A unified description of electron excitations in heavy ion collisions

Coulomb matrix elements can be described by interactions between rotation-free currents. This formalism unifies the expressions for processes of different orders, corresponding to direct excitations and inclusion of correlations induced by electron-electron interactions. The methods employed are shown to be applicable for relativistic electron wavefunctions as well.

Angular Distribution of Electrons in Bound Muon Decay

A general formalism of the angular distribution of emitted electrons in the bound muon decay is made in terms of the multi pole series of the relativistic electron wave functions. The effect of the finite size nucleus to wave functions is properly taken into account for the bound muon and emitted electron. In order to show the importance of this effect, energy spectra and asymmetry coefficients are given numerically in the cases of 160, 40Ca, 90Zr and 2°Pb.

Internal electron-positron pair production from electric monopole transitions

NuclearE0-transitions above the pair production threshold of ω=2mc2∼1 MeV are considered. The rationη of the electron-positron pair creation probability to theK-shell conversion probability is calculated. For92U and ω=1,450 keV a typical value ofη=2·10− is obtained. Good agreement with experimental data is found. Our numerical results for finite size nuclei are compared with analytical expressions utilizing relativistic electron wavefunctions for point nuclei. The energy distributions of emitted positrons are presented for various nuclear charge values and transition energies. The possibility of monoenergetic positron emission is discussed. Of particular interest is the relation of the theoretical results to the recently observed structure in the positron spectra, which have been measured in high energy collisions of very heavy ions.

Electrons from quasi-atoms formed in heavy-ion collisions

Electrons emitted in heavy-ion collisions of oxygen and sulfur projectiles with heavy targets were measured. Electron energies up to five times the K-shell binding energy of the target atom were observed. The high-energy part can be explained well within the framework of the Born approximation using relativistic electron wave functions for the united atoms. It is argued that the high-energy shape of the spectra is directly related to the form factor of inner shell electrons of the united atoms.

Coulomb deflection effects in ion-induced K-shell ionisation

Assuming a hyperbolic Kepler orbit for the projectile, an analytical expression for the monopole projectile path factor is derived in the semiclassical approximation. The case of an attractive potential is also included. An improved Bang-Hansteen deflection factor is found to be approximately valid for arbitrary electron wavefunctions. The method is applied to calculations of the monopole contribution to K-shell ionisation using relativistic electron wavefunctions.

K-shell ionization probabilities of Pb at small impact parameters by 25-, 40-, and 74-MeVαparticles

The absolute $K$-shell ionization probabilities of Pb at very small impact parameters by 25-, 40-, and 74-MeV $\ensuremath{\alpha}$ particles are measured in an $\ensuremath{\alpha}$-x-ray coincidence experiment, using the slight inelasticity of the scattered $\ensuremath{\alpha}$ particles. Ionization probabilities are (1.9 \ifmmode\pm\else\textpm\fi{} 0.4) \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$, (4.9 \ifmmode\pm\else\textpm\fi{} 1.7) \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$, and (11 \ifmmode\pm\else\textpm\fi{} 3) \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ at 25, 40, and 74 MeV, respectively. Semiclassical calculations, using relativistic electron wave functions and corrections for Coulomb deflection effects, agree very well with the data. Possible quantitative studies of atomic distortion effects are suggested.

Internal Ionization during Beta Decay

In nuclear $\ensuremath{\beta}$ decay, atomic excitation, including ionization, can occur because of (i) imperfect overlap of parent and daughter orbital electron wave functions, and (ii) scattering of atomic electrons by the $\ensuremath{\beta}$ particle (direct collisions). In order to evaluate the relative importance of these two processes and test the possible role of nuclear matrix elements in orbital-electron ejection, published work on the subject has been reviewed. New measurements have been performed determining the total probability of $K$ ionization in the $\ensuremath{\beta}$ decay of ${\mathrm{Tc}}^{99}$ [(4.8\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ per $\ensuremath{\beta}$] and ${\mathrm{Pm}}^{147}$ [(9.3\ifmmode\pm\else\textpm\fi{}1.4)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ per $\ensuremath{\beta}$]. The $K$ ionization probability as a function of $\ensuremath{\beta}$ energy has also been measured; a strong energy dependence is found, in disagreement with the traditional wave-function overlap theory. This theory is improved by including phase-space considerations, using relativistic electron wave functions in the calculation of the pertinent matrix elements, and paying careful attention to the antisymmetrization of the final-state wave function. The predictions of the modified wave-overlap theory agree very well with experiment. Direct collisions appear to play only a minor role. The calculations indicate that the ionization probability is inhibited in forbidden decay. A possible change in the shape factor for transitions accompanied by internal ionization is predicted.