Simple Perturbation Theory Research Articles

Absorption-edge singularities for a nonequilibrium Fermi sea. II. Second-order diagrammatic expansion.

The diagrammatic expansion of absorption-edge singularities for a nonequilibrium Fermi sea is performed, up to second order in the conduction-electron--deep-hole final-state interaction. To this end, we use a nonstandard electron Green's function that we have adequately modified to take the nonequilibrium Fermi sea into account. This second-order diagrammatic expansion allows us to validate this Green's function, which we will use in a nonperturbative solution of the problem. This approach also puts the absorption in a different light, when compared to the physical picture provided by simple second-order perturbation theory (paper I). Finally, it allows us to determine securely the part of the critical exponent associated with excitations of electron-hole pairs in the Fermi sea.

Absorption-edge singularities for a nonequilibrium Fermi sea. I. Second-order perturbation theory.

The absorption threshold singularity induced by the conduction-electron--deep-hole final-state interaction is recovered from the Fermi golden rule using a simple second-order perturbation theory. This approach allows a direct physical understanding of the various contributions to the power-law critical exponent. We use this approach to get the absorption singularities for a nonequilibrium Fermi sea (${\mathrm{\ensuremath{\mu}}}_{1}$,${\mathrm{\ensuremath{\mu}}}_{2}$) instead of (0,\ensuremath{\mu}). We find a diverging behavior at ${\mathrm{\ensuremath{\mu}}}_{2}$ and a converging singularity at ${\mathrm{\ensuremath{\mu}}}_{1}$. Unexpectedly, the critical exponents of these singularities disobey Hopfield's well accepted rule of thumb.

Si 2p core-level chemical shifts at the H/Si(111)-(1 x 1) surface.

We calculate the core-level shift of the Si 2[ital p] levels for atoms near the H/Si(111)-(1[times]1) surface. We show that a simple first-order perturbation theory using pseudopotentials and the local-density approximation gives good results for the photoemission spectra of the core electrons. The electric-dipole matrix elements for the levels are also calculated. The results are in good agreement with a recent high-resolution angle-resolved photoemission-spectroscopy measurement and allow us to interpret some previously unexplained experimental peaks.

Theory of phase equilibria and closed-loop liquid–liquid immiscibility for model aqueous solutions of associating chain molecules: Water +alkanol mixtures

In this work we examine the phase equilibria exhibited by a number of model binary mixtures of water+alkanols. The water molecules are modeled as hard spheres with four off-center square-well hydrogen-bonding sites representing the two hydrogen atoms and the two electron lone pairs on the oxygen atom. Dispersion forces are included and are treated within the mean-field approximation. Each alkanol is modeled as a chain molecule formed from fused hard spheres with dispersion forces and two hydrogen-bonding sites. Thus water–water, water–alkanol, and alkanol–alkanol association is allowed. A simple thermodynamic perturbation theory is used to develop an augmented van der Waals equation of state which treats the asymmetry in the attractions which arise because of the directional hydrogen-bonding sites, and the asymmetry in the repulsions caused by the molecular shape of the chains. The effects of chain length and association on the global phase equilibria are studied. As well as the usual gas–liquid phase separation, liquid–liquid and gas–gas phase separation, positive, negative, and double azeotropes, and Bancroft points are found. Closed-loop liquid–liquid immiscibility corresponding to the type VI classification of Scott and van Konynenburg is seen for water+alkanol mixtures with intermediate chain lengths and bonding energies. We also discuss our serious reservations about the type VI behavior claimed to have been found in some recent work. Although the potential parameters used in our work were not chosen to model any specific systems, it is gratifying to see that the theory can reproduce the types of phase behavior exhibited by aqueous solutions of alkanols.

Simple theory of a nonlinear diocotron mode

A simple perturbation theory of the frequencies and density distortions of the l=1 diocotron mode on pure electron plasma columns is presented. This mode is the dynamical state of an off-axis electron column which is orbiting around the containment axis at the E×B drift velocity. The theory predicts shifts in the orbit frequency and quadrupole distortions of the plasma surface, both varying as the offset squared, in good agreement with experiments. The results also apply to a two-dimensional vortex patch in an inviscid fluid with circular boundaries.

Halfway house variational continuum distorted wave theory: high energy cross sections in the distorted wave perturbation approximation

The total cross section for charge transfer at high but non-relativistic energy is investigated within a new time-independent wave version of the halfway-house phase integral variational continuum distorted wave formalism, using simple first order perturbation theory. Following Crothers, Dube and Brown this theory satisfies all known short and long range Coulomb boundary conditions which is essential if intermediate elastic divergences are to be avoided. The discussion is specialised to the particular case ofls →ls capture. Numerical results for this amplitude are obtained by a double numerical integration. Wave results derived for the capture process $$P^{Z_P^ + } + (T^{Z_T^ + } + e^ - ) \to (P^{Z_P^ + } + e^ - ) + T^{Z_T^ + } $$ in the caseZT = 1 =ZP are compared with various theoretical impact parameter models and experimental data. Representative distributions of the total cross sections are presented as a function of the impact energy.

Theory of phase equilibria for model aqueous solutions of chain molecules: water + alkane mixtures

This work is the first in a series of theoretical studies of aqueous solutions. We begin by examining the phase equilibria exhibited by a number of model binary mixtures of water and alkanes. The water molecules are modelled as hard spheres with four off-centre square-well hydrogen-bonding sites representing the two hydrogen atoms and the two electron lone pairs on the oxygen atom. Dispersion forces are included, but these are treated within the mean-field approximation. The alkanes are modelled as chain molecules formed from fused hard spheres, and dispersion forces corresponding to each hard-sphere site are also included. A simple thermodynamic perturbation theory is used to develop an augmented van der Waals equation of state which incorporates the asymmetry in the attractions, which arise because of the directional hydrogen-bonding sites, and the asymmetry in the repulsions caused by the molecular shape of the chains. We are the first to use a continuum equation of state to study the effects of hydrogen bonding and chain length on the phase equilibria of model aqueous solutions. The phase diagrams and critical lines for systems of increasingly longer chain molecules are determined to give a global picture of the phase equilibria. As well as the usual gas–liquid phase separation, liquid–liquid and gas–gas phase separation, positive, negative and double azeotropes, and Bancroft points are found. The double azeotropy is of a type which has not been obtained previously using equations of state of the van der Waals type. The extent of hydrogen bonding in different regions of the phase diagram is also used to examine the nature of the liquid–liquid phase equilibria. This confirms that the large degree of liquid–liquid immiscibility found in these systems is a consequence of strong water–water hydrogen bonding, and is not due to unfavourable intermolecular dispersive interactions between the water and the alkanes. Although the model systems are not chosen to reproduce the quantitative features of the phase equilibria of water + alkane mixtures, the theory reproduces the important qualitative trends remarkably well.

The freezing of soft spheres from a simple hard-sphere perturbation theory

It is shown that a simple perturbation expansion around the free energy of a (BCC/FCC) hard-sphere crystal leads to accurate predictions for the (BCC/FCC) freezing of soft spheres interacting via the inverse-power potential, V(r)= epsilon ( sigma /r)n, for all values of n including the extreme case of the one-component plasma (n=1). In particular, the authors find that for 1<or=n<or=6 there is both a fluid-BCC and BCC-FCC transition whereas for n>6 the BCC phase is unstable and only the fluid-FCC transition survives, in agreement with the computer simulations.

Computer simulation of fluid–fluid phase coexistence in mixtures of nonadditive soft disks

Fluid–fluid phase equilibrium in binary mixtures of positively nonadditive soft disks is studied using both molecular dynamics and Gibbs ensemble Monte Carlo simulations. The molecular dynamics simulations are unable to provide quantitative results for the phase transition, but they do indicate that some sort of ordering takes place as the temperature is lowered. In contrast, the Gibbs ensemble simulations demonstrate unambiguously the presence of a first-order transition and give quantitative results for the coexistence curve; however, convergence is very slow for mixtures that are not symmetric. Comparison with the Gibbs ensemble results indicates that a simple first-order perturbation theory provides good results far from the consolute point. Closer to the consolute point, the theory does not reproduce the flatness of the coexistence curve; this limitation is intrinsic in any mean-field theory. Comments are made regarding the relative merits of molecular dynamics and Gibbs ensemble simulations for studying fluid–fluid phase coexistence.

Self locking of fluxon oscillations in series arrays of niobium Josephson tunnel junctions coupled to a linear resonator

Spontaneous phase locking of a controlled number from one to ten of all-niobium Josephson fluxon oscillators is obtained at 10 GHz by coupling the oscillator arroy to a linear resonator. The locking range in bias current varies linearly with the number of junctions; the output power, up to 10 nW in a 50 ohm load, varies quadratically. A simple particle-map perturbation theory model efficiently captures the essential experimental phenomenology.

On the Effect of Three-Body Interactions and Proximity of the Fermi Level to the Brillouin Zone Faces on Elastic Modules of Simple Metals

A discussion is made in the model of the local pseudo-potential V(q) ≡ Vq on effects going beyond the scope of the second-order terms of the Vq perturbation theory for describing the static properties of simple metals. It is shown that the allowance for third order in Vq hardly improves the description of static properties (energy, pressure, elastic modules, etc.), being nevertheless apparently essential for Ba and Al. The effect of non-sphericity of the Fermi surface ignored in the third order in Vq is discussed in the two-wave approximation. The ranges of parameters in which the simple perturbation theory in Vq is not adequate are found for the b.c.c., f.c.c., and h.c.p. structures of simple metals alloys in the rigid-band approximation. [Russian Text Ignored].

Calculated thermodynamic properties and phase transitions of solid N2 at temperatures 0≤T≤300 K and pressures 0≤P≤100 GPa

Thermodynamic properties of solid nitrogen are calculated over a variety of isotherms and isobars using a constant pressure Monte Carlo method with deformable, periodic boundary conditions. Vibron frequencies are calculated using a simple perturbation theory. In addition, pressure–volume relations, thermal expansion coefficients, structures, and phase transition pressures and temperatures are determined. In particular, the nature of the orientational disorder in the plastic crystal phases is examined by calculating a variety of orientational order parameters.

The use of high-order TM0m0 modes of a resonant cavity for magnetized plasma density measurements

A study of the application of high-order TM0m0 modes of a cylindrical cavity to the measurement of the density of a magnetized plasma column is presented. It is shown theoretically that judiciously chosen high-order modes have the potential advantages of both a wide operational range of densities, and a wide range for which a simple perturbation theory is valid. Furthermore, an experiment is described which shows that the TM060 mode can be excited with a sufficiently high Q value to allow accurate determination of the resonant frequencies, and hence plasma density. Favorable comparison between densities in the range 1010–1012 cm−3 measured by means of the resonant cavity technique and microwave interferometry is presented.

Effects of lateral viscosity variations on long‐wavelength geoid anomalies and topography

Models for the Earth's geoid and dynamically supported topography usually account only for radial (vertical) variations in mantle rheology and flow structure. However, expected lateral variations in viscosity, due to temperature‐ or stress‐dependent rheology, can degrade our ability to infer mantle structure from geoid models based on seismically determined density contrasts in the mantel. Theoretically, the main difficulty is that the harmonic components of the flow‐stress‐deformation field are no longer independent, and exact analytical methods cannot account for this “mode coupling.” For large thermal density contrasts of small lateral dimension (e.g., subducted slabs), it is difficult to form any general theory for these effects. However, we believe that the largest geoid anomalies which occur at the lowest harmonic degrees (l = 2,3) are due to a global pattern of relatively mild temperature variations. For convection with mainly broadscale thermal and rheological changes, some relatively straightforward ideas can be developed. Combining results from simple perturbation theory and numerical models for convective flow, we find the following: (1) If we are correct in inferring viscosity variations of less than one order of magnitude globally, the longest‐wavelength geoid anomalies (l = 2,3) are not seriously contaminated by lateral viscosity variations. (2) Contamination of the higher degrees (l≥4) is expected, and, in particular, self‐coupling between l = 2 heterogeneity and viscosity variations may be observable at degree 4; the most severe contamination occurs at the doubled harmonic (l = 4) of the dominant heterogeneity harmonic (l = 2). (3) Perturbation techniques are adequate to model global‐scale lateral viscosity variations of an order of magnitude or less. (4) The steady state surface deformation problem at the heart of our geoid models may be more robust with respect to broadscale lateral viscosity variations than are models for transient deformations (e.g., postglacial rebound). (5) For loading by broadscale density heterogeneity, the effects of stress‐dependent rheology are probably secondary to the effects of radial viscosity stratification.

Density-functional theory of crystal-melt interfaces.

The equilibrium structure and energetics of the crystal-melt interfaces of simple systems are studied using the nonlocal weighted density-functional approximation (WDA) to the Helmholtz free energy. The WDA, previously used to accurately predict bulk phase coexistence at the melting point, is combined with a new flexible, two-parameter parametrization of the crystal-melt interfacial region to predict interfacial properties. The parametrization allows for variations in the width of the interface and in the rate of broadening of the sharp density peaks of the crystal through the interface at fixed width, generating physically appealing profiles similar to those observed in simulation studies. The WDA in tandem with this parametrization thus avoids the use of the perturbation and/or square-gradient expansions previously used to describe both the crystal and the interface. Applying the approach to the model hard-sphere system (diameter \ensuremath{\sigma}), the (100) and (111) fcc-liquid interfaces are found to be four atomic layers in width and nearly isotropic in surface free energy \ensuremath{\gamma}, with \ensuremath{\gamma}(100)=0.66kT/${\ensuremath{\sigma}}^{2}$ and \ensuremath{\gamma}(111)=0.63kT/${\ensuremath{\sigma}}^{2}$. These results and the general interfacial structure are in qualitative agreement with simulation studies on the similar soft-sphere (${r}^{\mathrm{\ensuremath{-}}12}$) potential system. Using a simple hard-sphere perturbation theory, the crystal-liquid phase coexistence and (111) interface of the Lennard-Jones system are also examined. Both the freezing transition and the interfacial properties are dominated by the hard-core interactions, and the predicted value of \ensuremath{\gamma}(111)=0.43\ensuremath{\varepsilon}/${\ensuremath{\sigma}}^{2}$ near the triple point is in reasonable agreement with the recent simulation result of 0.35\ensuremath{\varepsilon}/${\ensuremath{\sigma}}^{2}$. A brief comparison of the present ``liquid'' and the usual self-consistent phonon approaches to the crystal thermodynamics brings out some previously unrecognized similarities.

Perturbative analysis of long Josephson junctions having uniform bias and spatially varying penetration depth.

The effect that a uniform bias current has on the basic theoretical structure of a simple perturbation theory relating to single fluxon motion on long Josephson junctions is discussed. By including the bias directly into the general framework of the theory, it is possible to arrive at a fundamental limitation on the maximum value for the bias above which the validity of the usual power balance expression relating the loss factors, and the voltage to the applied current cannot be justified. As part of this effort, an interpretive generalization of the original theory is provided. This theory is then used to illustrate the way in which a step-wise variation in the London penetration depth affects the propagation of solitons on long junctions, and the influence that this has on the junction I-V curve. In the process of developing this theory and applying it to this problem, a formal connection was found between certain aspects of this approach and the technique outlined by McLaughlin and Scott. The results of the theoretical calculations are also shown to be in good agreement with experimental data.

Viscous modulation of the Lamb dipole vortex

A simple analytical singular perturbation theory is developed to describe the viscous adiabatic decay of the two-dimensional Lamb dipole vortex. The vortex parameters (translation speed, radius, and wavenumber) evolve so as to satisfy leading-order globally averaged energy and enstrophy balances. These transport equations are shown to be solvability conditions for the asymptotic expansion. Extensions of the asymptotic procedure to other isolated vortex problems are discussed.

Interpretation of n-donor specific solvent effect on electronic spectra of copper acetate and chloride complexes by intermolecular perturbation theory

A simple second order intermolecular perturbation theory is proposed in the Fukui approximation to account for the electronic spectral behavior of electron acceptor solutes dissolved in pure n-donor solvents. This double perturbation theory considers explicitly both the solvating and complexing effects of the solvent. Using the electronic spectral data of mono- and binuclear copper acetate and chloride complexes obtained in pure pyridine derivatives it is shown that the predicted shifts brought about by increasing n-donor ability of solvent are fully corroborated by experiment. The sensitivity of mononuclear copper complexes to steric effects is also confirmed and it is shown that binuclear ones are not sensitive to this effect. New ionization potential values have also been determined using a simple interpolation–standardization procedure.

The onion method for multiple perturbation theory

We develop a method of successive approximations for molecular scattering theory. This consists of a recipe for removing from the Schrödinger equation, one by one, the wave functions of a set of approximate solutions. The radial wave function is expressed as a linear combination of the well-behaved and singular solutions of the first approximation, and a set of coupled differential equations is obtained for the coefficients of the approximate solutions. A similar set of coefficients is obtained for the next approximation, and the exact coefficients are expressed in terms of the approximate coefficients to yield a set of second-level coefficients. The process can be continued like pealing off the layers of an onion. At each stage the coupled differential equations for the coefficients is equivalent to the Schrödinger equation. Finally, one can either ignore the remaining coefficients or approximate the coupled equations by a simple perturbation theory.

Simulation of random walks in field theory

The numerical simulation of random walks is considered using the Monte Carlo method proposed by Janse van Rensburg (1987). The algorithm is tested and then generalised to generate Edwards random walk (1965). The renormalised masses of the Edwards model (which corresponds to an N to 0, g0 mod phi 2 mod 2 theory) are calculated and the results are compared with those obtained from a simple perturbation theory calculation for small values of the bare coupling constant. The efficiency of this algorithm is discussed and compared with an alternative approach.