Roughness Correlation Function Research Articles

Transport equation and diffusion in ultrathin channels and films

A rigorous perturbative transport equation for ballistic particles in thin films with random rough walls is derived by the diagrammatic Keldysh technique for both quasiclassical and quantized motion across the film. The derivation is based on canonical Migdal transformation that replaces a transport problem with random rough walls by an equivalent problem with flat boundaries and randomly distorted bulk. The rigorous derivation requires a modification of our previously used transformation to avoid non-Hermitian perturbations. The unusual nondiagonal structure of the effective scattering operator makes the transport equation different from the standard Waldmann-Snider equation when the distance between quantized levels for the motion across the film is comparable to the wall-induced perturbation. Outside of this anomalous quantum resonance region, the transport equation is similar to that for scattering by bulk impurities. The magnitude of the anomaly is calculated for degenerate particles and Gaussian correlations of surface inhomogeneities. The transport problem is solved analytically for the single-band occupancy and in the limiting cases of very large and very small correlation radii of inhomogeneities for an arbitrary correlation function of surface roughness. Elsewhere, the transport equation is analyzed numerically for the Gaussian correlation function. Transport coefficients are expressed explicitly via the angular harmonics of the surface correlation function; in the anomalous region, the results contain certain supplemental correlators. The results reveal various effects of interwall correlations on transport including an oscillatory dependence on the number of occupied minibands. The transition from quantum to quasiclassical description of ballistic motion across the (thick) film can be hindered by residual interwall interference effects similar to those in classical optic problems for thick films without bulk attenuation. Erroneous matrix elements in our previous calculations have been corrected.

An improved formulation of coherent forward scatter from random rough surfaces

The operator expansion method is known to give accurate numerical results for scattering from individual surfaces that are too complicated for other methods. It is less widely appreciated that the method can be applied to random surfaces as well. The simplest application is the modelling of mean forward scatter from a homogeneous Gaussian ensemble of surfaces. To leading order in the admittance operator, the formula for the scalar Dirichlet boundary includes an exponential form in the roughness correlation function. The scalar Neumann boundary adds terms involving the gradients of the exponential form. These factors modestly alter the magnitude and advance the phase of the coherent scatter relative to the conventional one-point (Kirchhoff) approximation when the significant surface correlation scales are comparable to the radiation wavelength. Narrow troughs in the surface undulations ‘repel’ the radiation and effectively elevate and flatten the mean surface. These results are reliable over a wide range of surface amplitudes and correlation scales, provided the slope times Rayleigh height (Dirichlet problem) and slope (Neumann problem) are not large.

LOCALIZATION OF ULTRA-COLD PARTICLES OVER ROUGH SUBSTRATES

Localization and diffusion parameters are calculated for particles adsorbed over inhomogeneous substrates for discrete and quasicontinuous spectra of the adsorbed states. The results are expressed via the angular harmonics of the correlation function of surface roughness. The problem is solved analytically in the limiting cases of longwave particles and large correlation radii of surface inhomogeneities. Elsewhere, the problem is solved numerically for Gaussian correlation of inhomogeneities. Applications to electrons on helium films, mobile adsorbed hydrogen atoms and molecules, ultra-cold neutrons in gravitational or magnetic field, etc., are discussed.

Probing interface roughness by X-ray scattering

X-ray scattering at glancing angles can be exploited to probe interface roughness. Various theories for this technique will be reviewed. The applicability of the theories is shown to depend on the relevant length scales of sample and X-rays. Approximations are discussed and improvements of the theory are suggested. Both specular reflection, diffuse scattering and absorption of X-rays will be discussed. It will be shown that relevant roughness parameters, like root-mean-square roughness, lateral and perpendicular correlation lengths and the degree of jaggedness can be extracted from the experiments. Possible forms for the roughness correlation function are discussed. As an example, it is shown how the interface roughness of an oxidic multilayer has been probed by X-ray scattering.

Determination of electrophysical properties of a layered structure with a statistically rough surface via an inversion method.

The existing methods for nonevasive determination of the electrophysical properties of plane-layered media are applicable only to the structures with an even boundary. Meanwhile, the irregular boundaries are often encountered in thin-film research, plasma physics, remote sensing. Here, an approach to reconstruct the permittivity profile of a one-dimensional inhomogeneous medium with a slightly rough surface is proposed. What differentiates it from the existing approaches is that it makes use of the noncoherent scattering data due to random roughness. It is assumed that the unperturbed position of medium's boundary, as well as the correlation function of random roughness, are known a priori. The available data consist of the backscattering cross-section (BCS) measurements at a discrete set of frequencies for an incident linearly polarized time-harmonic plane wave. The profile reconstruction is performed by successive minimization of a squared error of BCS via the Newton-Kantorovich iterative technique. The modeled values of BCS are calculated via the small perturbation method. The numerical experiments show that the information concerning the underlying medium, which is contained in the noncoherent scattering data due to surface roughness, is quite sufficient to permit feasible reconstruction of the medium's electrophysical properties.

Determination of volume and surface scattering from saline ice using ice sheets with precisely controlled roughness parameters

Experiments were performed at the U.S. Army Cold Regions Research and Engineering Laboratory (CRREL) in Hanover, NH, to precisely determine the relative contributions of surface and volume scattering from saline ice that has well-known surface roughness characteristics. The ice growth phase of the experiment made use of two 6-ft diameter tanks and a 6-ft diameter mold with known roughness statistical parameters of rms height=0.25 cm and Gaussian correlation (correlation length=2.0 cm). One tank was used for growing a moderately thick saline ice sheet with very smooth surface, and the other was used for growing a thin layer of freshwater ice over the surface mold. The latter resulted in a layer with one statistically known rough boundary and one smooth boundary. Wide-bandwidth, multiple incidence angle backscattering measurements were performed, first on the bare saline ice sheet and then on the same sheet after the thin freshwater ice sheet was placed on top of it. Results indicate that the surface scattering dominates over saline ice volume scattering at all frequencies for low incidence angles for both the very smooth and Gaussian rough surfaces. The significance of volume scattering depends strongly on angle of incidence, frequency, volume scattering albedo, surface roughness, and surface correlation function.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Influence of roughness profile on reflectivity and angle-dependent X-ray fluorescence

The theory of X-ray scattering from rough interfaces using the second-order distorted-wave Born approximation is reviewed. For specular reflectivity a new expression is given which smoothly connects the Nevot-Croce and Debye-Waller expressions. The shape of the reflectivity curve depends not only on the average roughness, but also on the lateral correlation of the roughness profile. Using the same theory, the several ways in which roughness can influence angle-dependent X-ray fluorescence are evaluated. Finally, possible shapes for the roughness correlation function are discussed

Optimum target detection in an uncertain reverberation-limited environment

In a reverberation-limited environment, can imperfect knowledge of the spatial correlation properties of the underlying seafloor geomorphology impact active target detection performance? This work seeks to address this question by examining the dependence of the joint probability density function of the scattered acoustic field at low to moderate incidence angles on the seafloor roughness correlation function. The objective is the development of an optimum decision theoretic approach to the problem of high-frequency active detection of a target suspended in the water column in the vicinity of the seafloor. Acoustic modeling is a geologically meaningful seafloor scattering environment is performed by applying the 3-D Helmholtz–Kirchhoff formulation to the anisotropic, power-law description of seafloor relief due to Goff and Jordan. The generality of the Bayesian approach permits uncertainty with respect to geoacoustic environmental properties to be defined probabilistically and incorporated into the detection algorithm. The method attempts to exploit the spatial coherence of the scattered sound field in constructing the joint pdf of the scattered pressure field measured at an N element array of sensors. Upper bounds on detection performance in terms of ROC curves are obtained as a function of environmental uncertainty. [Work supported by ONR, Ocean Acoustics.]

Roughness effect on the frequency of a quartz-crystal resonator in contact with a liquid.

The problem of coupling of the shear oscillations of a quartz crystal bounded by a rough surface with damped waves in a liquid has been investigated. The shift of the resonance frequency of a quartz crystal due to either random roughness or periodical corrugation has been considered. A relation between the shift and a correlation function of roughness has been found. It has been shown that the roughness effect is most pronounced in the low viscosity limit. It has been demonstrated that the results of quartz-crystal microbalance measurements can provide valuable information on the average height and the correlation length of surface roughness.

Broadening of electron states in a quantum dot with a rough boundary in an external magnetic field

The authors investigate the influence of the quantum-dot rough boundary on its electronic energy spectrum in an external magnetic field. The case in which roughness amplitude a is small in comparison to the quantum-dot radius R is considered. The authors show that the ratio of damping edge states to their energy level separation is dependent on binary roughness correlation function and changes in a linear or quadratic way with respect to a/R.

Anomalous impedance and roughness at a metallic electrode

We study the frequency dependence of the electrical impedance at a rough metallic electrode. Using the Rayleigh-Fano approximation to treat the boundary conditions at a rough surface, we get an impedance which depends on the roughness correlation function and exhibits anomalous behaviour under suitable condition. Explicit forms are considered and discussed.

An addition to the theory of the Rayleigh surface waves attenuation by surface roughness

The problem of the attenuation of Rayleigh surface waves by surface roughness is solved in the first Born approximation of perturbation theory. A technique of the Green function developed by Maradudin and Mills [3] has been used on account of a principal correction whose mathematical treatment is given. When the correlation function of surface roughness has a Gaussian from, the attenuation coefficient in the short-wavelength limit ℷ«ɑ (λ is the wavelength of Rayleigh wave, α is the transverse correlation length of the surface roughness) does not depend on the frequency, it tends to the constant value. The results of this paper coincide with the results for the attenuation constant obtained by the Rayleigh method by Eguiluz and Maradudin [6]. It is shown that the scattering of the Rayleigh wave into other Rayleigh waves in the region ℷ«ɑ is a much more effective attenuation mechanism than scattering into bulk waves. A strong dependence of the attenuation constant on the properties of an elastic medium is found.

Surface roughness effects on energy losses of fast electrons crossing metallic samples

The unevenness of the surface of a metallic sample affects the surface plasmon spectrum and thence the energy losses of fast charged particles crossing the sample. A theory of this effect is presented in which the metal is characterized by a simple local dielectric constant and the interaction with the probe particle is treated semi-classically. By a classical Green’s function method, a closed expression is obtained for the loss spectral profile as a function of the momentum transfer, without assumption on the roughness shape. The geometry of a rough surface is then considered in detail. The loss spectral profile is calculated for some examples of weak periodic corrugations. For a general weak roughness, the width and the mean shift of the loss profile are given in terms of the roughness correlation function. Finally, the case of strong roughnesses is examined for small and large values of the momentum transfer.