Showing papers on "Symmetry (physics) published in 1978"

Molecular symmetry. II. Gradient of electronic energy with respect to nuclear coordinates

Abstract: Symmetry methods employed in the HONDO ab initio polyatomic SCF program are extended to the analytic computation of energy gradients Validity of the Hellmann–Feynman theorem is not assumed, ie, all two‐electron contributions to the gradient are included explicitly The method is geared to the efficient computation of entire blocks of two‐electron integrals Just one of a set of symmetrically related blocks must be computed The gradient contribution from each unique block is multiplied by q4, the number of equivalent blocks, and added into a ’’skeleton gradient vector,’’ all other blocks are simply omitted After processing molecular integrals, the true gradient vector is generated by projecting the symmetric component out of the skeleton vector The analysis is based on Eqs (26) and (33) which are valid for many variational wavefunctions including restricted closed shell and unrestricted open shell self‐consistent field functions We also extend the use of translational symmetry proposed previously b

Molecular symmetry. II. Gradient of electronic energy with respect to nuclear coordinates

Abstract: Symmetry methods employed in the HONDO ab initio polyatomic SCF program are extended to the analytic computation of energy gradients Validity of the Hellmann–Feynman theorem is not assumed, ie, all two‐electron contributions to the gradient are included explicitly The method is geared to the efficient computation of entire blocks of two‐electron integrals Just one of a set of symmetrically related blocks must be computed The gradient contribution from each unique block is multiplied by q4, the number of equivalent blocks, and added into a ’’skeleton gradient vector,’’ all other blocks are simply omitted After processing molecular integrals, the true gradient vector is generated by projecting the symmetric component out of the skeleton vector The analysis is based on Eqs (26) and (33) which are valid for many variational wavefunctions including restricted closed shell and unrestricted open shell self‐consistent field functions We also extend the use of translational symmetry proposed previously b

New symmetry in sd boson model of nuclei - group o(6)

Abstract: We suggest that, within the framework of the interacting-boson model, the group O(6) of orthogonal transformations in six dimensions may be useful in classifying nuclear spectra at the end of major shells. We derive analytic expressions for the energy levels and electromagnetic transition rates in this limit.

Generalization of Dirac’s monopole to SU2 gauge fields

Abstract: Dirac’s monopole is generalized to SU2 gauge fields in five‐dimensional flat space or four‐dimensional spherical space. The generalized fields have SO5 symmetry. The method used is related to the concept of orthogonal gauge fields which is developed in an appendix. The angular momenta operators for the SO5 symmetrical fields are given.

Theory of indirect interaction between chemisorbed atoms

Abstract: Progress in the development of the theory of two chemisorbed atoms has been hindered by the very low symmetry of the problem. Mono-layers of adatoms are simplest since they have the full two-dimensional symmetry of the substrate. Going to a (2 × 1) or a (2 × 2) adlayer, which doubles the size of the surface primitive cell, quadruples the size of a secular matrix, raising computer time even more. At the other end of the scale, a single adatom (in a symmetric site) will at least have the point symmetry of the substrate. With jellium as a substrate, this increases to full rotational and translational symmetry. Conserved quantities (“good quantum numbers”), which make calculations simpler, are associated with these symmetries. As a result, a variety of elaborate many-body techniques have been successfully applied to these systems; there are several excellent recent reviews.1,2 For two adatoms on a surface, there is little or no symmetry, typically just a twofold rotation or mirror plane (often leadin...

The localisation of energy in general relativity

Abstract: The logic of gravitational field energy localisation for static or quasi-static fields is discussed. A particular form of localisation in the case of spherical symmetry is justified by physical considerations. This form coincides with that general form presented by Moller (1952) for the case of the Schwarzschild constant-matter-density fluid but differs when one considers other equations of state.

Theory of the coupling between conduction electrons and moments of 3d and 4f ions in metals

Abstract: The theory of the effective coupling between conduction electrons and the magnetic (or electric) moments of 3d and 4f ions in metals is reviewed, with emphasis on a systematic treatment of the generalized coupling forms which usually apply for real ions with orbital degrees of freedom. With the help of the irreducible-tensor method, the coupling is expanded in forms consistent with general symmetry requirements, and the effects of intra-ionic level structure of LS or LSJ type are taken into account. Coupling contributions from the direct and exchange parts of the Coulomb interaction between ionic and conduction electrons, and from the Anderson-Clogston-Schrieffer-Wolff virtual-mixing mechanism, are evaluated. The symmetry analysis is performed according to the full rotation group, meaning that the conduction band is represented as a free-electron band and effects of crystal-field splitting are not explicitly considered.

Modes of a symmetric slab optical waveguide in birefringent media - Part I: Optical axis not in plane of slab

Abstract: We derive the eigenvalue equation and the expressions for the field distribution for the TM modes of an anisotropic symmetric slab waveguide; the TE modes are identical to the isotropic case. It is assumed that the field components do not depend on the y coordinate and that the c axis of the uniaxial crystal lies in the x-z plane. We find that, in spite of the symmetry of the waveguide, even and odd modes do not exist because the phase fronts of the modes are tilted with respect to the waveguide axis. However, the absolute magnitude of the H y component of the field has even or odd symmetry. The interesting case of an anisotropic slab waveguide with the optical axis located in the plane of the slab will be presented in Part II of this paper.

Group Symmetries in Nuclear Structure

Abstract: 1: Introduction.- 2: Classification of Symmetries.- 2.1. Space-Time (Geometrical) Symmetries.- 2.2. Exact Dynamical Symmetry (Unknown Origin).- 2.3. Almost Exact Dynamical Symmetry (Unknown Origin).- 2.4. Approximate Dynamical Symmetry.- 2.5. Dynamical Symmetries in Vector Spaces ("Model" Symmetries).- 2.6. Shape Symmetries.- 3: Symmetries and Groups.- 3.1. Groups and Representations of Groups.- 3.2. ?-Particle Model in Light Nuclei.- 3.3. Summary.- 4: Lie Groups and Their Algebras.- 4.1. Definition of a Lie Group.- 4.2. Infinitesimal Operators of a Lie Group.- 4.3. Representations of Lie Groups and Labeling of States.- 4.4. Representations of Lie Groups: Irreducible Tensors.- 4.5. Outer Product and Littlewood Rules.- 4.6. Matrix Groups and Their Representations.- 4.7. Two Theorems Concerning Goodness of Symmetry.- 5: Manifestation of Symmetries.- 5.1. Relationship between Energies.- 5.2. Symmetry Effect in Nuclear Reactions.- 5.3. Selection Rules.- 5.4. The Goodness of Symmetries.- 6: Spectral Distribution Methods.- 6.1. Introduction.- 6.2. The Method.- 6.3. Evaluation of Moments.- 6.4. Normality of the Distribution.- 6.5. Application of Distribution Method to Nuclear Spectroscopy.- 7: The Unitary Group and Its Subgroups.- 7.1. Introduction.- 7.2. Subgroups of U(N).- 7.3. Unitary Decomposition of Operators.- 7.4. Method of Separation.- 7.5. Number Nonconserving Operators.- 7.6. Decomposition by Contraction.- 7.7. Extension to Many Orbits: Configuration Averages.- 7.8. Unitary Group and Hartree-Fock Approximation.- 7.9. Application of Configuration Distributions.- 8: Angular Momentum and Isospin.- 8.1. Introduction.- 8.2. Multipole Sum-Rule Methods.- 8.3. Isospin Distributions.- 8.4. Strength Distributions.- 8.5. Mixing of Isospin Symmetry in Nuclei.- 8.6. Isobaric Mass Formula.- 8.7. Angular Momentum Averaging.- 9: Space-Symmetry Group-Wigner Supermultiplet Scheme.- 9.1. The Group SU(4) and the Supermultiplet Scheme.- 9.2. Casimir Operators of SU(4) and the Space Exchange Operator M.- 9.3. Evidence for Space Symmetry.- 9.4. ?-Particle Spectroscopy.- 9.5. ? Decay and Magnetic Moments of f7/2 Shell Nuclei.- 9.6. Muon Capture in Nuclei.- 9.7. SU(4) Classification of Nuclear Interaction.- 9.8. Study of SU(4) Symmetry Using Spectral Distribution Method.- 9.9. The "Goodness" of SU(4) Symmetry.- 9.10. SU(4)-ST Averaging.- 10: SU(3) Symmetry.- 10.1. Introduction.- 10.2. Brief Summary of Rotational Features in Light Nuclei.- 10.3. Search for the Intermediate Group G.- 10.4. Classification of States within an SU(3) Representation.- 10.5. States in the Projected Representation.- 10.6. Shell Model Calculation in the SU(3) Basis.- 10.7. SU(3) Classification of Interactions in the ds Shell.- 10.8. Mixing of SU(3) Symmetry in the ds Shell.- 10.9. Pseudo-LS and Pseudo-SU(3) Coupling Schemes.- 10.10. Configuration Mixing across Major Shells.- 10.11. "Macroscopic" SU(3) Symmetry.- 11: Seniority and Symplectic Symmetry.- 11.1. Introduction.- 11.2. Seniority in a Single j Shell.- 11.3. Representations of Sp(2j + 1).- 11.4. Casimir Operators and Their Eigenvalues.- 11.5. Goodness of Symmetry.- 11.6. Seniority in the j = 9/2 Shell.- 11.7. Symplectic Symmetry for the 1f7/2 Shell.- 11.8. Quasispin.- 11.9. Quasispin and Its Relation to Seniority.- 11.10. Multishell Seniority.- 11.11. Multishell Seniority Averaging.- 11.12. Multishell Seniority and the Two-Body Interaction.- 11.13. A New Truncation Scheme for Shell-Model Calculations.- 12: Summary and Final Remarks.- References.

The effect of symmetry on electron momentum distributions in solids

Abstract: A group-theoretical treatment of the symmetry properties of electron momentum distributions in cubic crystals is given. The contribution to the momentum density from a band not belonging to the totally symmetric representation possesses nodes in certain directions. A selection rule allows the location of these nodes. Tables are presented which summarise the consequences of the selection rule for simple cubic, FCC and BCC lattices, and some examples are discussed.

U-statistics for skewness or symmetry

Abstract: We propose certain U-statistics as intuitively reasonable measures of skewness and as criteria for asymptotically distribution-free tests of symmetry. Some comparisons are made with the moment-skewness b1 in terms of small-sample power and asymptotic relative efficiency.

Calculation of Microstrip Bends and Y-Junctions with Arbitrary Angle

Abstract: A method is described for calculating the frequency-dependent scattering parameters of microstrip bends and Y-junctions with arbitrary angles. Use is made of a waveguide model and an orthogonal series expansion for the fields around the discontinuity of the bend, so that the excitation and propagation of higher order modes can be considered. The transmission properties of the Y-junctions are derived from those of the bends by a symmetry consideration. Numerical results are given for two different substrates and are compared with experimental data. Neglecting radiation effects, they are in good agreement.

The symmetry of inner elastic constants

Abstract: The symmetry of inner elastic constants depends on the symmetry of the fundamental interlattice tensors from which they are built up. The number of independent interlattice tensors and their symmetries are determined by considering transformations of the interlattice index using a representation of the space group in which the not-purely-translational elements correspond to permutations. The method is explained and illustrated in detail. The non-zero components of the interlattice tensors are determined by standard methods and results are tabulated for the six species of tensor under consideration.

Time‐reversal symmetry in light‐scattering theory

Abstract: The general connections between Stokes and anti-Stokes cross sections are examined on the basis of the time-reversal symmetry of light-scattering experiments. The corresponding symmetry properties of the transition rates, susceptibilities and Green functions which occur in light-scattering theory are reviewed. A rigorous Stokes-anti-Stokes connection formula is derived and compared with an approximation in common use.

The symmetry of the electron-electron interaction operator in the dipole approximation

Abstract: The operator for the interaction of a charged quantum particle with an excited hydrogen-like atom or ion is considered in the dipole approximation. A new additional integral of motion and related symmetry group are found. A broader approximate symmetry group and expressions for the eigenvalues and eigenfunctions are proposed in the semiclassical limit of large total orbital momentum. The qualitative peculiarities of the spectrum are discussed and limiting cases are considered. A comparison with the results of other authors and with analytical and numerical data is made. The results are applicable to the doubly excited states of two-electron atoms, scattering of charged particles on hydrogen-like atoms or ions, and spectral line broadening etc.

Baryon spectroscopy in the non-relativistic quark model with a one-gluon exchange potential

Abstract: In the three-quark shell model with SU(6)(X)O(3) symmetry the baryonic states assigned to the (56,0+)0 and (70,1-)1 multiplets are discussed. Superposed on the confining potential, the short-range quark-quark interaction is taken to be Coulomb-like. In order to reproduce the large splitting between the two singlet Lambda states in the (70,1-)1 multiplet, an extra spin-orbit term originating from the confining potential is introduced, which is supposed to come from scalar coupling, SU(3) breaking is put in by taking different masses for the strange and non-strange quarks, and is treated in an exact way. General mass formulae for all (56,0+)0 and (70,1-)1 resonances are given and a fit to the spectrum has been performed.

Field-Flow Fractionation: Extensions to Nonspherical Particles and Wall Effects

Abstract: A rigorous analysis of the phenomenon of field-flow fractionation (FFF) is presented for particles which are both nonspherical in shape and of sufficient size (compared with apparatus dimensions) to be significantly influenced by wall effects. Calculations are presented for axially-symmetric particles in an arbitrary flow field. Orienting torques directed along the symmetry axis of the particle are also considered. The theory is compared with the experimental data of Berg, Purcell, and Stewart. Reasonably satisfactory agreement is observed.

Surface waves in anisotropic media: propagation in a homogeneous piezoelectric halfspace

Abstract: A decomposition of the elastic, piezoelectric and dielectric tensors has been developed for numerical computation of the characteristics of surface waves propagating in a homogeneous half-space. The decomposition also offers many theoretical insights not previously available. The particular form of anisotropic symmetry with respect to the direction of propagation critically affects the properties of the surface waves. These symmetries introduce singularities in the computations, which require special treatment, and distinctive polarizations of the elastic displacements and electric potential, which are diagnostic of the particular symmetry.

Rotation of an isolated triaxial ellipsoid suspended in slow viscous flow

Abstract: A freely suspended triaxial ellipsoid, or particle with two perpendicular symmetry planes, orients in a creeping flow at a rate given by differential equations in the three Euler angles, derived here for simple shear flow. The parameters for a given particle are three components of Bretherton's tensor; observations along two directions in space suffice to determine these. Any ellipsoid's two axis ratios can be chosen with the second deviating from unity less than the square root of the first. Numerical integration shows that, in contrast to the orbits of Jeffery, the particle has no fixed period of rotation about the fluid's vorticity axis. The longest axis of a rod-like triaxial particle crosses the plane perpendicular to the fluid vorticity, a motion impossible for an axisymmetric particle. In typical graphs for a quarter rotation, times required and changes in orientation are shown. Lack of axial symmetry implies that a suspension of such particles can be more disoriented.