The evolution of a crystal surface: Analysis of a one-dimensional step train connecting two facets in the ADL regime

We study the evolution of a monotone step train separating two facets of a crystal surface. The model is one-dimensional and we consider only the attachment-detachment-limited regime. Starting with the well-known ODEs for the velocities of the steps, we consider the system of ODEs giving the evolution of the “discrete slopes.” It is the l²-steepest-descent of a certain functional. Using this structure, we prove that the solution exists for all time and is asymptotically self-similar. We also discuss the continuum limit of the discrete self-similar solution, characterizing it variationally, identifying its regularity, and discussing its qualitative behavior. Our approach suggests a PDE for the slope as a function of height and time in the continuum setting. However, existence, uniqueness, and asymptotic self-similarity remain open for the continuum version of the problem.     

Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations

High resolution finite difference schemes for solving the nonlinear model Boltzmann equations are presented for the computations of rarefied gas flows. The discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model Boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space. Then a high order essentially nonoscillatory method due to Harten et al. (J. Comput. Phys. 71, 231, 1987) is adapted and extended to solve them. Explicit methods using operator splitting and implicit methods using the lower-upper factorization are described to treat multidimensional problems. The methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use. The computed results using model Boltzmann equations are found to compare well both with those using the direct simulation Monte Carlo results in the transitional regime flows and those with the continuum Navier-Stokes calculations in near continuum regime flows.     

Novel continuum modeling of crystal surface evolution

We propose a novel approach to continuum modeling of the dynamics of crystal surfaces. Our model follows the evolution of an ensemble of step configurations, which are consistent with the macroscopic surface profile. Contrary to the usual approach where the continuum limit is achieved when typical surface features consist of many steps, our continuum limit is approached when the number of step configurations of the ensemble is very large. The model can handle singular surface structures such as corners and facets. It has a clear computational advantage over discrete models.     

Leed studies of vicinal surfaces of silicon

Clean silicon surfaces inclined at small angles to (111), (100) and (110) planes were investigated by LEED. Surfaces oriented at low angles to the (111) plane contain steps with edges towards [2̄11] or [21̄1̄]. Steps with edges towards [2̄11] have a height of two interplanar distances d₁₁₁ at low temperatures. At 800°C the reversible reconstruction of this step array into the steps of monolayer height takes place. Steps with edges towards [21̄1̄] can be seen at low temperatures only. They are of monolayer height and disappear at annealing in vacuum. Surfaces oriented at low angles to the (100) plane contain steps with (100) terraces and have a height of about two interplanar distances d₁₀₀. Surfaces at low angles to (110) planes are facetted and contain facets of the (47 35 7) type. The information about surface self-diffusion of silicon may be obtained using the kinetic data of structural reconstructions on surfaces close to (111) at different temperatures.     

Discrete vs. continuum-scale simulation of radiative transfer in semitransparent two-phase media

The mathematical formulation of the continuum approach to radiative transfer modeling in two-phase semi-transparent media is numerically validated by comparing radiative fluxes computed by (i) direct, discrete-scale and (ii) continuum-scale approaches. The analysis is based on geometrical optics. The discrete-scale approach uses the Monte Carlo ray-tracing applied directly to real 3D geometry measured by computed tomography. The continuum-scale approach is based on a set of continuum-scale radiative transfer equations and associated radiative properties, and employs the Monte Carlo ray-tracing for computations of radiative fluxes and for computations of the radiative properties. The model two-phase media are reticulate porous ceramics and a particle packed bed, each composed of semitransparent solid and fluid phases. The results obtained by the two approaches are in good agreement within the limits of statistical uncertainty. The continuum-scale approach leads to a reduction in computational time by approximately one order of magnitude, and is therefore suited to treat radiative transfer problems in two-phase media in a wide range of engineering applications.     

A strong-coupling expansion for fermion field theories and the large-N limit of the gross-neveu model

An expansion in the fermion propagator is formulated for the N-species Gross-Neveu model in the large-N limit. Different regularisation schemes may be adopted and we compare two. We find that a continuum momentum cut-off is easiest to work with and automatically avoids spurious fermionic states which afflict a naive lattice formulation. Chiral symmetry is broken at zeroth order and the resulting expansion is inverse powers of g²N simplifies considerably for large N. In this limit the strong-coupling expansion may be summed to all orders. Extrapolation techniques, like Padé approximants, are not needed. Using a momentum cut-off we recover all the exact results previously derived by summing weak-coupling expansions.     

New High-Resolution Semi-discrete Central Schemes for Hamilton–Jacobi Equations

We introduce a new high-resolution central scheme for multidimensional Hamilton–Jacobi equations. The scheme retains the simplicity of the non-oscillatory central schemes developed by C.-T. Lin and E. Tadmor (in press, SIAM J. Sci. Comput.), yet it enjoys a smaller amount of numerical viscosity, independent of 1/Δt. By letting Δt↓0 we obtain a new second-order central scheme in the particularly simple semi-discrete form, along the lines of the new semi-discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws. Fully discrete versions are obtained with appropriate Runge–Kutta solvers. The smaller amount of dissipation enables efficient integration of convection-diffusion equations, where the accumulated error is independent of a small time step dictated by the CFL limitation. The scheme is non-oscillatory thanks to the use of nonlinear limiters. Here we advocate the use of such limiters on second discrete derivatives, which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives. Numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme.     

The multiple scattering and N-body approaches to nuclear reactions

The relationship between conventional multiple scattering approaches and the recently developed N-body approaches to nuclear reactions is considered with a view towards elastic scattering applications. Connectivity expansions in the N-body approach and multiple scattering expansions in the Watson approach are developed by a common technique so that a comparison of the physical content of each can be made. In the N-body case this leads to a new derivation of the equations of Bencze, Redish, and Sloan in both particle-labelled and partition-labelled form and this yields new insight into the minimal dimensionality of these equations and into the role of channel coupling schemes within this formulation. The relative simplicity and generality with which these results are obtained is designed to be easily understood by those unfamiliar with N-body formalisms. The two approaches are contrasted first for the three-particle problem and subsequently for the many-body problem. We argue that a strict adherence to the connected-kernel property which is advantageous for the three-particle problem may not be so advantageous for the many-body elastic scattering problem. Undesirable physical characteristics of the connectivity expansion for elastic scattering are identified and their rectification is discussed. The off-shell transformation associated with the N-body approach is examined critically. The origin of the multiplicity of N-body coupling schemes is elucidated. It is shown that a modified concept of connectivity, called inclusive connectivity, can be introduced to guide expansions which can be truncated in a physically meaningful way. The inclusive connectivity expansion is seen to be identical to the spectator expansion for an elementary projectile but differs in the case of a composite projectile. Extant elastic scattering optical potential formulations based on the two concepts of connectivity are compared and contrasted. We show that connected kernel integral equations of the few-body type are required for computation of the individual low-order terms of the inclusive connectivity expansion of the optical potential.     

On the Continuum Limit for Discrete NLS with Long-Range Lattice Interactions

We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice ${h\mathbb{Z}}$ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on ${\mathbb{R}}$ with the fractional Laplacian (?Δ) ^α as dispersive symbol. In particular, we obtain that fractional powers ${\frac{1}{2} < \alpha < 1}$ arise from long-range lattice interactions when passing to the continuum limit, whereas the NLS with the usual Laplacian ?Δ describes the dispersion in the continuum limit for short-range or quick-decaying interactions (e. g., nearest-neighbor interactions). Our results rigorously justify certain NLS model equations with fractional Laplacians proposed in the physics literature. Moreover, the arguments given in our paper can be also applied to discuss the continuum limit for other lattice systems with long-range interactions.     

Ethylene dissociation on flat and stepped Ni(1 1 1): A combined STM and DFT study

The dissociative adsorption of ethylene (C₂H₄) on Ni(1 1 1) was studied by scanning tunneling microscopy (STM) and density functional theory (DFT) calculations. The STM studies reveal that ethylene decomposes exclusively at the step edges at room temperature. However, the step edge sites are poisoned by the reaction products and thus only a small brim of decomposed ethylene is formed. At 500 K decomposition on the (1 1 1) facets leads to a continuous growth of carbidic islands, which nucleate along the step edges.DFT calculations were performed for several intermediate steps in the decomposition of ethylene on both Ni(1 1 1) and the stepped Ni(2 1 1) surface. In general the Ni(2 1 1) surface is found to have a higher reactivity than the Ni(1 1 1) surface. Furthermore, the calculations show that the influence of step edge atoms is very different for the different reaction pathways. In particular the barrier for dissociation is lowered significantly more than the barrier for dehydrogenation, and this is of great importance for the bond-breaking selectivity of Ni surfaces.The influence of step edges was also probed by evaporating Ag onto the Ni(1 1 1) surface. STM shows that the room temperature evaporation leads to a step flow growth of Ag islands, and a subsequent annealing at 800 K causes the Ag atoms to completely wet the step edges of Ni(1 1 1). The blocking of the step edges is shown to prevent all decomposition of ethylene at room temperature, whereas the terrace site decomposition at 500 K is confirmed to be unaffected by the Ag atoms.Finally a high surface area NiAg alloy catalyst supported on MgAl₂O₄ was synthesized and tested in flow reactor measurements. The NiAg catalyst has a much lower activity for ethane hydrogenolysis than a similar Ni catalyst, which can be rationalized by the STM and DFT results.     

Continuum symmetries of lattice models with staggered fermions

The validity of the flavour interpretation of staggered fermions is discussed in terms of the discrete symmetries of the interaction terms. Some aspects of the embedding of these symmetries in the symmetry group of the continuum limit are clarified. An explicit calculation, at first non-trivial order in 1/N, of the four-point function for a latticized Gross-Neveu model yields the same result in the continuum limit as the continuum theory for 2N fermions. A proof is then given that flavour and C, P, and T symmetries are restored in the continuum limit of 2-point correlation functions, for interactions, including the case of 4-dimensional QCD, which respect the discrete symmetries of the free action.     

Capillarity-Induced Surface Morphologies

Surface energetics is reviewed including expressions for the chemical potential of a curved surface element and the Legendre transform relation between the projected surface free energy as a function of orientation and the Wulff equilibrium shape. A well known equation is derived describing surface evolution by surface diffusion, assuming local equilibrium. Solutions are reviewed including a decaying sinusoid and a developing thermal groove. Breakdown of local equilibrium is considered. The structure, energetics and dynamics of steps on a vicinal surface are discussed. Facet sizes on the Wulff shape and the surface profile at the edge of a facet are related to the step self and interaction free energies respectively. Fourier analysis of step fluctuations is described, revealing the underlying transport processes. Analysis of the decay of a sinusoidal profile on a vicinal surface in terms of step behavior is given. Finally, examples are reviewed of surface evolution below the roughening temperature T _R in which case facets move by the lateral spreading of steps. Results differ greatly from those of the continuum theory applicable above T _R.     

Local conserved currents for CPN σ-models

We present a new method to derive an infinite series of conserved local charges for the two-dimensional CP^N σ-models. The generating relation for the conservation laws is a couple of first-order nonlinear differential equations. The method displays transparently the connection of the local charges with nonlocal dynamical charges of CP^N models previously found.     

Step structure on the fivefold Al-Pd-Mn quasicrystal surface, and on related surfaces

We compare step morphologies on surfaces of Al-rich metallic alloys, both quasicrystalline and crystalline. We present evidence that the large-scale step structure observed on Al-rich quasicrystals after quenching to room temperature reflects equilibrium structure at an elevated temperature. These steps are relatively rough, i.e., have high diffusivity, compared to those on crystalline surfaces. For the fivefold quasicrystal surface, step diffusivity increases as step height decreases, but this trend is not obeyed in a broader comparison between quasicrystals and crystals. On a shorter scale, the steps on Al-rich alloys tend to exhibit local facets (short linear segments), with different facet lengths, a feature which could develop during quenching to room temperature. Facets are shortest and most difficult to identify for the fivefold quasicrystal surface.     

Video solitons in a dispersive transmission line with the nonlinear capacitance of a p-n junction

Possible types of low-frequency electromagnetic solitary waves in a dispersive LC transmission line with a quadratic or cubic capacitive nonlinearity are investigated. The fourth-order nonlinear wave equation with ohmic losses is derived from the differential-difference equations of the discrete line in the continuum approximation. For a zero-loss line, this equation can be reduced to the nonlinear equation for a transmission line, the double dispersion equation, the Boussinesq equations, the Korteweg-de Vries (KdV) equation, and the modified KdV equation. Solitary waves in a transmission line with dispersion and dissipation are considered.     

Continuum radiative heat transfer modeling in media consisting of optically distinct components in the limit of geometrical optics

Continuum-scale equations of radiative transfer and corresponding boundary conditions are derived for a general case of a multi-component medium consisting of arbitrary-type, non-isothermal and non-uniform components in the limit of geometrical optics. The link between the discrete and continuum scales is established by volume averaging of the discrete-scale equations of radiative transfer by applying the spatial averaging theorem. Precise definitions of the continuum-scale radiative properties are formulated while accounting for the radiative interactions between the components at their interfaces. Possible applications and simplifications of the presented general equations are discussed.     

A study of the critical behavior of the O(N) linear and nonlinear σ models

A study of the large N behavior of both the O(N) linear and nonlinear σ models is presented. The purpose is to investigate the relationship between the disordered (ordered) phase of the linear and nonlinear sigma models. Utilizing operator product expansions and stability analyses, it is shown that for 2 ≤ d < 4, it is the dimensionless renormalized quartic coupling and $\text{λ}{}^1$ is the IR fixed point) limit of the linear σ model which yields the nonlinear σ model. It is also shown that stable large N linear σ models with λ < 0 (σ is the bare quartic coupling) can exist (at least in the context of no tachyonic states being present). A criteria valid for all dimensionalities d, less than four, is derived which determines when λ < 0 models are tachyonic free. Arguments are given showing that the d = 4 large N linear (for λ > 0) and nonlinear models are trivial. This result (i.e., triviality) is well known but only for one and two component models. Interestingly enough, the λ < 0, d = 4 linear σ model remains nontrivial and tachyonic free.     

Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws

In this article, we propose a new class of finite volume schemes of arbitrary accuracy in space and time for systems of hyperbolic balance laws with stiff source terms. The new class of schemes is based on a three stage procedure. First a high-order WENO reconstruction procedure is applied to the cell averages at the current time level. Second, the temporal evolution of the reconstruction polynomials is computed locally inside each cell using the governing equations. In the original ENO scheme of Harten et al. and in the ADER schemes of Titarev and Toro, this time evolution is achieved via a Taylor series expansion where the time derivatives are computed by repeated differentiation of the governing PDE with respect to space and time, i.e. by applying the so-called Cauchy–Kovalewski procedure. However, this approach is not able to handle stiff source terms. Therefore, we present a new strategy that only replaces the Cauchy–Kovalewski procedure compared to the previously mentioned schemes. For the time-evolution part of the algorithm, we introduce a local space–time discontinuous Galerkin (DG) finite element scheme that is able to handle also stiff source terms. This step is the only part of the algorithm which is locally implicit. The third and last step of the proposed ADER finite volume schemes consists of the standard explicit space–time integration over each control volume, using the local space–time DG solutions at the Gaussian integration points for the intercell fluxes and for the space–time integral over the source term. We will show numerical convergence studies for nonlinear systems in one space dimension with both non-stiff and with very stiff source terms up to sixth order of accuracy in space and time. The application of the new method to a large set of different test cases is shown, in particular the stiff scalar model problem of LeVeque and Yee [R.J. LeVeque, H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms, Journal of Computational Physics 86 (1) (1990) 187–210], the relaxation system of Jin and Xin [S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Communications on Pure and Applied Mathematics 48 (1995) 235–277] and the full compressible Euler equations with stiff friction source terms.