Green function for the diffusion limit of one-dimensional continuous time random walks

It is shown how the fractional probability density diffusion equation for the diffusion limit of one-dimensional continuous time random walks may be derived from a generalized Markovian Chapman-Kolmogorov equation. The non-Markovian behaviour is incorporated into the Markovian Chapman-Kolmogorov equation by postulating a Levy like distribution of waiting times as a kernel. The Chapman-Kolmogorov equation so generalised then takes on the form of a convolution integral. The dependence on the initial conditions typical of a non-Markovian process is treated by adding a time dependent term involving the survival probability to the convolution integral. In the diffusion limit these two assumptions about the past history of the process are sufficient to reproduce anomalous diffusion and relaxation behaviour of the Cole-Cole type. The Green function in the diffusion limit is calculated using the fact that the characteristic function is the Mittag-Leffler function. Fourier inversion of the characteristic function yields the Green function in terms of a Wright function. The moments of the distribution function are evaluated from the Mittag-Leffler function using the properties of characteristic functions and a relation between the powers of the second moment and higher order even moments is derived. (C) 2004 Elsevier B.V. All rights reserved.

The role of Mittag-Leffler functions in anomalous relaxation

The inertia-corrected Debye model of rotational Brownian motion of polar molecules was generalized by Coffey et al. [Phys. Rev. E, 65, 32 102 (2002)] to describe fractional dynamics and anomalous rotational diffusion. The linear- response theory of the normalized complex susceptibility was given in terms of a Laplace transform and as a function of frequency. The angular-velocity correlation function was parametrized via fractal Mittag-Leffler functions. Here we apply the latter method and complex-contour integral- representation methods to determine the original time-dependent amplitude as an inverse Laplace transform using both analytical and numerical approaches, as appropriate. (C) 2004 Elsevier B.V. All rights reserved.

An exactly solvable four transition point model

Heavy particle collisions, in particular low-energy ion-atom collisions, are amenable to semiclassical JWKB phase integral analysis in the complex plane of the internuclear separation. Analytic continuation in this plane requires due attention to the Stokes phenomenon which parametrizes the physical mechanisms of curve crossing, non-crossing, the hybrid Nikitin model, rotational coupling and predissociation. Complex transition points represent adiabatic degeneracies. In the case of two or more such points, the Stokes constants may only be completely determined by resort to the so-called comparison- equation method involving, in particular, parabolic cylinder functions or Whittaker functions and their strong-coupling asymptotics. In particular, the Nikitin model is a two transition-point one-double-pole problem in each half-plane corresponding to either ingoing or outgoing waves. When the four transition points are closely clustered, new techniques are required to determine Stokes constants. However, such investigations remain incomplete, A model problem is therefore solved exactly for scattering along a one-dimensional z-axis. The energy eigenvalue is b(2)-a(2) and the potential comprises -z(2)/2 (parabolic) and -a(2) + b(2)/2z(2) (centrifugal/centripetal) components. The square of the wavenumber has in the complex z-plane, four zeros each a transition point at z = +/-a +/- ib and has a double pole at z = 0. In cases (a) and (b), a and b are real and unitarity obtains. In case (a) the reflection and transition coefficients are parametrized by exponentials when a(2) + b(2) > 1/2. In case (b) they are parametrized by trigonometrics when a(2) + b(2) <1/2 and total reflection is achievable. In case (c) a and b are complex and in general unitarity is not achieved due to loss of flux to a continuum (O'Rourke and Crothers, 1992 Proc. R. Sec. 438 1). Nevertheless, case (c) coefficients reduce to (a) or (b) under appropriate limiting conditions. Setting z = ht, with h a real constant, an attempt is made to model a two-state collision problem modelled by a pair of coupled first-order impact parameter equations and an appropriate (T) over tilde-tau relation, where (T) over tilde is the Stueckelberg variable and tau is the reduced or scaled time. The attempt fails because (T) over tilde is an odd function of tau, which is unphysical in a real collision problem. However, it is pointed out that by applying the Kummer exponential model to each half-plane (O'Rourke and Crothers 1994 J. Phys. B: At. Mel. Opt. Phys. 27 2497) the current model is in effect extended to a collision problem with four transition points and a double pole in each half-plane. Moreover, the attempt in itself is not a complete failure since it is shown that the result is a perfect diabatic inelastic collision for a traceless Hamiltonian matrix, or at least when both diagonal elements are odd and the off-diagonal elements equal and even.

Semiclassical-quantal approach to the near-threshold ionization of hydrogen

A recent improved version of the semiclassical-quantal approach has been applied to the e(-)-H near-threshold ionization for theta (12) = 180 degrees geometry. It is found, that unlike other sophisticated theoretical methods such as distorted wave theory or convergent close-coupling calculation, the present relatively simpler approach produces correct behavior and numerical values for the triple-differential cross sections. We compare our results with recent absolute measurements and accurate numerical calculations at 2 eV and 4 eV above the threshold at constant theta (12) geometry.

Electron impact ionization of He above the threshold

Near-threshold ionization of He has been studied by using a uniform semiclassical wavefunction for the two outgoing electrons in the final channel. The quantum mechanical transition amplitude for the direct and exchange scattering derived earlier by using the Kohn variational principle has been used to calculate the triple differential cross sections. Contributions from singlets and triplets are critically examined near the threshold for coplanar asymmetric geometry with equal energy sharing by the two outgoing electrons. It is found that in general the tripler contribution is much smaller compared to its singlet counterpart. However, at unequal scattering angles such as theta (1) = 60 degrees, theta (2) = 120 degrees the smaller peaks in the triplet contribution enhance both primary and secondary TDCS peaks. Significant improvements of the primary peak in the TDCS are obtained for the singlet results both in symmetric and asymmetric geometry indicating the need to treat the classical action variables without any approximation. Convergence of these cross sections are also achieved against the higher partial waves. Present results are compared with absolute and relative measurements of Rosel et al (1992 Phys. Rev. A 46 2539) and Selles et al (1987 J. Phys. B. At. Mel. Phys. 20 5195) respectively.

Strong configuration mixing among the fine-structure levels of Cl+

A large-scale configuration interaction (Cl) calculation using Program CIV3 of Hibbert is performed for the lowest 62 fine- structure levels of the singly charged chlorine ion. Our calculated energy levels agree very well with most of the NIST results and confirm the identification of the lowest P-1(o) as actually 3s(2)3p(3)(D-2(o))3d P-1(o) rather than the generally employed 3s3p(5) P-1(o) in measurements and calculations. Discrepancies in the energy positions of some symmetries are found and discussed. Some large oscillator strengths for allowed and intercombination transitions in both length and velocity gauges are presented. Their close agreement gives credence to the accuracy of our CI wavefunctions.

Semiclassical analytical approach to the description of quasimolecular optical transitions

A formula was obtained that describes asymptotically forbidden quasimolecular optical transitions in the frame of the semiclassical approach. It is particularly relevant for the weak extrema in the difference between the ground- and excited- state interaction potentials. When averaged over impact parameters and velocity distribution the formula agreed reasonably well with the recent experimental data for the Ca(4(1)S --> 3(1)D) + He transition.

Energy dependence for pair production with electron capture in relativistic heavy-ion collisions

In 'Charge transfer from the negative-energy continuum: alternative mechanism for pair production in relativistic atomic collisions', Eichler (1995 Phys. Rev. Lett. 75 3653) proposes an alternative mechanism for capture by pair production, and from it derives an analytic expression for the total cross section with a surprisingly strong energy dependence. We show that, in fact, there is no alternative mechanism; rather the above mechanism may be more transparently viewed as an ionization-like transition in one centre with inclusion of continuum distortion by the second centre. We further show that to Centre the initial and final states on the target and projectile leads to confusion in the momentum transfer vectors, and hence, respectively that the alleged high-energy behaviour is erroneous.

Shifts in electron capture to the continuum at low collision energies: Enhanced role of target postcollision interactions

Measurements of electron velocity distributions emitted at 0degrees for collisions of 10- and 20-keV H+ incident ions on H-2 and He show that the electron capture to the continuum cusp formation, which is still possible at these low impact energies, is shifted to lower momenta than its standard position (centered on the projectile velocity), as recently predicted. Classical trajectory Monte Carlo calculations reproduce the observations remarkably well, and indicate that a long-range residual interaction of the electron with the target ion after ionization is responsible for the shifts, which is a general effect that is enhanced at low nuclear velocities.

A reactive force field simulation of liquid-liquid phase transitions in phosphorus

A force field model of phosphorus has been developed based on density functional (DF) computations and experimental results, covering low energy forms of local tetrahedral symmetry and more compact (simple cubic) structures that arise with increasing pressure. Rules tailored to DF data for the addition, deletion, and exchange of covalent bonds allow the system to adapt the bonding configuration to the thermodynamic state. Monte Carlo simulations in the N-P-T ensemble show that the molecular (P-4) liquid phase, stable at low pressure P and relatively low temperature T, transforms to a polymeric (gel) state on increasing either P or T. These phase changes are observed in recent experiments at similar thermodynamic conditions, as shown by the close agreement of computed and measured structure factors in the molecular and polymer phases. The polymeric phase obtained by increasing pressure has a dominant simple cubic character, while the polymer obtained by raising T at moderate pressure is tetrahedral. Comparison with DF results suggests that the latter is a semiconductor, while the cubic form is metallic. The simulations show that the T-induced polymerization is due to the entropy of the configuration of covalent bonds, as in the polymerization transition in sulfur. The transition observed with increasing P is the continuation at high T of the black P to arsenic (A17) structure observed in the solid state, and also corresponds to a semiconductor to metal transition. (C) 2004 American Institute of Physics.

Equilibrium polymerization of cyclic carbonate oligomers. II. Role of multiple active sites

Ring opening polymerization of bisphenol A polycarbonate is studied by Monte Carlo simulations of a model comprising a fixed number of Lennard-Jones particles and harmonic bonds [J. Chem. Phys. 115, 3895 (2001)]. Bond interchanges produced by a low concentration (0.10%less than or equal toc(a)less than or equal to0.36%) of chemically active particles lead to equilibrium polymerization. There is a continuous transition in both 2D and 3D from unpolymerized cyclic oligomers at low density to a system of linear chains at high density, and the polymeric phase is much more stable in three dimensions than in two. The steepness of the polymerization transition increases rapidly as c(a) decreases, suggesting that it is discontinuous in the limit c(a)-->0. The transition is entropy driven, since the average potential energy increases systematically upon polymerization, and there is a steady decline in the degree of polymerization as the temperature is lowered. The mass distribution functions for open chains and for rings are unimodal, with exponentially decaying tails that can be fitted by Zimm-Schulz functions and simpler exponential forms. (C) 2002 American Institute of Physics.

Equilibrium polymerization of cyclic carbonate oligomers

A model of the polymerization of ring oligomers of bisphenol A polycarbonate (BPA-PC) is used to investigate the influence of dimensionality (2D or 3D), density and temperature on the size distribution of the polymer chains. The polymerization step is catalyzed by a single active particle, conserves the number and type of the chemical bonds, and occurs without a significant gain in either potential energy or configurational entropy. Monte Carlo and molecular dynamics simulations show that polymerization of cyclic oligomers occurs readily at high density and is driven by the entropy associated with the distribution of interparticle bonds. Polymerization competes at lower densities with long range diffusion, which favors small molecular species, and is prevented if the system is sufficiently dilute. Polymerization occurs in 2D via a weakly first order transition as a function of density and is characterized by low hysteresis and large fluctuations in the size of polymer chains. Polymerization occurs more readily in 3D than in 2D, and is favored by increasing temperature, as expected for an entropy-driven process. (C) 2001 American Institute of Physics.

Light emission from scanning tunnelling microscope on polycrystalline Au films - what is happening at the single-grain level?

When operated with a metallic tip and sample the scanning tunnelling microscope constitutes a nanoscale, plasmonic light source yielding broadband emission up to a photon energy determined by the applied bias. The emission is due to tunnelling electron excitation and subsequent radiative decay of localized plasmon modes, which can be on the lateral scale of a single metal grain (similar to 25 nm) or less. For a Au-tip/Au-polycrystalline sample under ambient conditions it is found that the intensity and spectral content of the emitted light are not dependent on the lateral grain dimension, but are predominantly determined by the tip geometry. However, the intensity increases strongly with increasing film thickness (grain depth) up to 20-25 nm or approximately the skin depth of the Au film. Photon maps can show less emissive grains and two classes of this occurrence are distinguished. The first is geometrical in origin - a double-tip structure in this case - while the second is due to a contamination-induced lowering of the local work function that causes the tunnel gap to increase. It is suggested that differences in work-function lowering between grains presenting different crystalline facets, combined with an exponential decay in emitted light intensity with tip - sample distance, leads to grain contrast. These results are relevant to tip-enhanced Raman scattering and the fabrication of micro/nano-scale planar, light-emitting tunnel devices.

Finite temperature dense matter studies on next-generation light sources

The construction of short-pulse tunable soft x-ray free electron laser sources based on the self-amplified spontaneous emission process will provide a major advance in capability for dense plasma-related and warm dense matter (WDM) research. The sources will provide 10(13) photons in a 200-fs duration pulse that is tunable from approximately 6 to 100 nm. Here we discuss only two of the many applications made possible for WDM that has been severely hampered by the fact that laser-based methods have been unavailable because visible light will not propagate at electron densities of n(e) greater than or equal to 10(22) cm(-3). The next-generation light sources will remove these restrictions.

Electron impact ionization of hydrogen-like molybdenum ions

The electron impact ionization cross sections of hydrogen-like molybdenum ions were measured with an electron beam ion trap at the electron energies of 49.4, 64.4 and 79.6 keV The results are 2.82(22) x 10(-23), 3.13(29) x 10(-23) and 3.23(51) x 10(-23) cm(2), respectively. These results are compared with the experimental results measured previously. The agreement with the results obtained with, scaling formulae is also discussed.