Odd Rings Research Articles

Instability of hydrogenated amorphous carbon films towards defect creation at high disorder

We report a complete characterization of hydrogenated amorphous carbon (a-C:H) thin films through Raman spectroscopy, photothermal deflection spectroscopy, and mechanical studies. The interpretation of the optical absorption data within the Dasgupta’s model offers an estimate of the disorder intensity through a characteristic energy scale of the model. The vibrational data (Raman spectra) and the stress data together with the optical absorption data give strong evidence for a sharp transition between a low disorder state dominated by aromatic rings and a strong disorder state likely in keeping with the creation of some topological defects, such as odd rings.

On the Existence of Stable Roommate Matchings

This paper identifies a condition called “no odd rings” that is sufficient for the existence of stable roommate matchings in the weak preferences case. It shows that the process of allowing randomly chosen blocking pairs to match converges to a stable roommate matching with probability one as long as there are no odd rings. This random-paths-to-stability result generalizes that of Roth and Vande Vate (1990, Econometrica58, 1475–1480) and may not hold if there are odd rings. The “no odd rings” condition can also be used to prove a number of other sufficient conditions that are more economically interpretable. Journal of Economic Literature Classification Numbers: C78, D71.

Derization problem in conjugated polymers

Abstract Applying the self-consistent solitonic approach [1] to the extended Peierls-Hubbard model on odd rings, the interplay of el-el interactions U, V , various off-diagonal interactions, and external dimerization for the dimerization amplitude is studied by exact diagonalizations in the adiabatic limit. The applicability of an effective spin-Peierls Hamiltonian in the intermediate correlation regime U ~ 3t typical for conjugated polymers and an approximate analytical solution based on the Bethe-ansatz solution for the spin velocity and the continuum model solution for the spin-Peierls problem are discussed.

MINDO/3 study of the interaction of small carbon clusters

Interaction of small carbon clusters is studied using the MINDO/3 method. Energetic barriers for coagulation and isomerization are estimated. It is shown that coagulation of two linear chains leads to formation of a chain with possible closure to a monocyclic ring, while a linear chain and a ring results in a monocyclic ring. The barriers for these processes are less than 0.5 eV for coagulation and 2 eV for isomerization. Coagulation of two rings leads to a bicyclic cluster with a barrier of 0.2–0.7 eV. Binding of two even rings is relatively weak (1.5–2.0 eV). Odd and even rings may form a non-planar bicyclic cluster with an energy gain of 3.0–4.0 eV. Most stable structures result from two odd rings with binding energies of ≈ 5 eV.

Odd rings and tunneling modes in glasses

Tunneling modes, elementary excitations of a large class of glasses, are local dynamical signatures of disorder. There is one tunneling mode per odd loop (a necklace through odd rings in the network), decoupled from other loops, and tunneling is forced by gauge invariance of the structure, the symmetry of disorder. Absence of tunneling modes in trivalent glasses (a-As or B 2O 3) is a notable consequence of this model. It is possible to decompose topologically a tetravalent network into a trivalent network, and to add a large proportion of boroxol rings with their characteristic but different dynamical signature.

Glass: Elementary excitations and geometry; from bonds to tunnelling modes

Abstract Ground-state and elementary excitations (tunnelling modes) in glass are obtained from an analysis of its symmetry: a local gauge invariance. Glass is represented as a discrete fibre bundle. The base space is a continuous random network with tetravalent silicon atoms as vertices and chemical bonds as edges. The connection is given by the elasticity of the network. The bundle is non-trivial, and the elastic connection is entangled in one of two ways around odd rings in the network. To restore gauge invariance, tunnelling must occur between the two possible configurations. Entanglement, and elementary excitations are labelled by permutations of the covalent bonds incident on an atom.

Generalized Wheland polynomial for large graphs

We report the generalized Wheland polynomial for acyclic graphs depicting polyenes havingn = 10 carbon atoms. We consider the problem of deriving generalized Wheland polynomials for larger chains by recursion. The recursion Wh(n + l;x) =, Wh(n; x) + (1 −x)Wh(n − 1;x) allows one to find the next larger generalized Wheland polynomial for a chain having an even number of vertices by knowing generalized Wheland polynomials of chains having fewer vertices. The recursion, however, does not allow one to predict the generalized Wheland polynomial for a chain having an odd number of vertices from smaller chains! Here we report a procedure which allows one to derive the generalized Wheland polynomial for a chain having an odd number of vertices. This is achieved by combining the coefficients for rings having the same number of vertices. The generalized Wheland polynomials for odd rings are simply related to the generalized Wheland polynomials for smaller chains and can be derived from the information on smaller chains. This makes it possible to extend the recursion for generalized Wheland polynomials for arbitrarily largen.

Entanglements in glass

Ground state and elementary excitations (tunnelling modes) in glass are obtained from an analysis of its symmetry, a local gauge invariance. The configuration of glass is represented as a discrete fiber bundle. The base space is a continuous random network, standard model of the structure of covalent glasses. The connection is determined naturally by the elasticity of the network. The bundle is non-trivial, the elastic connection is entangled in one of two ways. Sources of non-triviality are closed loops, threading through odd rings in the network. To restore gauge invariance, tunnelling must occur between the two possible configurations about an odd loop. Entanglement and elementary excitations are labelled by permutations of the covalent bonds incident on an atom.

The Hubbard model for alternant and nonalternant systems

In this paper we consider the effects of two-electron, one-center interactions when added to the one-electron, two-center molecular orbital model. There are, therefore, two parameters considered: The standard Hückel β (two-center, one-electron) term and the Hubbard one-center, two-electron term U. It is shown how the change in the ground state as one changes the U/β ratio is highly dependent on the presence or absence of odd member rings.

Tetrahedrally bonded random networks without odd rings

Models of random networks having only even rings have been built by a modification of the method of Wooten and Weaire. Present models (built with a rather short annealing schedule) are very distorted. The bond-switching transformation used to disorder the network has interesting topological consequences.

Elementary excitations in continuous random networks

The elastic elementary excitations of covalent glasses are investigated in order to determine the influence of the structure (in particular to odd rings and closed lines there of) on the elasticity. The structure is a tetravalent continuous random network and the elasticity is described by the Keating Hamiltonian. We show that the density of bond stretching excitations at low frequency is enhanced by the presence of odd lines (for a simpler class of networks - circulants), and that there are two ground state elastic configurations per odd line, between which the system can tunnel.

Ground-state alternation and excitation energy of S=(1/2) linear Heisenberg antiferromagnets.

The ground-state energy per site, \ensuremath{\epsilon}(\ensuremath{\delta}), and the singlet-triplet gap, \ensuremath{\Delta}(\ensuremath{\delta}), of the linear spin-(1/2) alternating Heisenberg antiferromagnet are found by extrapolation of direct solutions to N\ensuremath{\le}26 sites for even rings and N\ensuremath{\le}21 sites for odd rings, whose orbital degeneracy leads to qualitatively different critical behavior. The \ensuremath{\epsilon}(\ensuremath{\delta}) accuracy of ${<10}^{\mathrm{\ensuremath{-}}4}$ (in units of J) decisively rules out power-law behavior for any alternation \ensuremath{\delta}>0.001 and is consistent with ${\ensuremath{\delta}}^{4/3}$/\ensuremath{\Vert}ln\ensuremath{\delta}\ensuremath{\Vert} for \ensuremath{\delta}<0.05. The gap \ensuremath{\Delta}(\ensuremath{\delta}) is slightly sublinear in \ensuremath{\delta}. Direct ground-state solutions are found via diagrammatic valence-bond methods and provide sensitive tests for theoretical approaches to linear spin systems.

Another characterisation of Pfaffian bipartite graphs

AbstractIt has been known for over twenty years that every planar graph is Pfaffian. Recently a characterisation of planar graphs in terms of strict maximal odd rings has been discovered. This paper attempts to elucidate the connection between the Pfaffian property and planarity by characterising Pfaffian bipartite graphs in terms of maximal odd rings.

Topological disorder an amorphous semiconductors: a real-space renormalisation for Husimi cacti alloys

Decimation techniques have already been used successfully to describe the densities of states of several types of disordered linear chains. In this paper the decimation procedure is extended for the case of Husimi cacti. These lattices are widely used as approximations to the continuous-random-network model for an ideal amorphous tetrahedral bonded semiconductor. The densities of states for disordered Husimi cacti for different concentrations of sixfold and fivefold rings are calculated. The effects of the presence of odd rings on the densities of states and in particular on the band edges are discussed.

Continuum Theory of Excitation Spectra of Topologically Disordered Systems

We propose a continuum treatment of random networks with odd rings near the antibonding .bound. An analogy with nematic liquid crystals is demonstrated in which Rivier lines play the role of disclination lines. The state density is expressed in terms of fluctuations of gauge fields associated with Rivier lines. There has been a great interest lately on the effects of topological disorder on the spectra of elementary excitations in amorphous materi­ als. , ),2) An important aspect characterizing such topological disorder is the existence of closed lines threading through odd rings. 3 ) The so­ called Rivier lines are known to lead to drastic reductions in density of states near the antibond­ ing limit.I),2) Here we present a continuum treat­ ment which is hoped to provide further insights into this important problem. We adopt a simple one· electron problem on a random network consisting of N sites which is described in Ref. 1). The Schrodinger equation for a set of amplitudes {aJ where j designates a site on the network is given in some dimensionless units by

Critical exponents for alternation in Heisenberg antiferromagnetic chains

The ground-state energy per site, λ(δ), and singlet-triplet gap, ΔE(δ), of regular ( δ=0) and alternating (δ ≤ 0.01) Heisenberg antiferromagnetic chains are obtained by extrapolating exact numerical solutions of even and odd rings and chains of N ≤ 20 spins. The critical exponents of 1.72 ± 0.02 for λ(δ) and 0.9 ± 0.1 for ΔE(δ) are compared with recent theoretical results for spin-Peierls transitions. Interactions among spinless fermions alter the instability to alternation.

Regular odd rings and non-planar graphs

In a previous paper we have announced that a graph is non-planar if and only if it contains a maximal, strict, compact, odd ring. Little has conjectured that the compactness condition may be removed. Chernyak has now published a proof of this conjecture. However, it is difficult to test a ring for maximality. In this paper we show that for odd rings of size five or greater, the condition of maximality may be replaced by a new one called regularity. Regularity is an easier condition to diagnose than is maximality.

Density of states near the band edges in amorphous germanium

We investigate the density of states near the edges of the band gap in tetrahedrally bonded amorphous semiconductors for a number of proposed structural models. This is made possible by using a cluster calculation method previously developed by us. Several techniques have been introduced to eliminate surface effects. The density of states near the band edge is shown to be sensitive to the characteristics of the model structure. Contrary to the usual concept that disorder introduces exponential tailing of the density of states into the energy gap, the presence of odd number rings (especially 5-member rings) makes the upper edge of the valence band drop faster with only a small tailing region. Large exponential tailing into the gap is induced by strain, i.e., stretching of bond lengths and distortion of bond angles.

Elementary excitations of topologically disordered systems

We have examined the elementary excitation spectra of continuous random networks having only even rings or both even and odd rings. We have shown that odd rings introduce localized states and suppress the density of states near a spectral bound associated with sign alteration in the spatial dependence of the state. While this is a general conclusion, we show it explicitly only for the nearest-neighbor tight-binding s-band and related magnon and phonon models. In the latter case, phonon softening or phonon hardening can result. In the former case, we show that frustration leads to localized magnons.

An amorphous Ising antiferromagnetic model

A simple model which illustrates the frustration effects found in amorphous Ising antiferromagnets is introduced. It is shown that the phase diagram of this model may be obtained from the phase diagram of a suitably constructed spin glass reference model, and the implications of this are discussed. The phase diagram of a model with an annealed distribution of even and odd membered rings of spins is explicitly obtained. The difficulties introduced by a non-zero external magnetic field are briefly discussed.