Use Of Group Theory Research Articles

Mass spectrum of N∗ and source optimization

We have computed correlation functions for nucleons and extracted the masses for positive- and negative-parities. Use of group theory plays an important role in obtaining sources that have good overlap to higher spin states and minimum contamination from unwanted states. In the simulation three distinct sources and corresponding sinks that transform according to the G 1 irreducible representations are tested and used to form matrices of correlation functions. Diagonalizations give us clear mass splittings between low-lying states and excited states for both parities.

A REMPI investigation of the minimum energy conformations of diphenyl ether

The structure and low-frequency vibrations of jet-cooled molecules of diphenyl ether are studied with the resonance-enhanced two-photon ionization technique. The origin for the S 1←S 0 transition is assigned at 35,873 cm −1 and, within 400 cm −1, the vibrational progressions suggest the existence of different conformations proposed for this molecule. A potential energy surface for the ground state, obtained with the use of group theory, is checked against the conformational energy surface given by molecular mechanics simulations.

Optimal basis states for a microscopic calculation of intrinsic vibrational wave functions of deformed rotational nuclei

The primary achievement of the symplectic model is to give a realistic microscopic shell-model expression of the nuclear collective model. However, in applications of the model, one has to contend with the fact that its Hilbert spaces, like those of the shell model, are infinite dimensional. This means that truncation of the model Hilbert space to a finite-dimensional subspace is inevitable. Nevertheless, it is in principle possible to get results to any desired accuracy if a sequence of increasingly large, but finite-dimensional subspaces of the full Hilbert space can be determined so that truncation to the subspaces of the sequence leads to rapidly convergent results. We show in this paper that generator coordinate (also called coherent state) bases can be constructed and optimized to give extraordinarily rapid convergence. This makes it possible to perform accurate symplectic model calculations with realistic microscopic Hamiltonians by a method that is essentially an angular-momentum projected, multi-determinant, Hartree–Fock calculation for which powerful but highly practical methods, that make a minimal use of group theory, have been developed. It is shown that the techniques developed give a natural representation of the intrinsic states of rotational bands as beta- and/or gamma-vibrational wave functions. The techniques are illustrated by computation of beta-vibrational wave functions for the ground-state and one-phonon excited rotational bands of 8Be for a microscopic Hamiltonian with a Brink–Boeker two-body interaction.

Remarks on the use of group theory in quantum optics

The relationships between certain important nonclassical states of the quantized field and the coherent states associated with the SU(2) and SU(1,1) Lie groups and associated Lie algebras is briefly reviewed. As an example of the utility of group theoretical methods in quantum optics, a method for generating maximally entangled photonic states is discussed. These states may be of great importance for achieving Heisenberg-limited interferometry and in beating the diffraction limit in lithography.

Colored L-l filters and their application in speech pitch detection

This paper develops colored L-l filters with tap bias and evaluates their performance in nonlinear system modeling and speech pitch detection. L-l filters are nonlinear weighted sum based filters that jointly exploit the temporal and rank orderings of observation samples. The use of joint ordering allows L-l filters to effectively process nonstationary signals with heavy tailed distributions. Utilizing this method of incorporating temporal and rank order information, however, requires N/sup 2/ parameters to describe a window size N filter. This polynomial growth in the number of filter parameters limits the use of L-l filters to applications that can be satisfactorily addressed with moderate sized windows. It is shown here that performance increases in temporal-rank order filters can often be achieved by expanding the window size. A method for reducing the L-l filter parameter space, and thus relaxing the window size limitations, is developed through the use of group theory based coloring, or quantization, of rank order information. Additionally, the modeling ability of colored L-l filters is improved through the simple inclusion of tap biases. Optimization procedures are developed for determining filter tap weights and biases as well as the rank order coloring. It is shown that the window size/rank order quantization tradeoff has advantages in many applications and that the inclusion of tap bias results in improved nonlinear system modeling. Finally, a method for utilizing colored L-l filters in the determination of glottal closure instants (GCIs) for speech pitch period identification is developed. Results presented show that this method is a more accurate GCI indicator than a commonly used wavelet approach and requires less processing time.

Quantum mechanics of periodic systems

The quantum description of a periodic system can be naturally embedded in an auxiliary problem with an extended configuration space. Standard methods applied to the solution of the extended Schrödinger equation yield the results usually obtained by the use of group theory or extensions of Floquet’s theorem. This method is used to prove Bloch’s theorem and to construct a multimode Floquet method for the semiclassical theory of lasers interacting with atoms.

Empirical Tight-Binding Applied to Silicon Nanoclusters

ABSTRACTThe calculation of the electronic structure of silicon nanostructures is used to discuss the accuracy of results obtained by the tight-binding method. We first show that the level of refinement of the tight-binding approximation must be adapted to the calculated property. For example, an accurate description of both the valence and conduction bands which can be achieved with a 3rd-nearest neighbor approximation is necessary to calculate the variation of the gap energy with the silicon crystallite size. The sp3s* model which gives a bad description of the conduction band underestimates the confinement energy but can give good results when it is used to determine the variation of the crystallite band gap with pressure. To study Si-III (BC-8) nanocrystallites, we show that a good description of the bulk band structure can be obtained with non-orthogonal tight-binding but due to the large number of nearest neighbors one must take analytical variations of the parameter with interatomic distances. The parameters involved in these expressions can be easily fitted to the bulk band structures using the k-point symmetry without requiring the use of group theory. Finally we discuss the effect of increasing the size of the minimal-basis set and we show that it would be possible to get the values of the tight-binding parameters from a first-principles localized states band structure calculation avoiding the fit to the energy dispersion curves.

An embedded ring approach to the vibrational dynamics of low-dimensional amorphous solids with applications to graphitic carbon materials

A theoretical approach has been developed to model the vibrational modes of amorphous, two-dimensional materials. The method considers that the vibrational density of states is composed primarily of states originating from embedded ring structures of medium-range order. The materials are modeled as continuous random networks comprised of a statistical distribution of symmetric, planar rings with four to eight members. The rings are treated as local structural units embedded in the material, similar to molecules within a solid. The ring potentials are approximated with a valence force model (bond-stretching and bond-angle-bending force constants) modified by a third harmonic, effective force constant coupling the rings to the surrounding network. The molecular dynamics of the rings are analyzed with the use of group theory, and the frequencies are calculated using a normal coordinate treatment. The utility of the embedded ring approach lies in its use of the material's ring statistics and ring mode frequencies, allowing determination of the structure of an amorphous material from its vibrational spectrum or prediction of vibrational spectra and density of states from structural models. The method is applied here to various graphitic carbon materials as a test case and predicts a set of force constants and Raman spectra consistent with previous data and structural models.

Use of group theory in symmetrical 3-D eddy-current problems

Geometrical symmetry often occurs in computational electromagnetics. However, it is generally not taken into account when the excitations do not share this symmetry. A rationale is required, and group theory gives the only valuable tool for this purpose. It provides a symmetric decomposition of any problem that allows its study on a reduced part of the initial geometry. This way of treating field problems generally leads to substantial computational savings. The symmetry concept is applied in the paper to 3-D linear eddy-current analysis, where the numerical scheme used is a mixed FEM-BEM method. Detailed examples are presented to demonstrate the efficiency of the use of symmetry.

An elementary construction of the color wave functions in the quark model

The color wave functions of the quark model for colorless multiquark configurations (color singlets) are constructed by an elementary method without an explicit use of group theory. It is shown that no normalizable color singlet wave functions exists for free quarks, free diquarks and free diquark–antiquarks. The color wave functions of the mesons and the baryons are derived.

Fast Monte Carlo Algorithms for Permutation Groups

We introduce new, elementary Monte Carlo methods to speed up and greatly simplify the manipulation of permutation groups (given by a list of generators). The methods are of a combinatorial character, using only elementary group theory. The key idea is that under certain conditions, "random subproducts" of the generators successfully emulate truly random elements of a group. We achieve a nearly optimal O(n3 logcn) asymptotic running time for membership testing, where n is the size of the permutation domain. This is an improvement of two orders of magnitude compared to known elementary algorithms and one order of magnitude compared to algorithms which depend on heavy use of group theory. An even greater asymptotic speedup is achieved for normal closures, a key ingredient in group-theoretic computation, now constructible in Monte Carlo time O(n2 logcn), i.e., essentially linear time (as a function of the input length). Some of the new techniques are sufficiently general to allow polynomial-time implementations in the very general model of "black box groups" (group operations are performed by an oracle). In particular, the normal closure algorithm has a number of applications to matrix-group computation. It should be stressed that our randomized algorithms are not heuristic: the probability of error is guaranteed not to exceed a bound ϵ > 0, prescribed by the user. The cost of this requirement is a factor of |log ϵ| in the running time.

Simple Models of Complex Nuclei: The Shell Model and Interacting Boson Model

applications to the structure of atomic nuclei. The author systematically develops these models from the elementary level, through an introduction to tensor algebra, to the use of group theory in spectroscopy. The book's extensive and detailed appendix includes a large selection of useful formulae of tensor algebra and spectroscopy. The serious graduate student, as well as the professional physicist, will find this complete treatment of the shell model to be an invaluable addition to the literature.

Hypergeometric Expansions of Heun Polynomials

The product of two Heun polynomials is expanded in terms of products of two Jacobi polynomials. This is done by making crucial use of group theory and the knowledge of separable coordinate systems on the n-sphere. The expansion presented includes as a special case hypergeometric function expansions of single Heun polynomials that have been derived previously by other methods.

Computational Use of Group Theory in Bifurcation Analysis of Symmetric Structures

An efficient computational procedure is proposed for identifying singular points in global bifurcation analysis of the static behavior of symmetric discrete structures such as symmetric truss domes. Assuming group equivariance of the system of equations describing the steady state, and making use of group representation theory, the proposed method decomposes the Jacobian matrix (or tangent stiffness matrix) into block-diagonal form, with possibly repeated occurrences of identical blocks, by means of a suitable “local” coordinate transformation. The “local” transformation is computationally favorable in that it requires a small amount of computation and preserves the sparsity of the original Jacobian matrix fairly well. A concrete procedure is described for symmetric truss structures which are equivariant to dihedral groups; an explicit formula is given to the number of diagonal blocks into which the Jacobian matrix splits, and an estimate of the required number of computations shows the efficiency of the proposed method.

Abnormal orientation variants and interfaces in ferroelastics and precautions to be taken for the use of group theory (example : rare earth sesquioxides)

In group-subgroup phase transitions the orientation variants of the low symmetry structure are related by symmetry operations of the high symmetry structure which have been lost in the phase transition. In the case of the ferroelastic A (hexagonal)→B (monoclinic) phase transition of rare earth sesquioxides, orientation variants can occur which are not related by such symmetry operations but by the symmetry operations which would correspond to a twin (without change of the lattice) of the hexagonal structure. This phenomenon can occur when it reduces the high strain or high interface energies existing during the ferroelastic phase transition. Precautions necessary for the use of group theory in the study of ferroelastic orientation variants will be given. Special interest has also been taken in the study, by high resolution electron microscopy, of interfaces between a normal orientation variant and an unexpected (or abnormal) orientation variant. The interface has a certain thickness with possible atomic rearrangement Dans le cas de la transformation ferroelastique A (hexagonal)→B(monoclinique) des sesquioxydes de terres rares, il peut se produire certaines variantes d'orientation qui ne sont pas reliees entre elles par les operations de symetrie de la structure de haute symetrie, perdue lors de la transformation, mais par des operations de symetrie qui correspondraient a la formation d'une macle par merihedrie de la structure hexagonale. Indication des precautions necessaires lors de l'utilisation de la theorie des groupes pour l'etude des variantes d'orientation ferroelastiques. Etude par microscopie electronique a haute resolution des interfaces existant entre variante d'orientation normale et variante d'orientation inattendue

The SU(3)×U(1) invariant breaking of gauged N = 8 supergravity

The SU(3)×U(1) invariant stationary point of N = 8 supergravity is described in some detail. This vacuum has N = 2 supersymmetry, and it is shown how the fields of N = 8 supergravity may be collected into multiplets of SU(3)×Osp(2, 4). A new kind of shortened massive multiplet is described, and the multiplet shortening conditions for this and other multiplets are used to determine, by the use of group theory alone, the masses of many of the fields in the vacuum. The remaining masses are determined by explicit calculation. The critical point realizes Gell-Mann's scheme for relating the spin- 1 2 fermions of the theory to the observed quarks and leptons.

Renormalisation group study of the two-dimensional Hubbard model

The Hubbard model in two dimensions is studied using a recently proposed renormalisation group method for fermion systems. The author finds that there are no phase transitions in the model and that the ground state is ordered antiferromagnetically. The author also discusses the use of group theory in the renormalisation for large cells.

Use of Group Theory in Charge Density Calculations of ABO3 Compounds

Group theory was used to simplify LMTO charge density calculations of complex structures. To profit by the full crystal symmetry also at atoms with a lower site symmetry, the angular distribution functions of local charges at different sites are written as components of a suitable tensor function. The method was explicitly applied to the perovskite structure.

Microcomputer‐aided instruction and research in group theory

AbstractA collection of programs which aid in the use of group theory has been developed for the Apple II. The programs automate much of the tedious algebra necessary to generate character tables and representation matrices from the properties of the operators chosen to generate the group. At the same time, the presentation of intermediate results provides a schematic guide to the procedures by which groups are constructed. The choice of generators is not limited to the reflections, rotations, and inversions, which are the familiar features of chemical group theory. Any generator representable as a permutation may be treated; thus general permutation groups and Longuet‐Higgins groups may be studied. As an illustration we describe the states of beryllium borohydride, which require a form of Longuet‐Higgins group theory.

Use of group theory for the description of electromagnetic scattering from molecular systems

Group theory is applied to the calculation of the multipolar amplitudes of the electromagnetic field scattered by a cluster of spheres, a model scatterer intended to approximate the properties of molecular systems. The multipolar amplitudes are the solution of a system of linear nonhomogeneous equations that group theory allows us to get in factorized form. The consequent computational simplifications improve the reliability of the calculations and allow us to study the scattering even from complex clusters. A few physically significant examples are discussed with particular regard to the dependence of the cross sections on the direction of the incident field.