Terms Of Angular Momentum Research Articles

General-relativistic equations of motion in terms of energy and angularmomentum

An attempt is made to describe the general-relativistic equations of motion for theSchwarzschild geometry in terms of the classical concepts of energy and angularmomentum. Using the customary terms, the geodesic equations can be viewed in away that is very helpful in providing the physical meaning of the mathematicaldevelopment.

Inverse dynamic investigation of voluntary trunk movements in weightlessness: a new microgravity-specific strategy

Present investigation faces the question of quantitative assessment of exchanged forces and torques at the restraints during whole body posture exercises in long-term microgravity. Inverse dynamic modelling and total angular momentum at the ankle joint were used in order to reconstruct movement dynamics at the restraining point, represented by the ankle joint. The hypothesis is that the minimisation of the torques at the interface point assumes a key role in movement planning in 0 g. This hypothesis would respond to an optimisation of muscles activity, a minimisation of energy expenditure and therefore an accurate control of body movement. Results show that the 0 g movement strategy adopted ensures that the integral of the net ankle moment between the beginning and the end of the movement is zero. This expected mechanical constraint is not satisfied when 0 g movement dynamics is simulated using terrestrial kinematics. This accounts for a significant imposed change of movement strategy. Particularly, the efficient compensation of the inertial effects of the segments in terms of total angular momentum at the ankle joint was evidenced. These results explain the exaggerated axial synergies, observed on kinematics and which moved centre of mass (CM) backward from its already backward initial positioning, as a tool for enhancing the compensation and achieving the desired minimisation of the torques exchanges at the restraints.

Dynamical equivalence and the departure‐point equation in semi‐Lagrangian numerical models

AbstractThe kinematics and Newtonian dynamics of a continuous fluid may be presented in a number of ways. Given the kinematic relation between velocity and position, the dynamics may be formulated in terms of momentum or in terms of angular momentum and the scalar product of velocity and position. It is also possible to carry out the formulation in terms of momentum, angular momentum and the scalar product of velocity and position, in which case the kinematic relation becomes a deducible property. The three formulations are dynamically equivalent in the sense that the flow evolution is demonstrably the same in each. It seems desirable to build this elementary symmetry into discrete models. Two‐time‐level, semi‐Lagrangian models are considered here, with emphasis on the departure‐point equation (from which trajectory origins at the earlier time‐level are determined). Once a rule has been chosen for the approximation of trajectory time‐averages of vector and scalar products, dynamical equivalence implies a particular form of the departure‐point equation. One such rule delivers a doubly‐implicit equation that is a simple time‐centred discretization of the kinematic relation. A second rule leads to a form that differs only slightly from the singly‐implicit equation of Gospodinov and co‐workers. It is noted that dynamical equivalence relates to the consistency and non‐ambiguity of results rather than to their accuracy. © Crown copyright, 2003. Royal Meteorological Society

Eigenvalues in gaps of perturbed periodic Dirac operators: numerical evidence

This paper presents a method for the numerical investigation of the distribution of the eigenvalues introduced into a spectral gap of a periodic Dirac system by a perturbation of the type of the angular momentum term. A number of examples illustrate the effectiveness of the method and show the remarkable accuracy of the strong coupling asymptotic formula even for small values of the perturbation coupling constant. Furthermore, the results shed some light on the spectrum in the exceptional gap of radially periodic three-dimensional Dirac operators.

Radiofrequency spectra of ethyl fluoride recorded with two-dimensional Fourier transform spectroscopy using a coaxial sample cell

We describe a new experimental method in two-dimensional radiofrequency-microwave spectroscopy by using a coaxial sample cell. This technique allows to determine simultaneously transition frequencies in the RF range from about 1–600MHz as well as in the MW range from about 1–4GHz and was applied to investigations on ethyl fluoride in the ground state of the methyl torsion. The results show an interesting feature of Q-branch a-type lines in the RF range which is due to linear angular momentum terms in the effective Hamiltonian for the torsional E-states. The experimental frequencies are in good agreement with the applied theory.

Large magnetocrystalline anisotropy in bilayer transition metal phases from first-principles full-potential calculations

The computational framework of this study is based on the local-spin-density approximation with first-principles full-potential linear muffin-tin orbital calculations including orbital polarization (OP) correction. We have studied the magnetic anisotropy for a series of bilayer CuAu(I)-type materials such as $\mathrm{Fe}X,$ $\mathrm{Mn}X(X=\mathrm{N}\mathrm{i},\mathrm{P}\mathrm{d},\mathrm{P}\mathrm{t}),$ CoPt, NiPt, MnHg, and MnRh in a ferromagnetic state using experimental structural parameters to understand the microscopic origin of magnetic-anisotropy energy (MAE) in magnetic multilayers. Except for MnRh and MnHg, all these phases show perpendicular magnetization. We have analyzed our results in terms of angular momentum-, spin- and site-projected density of states, magnetic-angular-momentum-projected density of states, orbital-moment density of states, and total density of states. The orbital-moment number of states and the orbital-moment anisotropy for $\mathrm{Fe}X$ $(X=\mathrm{N}\mathrm{i},\mathrm{P}\mathrm{d},\mathrm{P}\mathrm{t})$ are calculated as a function of band filling to study its effect on MAE. The total and site-projected spin and orbital moments for all these systems are calculated with and without OP when the magnetization is along or perpendicular to the plane. The results are compared with available experimental as well as theoretical results. Our calculations show that OP always enhances the orbital moment in these phases and brings them closer to experimental values. The changes in MAE are analyzed in terms of exchange splitting, spin-orbit splitting, and tetragonal distortion/crystal-field splitting. The calculated MAE is found to be in good agreement with experimental values when the OP correction is included. Some of the materials considered here show large magnetic anisotropy of the order of meV. In particular we found that MnPt will have a very large MAE if it could be stabilized in a ferromagnetic configuration. Our analysis indicates that apart from large spin-orbit interaction and exchange interaction from at least one of the constituents, a large crystal-field splitting originating from the tetragonal distortion is also a necessary condition for having large magnetic anisotropy in these materials. Our calculation predicts large orbital moment in the hard axis in the case of FePt, MnRh, and MnHg against expectation.

Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner

We report a new mechanism for trapping metallic particles in inverted optical tweezers. The particles are loosely confined, in three dimensions, to an annular region just below the beam waist where gravity is counterbalanced by the scattering force. When using a Laguerre–Gaussian mode the trapping efficiency is improved and additionally we are able to observe an off-axis rotation of the particles around the beam circumference which is induced by the orbital angular momentum of the beam. Since there is no mechanism by which the spin angular momentum can contribute to the scattering force the spin and orbital angular momentum terms are decoupled and therefore the polarisation state does not influence the motion of the particles.

Spiral Structures and Shocks in Accretion Discs in Close Binary Systems: the Role of the Stellar Mass Ratio.

In this work we investigated, in the Smooth Particle Hydrodynamics (SPH) framework, the development of spiral structures and shock fronts in the radial flow of accretion discs in close binary systems. These shock waves take place when the initially supersonic radial flow penetrating the disc bulk, reduces substantially its speed becoming suddenly subsonic. To this purpose, keeping constant the mass of the compact primary (M1 = 1 M⊙ ), the separation between the two components and the injection speed at the inner Lagrangian point L1 (close to the local sound speed), we carried out 2D SPH simulations for four values of the stellar mass ratio M2/M1. We worked out 2D models because the damping effect of the artificial viscosity is too strong in 3D. Furthermore, the 2D environment seems the most suitable in order to evidence shock fronts in highly compressible gases. The results show that spiral structures and shock fronts develop for low values of M2/M1, whilst they become less evident for higher values. This behaviour is interpreted in terms of initial specific angular momentum at L1, whose position, relative to the companions, depends on the stellar mass ratio.

Rovibrational Hamiltonians for general polyatomic molecules in spherical polar parametrization. II. Nonorthogonal descriptions of internal molecular geometry

A rovibrational kinetic energy operator for a two-vector embedded reference frame is derived for general polyatomic molecules of arbitrary structure. New compact expressions for T̂ in terms of angular momentum and linear momentum vector operators are explicitly given for tri-, tetra-, and pentatomic molecules. The necessary actions for the evaluation of kinetic energy matrix elements in a standard rotation-angular basis are summarized. The structure of kinetic energy operators in the body-fixed formulation is explained with the help of Cartesian components of angular momentum and linear momentum vector operators. Kinetic energy operators for tetratomic molecules in a bisector embedded body-fixed formulation and for sequentially bonded pentatomic molecules are derived to show how to use our general result for diverse coordinate systems.

Spherical anisotropic diffusion models for the Earth's core

Estimates of the molecular values of magnetic, viscous and thermal diffusion in the Earth's core suggest turbulent small-scale velocity, magnetic and temperature fluctuations. Instability arguments (Braginsky, S.I., Meytlis, V.P., 1990. Local turbulence in the Earth's core. Geophys. Astrophys. Fluid Dynam. 55 (1990) 71–87.) indicate that the viscous and thermal diffusions of the resulting turbulence are strongly anistropic, the directions of anisotropy being determined by the mean magnetic field, the rotation of the core, the mean temperature gradient and gravity. Physical principles and invariance arguments are used to constrain the forms of the turbulent viscous stress tensor to eight types and the turbulent heat flux vector to a single type. The models are interpreted in terms of angular momentum, kinetic energy and entropy exchange between the mean and fluctuating fields. For the development of the turbulence models in spherical geometries, general non-linear spectral expansions are derived for one of the turbulent viscous stress tensors, the turbulent heat flux vector and their divergences using vector and tensor spherical harmonic methods and poloidal–toroidal representations. The spectral expansions, which are complicated and would be difficult to derive using other methods, include all invariance terms and can be programmed directly from the explicit forms given, thus laying the foundation for future computational studies of core turbulence. They are particularly suitable for linearised problems, such as anisotropic convection and magnetoconvection. The spectral techniques employed are applicable to all eight turbulent viscous stress tensors.

Electron-pair momentum densities of group 14, 15, and 16 atoms in their low-lying electronic states

Electron-pair intracule (relative motion) h̄(υ) and extracule (center-of-mass motion) d̄(P) densities in momentum space are studied for the 3P, 1D, and 1S terms of five group 14 atoms with p2 configurations, the 4S, 2D, and 2P terms of five group 15 atoms with p3 configurations, and the 3P, 1D, and 1S terms of five group 16 atoms with p4 configurations. Common to all fifteen atoms, the intracule densities show that a low energy term has a greater probability of finding a pair of electrons with a large relative momentum υ than a high energy term. The Fermi hole effect in a high spin term appears naively in momentum space, and the average relative momentum 〈υ〉 is larger in a high angular momentum term. For the terms arising from the pm electronic configurations (m=2–4), the differences in the radial extracule densities are found to be almost isomorphic with the corresponding intracule ones. In a term with a high angular momentum, the average center-of-mass momentum 〈P〉 of an electron pair is always larger, and two electrons are less likely to have opposite momenta. The major origin of these differences in the electron-pair densities lies in the valence np orbitals, and the contribution of different electron-pair motions to the relative stability of terms can be explained by a rigorous relation between the kinetic energy Tnp and the second intracule 〈υ2〉np and extracule 〈P2〉np moments of the valence np subshell.

An F=ma analysis of the spinning skater and decaying satellite orbit

Spinning skaters and satellite orbits are usually discussed in terms of angular momentum and other powerful formalisms. This paper presents an alternative, more primitive analysis based directly on Newton’s second law.

The Many-Electron System in the Forward, Exchange and BCS Approximation

The nonrelativistic many-electron system in the forward, exchange and BCS approximation is considered. In this approximation, which is still quartic in the annihilation and creation operators, the model is explicitly solvable for arbitrary space dimension d. The partition function and the correlation functions are given by finite-dimensional integral representations. Renormalization effects as well as symmetry breaking can be seen explicitly. It is shown that the usual mean field approach, based on approximating the Hamiltonian by a quadratic expression, may be misleading if the electron-electron interaction contains higher angular momentum terms and the space dimension is d=3. The perturbation theory of the solvable model is discussed. There are cases where the logarithm of the partition function has positive radius of convergence but the sum of all connected diagrams has radius of convergence zero implying that the linked cluster theorem is not applicable in these cases.

“Electromagnetic” contribution to the nucleon spin

A possible resolution to the question of where the nucleon gets some portion of its internal angular momentum, not carried by the valence quarks, is given in terms of the electromagnetic field angular momentum of the nucleon. This field angular momentum is similiar in origin to that of the Thomson dipole (an electric charge and a magnetic monopole), but arises from the interaction of the quark electric charges and magnetic dipoles. Rough, order of magnitude calculations show that such a field angular momentum could provide some of the nucleon's spin. Under certain assumptions QCD should also exhibit a similiar field angular momentum coming from the interaction of color charges with color magnetic dipoles.

Ab Initio Modeling of the Metal−Support Interface: The Interaction of Ni, Pd, and Pt on MgO(100)

The interaction of Ni, Pd, and Pt on MgO has been investigated using cluster models and a variety of ab initio techniques. The case of a single Pd atom with acidic and basic sites of the MgO(100) surface has been chosen to test the adequacy of different ab initio wave functions and DFT techniques. Both approaches result in a similar description but only if large basis sets, including high angular momentum terms, and extensive treatment of the many-body effects are accounted for in the ab initio cluster model wave function. The DFT results appear to be more stable and less dependent on the basis sets. When properly compared, both approaches lead to the same conclusion. Pd interacts in a rather weak way with the acidic sites, but a moderately large interaction is predicted for interaction above the basic sites. Next, the interaction of Ni, Pd, and Pt on the basic sites of the MgO(100) surface is considered. The general trend is that the interaction energy increases from Ni to Pt. The nature of the bond is rationalized by analysis of the dipole moment curves and of the molecular orbitals.

General formulation of covariant helicity-coupling amplitudes

A general formulation is given for constructing covariant helicity-coupling amplitudes involving two-body decays with arbitrary integer spins. The decay amplitudes are given exclusively in terms of both definite orbital angular momentum and total intrinsic spin. A systematic method is developed for calculating the energy and momentum dependence of daughter particles in the decay amplitudes, and a general formula for arbitrary integer spins is given. A number of illustrative examples is worked out, among which is that of the Higgs boson decay into two gauge bosons.

The role of angular momentum in the magnetic damping of turbulence

Landau & Lifshitz showed that Kolmogorov's E∼t−10/7 law for the decay of isotropic turbulence rests on just two physical ideas: (a) the conservation of angular momentum, as expressed by Loitsyansky's integral; and (b) the removal of energy from the large scales via the energy cascade. Both Kolmogorov's original analysis and Landau & Lifshitz's reinterpretation in terms of angular momentum are now known to be flawed. The existence of long-range velocity correlations means that Loitsyansky's integral is not an exact representation of angular momentum, nor is it strictly conserved. However, in practice the long-range velocity correlations are weak and Loitsyansky's integral is almost constant, so that the Kolmogorov/Landau model provides a surprisingly simple and robust description of the decay. In this paper we redevelop these ideas in the context of MHD turbulence. We take advantage of the fact that the angular momentum of a fluid moving in a uniform magnetic field has particularly simple properties. Specifically, the component parallel to the magnetic field is conserved while the normal components decay exponentially on a time scale of τ=ρ/σB2. We show that the counterpart of Loitsyansky's integral for MHD turbulence is ∫x2⊥Q⊥dx, where Qij is the velocity correlation. When the long-range correlations are weak this integral is conserved. This provides an estimate of the rate of decay of energy. At low values of magnetic field we recover Kolmogorov's law. At high values we find E∼t−1/2, which is a result derived earlier by Moffatt. We also show that ∫x2⊥Q∥dx decays exponentially on a time scale of τ. We interpret these results in terms of the behaviour of isolated vortices orientated normal and parallel to the magnetic field.

Can hexadecapole deformation lead to ΔI = 2 staggering in superdeformed bands?

The possibility is investigated that a C 4ν -symmetric hexadecapole deformation causes the observed ΔI = 2 staggering of rotational bands in even and odd- A nuclei. Calculating the energy as a function of intrinsic angular momentum components by means of the tilted cranking model, a rotor Hamiltonian with C 4ν -symmetry is derived, but the magnitude of the fourth order terms is found to be too small to generate a ΔI = 2 staggering of the experimental size. Considering an odd- A system composed of a rotor and a high j-quasiparticle, the ΔI = 2 staggering emerges only when the rotor core contains a C 4ν -symmetric fourth order angular momentum term.

Relativistic Particles Coupled to Chern-Simons Term-Revisited

We consider the model of N relativistic spinless particles coupled to an abelian Chern-Simons term. Rewriting the action in a time reparametrized form by introducing an arbitrary parameter, parametrizing the world line of the particles, we make a classical constraint Hamiltonian analysis of the model. Subsequent to gauge fixing by equating the arbitrary parameter with time we identify the Hamiltonian of the system, which agrees with the Hamiltonian obtained by using Banerjee′s method of fixing the arbitrary Lagrange multiplier by using equations of motion. We exhibit the Poincaré invariance of the model, at the classical level, by constructing spacetime generators using either the canonical or symmetric definition of the energy-momentum tensor. A detailed comparison of the expressions of angular momentum obtained by both methods shows that both agree up to a boundary term. In presence of a rotationally symmetric vortex configuration this term can be interpreted as an anomalous angular momentum term. We also heuristically discuss the effect of gauge fixing on the transformation properties.

Dynamical evolution of star clusters around a rotating black hole with an accretion disc

The dynamical evolution of relativistic star clusters around a massive Kerr black hole with an accretion disc is examined, in the regime where the black hole dominates the potential and star-disc interactions dominate the evolution. A set of diagrams exhibiting the time development of the energy-dependent distribution function, f (∊), and the distributions of semimajor axes, a, eccentricities, e, and inclinations, i, of the model system are presented; plots of the latter three quantitities over time for a few illustrative orbits are also given. A simple approximation for the final radius of an orbit brought into the disc under star-disc interactions, namely |$a_\text f\approx a_0(1-e^2_0) \enspace\text {cos}^4(i_0/2)$|⁠) (or, in terms of angular momentum, |$L_\text f\approx(L_0+L_{z,0})/2$|⁠), is derived. It is found that the main effect of star-disc interactions on an isotropic cluster, besides the circularization and alignment of orbits, is to steepen an initial density profile ρ* ∝ r–n to the approximate ‘asymptotic’ profile |$\rho_\ast\propto r^{-2.5}$| when n ≲ 2.5 to leave the profile unchanged when n ≳ 2.5, and in both cases to increase the central density (by several hundred very close to the black hole); initially anisotropic clusters are found to exhibit similar patterns. The numerical results can be explained in terms of a simple analytic model. Relativistic effects are found to affect the cluster properties significantly only at very small radii (≲ 10 GM/c2); in particular, the location of the last stable orbit limits the cluster's inner extent. By significantly increasing the central stellar density, star-disc interactions could be self-limiting by causing stellar collisions to become important; the future evolution of the cluster in this case will depend on the relative balance of the collisional, alignment and stellar evolutionary time-scales.