Charge Ze Research Articles

Non-relativistic binding energies of heavy neutral atoms: Dependence of correlation energy on atomic number

As shown by Milne in the early days of quantum mechanics, the non-relativistic binding energy of a heavy neutral atom with nuclear charge Ze is proportional to Z 7/3 in the large Z limit. Correction terms of order Z 2 and Z 5/3 to Milne’s limiting large Z result were first calculated by Scott. Here, the question as to the power of Z at which correlation energy first appears in such a binding energy formula is addressed. It is shown that a recent approximation proposed by one of us (ÁN) supports an earlier extrapolation from large dimensionality D to D = 3 by Herschbach and co-workers, that correlation energy enters the large Z binding energy formula for the first time at o( Z 4/3).

Towards a differential equation for the nonrelativistic ground-state electron density of the He-like sequence of atomic ions

The early study of Schwartz [Ann. Phys. (N.Y.) 6, 156 (1959)] led to an explicit expression for the ground-state electron density {rho}(r) of He-like atomic ions with nuclear charge Ze in the limit of large Z. Much later, Gal, March, and Nagy [Chem. Phys. Lett. 305, 429 (1999)] derived a third-order linear homogeneous differential equation satisfied by the Schwartz limiting density. Our aim here has been to solve a still correlated model more closely related to the He atom itself. Motivated by semiclassical studies (as by Handke [Phys. Rev. A 50, R3561 (1994)]), we have solved a fully quantal model in which the Coulomb repulsion energy e{sup 2}/r{sub 12} of the two electrons at separation r{sub 12} is expanded in a one-center form about the atomic nucleus and only the s-wave term is retained. From the resulting analytical ground-state density, a differential equation has been derived, which is contrasted with that satisfied by the large-Z limiting form of Schwartz. Suggestions are finally made as to the way the density of the He atomic ion series may be characterized, the ionization potential I(Z) being proposed as crucial input data.

Binding energy for hydrogen-like atoms in the Nelson model without cutoffs

In the Nelson model particles interact through a scalar massless field. For hydrogen-like atoms there is a nucleus of infinite mass and charge Ze, Z > 0 , fixed at the origin and an electron of mass m and charge e. This system forms a bound state with binding energy E bin = me 4 Z 2 / 8 π 2 to leading order in e. We investigate the radiative corrections to the binding energy and prove upper and lower bounds which imply that E bin = me 4 Z 2 / 8 π 2 + c 0 e 6 + O ( e 7 ln e ) with explicit coefficient c 0 and independent of the ultraviolet cutoff. c 0 can be computed by perturbation theory, which however is only formal since for the Nelson Hamiltonian the smallest eigenvalue sits exactly at the bottom of the continuous spectrum.

A Note on Time - Independent Electric Field in Superconductors

We derive a set of coupled partial differential equations for the determination of the electric field and of the order parameter of the superconducting electrons. For this purpose, we propose an expression for the free energy of the superconducting electrons in the presence of an electric field, the minimization of which yields the above-mentioned equations. It is shown that for a superconductor at zero temperature the electric field of a test charge Ze decreases exponentially with distance from the charge and the London penetration depth plays the role of the Debye length.

Normal mode spectra of two-dimensional classical atoms confined by a Coulomb potential.

The normal mode spectra of two-dimensional finite clusters of charged particles (-e) confined by a Coulomb potential resulting from a displaced positive charge Ze are obtained. This is a classical two-dimensional model system for atoms. We obtain the frequencies of the normal modes as a function of the confinement charge Z and the number of particles N. The analysis of the lowest normal mode frequency reveals a good agreement with the experimental results obtained in a system with screened interaction between charged particles. The dependence of the normal mode spectra as a function of a perpendicular magnetic field is also discussed and we found that the shear-like character of the modes is enhanced in the presence of the magnetic field. For large values of the magnetic fields the normal modes fall into two bands, a low frequency band with frequency approximately 1/omega(c) and a high frequency band with frequency approximately omega(c).

Lensing of ultra-high energy cosmic rays in turbulent magnetic fields

We consider the propagation of ultra high energy cosmic rays through turbulent magnetic fields and study the transition between the regimes of single and multiple images of point-like sources. The transition occurs at energies around Ec Z 41 EeV(Brms/5 μG)(L/2 kpc)3/2(50 pc/Lc)1/2, where L is the distance traversed by the CR's with electric charge Ze in the turbulent magnetic field of root mean square strength Brms and coherence length Lc. We find that above 2Ec only sources located in a fraction of a few % of the sky can reach large amplifications of its principal image or start developing multiple images. New images appear in pairs with huge magnifications, and they remain amplified over a significant range of energies. At decreasing energies the fraction of the sky in which sources can develop multiple images increases, reaching about 50% for E > Ec/2. The magnification peaks become however increasingly narrower and for E < Ec/3 their integrated effect becomes less noticeable. If a uniform magnetic field component is also present it would further narrow down the peaks, shrinking the energy range in which they can be relevant. Below E Ec/10 some kind of scintillation regime is reached, where many demagnified images of a source are present but with overall total magnification of order unity. We also search for lensing signatures in the AGASA data studying two-dimensional correlations in angle and energy and find some interesting hints.

Cosmic ray acceleration to very high energy through the non-linear amplification by cosmic rays of the seed magnetic field

The maximum energy for cosmic ray acceleration at supernova shock fronts is usually thought to be limited to around 10 14 ±10 15 eV by the size of the shock and the time for which it propagates at high velocity. We show that the magnetic field can be amplified non-linearly by the cosmic rays to many times the pre-shock value, thus increasing the acceleration rate and facilitating acceleration to energies well above 10 15 eV. A supernova remnant expanding into a uniform circumstellar medium may accelerate protons to 10 17 eVand heavy ions, with charge Ze ,t oZ � 10 17 eV: Expansion into a pre-existing stellar wind may increase the maximum cosmic ray energy by a further factor of 10.

Structure of charged colloids under a wedge confinement

Monte Carlo simulations have been performed to study the influence of a wedge confinement by hard walls on the ordering of charged colloidal particles interacting through screened Coulomb repulsive potential in an aqueous medium. The density distribution of particles for a fixed wedge angle straight theta(0) is studied for different suspension parameters, viz., bulk volume fraction straight phi, salt concentration C(s), and charge Ze on the particles. The density distribution rho(straight theta) along the angular direction and that along the radial direction, rho(r), have been analyzed in different regions of the wedge. Simulations show the formation of layered structure along the angular direction and a large gathering of particles along the wall. The number of layers as well as the density of particles within the layer are found to change as the strength and range of the interaction are varied, whereas the density profiles calculated close to the vertex region showed no significant variation in the density. The radial density profiles rho(r) corresponding to the vertex region show one-dimensional (1D) ordering of particles parallel to the vertex at a distance that is close to a wedge height, h equal to diameter of the particle. This 1D ordering is found to be destroyed upon the addition of salt or lowering the straight phi. The reported experimental observations on the "vacuum phase" are discussed in the light of the present results.

Inverse stimulated bremsstrahlung of slow electrons under Coulomb scattering

A simple new analytic expression has been obtained for the rate of inverse stimulated bremsstrahlung of slow electrons scattered by the Coulomb charge Ze. The initial and final electron energies are assumed to be small compared to the Rydberg energy mZ2e4/2. Weak linearly polarized light of frequency is considered, and only one-photon emission and absorption of the light are taken into account. High-frequency (>>mv3/Ze2) and low-frequency (<<mv3/Ze2) limits are considered (v is the electron velocity). Both the quantum cases ~p2, >p2 and the classical limit <<p2 are considered. The rate is averaged over the direction of the final electron momentum and over the angle between the light polarization and the direction of the initial electron momentum. It was found that in both laser frequency limits the rate of one-photon absorption is larger than the rate of one-photon emission, analogous to the well known opposite Born limit of rapid electrons. Therefore, the energy of a slow electron increases with time (inverse stimulated bremsstrahlung). The rate of energy gain for slow electrons decreases with increasing laser frequency . The rates are averaged over Maxwell distribution of electrons. It was also found that the energy gain in the Born limit (rapid electrons) coincides with the energy gain for slow electrons if <<mv3/Ze2. The rates of electron-ion collisions in plasma are discussed in various limits.

Inhomogeneous Electron Liquid: H Atom Dirac Density Matrix in Limit of Infinite Number of Closed Shells

The Dirac density matrix ρ N () for the H atom with N closed shells is here treated in the limit N tends to infinity. Then ρα () is shown to be derivable exactly from the zero energy limit of the bound state only Green function Gb (Z) (; E). It is further demonstrated for nuclear charge Ze that the Laplace transform of Gb (Z) with respect to Z satisfies a useful equation of motion. Finally the s-wave only components of ρN and Gb (Z) are considered.

Nuclear Electron Capture in a Plasma

We consider the electron density at the position of an ion of charge Ze in a plasma under conditions approximating those in the core of the sun. Numerical calculations have shown that the plasma effects on the density, over and above the ordinary Coulomb factors that are obtained in the absence of electron-electron interactions, are well represented by a reduction factor, exp{- Z e^2 beta kappa_D }, where beta is the inverse temperature and kappa_D is the Debye wave length. Although this factor is the direct analogue of the Salpeter enhancement factor for the fusion rates in stars, the elementary considerations that establish it in the fusion case are not applicable to the determination of the electron density and the resulting electron capture rates. We show analytically, through a sum rule that leads to a well-defined perturbative approach, that in the limit of Boltzmann statistics the Salpeter factor indeed provides the leading correction. We estimate residual effects, both from Fermi statistics and from short range terms.

Equilibrium electric potential of spherical, cylindrical, and planar dust grains moving through a plasma

The charge on a dust grain in a plasma depends on the plasma parameters and on the grain's velocity through the plasma. Intuition suggests that at high velocity the grain should sweep up equal numbers of both the positive and negative plasma species, and so it should come to zero potential with respect to the plasma. This is in fact not always the case. We have derived the equilibrium potential ϕ of a small conducting sphere moving at speed w through a plasma of temperature T composed of electrons with charge −e and ions with charge Ze and mass ratio M = mi/me and find that for large w the potential goes to the finite value eϕ/kT = −1/2[(M − 1)/(M + Z)]. For a conducting cylinder of infinite length in the same plasma, eϕ/kT is slightly different owing to the different potential variation outside the grain, but for large cylinder speed the potential goes to the same nonzero limit as the sphere. The case of an infinite, conducting plane is quite different, and values depend on whether there is plasma on one side of the plane or on both. In the latter case, which is most analogous to those of the sphere and cylinder, ϕ does go to zero for large plane speed. We show equilibrium e|ϕ|/kT versus w curves for the three elementary shapes, in each instance for an O+ − e−, H+ − e−, and “5me”+ − e− plasma, and comment on characteristics. The last example is a plasma of electrons and ions having the artificially small mass ratio M = 5, of interest because results can be compared with computer simulations of particle dynamics in the presence of a charged grain.

Interactions in undersaturated and supersaturated lysozyme solutions: Static and dynamic light scattering results

We have performed multiangle static and dynamic light scattering studies of lysozyme solutions at pH=4.7. The Rayleigh ratio Rθ and the collective diffusion coefficient Dc were determined as function of both protein concentration cp and salt concentration cs with two different salts. At low salt concentrations, the scattering ratio Kcp/Rθ and diffusivity increased with protein concentration above the values for a monomeric, ideal solution. With increasing salt concentration this trend was eventually reversed. The hydrodynamic interactions of lysozyme in solution, extracted from the combination of static and dynamic scattering data, decreased significantly with increasing salt concentration. These observations reflect changes in protein interactions, in response to increased salt screening, from net repulsion to net attraction. Both salts had the same qualitative effect, but the quantitative behavior did not scale with the ionic strength of the solution. This indicates the presence of salt specific effects. At low protein concentrations, the slopes of Kcp/Rθ and Dc vs cp were obtained. The dependence of the slopes on ionic strength was modeled using a DLVO potential for colloidal interactions of two spheres, with the net protein charge Ze and Hamaker constant AH as fitting parameters. The model reproduces the observed variations with ionic strength quite well. Independent fits to the static and dynamic data, however, led to different values of the fitting parameters. These and other shortcomings suggest that colloidal interaction models alone are insufficient to explain protein interactions in solutions.

Electron density near the nucleus of a large atom.

The density of electrons on a distance scale 1/Z near the nucleus of a large atom with nuclear charge Ze is given (asymptotically as Z\ensuremath{\rightarrow}\ensuremath{\infty}) by the sum of the squares of all the hydrogenic bound-state functions (with nuclear charge Ze). This density function, which is an important limiting function in quantum chemistry, is investigated here in detail. Several analytic results are found: In particular, the asymptotic expansion for large r is derived and it is shown that the function falls off as ${\mathit{r}}^{\mathrm{\ensuremath{-}}3/2}$ for large r; this behavior coincides with the Thomas-Fermi density for small r. ``Shell structure'' is visible, but barely so.

Multi-electron ejection of inner-shell electrons through multiphoton excitation of clusters

The existence of coherent electronic motions induced by the presence of a strong external optical field enables the process of direct multi-electron ejection from an inner-shell to occur with high probability. Experimental evidence for this mechanism is present in the observed spectra of five cases studied (Kr(M), Kr(L), Xe(N), Xe(M) and Xe(L)). Analysis shows that the multiphoton interaction can enter a regime of strong-coupling (Z2a>or=1) in which the rate of multiple electron ejection can become comparable to the removal of a single electron. The enhanced coupling arises from the coherent motion of the (Z) field-ionization electrons whose interactions constructively interfere, thereby behaving like a quasi-particle with a charge Ze. This mechanism consistently explains (1) the unusually high values of Xe M-shell ionization observed in both Xe(M) (Xe36+) and Xe(L) (Xe45+) emissions and (2) the anomalous double 2p vacancy production seen in the Xe(L) spectrum. Clusters of heavy atoms appear as exceptionally favourable systems for the copious production of multiple vacancy hollow atom configurations.

Evidence for coherent electron motions in multiphoton X-ray production from Kr and Xe clusters

Studies of multiphoton-induced X-ray generation in Kr and Xe clusters give direct information concerning (1) the atom-specific energy transfer rate, (2) the dependence of the X-ray yield on the strength of the intra-cluster inelastic electron scattering cross section, and (3) the threshold intensity for X-ray generation. Measurements of these three classes of observables with subpicosecond ( approximately 300 fs) ultraviolet (248 nm) radiation at a maximum intensity of approximately 1019 W cm-2 all indicate that the non-linear coupling to the cluster has an anomalous strength with respect to that derivable from conventional single-particle interactions. Five cases have been examined (Kr(M), Kr(L), Xe(N), Xe(M) and Xe(L)), spectrally spanning the range from approximately 80 eV to approximately 5 keV. In order to reconcile theoretical estimates with these experimental findings, three generalizations of the original formulation of the interaction were necessary. The most important involves an enhancement in the coupling arising from the coherent motion of the (Z) field-ionized electrons induced by the external driving field. The coherently energized electrons act like a quasi-particle possessing a charge Ze and a mass Zm, thereby presenting a sharply augmented coupling resembling that associated with energetic ion-atom collisions. The second involves the process of multiple electron ejection from an inner-shell, a mechanism that was found imperative to interpret the high level of ionization observed in the Xe M-shell. The third modification concerns multiple transits of the driven electrons in the cluster. The inclusion of these considerations consistently brings the theoretical analysis into agreement with the three measured properties. These results also indicate that energy deposition rates exceeding approximately 1 W/atom are feasible in appropriately designed molecules incorporating heavy atoms. The limiting magnitude of the excitation energy Delta Emax characteristic of the coherent coupling is estimated to be in the range Zamc2<or= Delta Emax<or=Z2amc2, a bound that can considerably exceed the K-shell binding energy of the heaviest atoms.

Asymptotics of heavy atoms in high magnetic fields: I. Lowest landau band regions

AbstractThe ground state energy of an atom of nuclear charge Ze in a magnetic field B is evaluated exactly to leading order as Z → ∞. In this and a companion work (see [28]) we show that there are five regions as Z → ∞: B &lt; Z4/3, B ∼ Z4/3, Z4/3 &lt; B &lt; Z3, B ∼ Z3, B &gt; Z3. Regions 1, 2, 3, and 4 (and conceivably 5) are relevant for neutron stars. Different regions have different physics and different asymptotic theories. Regions 1, 2, and 3 are described by a simple density functional theory of the semiclassical Thomas‐Fermi form. Here we concentrate mainly on regions 4 and 5 which cannot be so described, although 3, 4, and 5 have the common feature (as shown here) that essentially all electrons are in the lowest Landau band. Region 5 does have, however, a simple non‐classical density functional theory (which can be solved exactly). Region 4 does not, but, surprisingly, it can be described by a novel density matrix functional theory. © 1994 John Wiley &amp; Sons, Inc.

Fragmentation of Highly Charged Metallic Clusters

ABSTRACTMultiply charged metal clusters undergo fission at a certain size. This critical size can be predicted by the liquid drop model when some modifications are taken into consideration. In this work we revise the asymmetric liquid drop model (ALD) and modify it for the alkali metals. This modification addresses those fragmentation channels in which a parent cluster with charge Ze fissions into two fragments. One of the fragments is small and singly charged whereas the second fragment is large and carries the rest of the charge. A different energetic balance equation is presented in which the ionization energy of a single atom and the energy of formation of a small cluster are included. Results and comparison to experiments is provided for Na and Cs clusters. Prediction of the critical size of Na and Cs clusters with Ze&gt;7 is part of the discussion.

Heavy atoms in the strong magnetic field of a neutron star.

The ground-state energy of an atom of nuclear charge Ze and in a magnetic field B is evaluated exactly in the asymptotic regime Z\ensuremath{\rightarrow}\ensuremath{\infty}. We show rigorously that there are five regions as Z\ensuremath{\rightarrow}\ensuremath{\infty}: B\ensuremath{\ll}${\mathit{Z}}^{4/3}$, B\ensuremath{\approxeq}${\mathit{Z}}^{4/3}$, ${\mathit{Z}}^{4/3}$\ensuremath{\ll}B\ensuremath{\ll}${\mathit{Z}}^{3}$, B\ensuremath{\approxeq}${\mathit{Z}}^{3}$, B\ensuremath{\gg}${\mathit{Z}}^{3}$. Different regions have different physics and different asymptotic therories. Regions 1,2,3,5 are described exactly by a simple density-functional theory, but only in regions 1,2,3 is it of the semiclassical Thomas-Fermi form. Region 4 cannot be described exactly by any simple density-functonal theory; surprisingly, it can be described by a simple density matrix functonal theory.

Electron density in the ground states of atoms and molecules

The ground-state energy E of an atom or molecule is known to be a unique functional of the electron density ϱ( r). Unfortunately the functional, to date, is unknown except in the limiting cases of large and small numbers of electrons N. In this article, therefore, emphasis is first placed on the fact that, for an atomic ion having nuclear charge Ze, the Hamiltonian is fixed by specifying Z and N and hence E[ϱ] becomes E( Z, N). The 1/ Z expansion of this quantity is first set out and linked with the Thomas-Fermi statistical limit, in which both Z and N become large such that 0 < N/Z ⩽ 1. Partial summation of the 1/ Z expansion then leads to essential corrections in the Thomas-Fermi energy from (i) density gradients and (ii) exchange. The resulting formula is brought into close contact with numerical Hartree-Fock energies. This theory is then extended to apply to simple homonuclear diatomic molecules. Using scaling properties at the internuclear equilibrium distance, the above formula for atomic ions is generalized to such neutral molecules. By subtraction, the structure of the dissociation energy D is exposed in relation to (a) the number of electrons N and (b) the lowest-order density gradient kinetic energy correction T 2. In particular, D/N 2 will be shown to depend on the one-sixth power of T 2 ; contact will also be made with an empirical relation previously exposed. Finally, a discussion will be given of the effects on the ground states, especially for atomic ions, of both intense magnetic and electric fields; the electron density theory is again used, but now working within a statistical-mechanical context.