Symmetry of the wave functions in the band theory of ferromagnetic metals.
Wave functions symmetry in band theory of ferromagnetic materials, discussing fcc, bcc and hcp structures and character tables
Falicov, L. M.↗
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Results for “Symmetry functions”
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Symmetric basis functions for networks with arbitrary geometrical symmetries
Basis functions for arbitrary time variable network symmetries for large scale integration, using group theory
Rubin, H.↗
Derivation and application of the reciprocity relations for radiative transfer with internal illumination
A Green's function formulation is used to derive basic reciprocity relations for planar radiative transfer in a general medium with internal illumination. Reciprocity (or functional symmetry) allows an explicit and generalized development of the equivalence between source and probability functions. Assuming similar symmetry in three-dimensional space, a general relationship is derived between planar-source intensity and point-source total directional energy. These quantities are expressed in terms of standard (universal) functions associated with the planar medium, while all results are derived from the differential equation of radiative transfer.
Cogley, A. C.↗
The resolution of a perturbed wave function into its symmetry components
Resolving perturbed wave functions into unperturbed Hamiltonians
Harriss, D. K.↗
Symmetry Breaking and the B3LYP Functional
The infrared spectra of six molecules, each of which contains a five-membered ring, and their cations are determined using density functional theory (DFT); both the B3LYP and BP86 functionals are used. The computed results are compared with the experimental spectra. For the neutral molecules, both methods are in good agreement with experiment. Even the Hartree-Fock (HF) approach is qualitatively correct for the neutrals. For the cations, the HF approach fails, as found for other organic ring systems. The B3LYP and BP86 approaches are in good mutual agreement for five of the six cation spectra, and in good agreement with experiment for four of the five cations where the experimental spectra are available. It is only for the fluoranthene cation, where the BP86 and B3LYP functionals yield different results; the BP86 yields the expected C2v symmetry, while the B3LYP approach breaks symmetry. The experimental spectra supports the BP86 spectra over the B3LYP, but the quality of the experimental spectra does not allow a critical evaluation of the accuracy of the BP86 approach for this difficult system.
Bauschlicher, Charles W., Jr.↗
Symmetry and strong interaction
Connection between internal and space-time symmetries in high energy interactions - theoretical physics
SPACE-TIME FUNCTION↗
Electronic energy states
One-electron wave functions are reviewed and approximate solutions of two-electron systems are given in terms of these one-electron functions. The symmetry effects associated with electron spin are reviewed and the effects of electron exchange on energy levels of the two-electron system are given. The coupling of electronic orbital and spin angular momentum is considered next and the Lande interval rule for Russell-Saunders or LS coupling is derived. The configurations possible for various multi-electron LS couplings are enumerated (examples from the first two rows of the periodic table are given), and the meaning of the spectroscopic nomenclature is discussed, particularly with respect to the degeneracies of the electron states involved. Next the nomenclature, symmetries, and degeneracies for electron states of diatomic molecules are discussed, and some examples for N2, O2, and NO are presented. The electronic partition functions and derivative thermodynamic properties are expressed in terms of these energies and degeneracies, and examples are given for some of the simple gas species encountered in the earth's atmosphere.
Source record↗
Computation of three-dimensional turbulent boundary layers with heat transfer in a plane of symmetry using embedded wall-layer functions
In the calculation of turbulent boundary layers, a large number of mesh points are required to adequately resolve the intense variation in the velocity and enthalpy in the near-wall region. A substantial reduction in computational effort may be realized by representing the velocity and enthalpy profiles in the wall layer by analytical embedded functions. The effectively inviscid flow in the outer part of the boundary layer may then be resolved by employing a relatively coarser mesh. To obtain complete profiles, the outer numerical solution is matched asymptotically to the inner wall-layer analytical solution. To date, this approach has been restricted to two-dimensional flows; in the present study, a method which may be utilized for turbulent boundary layers with heat transfer in a plane of symmetry is developed as a first step in the application of the embedded-function method to full three-dimensional flows. The present method uses only about half as many mesh points as that required in a conventional procedure, which calculates the flow all the way to the wall, but there is no degradation in accuracy of the computed results.
Degani, A. T.↗
Full CI benchmark calculations for several states of the same symmetry
Full CI (FCI) wave functions are used to compute energies for several electronic states of the same symmetry for SiH2, CH2, and CH2(+). It is found that CASSCF/multireference CI wave functions yield results very similar to FCI, irrespective of whether the CASSCF MOs are optimized independently for each state or using an average of the CASSCF energies for all desired states. The ionization potentials and excitation energies obtained from the FCI calculations should help calibrate methods (such as Green's function approaches, equations of motion and propagator methods, and cluster expansions) in which energy differences are computed directly.
Bauschlicher, Charles W., Jr.↗
Molecular Static Third-Order Polarizabilities of Carbon-Cage Fullerene and Their Correlation with Three Geometric Properties: Symmetry, Aromaticity, and Size
The static third-order polarizabilities (gamma) of C60, C70, five isomers of C78 and two isomers of C84 were analyzed in terms of three properties, from a geometric point of view: symmetry, aromaticity and size. The polarizability values were based on the finite field approximation using a semiempirical Hamiltonian (AM1) and applied to molecular structures obtained from density functional theory calculations. Symmetry was characterized by the molecular group order. The selection of 6-member rings as aromatic was determined from an analysis of bond lengths. Maximum interatomic distance and surface area were the parameters considered with respect to size. Based on triple linear regression analysis, it was found that the static linear polarizability (alpha) and gamma in these molecules respond differently to geometrical properties: alpha depends almost exclusively on surface area while gamma is affected by a combination of number of aromatic rings, length and group order, in decreasing importance. In the case of alpha, valence electron contributions provide the same information as all-electron estimates. For gamma, the best correlation coefficients are obtained when all-electron estimates are used and when the dependent parameter is ln(gamma) instead of gamma.
Moore, C. E.↗
Flow field measurements for cylindrical configurations in a hypersonic wind tunnel: Windward and leeward flow fields
Windward and leeward measurements were made for a variety of simulated infinite cylinders exposed to hypersonic streams over an angle of attack from 30 deg to 90 deg. For the range of conditions included in the study, the following conclusions are made: (1) Swept cylinder theory provides a reasonable correlation of the measured laminar heat transfer rates from the plane of symmetry. (2) The boundary layer transition criteria in the plane of symmetry are a function of the transverse curvature. (3) Relaminarization of the circumferential boundary layer for a right circular cylinder was observed at the highest Reynolds number tested. (4) The effect of leeside geometry on the average heat transfer rate can be correlated with a single geometric parameter which is dependent on the location of separation. (5) The relationship of leeward heating to angle of attack is virtually linear for each cross section. (6) No systematic effect of free stream Reynolds number was observed.
Bertin, J. J.↗
On the characteristics of caloric nystagmus in healthy persons
The asymmetry of reflex activity of labyrinths and directional preponderance of the reaction were studied on healthy persons subjected to caloric tests. Calorization with hot water was accompanied by less pronounced reactions in all parameters of nystagmus than analogous indices at cold water stimulation. The symmetry of labyrinth function shifted to the right in individuals with greater activity of the left central vestibular formations, analogous to right handedness behavior. It is concluded that asymmetry of reflex nystagmus in healthy persons can be due to a certain preponderance of functional activity in structures of the left hemisphere of the brain.
Bodo, D.↗
Missile impact craters (White Sands Missile Range, New Mexico) and applications to lunar research: Contributions to astrogeology
Craters in natural materials at White Sands Missile Range, N. Mex., were produced by the impact of high-velocity to hypervelocity missiles traveling along oblique trajectories with kinetic energies between 2.1 and 81 × 1014 ergs. The oblique impacts produce craters 2 to 10 m across with morphologies and ejecta that are bilaterally symmetrical with respect to the plane of the missile trajectory. Rims are high and the amount of ejecta large in down-trajectory and lateral directions, whereas rims are low to nonexistent and ejecta thin to absent up-trajectory. Symmetry development and modifications of the symmetry are a function of target material, local topography, and angle of impact. Seven mappable units can be recognized in and around the craters. Three of these are ejecta: thick ejecta near the crater, thin to discontinuous ejecta at greater distances, and scattered ejecta at the greatest distances to the limit of throwout. These ejecta units may be absent on the up-trajectory side; if present, they are rarely as thick or continuous as on other sides of the crater. Three units are target materials: undeformed target material exposed in local patches through thin to discontinuous ejecta and everywhere between the fragments of scattered ejecta, tilted and broken target material exposed in upper crater walls, and shattered and fractured target material exposed on the up-trajectory crater wall. The seventh unit is slope material composed of talus and fallback within the crater. Development, character, and exposure of these units varies chiefly with the target material. Ejecta from the craters is chiefly broken but relatively undeformed target material that may range in size from very fine grained debris to large blocks. Where the target is porous, significant amounts of the ejecta are composed of sheared and compressed fragments, some coated with dark layers of mixed projectile pieces, powder, and fused metal mixed with crushed target material. For layered targets, the original stratigraphic sequence is crudely preserved and in inverted order in thick ejecta. Secondary impact craters are produced by the impact of ejected fragments when the surrounding surface materials are sufficiently weak. A wide variety of secondary impact crater relations may result. Secondary craters nearest the primary crater have blocks in them that are larger than or the same size as the crater they produced. Farther from the primary crater, the fragments are generally smaller than the secondary crater and are ejected from it. Excavation of four craters revealed a mixed breccia beneath the crater floor composed of missile pieces, sheared and compressed target material, and crushed debris. Banded, disaggregated target material and nonmixed breccia surrounded the mixed breccia, and these breccias were surrounded by a zone of conjugate fractures. Beneath the ejecta on the lateral and down-trajectory crater flanks, the target materials were tilted upward and broken. Up-trajectory, open fractures and downward displacement occurred in two of the craters. No displacement was observed for the other two. Beneath the down-trajectory rims of craters with distinct layering, overturned synclines were observed. Missile breakup and behavior during cratering are a function of target and missile properties. Missile breakup depends on missile velocity and is most extensive at high velocities, where the missile is fragmented, powdered, and partly fused. Burial of missile or its fragmented, powdered, and fused remains is greatest for porous targets and least for dense cohesive targets. For very porous targets, camouflet structures containing the fragmented missile may form. Least squares fit to the data on craters in dry to moist targets indicate V(a) = 10^(-11.433)E(p)^(1.205) where V(a) is the volume of the apparent crater and E(p) is the kinetic energy of the missile. This equation is consistent with expectations of the equations relating apparent depth and radius to kinetic energy. Extrapolation of displaced masses and kinetic energies for laboratory impacts with sand and rock converge near 10^(15) to 10^(16) ergs, where the extrapolations are near the data on missile impact craters, corrected for impact angle. Displaced masses of craters produced by missile impacts and by chemical explosives with small scaled depths of burial are about the same when the kinetic energies of the missiles (corrected for angle of impact) are equal to the TNT equivalent energy of the explosive. The problem of equivalent scaled depth of burst for an impact crater is complicated and not entirely resolved, however. Both missile impact craters and chemical explosive craters in water-saturated targets are larger than their counterparts in dry to moist materials. Data collected during the study of missile impact craters have helped resolve a number of problems in lunar research: (1) the soillike nature of lunar surface materials was predicted, (2) sizes of craters produced by artificial impacts were correctly predicted, (3) certain features imaged by Surveyor were found to be analogous to features associated with missile impact craters, {4) missile impacts were used in support of the Apollo passive seismic experiment, (5) craters seen in Apollo orbital photographs were found to be similar to some missile impact craters, (6) missile impact craters supplied data on sample collection and crater phenomenology used in training astronauts, and (7) some returned lunar samples are similar to coated, sheared, and compressed fragments ejected from missile impact craters.
H. J. Moore↗
Fuel spray measurements with an imaging-type spray analyzer
The application of an imaging-type, automated spray analysis system to measurements of fuel sprays requires consideration of a number of significant factors both in its use and in the interpretation of the data. These factors are reviewed and a procedure developed for one such instrument is described. The procedure involves optical scans along the optic axis of the system and an appropriate mathematical inversion scheme for interpretation of the results. The system is nominally capable of yielding the local size distribution function and the total number density at points along the optic axis for droplets from about 10 to 500 microns. No assumptions regarding the form of the size distribution function or the symmetry of the spray are required.
Skifstad, J. G.↗
Hamiltonian indices and rational spectral densities
Several (global) topological properties of various spaces of linear systems, particularly symmetric, lossless, and Hamiltonian systems, and multivariable spectral densities of fixed McMillan degree are announced. The study is motivated by a result asserting that on a connected but not simply connected manifold, it is not possible to find a vector field having a sink as its only critical point. In the scalar case, this is illustrated by showing that only on the space of McMillan degree = /Cauchy index/ = n, scalar transfer functions can one define a globally convergent vector field. This result holds both in discrete-time and for the nonautonomous case. With these motivations in mind, theorems of Bochner and Fogarty are used in showing that spaces of transfer functions defined by symmetry conditions are, in fact, smooth algebraic manifolds.
Byrnes, C. I.↗
Geometric interpretations of the Discrete Fourier Transform (DFT)
One, two, and three dimensional Discrete Fourier Transforms (DFT) and geometric interpretations of their periodicities are presented. These operators are examined for their relationship with the two sided, continuous Fourier transform. Discrete or continuous transforms of real functions have certain symmetry properties. The symmetries are examined for the one, two, and three dimensional cases. Extension to higher dimension is straight forward.
Campbell, C. W.↗
Molecular dynamics and quasidynamics simulations of the annealing of bulk and near-surface interstitials formed in molecular-beam epitaxial Si due to low-energy particle bombardment during deposition
The relaxation, diffusion, and annihilation of split and hexagonal interstitials resulting from 10 eV Si irradiation of (2x1)-terminated Si(100) are investigated. Molecular dynamics and quasidynamics simulations, utilizing the Tersoff many-body potential are used in the investigation. The interstitials are created in layers two through six, and stable atomic configurations and total potential energies are derived as a function of site symmetry and layer depth. The interstitial Si atoms are allowed to diffuse, and the total potential energy changes are calculated. Lattice configurations along each path, as well as the starting configurations, are relaxed, and minimum energy diffusion paths are derived. The results show that the minimum energy paths are toward the surface and generally involved tetrahedral sites. The calculated interstitial migration activation energies are always less than 1.4 eV and are much lower in the near-surface region than in the bulk.
Kitabatake, M.↗
High-performance multiprocessor architecture for a 3-D lattice gas model
The lattice gas method has recently emerged as a promising discrete particle simulation method in areas such as fluid dynamics. We present a very high-performance scalable multiprocessor architecture, called ALGE, proposed for the simulation of a realistic 3-D lattice gas model, Henon's 24-bit FCHC isometric model. Each of these VLSI processors is as powerful as a CRAY-2 for this application. ALGE is scalable in the sense that it achieves linear speedup for both fixed and increasing problem sizes with more processors. The core computation of a lattice gas model consists of many repetitions of two alternating phases: particle collision and propagation. Functional decomposition by symmetry group and virtual move are the respective keys to efficient implementation of collision and propagation.
Lee, F.↗