Angular Momentum Fluctuations Research Articles

Absence of quasiperiodicity in a kicked quantum rotator and a new localization-delocalization transition

It is found that the angular momentum fluctuations of a solvable kicked quantum rotator become unbounded in time if the kicking potential is discontinuous. This differs from the quasiperiodic behavior found for regular potentials and offers a possible route to quantum chaos. With use of a recently found mapping of the kicked rotator to an Anderson problem of conduction in a one-dimensional disordered lattice, our result implies a new localization-delocalization transition if the hopping matrix element decays more slowly than ${n}^{\ensuremath{-}1}$, where $n$ is the number of lattice sites connected by it.

Atmospheric angular momentum fluctuations, length-of-day changes and polar motion

Variations in the distribution of mass within the atmosphere and changes in the pattern of winds, particularly the strength and location of the major mid-latitude jet-streams, produce fluctuations in all three components of the angular momentum of the atmosphere on timescales upwards of a few days. In a previous study (Hide et al . 1980) it has been shown that variations in the axial component of atmospheric angular momentum during the Special Observing Periods in 1979 of the First GARP Global Experiment (FGGE, where GARP is the Global Atmospheric Research Program) are well correlated with changes in length-of-day. This would be expected if the total angular momentum of the atmosphere and ‘solid’ Earth were conserved on short timescales (allowing for lunar and solar effects) but not if angular momentum transfer between the Earth’s liquid core and solid mantle, which is accepted to be substantial and even dominant on timescales upwards of several years, were significant on timescales of weeks or months. Fluctuations in the equatorial components of atmospheric angular momentum should contribute to the observed wobble of the instantaneous pole of the Earth’s rotation with respect to the Earth’s crust, but this has not been shown conclusively by previous studies. In this paper we re-examine some aspects of the underlying theory of non-rigid body rotational dynamics and angular momentum exchange between the atmosphere and solid Earth. Since only viscous or topographic coupling between the atmosphere and solid Earth can transfer angular momentum, no atmospheric flow that everywhere satisfied inviscid equations (including, but not solely, geostrophic flow) could affect the rotation of a spherical solid Earth. Currently available meteorological data are not adequate for evaluating the usual wobble excitation functions accurately, but we show that partial integration leads to an expression involving simpler functions ─ here called ‘equatorial angular momentum functions’ ─ which can be reliably evaluated from available meteorological data. The length-of-day problem is treated in terms of a similar ‘axial angular momentum function’ ; and ‘effective angular momentum functions’ are defined in order to allow for rotational and surface loading deformation of the Earth. Daily values of these atmospheric angular momentum functions have been calculated from the ‘initialized analysis global database’ of the European Centre for Medium-Range Weather Forecasts (ECMWF). They are presented for the period 1 January 1981─30 April 1982, along with the corresponding astronomically observed changes in length-of-day and polar motion, published by the Bureau International de l’Heure (BIH). Changes in length-of-day during this period can be accounted for almost entirely by angular momentum exchange between the atmosphere and solid Earth, and the existence of a persistent fluctuation in this exchange, with a timescale of about 7 weeks, is confirmed. We also demonstrate that meteorological phenomena provide an important contribution to the excitation of polar motion. Our work offers a theoretical basis for future routine determinations of atmospheric angular momentum fluctuations for the purposes of meteorological and geophysical research, including the assessment of the extent to which movements in the solid Earth associated with very large earthquakes contribute to the excitation of the Chandlerian wobble.

Microscopic derivation of quantum fluctuations in nuclear reactions: Angular momentum fluctuations

Starting with a separable coupling between the intrinsic coordinates of a nuclear system and several collective variables, damping and diffusion of the collective system is studied by first-order zero-temperature time-dependent perturbation theory. Attention is focused on the relative coordinates between heavy ions and closed expressions are obtained for the mean values and the fluctuations of the coordinates, momenta and angular momentum components, and the asymptotic behaviour is analyzed. Application to the quasi-elastic region of heavy-ion reactions is indicated.

Remarks about angular momentum fluctuations and the particle-plus-rotor model as compared to the self-consistent cranking model

Remarks about angular momentum fluctuations and the particle-plus-rotor model as compared to the self-consistent cranking model

The Hartree-Fock-Bogoliubov theory of high-spin states

The HFB — cranking (HFBC) descriptions of backbending, Coriolis antipairing, pair realignment and gapless superconductivity are reviewed. The validity of the HFBC theory in the backbending region is considered by discussing (a) the significance of angular momentum fluctuations in cranked intrinsic states, (b) the HFB spectra for exactly soluble particle-plus-cotor models, and (c) the possibility that HFBC wave functions contain spurious linear combinations.

Hamamoto's Remedy for the Band Crossing Anomalies in the Cranking Model

Hamamoto's remedy for the band crossing anomalies is investigated using a schematic model without pairing interaction. It is shown that the remedy is applicable only in the close vicinity of the critical angular frequency, and that an anomalously large angular momentum fluctuation still remains in the band crossing region. It was pointed out by Hamamotou that the Hartree-Fock-Bogoliubov cranking (HFBC) model, which has been used to understand systematically the backbending phenomena, had the anomalies in the band crossing region, which are inherent in the cranking model. Although she used an idealized model of i 13; 2 particles coupled to a two-dimensional rigid rotor and inter­ acting via the pairing force neglecting self-consistency, it has also been shown 2 l

Atmospheric angular momentum fluctuations and changes in the length of the day

Fluctuations in the angular momentum of the Earth's atmosphere correspond fairly well with equal and opposite fluctuations in the angular momentum of the Earth's ‘solid’ mantle. A decrease of 20% of the relative angular momentum of the atmosphere during the second half of May 1979 was accompanied by a speeding up of the rotation of the mantle causing the length of the day (l.o.d.) to decrease by 0.6 × 10−3 s. Although motions in the Earth's liquid core produce the more pronounced ‘decade’ variations in the l.o.d. there is no evidence that core motions produce l.o.d. variations on much shorter time-scales.

Angular momentum fluctuation energy in the cranking model

Angular momentum is approximately projected from Hartree-Fock-Bogoliubov cranked (HFBC) wave functions. At each J the projected energy is E proj ≈ E HFBC − (Δ J) 2/2 J HFBC . The spin-dependent fluctuation Δ J includes contributions from J y and J z as well as J x . There are no correlations in the three angular momentum components. Projected energies are calculated for 168, 170Yb and 174Hf. When compared to experimental energies, the projected spectra are less compressed than the HFBC spectra. At low spins the projected and experimental energies are in good agreement.

Discontinuities and angular momentum fluctuations in the self-consistent cranking model

Two main shortcomings of the cranking model in band crossing regions, namely the possible occurrence of discontinuities and the strongly varying angular momentum fluctuations, are investigated. It is found that because of these drawbacks the reliability of the cranking approximation in regions of band crossing is more than doubtful, and that even the suggestions to overcome the difficulties by including the so-called physical parts of the 2qp trajectory do not improve the situation, at least as far as the wave functions are considered.

Angular momentum distributions of cranked Hartree-Fock-Bogoliubov wave functions

The angular momentum fluctuations of cranked Hartree-Fock-Bogoliubov wave functions are shown to be quite large and strongly varying in the regions of band crossing. The distribution of angular momenta is explicitly calculated using angular momentum projection. The peculiar behaviour of those distributions around the band crossing is discussed.

Band-crossing anomalies in the cranking model

The problems with the cranking model in the vicinity of a band crossing recently pointed out by Hamamoto are reexamined. The sharp peak in the angular-momentum fluctuation and related peculiarities obtained previously in a non-self-consistent cranking calculation applied to an i 13 2 particles + rotor model are found to persist even when the calculation is done self-consistently, contrary to the expectations of Sorensen. A new feature is also found: vanishing interaction between crossing bands for some nucleon numbers. A simple analytic model is introduced which can qualitatively simulate such peculiarities and suggests a possible explanation for them. This model indicates that a check on the stability of self-consistent cranking solutions may sometimes help to eliminate spurious segments of rotational trajectories. Finally, angular-momentum fluctuations resulting from realistic HFB cranking calculations are presented for the first time. A method based on the RPA is suggested for correcting the fluctuation problem for the yrast band.

Momentum Balancing of Four-Bar Linkages

The primary objective in this work is the determination of conditions for reducing the angular momentum fluctuations (i.e., vibration) transmitted to the frame of a completely force balanced four-bar linkage. This approach leads to relatively simple design equations for determining the inertial properties of the links for good momentum balancing. The essence of this procedure is that it yields analytical results as opposed to the computer-based search techniques required by most previously published methods, which are based on reducing the shaking forces and moments rather than the momentum fluctuations. Furthermore, this method allows for off-line mass distributions (i.e., the center of mass of the link is not on the line of pivots) and, as we show in the paper, this can result in better momentum balancing than the in-line case to which most previous works have been restricted. Some numerical examples are given and the results are compared to similar results obtained by minimizing the RMS shaking moment.

Light Scattering from Shear Waves: The Role of Angular Momentum Fluctuations in Light Scattering

Recent light-scattering experiments which have investigated the splittings in the depolarized Rayleigh spectrum of certain nonassociated liquids are discussed in the context of generalized hydrodynamics. It is shown that a purely hydrodynamic or viscoelastic theory does not account for important features of the observed spectrum. However, if angular momentum fluctuations are considered, many of the features of the observed spectra can be accounted for. Moreover, a theory due to Rytov is discussed and studied in the context of generalized hydrodynamics.

Hydrodynamics and Collective Angular-Momentum Fluctuations in Molecular Fluids

Generalized hydrodynamics for classical fluids composed of structured molecules is discussed from a fundamental microscopic viewpoint. The analysis is concentrated on collective fluctuations of the intrinsic molecular angular momentum, and their coupling to the conserved densities of particle number, linear momentum, and energy. It is found from symmetry that the transverse components of linear and angular momentum are dynamically coupled, while the longitudinal angular momentum density moves independently of the other variables. By using the Zwanzig-Mori projection-operator technique, we derive closed and rigorous equations of motion for the set of fluctuations considered. For the case where the intrinsic angular momentum is not far from being separately conserved, approximations can be motivated which reduce these equations to simple relaxation equations, valid for small $k$ and long times. General, Kubo-type expressions for the relaxation coefficients are given. Finally, sum-rule considerations are presented from which these coefficients can be approximately calculated.

ANGULAR MOMENTUM AS A PARAMETER IN THE INVESTIGATION OF CYCLONE-SCALE CIRCULATIONS

Abstract Angular momentum relative to a fixed vertical axis is used as a measure of rotational flow on the scale of cyclones and anticyclones. An exact equation derived by Starr for the time rate of change of this momentum is expressed in an approximate form, and a long-period observational study is made in two regions to determine the validity and prognostic implications of the resulting relationship. It is found that the fluctuations of angular momentum can be accounted for primarily by a single effect—the horizontal transport of momentum across the boundaries of the regions. Moreover, a sufficient lag relationship is found to exist so that the transport serves as a predictor of the 12-hour change of momentum.

On Dielectric Constants and Magnetic Susceptibilities in the New Quantum Mechanics Part III—Application to Dia- and Paramagnetism

1.General mathematical theory.---Modifications are given in the general derivation of the Langevin-Debye formula by means of quantum mechanics published in part I which are required by the moment now being magnetic rather than electric. The main additions are the appearance of a diamagnetic term, and allowance for the fact that the magnetic moment consists of two parts arising respectively from orbital electronic motions and from the electrons' internal spins. These parts cannot always be treated as rigidly coupled to form a permanent resultant moment. The Hamiltonian function used for the internal spin is that of a spherical top.2.Diamagnetism.---The writer's previous claim that Pauli's formula for the diamagnetic susceptibility can be applied to molecules as well as atoms is shown to be invalid because of the anomalous fact that the square of the angular momentum, unlike the average angular momentum, does not vanish even in $S$ states when there is more than one nucleus. Pauli's formula is instead an upper limit to the diamagnetism in non-monatomic molecules and is a good approximation when the Schr\odinger wave function has nearly as much symmetry as in an atomic $S$ state.3.Paramagnetism of atoms.---Limiting values for the paramagnetic susceptibilities of atoms are $\ensuremath{\chi}=N\frac{[4s(s+1)+k(k+1)]{\ensuremath{\beta}}^{2}}{3kT}$ and $\ensuremath{\chi}=[\frac{N{g}^{2}j(j+1){\ensuremath{\beta}}^{2}}{3kT}]+N\ensuremath{\alpha}$ where $\ensuremath{\beta}$ is the Bohr magneton and $\ensuremath{\alpha}$ is a constant. These two formulas are rigorously applicable when the multiplet intervals are respectively very small or very large compared to $\mathrm{kT}$, and are valid regardless of whether the magnetic field is strong enough to change the quantization by producing a Paschen-Back effect. The formula for small multiplets yields susceptibilities slightly different from those given by Laporte and Sommerfeld's expression for this case, and is much simpler, as they overlooked the contribution of the portion of the magnetic moment which is perpendicular to the invariable axis.4.Paramagnetism of molecules.---The susceptibility is calculated on the basis of the Hund theory of molecular quantization. Formulas are given applicable to his couplings of type (a) and type (b) provided in the former the multiplet intervals are either very large or very small compared to $\mathrm{kT}$. The experimental susceptibilities for ${\mathrm{O}}_{2}$ and Cl${\mathrm{O}}_{2}$ are in accord with the assumption that the normal states are respectively $^{3}S$ and $^{2}S$ terms. In the particular case of $S$ terms the numerical results are the same in the atomic and molecular formulas, but, unlike previous theories, it is not necessary to suppose the orbits are as freely oriented in molecules as in atoms. Polyatomic molecules may have lower paramagnetic susceptibilities than diatomic ones because the dissymmetry causes large fluctuations in electronic angular momentum.5.Paramagnetism of nitic oxide.---Spectroscopists have recently found that the normal states of the NO molecule are $^{2}P$ terms separated by 120.9 ${\mathrm{cm}}^{\ensuremath{-}1}$. This permits an unambigous calculation of the susceptibility of NO which agrees with the experimental value within 1.5 per cent. Deviations from Curie's law are calculated which result from the doublet interval being comparable with $\mathrm{kT}$. These deviations should be detectable experimentally if the susceptibility of NO could be measured over a wide temperature range.