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We consider 2D lattices which are disordered by an external field, in this case by a dense, random distribution of attractive pinning centers of an underlying substrate. The preferred configuration of the 2D system with screened-Coulomb two-body interactions and a fixed number of particles is a triangular lattice configuration. The pinning centers strongly affect the triangular lattice configuration. By keeping the location and the density of the pinning centers fixed, we study the effects of pinning strength on the structural properties of the lattice. We find that the strongly disordered lattice prefers to form irregular domains of size s which are composed entirely of topological defects. The distribution of sizes of these domains, D(s) suggest that s-τ for sufficiently large s. Our results are consistent with τ ≥ 2 in the solid phase and τ &lt; 2 in the liquid phase. We present a general argument in support of our findings and suggest that the physics which dictates the structure of 2D lattices on substrates with randomly distributed pinning centers is the same as that for a broad range of problems in which an external field strongly influences the behavior of an interacting physical system.

Department of Bioscience & Bioengineering

World University root node

Materials Research Society Symposium - Proceedings

2D lattices on substrates with randomly distributed pinning centers: a possible scaling law for domain sizes

Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T→∞ when the next-nearest-neighbor interaction is ≳1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., <1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. © 1996 The American Physical Society.

Physical Review B - Condensed Matter and Materials Physics

Relaxation in S=1/2 quantum spin chains: The role of second neighbor interactions

We present a careful simulational study of the onset of instabilities in a toy model grain pile as it is tilted through a threshold angle. Our simulations on a system of 1196 grains reveal qualitative differences in the number of grains undergoing translation over extended length scales below and at the threshold angle. The results are consistent with available experimental data and with the mean field treatments of self-organized critical behavior.

Physica A: Statistical Mechanics and its Applications

Metastability and instability in model grain piles: Scaling below and at the threshold

We present a detailed simulational study of the vertical propagation of weak and strong impulses in deep gravitationally compacted granular columns [see R. S. Sinkovits and S. Sen, Phys. Rev. Lett. 74, 2686 (1995)]. The intergrain potential is assumed to be [Formula Presented], [Formula Presented], where [Formula Presented] is the overlap between the grains. Due to gravitational compaction, the magnitude of the overlap between the grains increases progressively with increasing depth. Therefore the sound velocity increases as an impulse travels vertically downward into a granular column. For weak impulses, our large scale simulational studies show that the sound velocity [Formula Presented], where [Formula Presented] is the depth at which [Formula Presented] is measured. This result, which has been obtained from particle dynamical studies, is in perfect agreement with the predictions based upon elasticity theory. We then extend our analysis to show that (i) for columns with small void fractions, [Formula Presented], [Formula Presented] and (ii) for large amplitude impulses, the velocity of the perturbation [Formula Presented] behaves very differently compared to [Formula Presented] at shallow depths with [Formula Presented] as [Formula Presented]. We also present a detailed numerical study of the velocity power spectra of the individual grains as a function of depth [Formula Presented]. We close with a discussion of the effects of both light and heavy impurities on the vertical sound and shock propagation. © 1996 The American Physical Society.

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Sound propagation in impure granular columns

Using extensive numerical and some exact asymptotic analyses, we argue that the canonical ensemble relaxation of a classical particle in an anharmonic multi-well potential landscape with leading quartic anharmonicity exhibits a 1/t decay to equilibrium. In addition, we provide numerical evidence of algebraic decay of form 1/tθ, 0 &lt; θ &lt; 1, for stronger leading anharmonicities. Our results have important consequences for relaxation studies in several problems of considerable interest in condensed matter physics such as in the Krumhansl-Schrieffer model of displacive structural phase transitions, and in various complex systems.

Relaxation of classical particles in anharmonic multi-well potentials

Relaxation phenomena in a class of nondissipative systems with two highly disparate time scales (i.e., Brownian systems) have unique commonalities quantifiable via two scalars. It is shown that the exactly solvable problem of the dynamics of a weakly linked impurity spin in a s=1/2 XY chain belongs to this dynamical universality class and so does that of a heavy mass in an infinite harmonic oscillator chain and a spinless quasi two-dimensional attractive Fermi gas in the long-wavelength limit. The case of the strongly linked impurity in the XY chain is also discussed along with the corresponding limits in the harmonic oscillator chain and the electron-gas problems. © 1996 The American Physical Society.

Dynamical universality class of Brownian motion and exact results for a single-impurity s=1/2 XY chain

We report the first analytic study of real time and frequency dependent behaviour at T= infinity of the 2D square lattice transverse Ising model. Our study, along with known results in the 1D case, in the mean field or infinity D case and in other studies, helps understand the experimentally obtained longitudinal dynamic structure factor of the induced moment ferromagnet LiTbF4 studied by Youngblood et al., and is consistent with the studies of Kotzler et al.

Journal of Physics: Condensed Matter

Dynamic structure factor of the transverse Ising model

Atomistic simulations of the vertical propagation of perturbations in deep gravitationally compacted granular columns characterized by the intergrain potential Vδδn, with n2 and δ the grain overlap, are shown to recover the results of elasticity theory for weak perturbations. For strong perturbations, the sound velocity cstrong→cweak as z→ with the deviations from cweak best expressed via a certain recursion relation in z. We predict that voids in real granular columns lead to c1-ε, when the void fraction ε is small, and show that the velocity power spectrum of a grain resembles that of a harmonic oscillator chain as z→. © 1995 The American Physical Society.

Physical Review Letters

Nonlinear dynamics in granular columns

We calculate the interaction of a single helium atom with the low-temperature, orientationally ordered, simple cubic phase of a C60 fullerite crystal using anisotropic, helium-carbon pair potentials. The results of our study predict that helium is readily bound to sites on or near the surface of the fullerite crystal, but that intercalation of helium to the interior of the crystal at low temperatures is minimal due to the large energy barriers separating sites of octahedral and tetrahedral coordination. © 1995 The American Physical Society.

Physical Review B

He adsorption and intercalation in C60 fullerite crystals

Two-dimensional lattices in random potentials characterized by weakly attractive random pinning sites were studied via molecular-dynamics simulations. For a sufficiently high density of pinning sites, a "fragile" glasslike (i.e., amorphous solid) phase is found to be stable at temperatures slightly below the melting temperature. Simulations suggest that this phase can be replaced by a more stable phase, namely, an appproximately long-range-ordered solid phase, as the pins are made less dense and/or with weaker strength. The dynamics of the system depends sensitively on the density of pinning sites. This is probed by studying the velocity autocorrelation function of the particles in a random pinning potential as a function of the density of pinning sites. © 1995 The American Physical Society.

Structure and dynamics of two-dimensional lattices in random pinning potentials

This work reports a significantly improved estimation of the on-site dynamical spin pair correlation function in the s = 1 2 isotropic Heisenberg chain at T = ∞ with respect to an earlier study (Phys. Rev. B 46 (1992) 14617) which has received some attention in the literature. The calculations have been performed using a recently developed technique for estimating unsolvable infinite continued fractions which are relevant for calculating the relaxation functions. This improvement became possible due to an important advance in the machine based computation of quantum mechanical commutators by M. Böhm and H. Leschke (Physica A 199 (1993) 116). The work reported here builds on the results of Böhm and Leschke and provides new predictions on the behavior of the on-site dynamical spin pair correlation function. This work also provides insights into possible ways to qualitatively understand the complex relaxation at high temperatures associated with a hermitian operator A(t) in a system described by a hermitian Hamiltonian H = H0 + H1, where none of these three operators commutes with one another. © 1995.

Relaxation in the s = 1 2 isotropic Heisenberg chain at T = ∞: Towards a simple intuitive interpretation

The solution of the equation of motion for a particle in a Duffing potential, V(x) = α1x2/2 + α2x4/4 (α1, α2 &gt; 0) for arbitrary anharmonicity strength is characterized by the presence of odd frequencies which implies that velocity and position autocorrelation functions of such an oscillator in a microcanonical ensemble are also characterized by odd frequencies. It is, however, non-trivial to determine whether such "discrete" frequencies also characterize the autocorrelation functions in a canonical ensemble as discussed recently by Fronzoni et al. (J. Stat. Phys. 41 (1985) 553). We recover and extend upon the results of Fronzoni et al. to show analytically, via Mori-Lee theory, that "essentially discrete" (i.e. well-defined peaks with finite but "small" width) temperature-dependent frequencies characterize the autocorrelation functions in a canonical ensemble. © 1995.

Relaxation in a Duffing potential

Physical properties of a large class of intercalation compounds can be modeled by a two-dimensional (2D) repulsive screened Coulomb system subjected to a substrate corrugation potential. Using molecular dynamics simulation, we have investigated structural, thermodynamic (mainly 2D melting), and dynamic properties of these systems for different intercalant concentrations and corrugated strengths. For parameters appropriate to RbC24, a stage-2 graphite intercalation compound (GIC), we obtain an incommensurate ground state structure consisting of nano domains of triangular plaquettes separated by thick domain walls. A small decrease in the density gives a commensurate periodic domain wall ground state. In contrast to the corrugation-free systems, melting in these systems is continuous. Dynamics in the presence of corrugation show characteristics of a marginal fluid where solid and fluid-like dynamics coexist. © 1994, Taylor & Francis Group, LLC. All rights reserved.

Molecular Crystals and Liquid Crystals Science and Technology. Section A. Molecular Crystals and Liquid Crystals

Structure, melting and dynamics of screened coulomb systems in 2d-role of substrate corrugation

A Mori-Lee theoretic study of spin relaxation in the neighborhood of a magnetic impurity in the s=1/2 nearest-neighbor (NN) Heisenberg chain at high temperatures is presented. Our calculations suggest that in the limit of zero impurity coupling, the spectral function for the ''surface spin'' is consistent with a set of discrete frequencies given approximately by wn[n(w0+1/4)-1/4] against a shallow continuum background, where w03J/4 with 2J being the coupling between NN spins. The frequencies broaden as the impurity coupling is increased to the NN coupling. © 1994 The American Physical Society.

Relaxation in the neighborhood of a magnetic impurity in the s=1/2 Heisenberg chain at high temperatures

Marginal stability and avalanches at angles ΘΘaval are associated with granular media (GM) subject to special conditions. We study the Newtonian dynamics of random size-mismatched hard-core-like disks in a two-dimensional box which confirm claims that avalanches in GM are restricted to a few topmost boundary layers. We find that the velocity profile of the top layer grains, which obey -tγ, typically with 3.5≤γ, signal the onset of an avalanche in GM. Our studies suggest that the dynamics of a single particle in a cosine potential in the presence of a linear field well describes the onset of motion of a top layer grain. © 1994 The American Physical Society.

Physical Review E

Onset of avalanches in granular media

The canonical ensemble position and velocity autocorrelation functions (X and V ACF's) and the power spectra are obtained for a classical particle at T ≈ Γ, Γ ≡barrier height in a symmetric double-well (DW) potential. We report the presence of well defined temperature dependent characteristics frequencies in the VACF and the XACF, both above and below the harmonic frequency in this strongly anharmonic regime. A striking feature of the DW relaxation is that the V (X)ACF's are weakly temperature dependent at T ≈ Γ. © 1994.

Classical relaxation in a double well

The authors pointed out the disagreement between their calculation of dynamical correlations and that obtained using the exact result as in [2] can be attributed to two of following reasons: (i) The linear ansatz overestimates the higher coefficients and, given the nonoscillatory growth rate of coefficients with respect to n, suppresses the oscillation [1] in a0(t) thus damping it out faster. (ii) The inverse Laplace transform calculation is typically accurate to 10-4 or so unless a higher accuracy is specifically being sought [1]. © 1994 The American Physical Society.

Sen, Cai, and Mahanti reply

Dynamical correlations in simple quantum spin models can be conveniently studied via the continued fraction formalism in which the Laplace transformed dynamical two-point correlation can be written as C(z) = 1/(z + Δ1/(z + Δ2/(z + to ∞))), where Δn's are functions of static correlations. Very often, for systems without any natural dominant characteristic frequencies, as n increases, Δn ≊ κnα, α=0 or 1, κ being some constant, for large n for simple quantum spin systems. It is suggested that this property of Δn's could be related to some underlying "nearly noninteracting fermionic nature" of these quantum spin systems.

Journal of Applied Physics

A unique feature of some simple many body quantum spin systems

The Mori-Lee formalism for solving the Liouville (or Heisenberg) equation of motion for Hermitian systems demonstrates that the Laplace-transformed dynamical correlations in canonical ensembles can be written as (in)finite continued fractions. We show that a model-independent direct summation method allows accurate numerical evaluation of all known classes of these (in)finite continued fractions that arise in dynamics problems and thus provides a powerful technique to study the dynamics of many-body and few-body systems. Some studies on dynamical correlations in s=1/2 quantum spin chains are cited as applications of the method presented. © 1993 The American Physical Society.

Dynamical correlations and the direct summation method of evaluating infinite continued fractions

Calculations are presented for the on-site dynamical spin pair correlation of the s=1/2 Heisenberg chain at T=∞. The result for the isotropic chain is consistent with short time analysis. The long time behavior may be interpreted as compatible with exponential decay as well as with power law decay interpretations. The chain with small Ising-like anisotropy also exhibits exponential-like long time decay as found in recent Mössbauer effect spectroscopy work. Preliminary calculations are also presented for the mathematically challenging case of extreme Ising-like anisotropy.