E. Z. Liverts

Quasilinearization approach to the resonance calculations: the quartic oscillator

We pioneered the application of the quasilinearization method (QLM) to resonance calculations. The quartic anharmonic oscillator (kx 2 /2) + x 4 with a negative coupling constant was chosen as the simplest example of the resonant potential. The QLM has been suggested recently for solving the bound state Schrodinger equation after conversion into Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries. Comparison of our approximate analytic expressions for the resonance energies and wavefunctions obtained in the first QLM iteration with the exact numerical solutions demonstrate their high accuracy in the wide range of the negative coupling constant. The results enable accurate analytic estimates of the effects of the coupling constant variation on the positions and widths of the resonances.

Boundary solutions of the two-electron Schrödinger equation at two-particle coalescences of the atomic systems

The limit relations for the partial derivatives of the two-electron atomic wave functions at the two-particle coalescence lines have been obtained numerically using accurate correlation function hyperspherical harmonic method wave functions. The asymptotic solutions of the proper two-electron Schroedinger equation have been derived for both electron-nucleus and electron-electron coalescence. It is shown that the solutions for the electron-nucleus coalescence correspond to the ground and singly excited bound states, including triplet ones. The proper solutions at small distances R from the triple coalescence point were presented as the second order expansion on R and ln R. The vanishing of the Focks logarithmic terms at the electron-nucleus coalescence line was revealed in the frame of this expansion, unlike the case of electron-electron coalescence. On the basis of the obtained boundary solutions the approximate wave function corresponding to both coalescence lines have been proposed in the two-exponential form with no variational parameters.

Analytic presentation of a solution of the Schrödinger equation

High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into the nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.

Wavefunctions of Helium-Like Systems in Limiting Regions ¶

We find an approximate analytic form for the solution ψ(r1, r2, r12) of the Schrödinger equation for a system of two electrons bound to a nucleus in the spatial regions r1 = r2 = 0 and r12 = 0, which are of great importance for a number of physical processes. The forms are based on the well-known behavior of ψ(r1, r2, r12) near the singular triple coalescence point. The approximate functions are compared to the locally exact ones obtained earlier by the correlation function hyperspherical harmonic (CFHH) method for the helium atom, light helium-like ions, and the negative ion of hydrogen H−. The functions are shown to determine a natural basis for the expansion of CFHH functions in the considered spatial region. We demonstrate how these approximate functions simplify calculations of high-energy ionization processes.

Qualitative difference between the angular anisotropy parameters in fast electron scattering and photoionization

It is demonstrated for the first time that in spite of well known big similarities between atomic ionization by photons and fast electrons, a qualitative difference exists in angular anisotropy parameters of electrons knocked out in these processes. The difference is disclosed here and attributed to distinction between normal (transverse) and virtual (longitudinal) photons. Formulas are derived for dipole and non-dipole angular anisotropy parameters in fast electron-atom scattering. The ratio of quadrupole-to-dipole matrix elements is determined by the parameter ωR/v < 1, where ω is the transferred in collision energy, R is the ionized shell radius, and v is the speed of projectile. This factor can be much larger than that in the case of photoionization, where one has the speed of light c that is much higher than v. We illustrate general formulas by concrete results for outer s subshells of noble gas atoms Ar and Xe. Even for very low momentum transfer q, in the so-called optical limit, the deviation from photoionization case is prominent and instructive.

Back-to-back emission of the electrons in double photoionization of helium

We calculate the double differential distributions and distributions in recoil momenta for the high energy non-relativistic double photoionization of helium atom. We show that the results of recent experiments provide the pioneering experimental manifestation of the quasifree mechanism for the double photoionization which was predicted long ago in our papers. This mechanism provides a surplus in distribution over the recoil momenta at small values of the latter, corresponding to nearly “back-to-back” emission of the photoelectrons. Also, in agreement with previous analysis we demonstrate that this surplus is due to the quadrupole terms of the photon-electron interaction. We present the characteristic angular distribution for the back-to-back electron emission. The confirmation of the quasifree mechanism existence opens a new area for exciting experiments, which are expected to increase our understanding of the electron dynamics and of the bound states structure.

Photoionization of onion-type atoms

The photoionization of a two-shell (“onion“) endohedral A@CN1@CN2 is considered. The effect of the fullerenes shells upon photoelectron from atom A is taken into account substituting the action of the fullerene by two zero-thickness “bubble potentials.“ The fullerenes shells polarization is included assuming that the radius of the outer shell R2 is much bigger than the inner R1 and both much exceed the atomic radius r. The interaction between shells CN1 and CN2 is taken into account in the Random Phase Approximation (RPA). The effect of photoelectron scattering by both “bubble potentials“ is included in the RPA frame. As concrete examples, two endohedrals Ar@C60@C240 and Xe@C60@C240 are considered. We investigate 3p4 Ar and 5p Xe subshells.

Calculation of the photoionization with deexcitation cross sections of he and helium-like ions

We discuss the results of the calculation of the photoionization with deexcitation of excited He and helium-like ions Li+ and B3+ at high but nonrelativistic photon energies θ. Several lower 1S and 3S states are considered. We present and analyze the ratios Rd+* of the cross sections of photoionization with deexcitation, σ(d)+*(θ), and of the photoionization with excitation, σ+*(θ). The dependence of Rd+* on the excitation of the target object and the charge of its nucleus is presented. In addition to theoretical interest, the results obtained can be verified using long-lived excited states such as 23S of He.

Two‐electron processes in atom‐photon interactions

Two‐electron processes in atom‐photon interactions have been studied. Various mechanisms of double photoionization, angular correlations, total cross sections, energy distributions and results for non‐Helium like systems are discussed.