Critical electron state in heavy-ion collisions

The results of a calculation of the electron wave function for the critical state (? = ?m_ec²) and of some related quantities are presented. The effect of the finite nuclear size on the critical internuclear distance R_cr is taken into account. The calculations are performed by means of the variational method for nuclei with charges 85 ? Z ? 100. The error of the WKB method for strong Coulomb fields is found to be a few percent.     

Coulomb Problem for <Emphasis Type="Italic">Z</Emphasis> &gt; Z<Subscript>cr</Subscript> in Doped Graphene

The dynamics of charge carriers in doped graphene, i.e., graphene with a gap in the energy spectrum depending on the substrate, in the presence of a Coulomb impurity with charge Z is considered within the effective two-dimensional Dirac equation. The wave functions of carriers with conserved angular momentum J = M + 1/2 are determined for a Coulomb potential modified at small distances. This case, just as any two-dimensional physical system, admits both integer and half-integer quantization of the orbital angular momentum in plane, M = 0, ±1, ±2, …. For J = 0, ±1/2, ±1, critical values of the effective charge Z_cr(J, n) are calculated for which a level with angular momentum J and radial quantum numbers n = 0 and n = 1 reaches the upper boundary of the valence band. For Z < Z_cr (J, n = 0), the energy of a level is presented as a function of charge Z for the lowest values of orbital angular momentum M, the level with J = 0 being the first to descend to the band edge. For Z>Z_cr (J, n = 0), scattering phases are calculated as a function of hole energy for several values of supercriticality, as well as the positions ε₀ and widths γ of quasistationary states as a function of supercriticality. The values of ε₀* and width γ* are pointed out for which quasidiscrete levels may show up as Breit–Wigner resonances in the scattering of holes by a supercritical impurity. Since the phases are real, the partial scattering matrix is unitary, so that the radial Dirac equation is consistent even for Z > Z_cr. In this single-particle approximation, there is no spontaneous creation of electron–hole pairs, and the impurity charge cannot be screened by this mechanism.     

Nonperturbative effects of vacuum polarization for a quasi-one-dimensional Dirac–Coulomb system with <Emphasis Type="Italic">Z</Emphasis> &gt; <Emphasis Type="Italic">Z</Emphasis><Subscript><Emphasis Type="Italic">cr</Emphasis></Subscript>

Major nonperturbative properties of the vacuum charge density ρ _VP (x) and vacuum-polarization energy ξ _VP are considered for an overcritical Coulomb source with Z > Z _cr in 1 + 1 D. We demonstrate that, for a broad range of external-field parameters, vacuum energy may significantly deviate from the perturbative quadratic increase and even decrease towards deeply negative values due to vacuum polarization in the overcritical region.     

Electron shells in over-critical external fields

When the charge of a nucleus exceedsZ=Z _cr≈ 164–172 the energy of the 1s-electron level is lowered beyond the critical value of ?m _e c ². Then this bound level is degenerated with negative energy continuum solutions of the Dirac equation and becomes a resonance, whose shape varies and is approximately of Breit-Wigner type forZ?Z _cr?5. The physical meaning of this resonance can be understood most easily if the 1s-level is unoccupied (K-hole). In this case a positron may escape ifZ>Z _cr, a process, that can be interpreted as auto-ionization of the positron. This fundamentally new process of quantum electrodynamics of strong fields can be tested experimentally by scattering very heavy ions (Z≧80) on each other since in such collisions superheavy electronic molecules occur (superheavy quasimolecules).     

Dynamical screening of AMM and QED effects for large-<Emphasis Type="Italic">Z</Emphasis> hydrogen-like atoms

The effective interaction ΔU_AMM of the anomalous magnetic moment (AMM) of an electron with the Coulomb field of an extended nucleus is analyzed. As soon as the q² dependence of the electron formfactor F₂(q²)is taken into account from the beginning, the AMM is found to be dynamically screened at small distances of r ? 1/m. The ΔU_AMM effects on the low-lying electronic levels of a superheavy extended nucleus with Zα > 1are analyzed using the nonperturbative approach. The growth rate of the ΔU_AMM contribution with increasing Z is shown to be essentially nonmonotonic. At the same time, the energy shifts of electronic levels in the vicinity of the threshold of the lower continuum monotonically decrease in the region Z ?Z_cr,1s. The latter result is generalized to the whole self-energy contribution to energy shifts of electronic levels, thus also referring to the possible behavior of QED radiative effects with virtual-photon exchange, considered beyond the framework of the perturbative expansion in Zα.     

Mechanism of formation and decay of the compound nuclei150Gd produced by two entrance channels (16O+134Ba) and (40Ar+110Pd)

Excitation functions have been drawn for evaporation residues issued from¹⁵⁰Gd, a compound nucleus produced either by¹⁶O ions on¹³⁴Ba, or by⁴⁰Ar on¹¹⁰Pd. Absolute cross sections were measured for^145–147Gd,^145–147Eu,^141m Sm,^143m Sm,^140m Pm and^139m Nd. Complete fusion cross sections have been obtained and compared to calculated estimations based, at low energies, on the simple expressionσ _CF=πR _Fuss ² (1?V _Fus/E) whereR _Fus andV _Fus are respectively the distance and the potential for fusion atdV/dr=0. For high energies σ_CF=πR _cr ² (1?V_cr/E) with the model of critical distanceR _cr and critical potentialV _cr. Thresholds energies have been determined with a particular care. The analysis of the shape and the width of excitation functions, particularly for the emission of 4 and 5 neutrons, has been carried out and compared to calculated values. The difference between argon and oxygen induced reactions has been attributed to the difference inl population in the entrance channel.     

Interaction of dirac particle AMM with Coulomb field of a superheavy nucleus: Perturbative and nonperturbative aspects

The interaction ΔU_AMM of the Dirac particle anomalous magnetic moment (AMM) with the Coulomb field of a nucleus and its effect on the low-lying atomic levels are studied for Zα > 1 using both perturbative and essentially nonperturbative approaches. The Zα dependence of particle wavefunctions (WFs) is fully taken into account from the beginning. In deriving the ΔU_AMM contribution, the nucleus is viewed either as a uniformly charged extended Coulomb source or as a distributed system formed by pointlike u and d quarks. When estimated nonperturbatively, the ΔU_AMM-induced effects in the Dirac equation framework prove to be identical for these two cases. At the same time, the ΔU_AMM-induced effect is specific in that its perturbative and nonperturbative estimates are very different for Δg ? const and practically identical as soon as the dynamical screening of AMM at short distances is taken into account in the Dirac equation.     

Chaos in a supercritical atom

Stochastization of a supercritical atom (with a nuclear charge number Z in excess of 137) under the effect of a periodic perturbation is investigated. The Hamiltonian for a relativistic electron in the Coulomb field of a Z>137 charge is obtained. A simple analytic formula is derived for the critical external-field strength corresponding to the onset of stochastization. The diffusion coefficient is evaluated.     

Perturbativity and nonperturbativity in large-<Emphasis Type="Italic">Z</Emphasis> effects for hydrogen-like atoms

The effects induced by the interaction ΔU_AMM of the anomalous component of the electron magnetic moment with the Coulomb field of a nucleus at Zα > 1 are examined for the lower levels of a hydrogen-like atom within purely perturbative and essentially nonperturbative in α/π approaches. Owing to the specific features of the ΔU_AMM operator for a point-like Coulomb source, the distribution of the charge over the volume of the nucleus is set in the nonperturbative case at the quark structure level using valence (constituent) u and d quarks. It is demonstrated that such nonperturbative analysis turns out to be nontrivial as well as the manner in which its results are altered in the transition from the central problem with spherical symmetry to the inclusion of the peripheral contribution. It is also demonstrated that these methods yield matching (with an accuracy no worse than 0.1%) results for the shift of levels 1s_1/2 and 2p_1/2 due to ΔU_AMM at all values of Z through to critical ones. Thus, it is confirmed that the perturbation theory in α/π provides correct results for superheavy atoms if the complete dependence of the wave function on Zα is taken into account from the beginning.     

Absorption of electromagnetic energy by slow electrons under scattering from Coulomb centers

Simple analytical expressions are obtained for the rate of the inverse stimulated bremsstrahlung absorption under electron scattering from a Coulomb center with charge Z in the presence of the electromagnetic field. The initial and final values of electron energy are assumed to be small compared to the Rydberg energy Z ² (atomic units are used throughout). Single-photon processes of absorption and induced radiation of photon by electron are treated. It is assumed that the electromagnetic field frequency ω is rather low, so that the condition Zω/p ³ ? 1, where p is the electron momentum, and the condition ?ω ? p ² are valid. However, this frequency is assumed to be fairly high compared to the electron-Coulomb center collision frequency: ω ? v _nei. The dependences of the rates of photon absorption and induced radiation on the angle θ between the direction of incident electron and the electromagnetic field polarization vector (assumed to be linearly polarized) are obtained. It is demonstrated that, for any angles θ, the rate of photon absorption is higher than the rate of induced radiation and, therefore, the Marcuse effect for slow electrons (electromagnetic field amplification) is absent. It is further demonstrated that a slow electron on the average absorbs double ponderomotive energy per collision with an ion (Coulomb center) in Maxwellian plasma. This agrees both with the known results calculation for fast electrons and with the known results of the calculation based on the classical Boltzmann kinetic equation for plasma.     

Velocity distribution of plasma electrons in the negative H2- and He-glow with superimposed longitudinal magnetic field

The negative glow plasma has been found nearly field free in axial direction. Therefore plasma electrons in the stationary glow can thermalize down to the temperature of the neutral gas, whenever their diffusion—and recombination—lifetime is high enough. Applying Boltzmann's equation to this problem, the conditions of thermalization of plasma electrons are derived as a function of the outer parameters of the plasma: vessel diameter 2R, neutral gas pressurep and longitudinal magnetic fieldB. — If plasma electrons have a too short diffusion—and recombination—lifetime to be in thermal equilibrium with the neutral gas, the electron energy increases. For this case the distribution function of plasma electrons is derived using Boltzmann's equation. Approximating the calculated energy distribution by a Maxwellian distribution function, the electron temperature in the glow is obtained as a function of the parameters:R, p, B. OurT _e -measurements carried out in the H₂- and He-glows of different tube diameters, neutral gas pressures and magnetic fields agree closely with the theoretical results. TheT _e -measurements have been performed with Langmuir probes and by the method of reversal of the radial ambipolar electric field in a longitudinal magnetic field.     

Coulomb corrections to bremsstrahlung in the electric field of a heavy atom at high energies

We consider the differential and partially integrated cross sections for bremsstrahlung from high-energy electrons in an atomic field, with this field taken into account exactly. We use the semiclassical electron Green function and wavefunctions in an external electric field. It is shown that the Coulomb corrections to the differential cross section are very susceptible to screening. Nevertheless, the Coulomb corrections to the cross section summed over the final-electron states are independent of screening in the leading approximation in the small parameter 1/mr _scr (r _scr is the screening radius and m is the electron mass, ? = c = 1). We also consider bremsstrahlung from a finite-size electron beam on a heavy nucleus. The Coulomb corrections to the differential probability are also very susceptible to the beam shape, while the corrections to the probability integrated over momentum transfer are independent of it, apart from the trivial factor, which is the electron-beam density at zero impact parameter. For the Coulomb corrections to the bremsstrahlung spectrum, the next-to-leading terms with respect to the parameters mε (ε is the electron energy) and 1/mr _scr are obtained.     

High Z effects in accounting for radiative component of the electron magnetic moment in hydrogen-like atoms

The behavior of electron energy levels in hydrogen-like atoms is studied while taking into account the nonperturbative interaction between the radiative component of the magnetic moment of a free electron Δg _free and the Coulomb field of an atomic nucleus with charge Z, including those with Z > 137. It is shown that for Zα ? 1 the energy-level shift is rather effectively determined through the matrix elements of the corresponding Dirac-Pauli operator with relativistic Coulomb wave functions. At the same time, for superheavy nuclei with Z ～ 170, this shift, generated by Δg _free, is genuinely nonperturbative, behaves like ～Z ⁵ near the threshold of negative continuum, exceeds all the estimates of radiative corrections coming from vacuum polarization and electron self-energy known so far, and turns out to be at least of the same order as the effects of nuclear charge screening by filled electron shells.     

The multielectron dissociative ionization dynamics of N2 molecule in intense femtosecond laser fields with arbitrary polarization

The field-ionization Coulomb explosion model is extended to investigate the multielectron dissociative ionization process of N₂ molecule irradiated by an intense femtosecond laser field with an arbitrary polarization. The ionization process of N₂ molecule is found to be optimal at the critical internuclear distance R_c=7a.u., which is independent of the laser polarization state, the molecular explosion channel and the angle between the molecular axis and the direction of laser electric field. The kinetic energies of the ion fragments are identical in the cases of linear and circular polarizations at the same incident laser intensity. However,the probability of electron ionization is very sensitive to the above three parameters. At the critical distance R_c=7a.u. the angular dependence of the threshold intensity for the over-the-barrier ionization leads to the geometric alignment of molecules in the case of linear polarization. The threshold intensity in the case of circular polarization is apparently higher than that in the case of linear polarization, which can well explain the significant decrease of ionization in the case of circular polarization. The numerical calculations are compared with the experimental measurements.     

Three-dimensional structure of electron transfer components in photosystem II: “2 + 1” ESE of chlorophyll Z and tyrosine D

A“2 + 1” electron spin echo method has been applied to measure the orientation of the distance vectorR from the redox-active tyrosine residue Y_D to chlorophyll Z Chl_Z in a membraneoriented Mn-depleted preparation of photosystem II. The angle betweenR and the normaln to the photosystem II membranes was determined to be 50.0 ±5° with the value of the dipolar interaction parameterD ₀ = 2.0 MHz, the corresponding distance of 29.4 ±0.5 Å, determined in a nonoriented PS II sample.     

Critical temperature of metallic hydrogen sulfide at 225-GPa pressure

The Eliashberg theory generalized for electron—phonon systems with a nonconstant density of electron states and with allowance made for the frequency behavior of the electron mass and chemical potential renormalizations is used to study T _c in the SH₃ phase of hydrogen sulfide under pressure. The phonon contribution to the anomalous electron Green’s function is considered. The pairing within the total width of the electron band and not only in a narrow layer near the Fermi surface is taken into account. The frequency and temperature dependences of the complex mass renormalization ReZ(ω), the density of states N(ε) renormalized by the electron—phonon interactions, and the electron—phonon spectral function obtained computationally are used to calculate the anomalous electron Green’s function. A generalized Eliashberg equation with a variable density of electron states has been solved. The frequency dependence of the real and imaginary parts of the order parameter in the SH₃ phase has been obtained. The value of T _c ≈ 177 K in the SH₃ phase of hydrogen sulfide at pressure P = 225 GPa has been determined by solving the system of Eliashberg equations.     

Electron gas induced in SrTiO<Subscript>3</Subscript>

This mini-review is dedicated to the 85th birthday of Prof. L.V. Keldysh, from whom we have learned so much. In this paper, we study the potential and electron density depth profiles in surface accumulation layers in crystals with a large and nonlinear dielectric response such as SrTiO₃ (STO) in the cases of planar, spherical, and cylindrical geometries. The electron gas can be created by applying an induction D₀ to the STO surface. We describe the lattice dielectric response of STO using the Landau–Ginzburg free energy expansion and employ the Thomas–Fermi (TF) approximation for the electron gas. For the planar geometry, we arrive at the electron density profile n(x) ∝ (x + d)^–12/7, where d ∝ D₀^–12/7. We extend our results to overlapping electron gases in GTO/STO/GTO heterojunctions and electron gases created by spill-out from NSTO (heavily n-type doped STO) layers into STO. Generalization of our approach to a spherical donor cluster creating a big TF atom with electrons in STO brings us to the problem of supercharged nuclei. It is known that for an atom with a nuclear charge Ze where Z > 170, electrons collapse onto the nucleus, resulting in a net charge Zn < Z. Here, instead of relativistic physics, the collapse is caused by the nonlinear dielectric response. Electrons collapse into the charged spherical donor cluster with radius R when its total charge number Z exceeds the critical value Z_c ≈ R/a, where a is the lattice constant. The net charge eZ_n grows with Z until Z exceeds Z* ≈ (R/a)^9/7. After this point, the charge number of the compact core Z_n remains ≈ Z*, with the rest Z* electrons forming a sparse TF atom with it. We extend our studies of collapse to the case of long cylindrical clusters as well.     

Threshold dependence of the vibrational excitation of molecules on laser radiation intensity

The collisionless vibrational excitation of a polyatomic molecule in an IR laser radiation field has been theoretically studied. It has been shown that (i) the degree of vibrational excitation (namely, number 0000 of vibrational quanta of a molecular mode near-resonant with the IR laser field that are absorbed by the molecule) is low if laser pulse intensity P (energy flux density in the laser beam) is lower than a certain critical value P _cr; (ii) the degree of excitation abruptly increases after crossing the boundary where P = P _cr; (iii) this effect is attributed to two properties inherent in polyatomic molecules, namely, the anharmonicity of the vibrational mode interacting with the laser field and the energy exchange with other modes; and (iv) at P > P _cr, number 00000 is determined only by energy density Φ = Pτ_P, where τ_P is the laser pulse duration, 00000 monotonically increases with increasing Φ. The model is in good agreement with the experimental data.     

Scaling law for the fluid-solid phase transition in Yukawa systems (dusty plasmas)

Yukawa systems serve as models for plasmas and colloidal suspensions of charged particles. The state of these systems is determined by two dimensionless parameters: k = a/λ _D , which is the ratio of the mean interparticle distance to the Debye length λ _D , and Γ = Z _d ² e ²/aT _d , which is the ratio of the Coulomb potential energy to the particle temperature T _d (Z _d is the charge of each particle). We propose an empirical scaling law for the critical coupling parameter Γ _c needed for crystallization in Yukawa systems. The dependence of Γ _c on k is in good agreement with recent molecular dynamics simulations.     

Cascade of Quantum Transitions and Magnetocaloric Anomalies in an Open Nanowire

A sequence of magnetocaloric anomalies occurring with the change in a magnetic field H is predicted for an open nanowire with the Rashba spin–orbit coupling and the induced superconducting pairing potential. The nature of such anomalies is due to the cascade of quantum transitions related to the successive changes in the fermion parity of the nanowire ground state with the growth of the magnetic field. It is shown that the critical H_c values fall within the parameter range corresponding to the nontrivial values of the Z₂ topological invariant of the corresponding 1D band Hamiltonian characteristic of the D symmetry class. It is demonstrated that such features in the behavior of the open nanowire are retained even in the presence of Coulomb interactions.