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Showing papers on "Debye published in 2007"


Journal ArticleDOI
TL;DR: Two different techniques indicate that the interaction of water with anions is by an approximately linear hydrogen bond, suggesting that the dominant forces on ions in water are short range forces of a chemical nature.

571 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived a static potential for a heavy quark- antiquark pair propagating in Minkowski time at finite temperature, by defining a suitable gauge-invariant Green's function and computing it to first non-trivial order in Hard Thermal Loop resummed perturbation theory.
Abstract: We derive a static potential for a heavy quark- antiquark pair propagating in Minkowski time at finite temperature, by defining a suitable gauge-invariant Green's function and computing it to first non-trivial order in Hard Thermal Loop resummed perturbation theory. The resulting Debye- screened potential could be used in models that attempt to describe the "melting" of heavy quarkonium at high temperatures. We show, in particular, that the potential develops an imaginary part, implying that thermal e effects ects generate a finite width for the quarkonium peak in the dilepton production rate. For quarkonium with a very heavy constituent mass M, the width can be ignored for T <= g(2)M/12 pi, where g2 is the strong gauge coupling; for a physical case like bottomonium, it could become important at temperatures as low as 250 MeV. Finally, we point out that the physics related to the finite width originates from the Landau-damping of low-frequency gauge fields, and could be studied non-perturbatively by making use of the classical approximation.

427 citations


Journal ArticleDOI
TL;DR: In this article, the authors presented computationally simpler one-and two-pole Debye models that retain the high accuracy of the Cole-Cole models for two sets of frequency ranges: the entire measurement frequency range from 0.5 to 20 GHz, and the 3.1-10.6 GHz FCC band allocated for ultrawideband medical applications.
Abstract: The finite difference time domain (FDTD) method is widely used as a computational tool for development, validation, and optimization of emerging microwave breast cancer detection and treatment techniques. When expressed in terms of Debye parameters, dispersive breast tissue dielectric properties can be efficiently incorporated into FDTD codes. Previously, we experimentally characterized the dielectric properties of a large number of excised normal and malignant breast tissue samples from 0.5 to 20 GHz. We subdivided the large database of normal tissue data into three groups based on the percent adipose tissue present in a particular sample. In addition, we formed a group of all cancer samples that contained at least 30% malignant tissue. We summarized the data using one-pole Cole-Cole models that were rigorously fit to the median dielectric properties of the three normal tissue groups and one malignant tissue group. In this letter, we present computationally simpler one- and two-pole Debye models that retain the high accuracy of the Cole-Cole models. Model parameters are derived for two sets of frequency ranges: the entire measurement frequency range from 0.5 to 20 GHz, and the 3.1-10.6 GHz FCC band allocated for ultrawideband medical applications. The proposed Debye models provide a means for creating computationally efficient FDTD breast models with realistic wideband dielectric properties derived from the largest and most comprehensive experimental study conducted to date on human breast tissue.

267 citations


Journal ArticleDOI
TL;DR: In this paper, the behavior of correlators of Polyakov loops and other operators in N=4 supersymmetric Yang-Mills theory at non-zero temperature was examined and the implications for Debye screening in this strongly coupled non-Abelian plasma, and comparisons with available results for thermal QCD, were discussed.
Abstract: Using the AdS/CFT correspondence, we examine the behavior of correlators of Polyakov loops and other operators in N=4 supersymmetric Yang-Mills theory at non-zero temperature. The implications for Debye screening in this strongly coupled non-Abelian plasma, and comparisons with available results for thermal QCD, are discussed.

168 citations


Journal ArticleDOI
TL;DR: In this article, the behavior of correlators of Polyakov loops and other operators in = 4 supersymmetric Yang-Mills theory at non-zero temperature was examined using the AdS/CFT correspondence, and the implications for Debye screening in this strongly coupled non-Abelian plasma, and comparisons with available results for thermal QCD, were discussed.
Abstract: Using the AdS/CFT correspondence, we examine the behavior of correlators of Polyakov loops and other operators in = 4 supersymmetric Yang-Mills theory at non-zero temperature. The implications for Debye screening in this strongly coupled non-Abelian plasma, and comparisons with available results for thermal QCD, are discussed.

156 citations


Journal ArticleDOI
TL;DR: In this paper, the Debye-Gruneisen model combined with ab initio calculations was used to calculate the thermal expansivities and heat capacities of MX (M = Ti, Zr, Hf, V, Nb, Ta; X = C, N) carbides with NaCl structure.

146 citations


Journal ArticleDOI
TL;DR: Here the case of a finite Debye layer thickness is considered while restricting the pore geometry to that of a cylinder of length much larger than the diameter, and it is shown that the calculated transit times are consistent with recent measurements in silicon nanopores.
Abstract: In a previous paper [S. Ghosal, Phys. Rev. E 74, 041901 (2006)] a hydrodynamic model for determining the electrophoretic speed of a polyelectrolyte through an axially symmetric slowly varying nanopore was presented in the limit of a vanishingly small Debye length. Here the case of a finite Debye layer thickness is considered while restricting the pore geometry to that of a cylinder of length much larger than the diameter. Further, the possibility of a uniform surface charge on the walls of the nanopore is taken into account. It is thereby shown that the calculated transit times are consistent with recent measurements in silicon nanopores.

138 citations


Journal ArticleDOI
TL;DR: A complete set of integrals for the inner system of the PNP system can be treated as a singularly perturbed system that has two limiting systems: inner and outer systems (termed fast and slow systems in geometric singular perturbation theory).
Abstract: Ionic channels and semiconductor devices use atomic scale structures to control macroscopic flows from one reservoir to another. The one-dimensional steady-state Poisson-Nernst- Planck (PNP) system is a useful representation of these devices, but experience shows that describing the reservoirs as boundary conditions is difficult. We study the PNP system for two types of ions with three regions of piecewise constant permanent charge, assuming the Debye number is large, because the electric field is so strong compared to diffusion. Reservoirs are represented by the outer regions with permanent charge zero. If the reciprocal of the Debye number is viewed as a singular parameter, the PNP system can be treated as a singularly perturbed system that has two limiting systems: inner and outer systems (termed fast and slow systems in geometric singular perturbation theory). A complete set of integrals for the inner system is presented that provides information for boundary and internal layers. Application of the exchange lemma from geometric singular perturbation theory gives rise to the existence and (local) uniqueness of the solution of the singular boundary value problem near each singular orbit. A set of simultaneous equations appears in the construction of singular orbits. Multiple solutions of such equations in this or similar problems might explain a variety of multiple valued phenomena seen in biological channels, for example, some forms of gating, and might be involved in other more complex behaviors, for example, some kinds of active transport.

138 citations


Journal ArticleDOI
TL;DR: Two different volume fraction-dependent regimes for the electrophoretic mobility that can be explained in terms of the static properties of the ionic double layer are observed.
Abstract: Many important properties of colloidal dispersions are directly or indirectly determined by the electric charge of the colloidal particles. Phase stability is provided by the repulsive interaction between like charges, while the details of the static and dynamic behavior are the result of the interplay between electrostatic interactions between macroions, counterions, and added salt ions, the dielectric response of the solvent, and the solvent hydrodynamics. Depending on whether the concentration of the background electrolyte (relative to that of the “native” counterions) is large or small, one must expect quite different behavior both with respect to statics and dynamics. The salt-dominated regime (high salt concentration) has been studied extensively both for static [1] and dynamic properties [2, 3, 4, 5]. In this case, the large reservoir of salt ions results in a strong Debye screening of the electrostatic interactions between the macroions, such that the net charge density is essentially zero throughout the dispersion except for narrow ionic atmospheres around the colloids. An external electric field will therefore exert forces only within these layers, such that hydrodynamic interactions are also strongly screened in a system subjected to electrophoresis [6, 7]. Therefore, the problems both of ion cloud structure and of electrophoresis can be treated within a single-macroion framework [8], and the dependence of the electrophoretic mobility � = v/E (v denoting the colloid drift velocity and E the driving

126 citations


Journal ArticleDOI
TL;DR: The authors conclude that the Debye process corresponds to a transition among states which differ in energy only in the case of an external electric field.
Abstract: Dielectric relaxation and dynamic heat capacity measurements are compared for 2-ethyl-1-hexanol near its glass transition temperature Tg in order to further clarify the origin of the prominent Debye-type loss peak observed in many monohydroxy alcohols and other hydrogen-bonding liquids. While the dielectric spectrum e″ displays two distinct polarization processes that are separated by a factor of 2000 in terms of the peak frequency, the heat capacity cp″ shows only a single peak. The dielectric process with lower amplitude and higher peak frequency coincides with the calorimetric signal, whereas the large dielectric Debye signal is not associated with calorimetric modes. The authors conclude that the Debye process corresponds to a transition among states which differ in energy only in the case of an external electric field.

118 citations


Journal ArticleDOI
TL;DR: In this paper, an approach is presented for theoretical calculations of the Debye-Waller factors in x-ray absorption spectra, which are represented in terms of the cumulant expansion up to third order and account respectively for the net thermal expansion, mean-square relative displacements, and asymmetry of the pair distribution function.
Abstract: An approach is presented for theoretical calculations of the Debye-Waller factors in x-ray absorption spectra These factors are represented in terms of the cumulant expansion up to third order They account respectively for the net thermal expansion � (1) (T), the mean-square relative displacements � 2 (T), and the asymmetry of the pair distribution function � (3) (T) Similarly, we obtain Debye-Waller factors for x-ray and neutron scattering in terms of the mean-square vibrational amplitudes u 2 (T) Our method is based on density functional theory calculations of the dynamical matrix, together with an efficient Lanczos algorithm for projected phonon spectra within the quasiharmonic approximation Due to anharmonicity in the interatomic forces, the results are highly sensitive to variations in the equilibrium lattice constants, and hence to the choice of exchangecorrelation potential In order to treat this sensitivity, we introduce two prescriptions: one based on the local density approximation, and a second based on a modified generalized gradient approximation Illustrative results for the leading cumulants are presented for several materials and compared with experiment and with correlated Einstein and Debye models We also obtain Born-von Karman parameters and corrections due to perpendicular vibrations


Journal ArticleDOI
TL;DR: In this paper, thermal resistances of 1250 kinds of interface were computed at room temperature based on the phonon diffuse mismatch model, and the relationship between the interfacial thermal resistance with the ratio of average sound velocity was shown.
Abstract: The thermal resistances of 1250 kinds of interface were computed at room temperature based on the phonon diffuse mismatch model. The result shows that the ratio of Debye temperature and the ratio of average sound velocity can be approximately used to characterize the difference of two materials in terms of interfacial thermal resistance. The high interfacial thermal resistances are composed of high and low Debye temperature materials. The low interfacial thermal resistances are composed of both similar Debye temperature materials, and their Debye temperatures are very high. The relation between the interfacial thermal resistance with the ratio of average sound velocity is similar to that of the ratio of Debye temperature. [doi:10.2320/matertrans.MAW200717]

Journal ArticleDOI
TL;DR: In this article, a weighted sum of Debye functions is used to approximate more general complex permittivity functions and a combination of the particle swarm optimization method and linear least squares optimization was used to find the relaxation frequencies and weights in the expansion, which can then be accommodated in the FDTD method using one of the established methods.
Abstract: With appropriate modifications, the finite-difference time-domain (FDTD) method can be used to analyze propagation through linear isotropic dispersive media. Although materials characterized by the Debye permittivity model can be analyzed accurately and efficiently using well established methods, the treatment of other types of frequency dependence is more difficult. This paper proposes the use of a weighted sum of Debye functions to approximate more general complex permittivity functions. A combination of the particle swarm optimization method and linear least squares optimization is used to find the relaxation frequencies and weights in the expansion, which can then be accommodated in the FDTD method using one of the established methods. Two key advantages of the proposed approach are that the relaxation frequencies are bandlimited and the weights are always positive. These two characteristics help to maintain the accuracy and stability of the FDTD solution. It is also shown that the correlation between the imaginary parts of two Debye functions is the same as that between the real parts.

Journal ArticleDOI
TL;DR: In this article, the second order elastic constants of hexagonal structured (wurtzite) third group nitrides (GaN, AlN and InN) were calculated using the Lennard-Jones potential for the determination of ultrasonic attenuation.

Journal ArticleDOI
TL;DR: In this paper, a finite difference time domain model based on efficient two-pole Debye dispersion was proposed to analyze the electromagnetic interaction between a whole human body and ultra wide band radiation having a wide frequency spectrum.
Abstract: We have successfully developed a human body finite difference time domain model based on efficient two-pole Debye dispersion, and analyzed for the first time the electromagnetic interaction between a whole human body and ultra wide band radiation having a wide frequency spectrum. The two-pole Debye dispersion model is obtained for 50 individual human tissue properties from Gabriel's Cole-Cole data by least squares fitting over a wide frequency range from 100 MHz to 6 GHz. For validation, the model is exposed to radiation of a spread spectrum signal modulated by typical binary phase shift keying. Local energy absorption in a human body has been compared between the two-pole Debye model and a conventional model with frequency-independent permittivity and conductivity.

Journal ArticleDOI
TL;DR: In this article, the intramolecular form factor F(q) in dense polymer systems is discussed theoretically and numerically, and the Kratky plot (q2F(q)) vs wave vector q does not exhibit the plateau expected for Gaussian chains in the intermediate q range.
Abstract: We discuss theoretically and numerically the intramolecular form factor F(q) in dense polymer systems. Following Flory's ideality hypothesis, chains in the melt adopt Gaussian configurations, and their form factor is supposed to be given by Debye's formula. At striking variance to this, we obtain noticeable (up to 20%) non-monotonic deviations which can be traced back to the incompressibility of dense polymer solutions beyond a local scale. The Kratky plot (q2F(q) vs wave vector q) does not exhibit the plateau expected for Gaussian chains in the intermediate q range. One rather finds a significant decrease according to the correction δ(F-1(q)) = q3/32ρ that only depends on the concentration ρ of the solution, but neither on the persistence length or the interaction strength. The non-analyticity of the above q3 correction is linked to the existence of long-range correlations for collective density fluctuations that survive screening. Finite-chain size effects are found to decay with chain length N as .

Journal ArticleDOI
TL;DR: In this paper, the heavy-quark free energies, the effective running coupling geff(T), and the Debye screening mass mD(T) for various color channels of heavy quark pairs above the critical temperature were extracted from the line of constant physics at mPS/mV = 0.65 and 0.80.
Abstract: on a 16 3 × 4 lattice. From the line of constant physics at mPS/mV = 0.65 and 0.80, we extract the heavy-quark free energies, the effective running coupling geff(T) and the Debye screening mass mD(T) for various color channels of heavy quark–quark and quark–anti-quark pairs above the critical temperature. The free energies are well approximated by the screened Coulomb form with the appropriate Casimir factors at high temperature. The magnitude and the temperature dependence of the Debye mass are compared to those of the next-to-leading order thermal perturbation theory and to a phenomenological formula in terms of geff(T). We make a comparison between our results with the Wilson quark action and the previous results with the staggered quark action. The spatial string tension is also studied in the high temperature phase and is compared to the next-to-nextleading order prediction in an effective theory with dimensional reduction.

Journal ArticleDOI
TL;DR: In this paper, a pedagogical and qualitative introduction to the theory of equilibrium and thermalization of quark-gluon plasmas is given, assuming only that the reader is familiar with quantum field theory at zero temperature and with QCD as the theoretical theory of strong interactions.
Abstract: In these lectures, I will attempt a pedagogical and qualitative introduction to the theory of equilibrium and thermalization of quark-gluon plasmas. I assume only that the reader is familiar with quantum field theory at zero temperature and with QCD as the theory of the strong interactions. I focus on the limit of small alpha_s, which in principle should be relevant at extremely high temperature because of asymptotic freedom, and in any case provides a clean theoretical context in which to discuss a variety of phenomena. Topics discussed include the basic equilibrium formalism for finite-temperature quantum field theory, Debye screening, electric deconfinement, magnetic confinement, dimensional reduction, plasma waves, kinetic theory, hydrodynamic properties such as viscosity, the Landau-Pomeranchuk-Migdal effect, thermalization in (arbitrarily high energy) heavy ion collisions, and QCD plasma instabilities.

Journal ArticleDOI
TL;DR: In this paper, the authors considered superconductivity in boron (B) doped diamond using a simplified model for the valence band of diamond and treated the effects of substitutional disorder of B ions by the coherent potential approximation and those of the attractive force between holes by the ladder approximation under the assumption of instantaneous interaction with the Debye cutoff.
Abstract: We consider superconductivity in boron (B) doped diamond using a simplified model for the valence band of diamond. We treat the effects of substitutional disorder of B ions by the coherent potential approximation (CPA) and those of the attractive force between holes by the ladder approximation under the assumption of instantaneous interaction with the Debye cutoff. We thereby calculate the quasiparticle life time, the evolution of the single-particle spectra due to doping, and the effect of disorder on the superconducting critical temperature T c . We in particular compare our results with those for supercell calculations to see the role of disorder, which turns out to be of crucial importance to T c .

Journal ArticleDOI
TL;DR: The integral localized approximation is employed in the calculation of beam-shape coefficients (BSCs) and allows the study of the scattering characteristics of particles illuminated by the strongly focused beams.
Abstract: The Debye series has been a key tool for the understanding of light scattering features, and it is also a convenient method for understanding and improving the design of optical instruments aimed at optical particle sizing. Gouesbet has derived the Debye series formulation for generalized Lorenz-Mie theory (GLMT). However, the scattering object is a homogeneous sphere, and no numerical result is provided. The Debye series formula for plane-wave scattering by multilayered spheres has been derived before. We have devoted our work to the Debye series of Gaussian beam scattering by multilayered spheres. The integral localized approximation is employed in the calculation of beam-shape coefficients (BSCs) and allows the study of the scattering characteristics of particles illuminated by the strongly focused beams. The formula and code are verified by the comparison with the results produced by GLMT and also by the comparison with the result for the case of plane-wave incidence. The formula is also employed in the simulation of the first rainbow by illuminating the particle with one or several narrow beams.

Journal ArticleDOI
TL;DR: In this article, the fundamental equations of charge-carrier transport in semiconductor structures were discussed and a correct modeling of the recombination terms that does not violate the charge conservation law was discussed.
Abstract: In this work we present results concerning the fundamental equations of charge-carrier transport in semiconductor structures. We discuss a correct modeling of the recombination terms that does not violate the charge conservation law. We obtained that under stationary conditions and equal generation rates of electrons and holes the recombination rates of both kinds of carriers must be matched. Under low excitation conditions (linear regime) the recombination rate can be expressed as a linear combination of the excess concentrations of electrons and holes, and only in very precise situations can a lifetime be defined. Explicit calculation of the charge variation was carried out within the framework of the band-to-band and Shockley-Read-Hall statistics. It is shown that the bulk space charge established on distances of the order of Debye’s length cannot influence the current-voltage characteristics of a device in a linear regime. It was also shown that metal-p-type semiconductor contacts are essentially diff...

Journal ArticleDOI
TL;DR: In this paper, an asymptotic theory for the screening of the electric field of a dust particle or a spherical probe in a plasma with an external steady and/or internal (proportional to the electron density) gas ionization source has been developed.
Abstract: An asymptotic theory for the screening of the electric field of a dust particle or a spherical probe in a plasma with an external steady and/or internal (proportional to the electron density) gas ionization source has been developed for the first time. It has been established that the screening of the charge of a spherical body adsorbing the charge of the incident plasma particles is described by a superposition of two exponentials with different screening constants. The two exponentials are retained even in the absence of nonequilibrium fluxes on the macroparticle and only in the special case of an isothermal plasma does the screening become Debye one. The screening length is determined by the ratio of the electron-ion, βei, and Langevin, βL = 4πeμi (where μi is the ion mobility), recombination coefficients. If βL ≫ βei, then it is much larger than the electron Debye length. The ions in an isothermal plasma have been found to give the same contribution to the screening as the electrons if the electron-ion recombination coefficient exceeds the Langevin ion recombination coefficient by a factor of 2 or more, βei ≥ 2βL. The Vlasov equation is used to analyze the asymptotic behavior of the macroparticle potential in a collisionless plasma.

Journal ArticleDOI
TL;DR: In this paper, the interplay between the nanowire radius, the Thomas-Fermi and Debye screening lengths, and the length of the functionalization molecules is studied.
Abstract: The conductance change of nanowire field-effect transistors is considered a highly sensitive probe for surface charge. However, Debye screening of relevant physiological liquid environments challenge device performance due to competing screening from the ionic liquid and nanowire charge carriers. The authors discuss this effect within Thomas-Fermi and Debye-Huckel theory and derive analytical results for cylindrical wires which can be used to estimate the sensitivity of nanowire surface-charge sensors. They study the interplay between the nanowire radius, the Thomas-Fermi and Debye screening lengths, and the length of the functionalization molecules. The analytical results are compared to finite-element calculations on a realistic geometry.

Journal ArticleDOI
TL;DR: In this paper, the interplay between the nanowire radius, the Thomas-Fermi and Debye screening lengths, and the length of the functionalization molecules is studied.
Abstract: The conductance change of nanowire field-effect transistors is considered a highly sensitive probe for surface charge. However, Debye screening of relevant physiological liquid environments challenge device performance due to competing screening from the ionic liquid and nanowire charge carriers. We discuss this effect within Thomas-Fermi and Debye-Huckel theory and derive analytical results for cylindrical wires which can be used to estimate the sensitivity of nanowire surface-charge sensors. We study the interplay between the nanowire radius, the Thomas-Fermi and Debye screening lengths, and the length of the functionalization molecules. The analytical results are compared to finite-element calculations on a realistic geometry.

Journal ArticleDOI
TL;DR: In this article, the full-potential linearized augmented plane waves (FP-LAPW) method based on density functional theory (DFT) using the generalized gradient approximation (GGA) is applied to study the structural, mechanical, and electronic properties of BaTiO3, BaZrO3 and BaNbO3 cubic perovskites.

Journal ArticleDOI
TL;DR: In this paper, a pedagogical and qualitative introduction to the theory of equilibrium and thermalization of quark-gluon plasmas is given, assuming only that the reader is familiar with quantum field theory at zero temperature and with QCD as the theoretical theory of strong interactions.
Abstract: In these lectures, I will attempt a pedagogical and qualitative introduction to the theory of equilibrium and thermalization of quark-gluon plasmas. I assume only that the reader is familiar with quantum field theory at zero temperature and with QCD as the theory of the strong interactions. I focus on the limit of small αs, which in principle should be relevant at extremely high temperature because of asymptotic freedom, and in any case provides a clean theoretical context in which to discuss a variety of phenomena. Topics discussed include the basic equilibrium formalism for finite-temperature quantum field theory, Debye screening, electric deconfinement, magnetic confinement, dimensional reduction, plasma waves, kinetic theory, hydrodynamic properties such as viscosity, the Landau–Pomeranchuk–Migdal effect, thermalization in (arbitrarily high energy) heavy ion collisions, and QCD plasma instabilities.

Journal ArticleDOI
TL;DR: In this article, the heat capacities of bulk polycrystalline samples of Ti2AlN and solid solutions of Ti 2Al(N 0.5C0.5) in the 3-260k temperature range were reported.

Journal ArticleDOI
TL;DR: In this article, first principles for the total energy and elastic constants of the B2-type AlRE (RE=Sc, Y, lanthanide) have been performed at T = 0 K by using the projector augmented wave (PAW) method within the generalized gradient approximation (GGA).
Abstract: First-principles calculations for the total energy and elastic constants of the B2-type AlRE (RE=Sc, Y, lanthanide) have been performed at T =0 K by using the projector augmented wave (PAW) method within the generalized gradient approximation (GGA). The Debye temperatures, Gruneisen constants, the temperature dependences of the Gibbs free energy, coefficients of thermal expansion, heat capacities are obtained for the B2-AlRE within the Debye–Gruneisen model. The activation energy of self-diffusion, Poisson's ratio, Debye sound velocities are also evaluated for the B2-AlRE in the present work.

Journal ArticleDOI
TL;DR: In this article, the authors investigate a model for sound waves in a disordered environment, in which the local fluctuations of the elastic modulus are spatially correlated with a certain correlation length.
Abstract: We investigate a $d$-dimensional model ($d$ = 2,3) for sound waves in a disordered environment, in which the local fluctuations of the elastic modulus are spatially correlated with a certain correlation length. The model is solved analytically by means of a field-theoretical effective-medium theory (self-consistent Born approximation) and numerically on a square lattice. As in the uncorrelated case the theory predicts an enhancement of the density of states over Debye's $\omega^{d-1}$ law (``boson peak'') as a result of disorder. This anomay becomes reinforced for increasing correlation length $\xi$. The theory predicts that $\xi$ times the width of the Brillouin line should be a universal function of $\xi$ times the wavenumber. Such a scaling is found in the 2d simulation data, so that they can be represented in a universal plot. In the low-wavenumber regime, where the lattice structure is irrelevant there is excellent agreement between the simulation at small disorder. At larger disorder the continuum theory deviates from the lattice simulation data. It is argued that this is due to an instability of the model with stronger disorder.