Semi-infinite Models Research Articles

Critical behavior at dirty surfaces

The influence of quenched surface disorder — i.e. quenched disorder that is located at the bounding surface of a macroscopic system — on the surface critical behavior of such systems is investigated. To this end a class of semi-infinite continuum models of then-vector type with random surface interactions is studied. Both the case of surface transitions at a bulk critical point as well as that of surface transitions at a bulk tricritical point is considered. General irrelevance/relevance criteria of the Harris type are derived for both short-range and long-range correlated random surface interactions. These are used to assess the stability of the pure system critical behavior and to point out when random surface field or enhancement disorder is expected to be relevant.

A Numerical Study of Topographic Wave Reflection in Semi-Infinite Channels

Abstract The Potential vorticity equation describing topographic waves is approximately solved using the channel method of Stocker and Hutter. The domain of integration is a semi-infinite channel and models an estuary, the bathymetry of which is varied through a transverse and a longitudinal topography parameter. It is shown that in this domain the spectrum of topographic waves consists of a discrete and a continuous part. The former exhibits wave modes trapped at the closed end of the channel; these waves correspond to the bay modes in a rectangular basin. Resonances with a similar bay-trapped structure also occur in the continuous spectrum. Their dependence on the bay geometry is studied. A consistent explanation of the three topographic wave types found earlier in an enclosed basin is given in terms of topographic wave reflections.

Long-range semi-infinite models with critical behaviour controlled by a dynamical system

The surface critical behaviour of the semi-infinite ferromagnetic n-vector Kac-Baker models is obtained. Near the ordinary and special transition points the self-consistency equations (which couple all the unknowns) are shown to be a small perturbation of a two-step recursion relation. The critical behaviour is thus shown to be controlled by the behaviour of a dynamical system near the bifurcation of a new hyperbolic fixed point. This provides both the local critical behaviour and the scaled magnetisation profile. Thereby, the range of validity of the continuum approximation is established.

Nuclear spin response to inelastic proton scattering: Finite geometry and absorption effects

Recent experiments have clearly demonstrated the ability of inelastic proton scattering to act as a precise probe of the nuclear spin response. Previous calculations based on infinite or semi-infinite matter models differ markedly from experiment in their prediction of longitudinal to transverse response function ratios. The present work indicates that finite geometry and absorption effects are crucial in bringing predictions closer to experiment. The effect of including isobar excitations is also investigated and assessed critically.

Real-space renormalization-group investigation of the three-dimensional semi-infinite Blume-Emery-Griffiths model.

We investigate the three-dimensional semi-infinite Blume-Emery-Griffiths (BEG) model using the Migdal-Kadanoff renormalization-group method. The parameter space is six dimensional. We find 69 fixed points describing a large variety of critical behaviors and discuss various semi-infinite models which are special cases of the BEG model such as the Blume-Capel model. For this last model we determine the various generic types of phase diagrams according to the values of the interactions on the surface and in the bulk of the system.

Mean-field theory of magnetic transitions in semi-infinite Ising models.

Le modele semi infini de Ising, pour S=1/2 et avec un nombre arbitraire de couplages magnetiques de surface J mn , differents de ceux du volume, J, est resolu dans l'approximation du champ moyen. On obtient des expressions exactes pour les couplages critiques J mn,c , conduisant a une temperature de Curie en surface T cs superieure a la valeur T c en volume. On obtient la valeur de T cs dans le cristal semi infini par une methode de fraction continue. Application du modele a l'explication de resultats experimentaux recents obtenus sur la surface (0001) du gadolinium

Duality of ordinary and extraordinary surface critical behaviour in the two-dimensional Potts model

The dual relationship between semi-infinite two-dimensional q-state Potts models with free and fixed spin boundary conditions is used to study the surface critical behaviour at the bulk critical temperature with fixed boundary spins, i.e. the extraordinary transition. Both the magnetisation and energy densities have singularities of the same form mod T-Tc mod 2- alpha as the bulk free energy. At T=Tc both the spin-spin and energy-energy correlations decay as r-4 parallel to the surface.

Surface states of ternary semiconductor alloys: Effect of alloy fluctuations in one-dimensional models with realistic atoms.

Calculations of the local electronic surface densities of states (DOS) of semi-infinite one-dimensional chain models for ternary semiconductor alloys are presented to illustrate the effects of alloy fluctuations on semiconductor surface states. The atoms are modeled with a basis set of s and p levels which produces the correct level mixing and energy-band dispersion.Alloys with randomly occupied cation sites are considered. The embedded-cluster approach is employed to incorporate the alloy effects on states localized near the chain end. The cluster calculations are performed by including the contributions from all possible configurations of a cluster at the end of the semi-infinite chain. These results are compared with results obtained using the coherent-potential approximation (CPA) and the virtual-crystal approximation (VCA). The CPA is implemented with one self-energy appropriate for all cation sites in the bulk and different self-energies for each cation site near the surface. In practice the most accurate CPA results can be obtained by treating only the four cation sites nearest the surface differently from the bulk.Examples which are typical of the weak and strong alloy limits are considered. In the first limit the alloy constituents are similar direct-gap semiconductors. In the strong alloy limit one constituent has an inverted band structure, as HgTe does, and the other is a normal semiconductor such as CdTe. The cluster calculation for the surface DOS is sensitive to fluctuations in the environment local to the chain end. In the weak alloy limit, the surface DOS in the band gap has a bimodal distribution with the peaks located near the energies of the surface states of the constituents. In the strong alloy limit, the HgTe-like constituent of the HgCdTe-like alloy has no surface states while the surface states of the CdTe-like constituent lie above the alloy band gap.As a consequence, the alloy surface DOS has a resonance structure above the band edge and band tailing below the edge. The CPA reproduces qualitatively the structure of the cluster calculations. The VCA predicts well-defined surface-state peaks in the band gap. These peaks are related to the average of the bimodal distribution in the weak alloy limit but cannot be consistently identified with any resonance structure or band tailing in the strong alloy limit.

Properties of metal-semiconductor contacts—II: Numerical solutions for intrinsic material

Part I of this paper was concerned mainly with an explicit solution to the problem of contacts on intrinsic material. The principal result was the prediction of a linear log j vs V a dependence in the forward direction, with a slope of 2 qV a / kT, instead of the more familiar qV a / kT expected for the simplest extrinsic case, V a being the voltage applied to the contact barrier itself. The present paper deals with numerical solutions of the same transport equations, free from the approximations which are necessary in the course of analytic treatments to make the problem soluble. Some of the evaluated relationships refer to semi-infinite models, and some to models of severely restricted size. Solutions in the form of one-dimensional contours of potential, field and carrier concentration are provided, as well as the relationship between the injection ratio and current density. Inasmuch as the computations concern the injection region, they confirm (qualitatively and quantitatively) the previous calculations made by linearized analytic methods; inasmuch as they include the entire barrier region, they represent new solutions to a complementary aspect of the total problem.

Beam Elements on Two-Parameter Elastic Foundations

Two-parameter elastic foundations are more accurate than a one-parameter (Winkler) foundation and are simpler than semi-infinite elastic continuum foundation models. Two kinds of finite elements are formulated in this paper to analyze beams on one-or two-parameter foundations. Models include Winkler, Filonenko-Borodich, Pasternak, generalized, and Vlasov foundations. One of the two kinds of elements is based on the exact displacement function. The other is based on a cubic displacement function. The stiffness matrix and consistent load vector is derived for each. Numerical tests show that the element based on the exact displacement function can give exact numerical results even if the number of elements is very small. The element based on a cubic function may require a fine mesh to give acceptable results.

Approximate ladder of resonances in a semi-infinite crystal

Numerical evidence for approximate ladders of resonances in semi-infinite one-dimensional crystal models is reported. The accuracy of the results allows the authors to point out many features of such ladders and to compare them with theoretical speculations going back to Wannier and Zener.

Blunt fin-induced shock wave/turbulent boundary-layer interaction

This paper presents results from an experimental study of blunt fin-induced shock wave/turbulent boundarylayer interaction. Semi-infinite fin models with hemicylindrical, unswept leading edges were tested in Mach 3, high Reynolds number, turbulent boundary layers. All tests were made under approximately adiabatic wall conditions. The program had two fundamental objectives. The first was to examine the spanwise development of the disturbed flowfield and to determine its dependence on the configuration geometry and incoming flow conditions. To achieve this, streamwise surface pressure distributions were measured in the region extending from the centerline to 110 fin diameters outboard. The second objective was to determine the vertical extent of the interaction on the fin. This was carried out using a fin model whose leading edge and side face were instrumented with pressure taps. The results show that, on the test surface near the fin and on the fin itself, the leading-edge diameter plays a dominant role in determining the interaction's scale and characteristics.

Comparison of sharp and blunt fin-induced shock wave/turbulent boundary-layer interaction

In this paper, results are presented from an experimental study of fin-induced shock wave/turbulent boundary-layer interaction. Semi-infinite fin models, with sharp and hemicylindrically blunted leading edges, were tested at Mach 3 in two high Reynolds number, adiabatic wall, turbulent boundary layers. Detailed streamwise surface pressure distributions were measured at several spanwise stations for angles of attack from 0 to 12 deg. The objective was to examine the spanwise development of the interaction region and determine its dependence on leading edge geometry and the incoming-flow parameters. The results show that for blunt fins there is a region of the interaction in the vicinity of the centerline where the scale and characteristics of the disturbed flowfield are controlled primarily by the leading edge diameter. Outboard of this inner, or dominated, region the interaction properties and spanwise development are essentially the same as if the leading edge were sharp. Thus there is an outer region of the blunt fin-induced flowfield in which the properties are effectively independent of leading edge blunting. Nomenclature D = blunt fin leading-edge diameter h - fin height Lu = upstream influence measured relative to the freestream shock wave M = Mach number P = static pressure P2 = wall pressure downstream of interaction Re = Reynolds number X = coordinate in the plane of the test surface and aligned with the tunnel axis, with X= 0 at the fin leading edge Xs = coordinate in the X direction measured relative to the freestream shock wave Y = coordinate normal to the X axis in the plane of the test surface, with Y= 0 at the fin leading edge

On finite and semi-infinite steady state tubular reactor models.

Two contrasting models of one-dimensional axial dispersion tubular reactors are studied. One is a finite reactor model with zero concentration gradient at the exit and the other is a semi-infinite reactor model with its solution truncated at the exit. The comparison of concentration profiles and conversions between the two models for first-oder reactions shows that, for a reasonably wide range of Peclet and Damkohler numbers, the profiles as well as the conversions agree quite well and the choice of model is of little consequence. It is within the intermediate range of Damkohler numbers and low Peclet numbers that the two models give rise to considerably different profiles and conversions.

Effective-modulus approximation for surface-vibration modes

The effective-mass approximation enables one to obtain the energies and wave functions of impurity levels, in a slowly varying perturbing potential, associated with extrema in the electronic band structure of the unperturbed crystal. We have developed an analogous method for obtaining the dispersion relation and displacement field of acoustical or optical surface vibration modes associated with extremal points in the surfaces of constant frequency of the corresponding, infinitely extended crystal. By factoring out the rapidly varying part of the displacement field corresponding to each of the normal modes associated with such an extremal point, a set of partial difference equations is obtained for the slowly varying amplitudes. These equations are converted into a set of coupled partial differential equations which explicitly incorporate the stress-free boundary conditions at the crystal surface. The solution of these equations yields the surface-mode dispersion relation and displacement field. The method is illustrated by applying it to one- and three-dimensional semi-infinite models.

Equilibrium Ion Spacing and Ion-Pair Electrostatic Energy for a Finite Ionic Crystal with NaCl Structure: Point-Ion Model

The equilibrium ion spacing and energy per ion pair are calculated for a point-charge model of a slab-shaped ionic crystal with NaCl structure. The Madelung constant and the specific electrostatic surface energy are deduced. The results confirm previous work on infinite and semi-infinite models.

Effect of Size and Surface on the Specific Heat of Small Lead Particles

In dealing with small particles it is often assumed that their lattice properties can be described by semiinfinite models. These and recent experimental results1 on the low-temperature specific heat of small metal particles from 1.5 to 15°K indicate that this approach is limited. The enhancements are not proportional to the surface area and are not quadratic in temperature as predicted by semiinfinite theories.2; A decrease in the enhancements at lowest temperatures agrees qualitatively with the effect of finite size, which places a lower limit on the frequency spectrum. We have also observed an anomalous behavior in the electronic specific heat which probably results from the boundary conditions on the electronic wave functions which affect the quasiparticle states.

On the nature of tamm surface states in finite linear crystals

Working within the framework of finite crystal models, a comparative study is made of the various shortcomings of the molecular orbital (MO) theory of surface states based on semi-infinite models. Particular attention is paid to the removal of the degeneracy of surface states and the inconsistency of the boundary conditions encountered with semi-infinite models. The main conclusion drawn is that a physically more realistic MO theory of surface states is obtained by using finite models.

Molecular-orbital models of chemisorption

Small finite models of one- and two-dimensional crystals are shown to predict the same localized states (i.e., surface states) as semi-infinite models. An SCF—LCAO—MO calculation for a 6-atom linear chain where each atom has one valence electron of s-type indicates that simple Hückel models give a reasonably good description of the orbital changes that occur as the electron-attracting power (i.e., electronegativity) of an end atom is changed, to simulate chemical adsorption. The utility of small finite models is demonstrated in calculations of the effect of surface geometry on adsorption. Adsorption bond orders are greater for adsorption on a corner site than on a middle edge-atom. A similar result is found by using free valences. With adsorption bond orders it is essential to consider not only the contribution from any localized state, but also the contributions from all the occupied molecular orbitals.