Phenomenon Of Damping Research Articles

A global jacobi method for a symmetric indefinite problem Sx = λTx

A reliable Jacobi-like method for the problem Sx = λTx has been presented where S and T are real, symmetric but indefinite, and T has the same number of positive and negative eigenvalues as it occurs in standard problems of damped oscillations of structures. The method is quadratically convergent in all non-defective cases including also the cases of double defective real eigenvalues which correspond to the physically important phenomenon of critical damping. The symmetry is preserved in all stages which carries a 50% saving of both computing time and space. The experimental comparison with the QR method and the standard complex method of Eberlein has been made. The present method has shown to be particularly suitable for diagonally dominant matrices as they appear e.g. in subspace iterations.

Damping of nuclear excitations

The authors review the theory and the empirical evidence of damping of simple nuclear excitations. The excitations considered are the particle states and vibrational states. The particle damping phenomena include the fragmentation of single-particle levels, the systematics of neutron strength functions, and the optical absorption of elastic scattering. Information on the known collective vibrational states is summarized and compared with theory. A theoretical model that has found considerable success is based on a damping mechanism in which the simple excitations mix with the surface vibrations. This implies that the surface damping dominates for excitation energies below about 15 MeV. There is a close relation between the single-particle damping and the damping of collective vibrations. However, the vibrational damping is strongly suppressed by the coherence between the particle and the hole. While the model reproduces many of the observed features of the data rather well, it tends to underpredict the spreading width by as much as a factor of 2. Thus other degrees of freedom, not well understood at present, may play a role in the damping.

Simulation study of Bernstein modes

The properties of Bernstein modes were investigated through computer simulations using two-dimensional and two-and-one-half-dimensional (i.e., two spatial and three velocity coordinates) electrostatic models with fixed magnetic field. The measured discrete spectrum was found to agree with the linear dispersion relation for these modes. The quasi-periodic phenomenon of early phase-mixing damping and later recurrence, predicted by Baldwin and Rowlands, was observed. For large wavenumber k⊥, the initial damping rate is the same as that for Landau damping in an unmagnetized plasma; for small k⊥, however, it is much stronger. The recurrence peaks slowly damp in time at a rate proportional to k2⊥D, where D is the measured cross-field particle diffusion coefficient which is dominated by convective transport. Finally, splitting of the main spectral peaks and the appearance of subpeaks at half-integral multiples of the cyclotron frequency are observed and may be explained by nonlinear mode coupling.

Theory of parametric heating of plasma in the lower hybrid frequency range

Lower hybrid turbulence can be driven parametrically by a whistler pump wave. Destabilization occurring through the resonant interaction of the whistler, an electrostatic mode, and a particle, either electron or ion, which mechanism is akin to the well-known phenomenon of inverse nonlinear Landau damping, is considered. A nonlinear theory of turbulence is developed for a range of parameters characteristic of tokamaks on the basis of the wave kinetic equation for weakly turbulent plasma in a uniform magnetic field. This yields an estimate of the saturation level of the turbulence which is, in turn, responsible for the anomalous absorption of the pump wave electromagnetic energy. Both the anomalous absorption rate and the plasma heating rate are evaluated. The theoretical results are very promising: lower hybrid parametric heating should be realizable up to densities typically of order 3×1014 cm−3 (for larger densities, the poor tunneling of the pump wave through the evanescent region between the launching structure and the turning point leads to too high electric fields in the waveguide). A decisive advantage of this form of ratio frequency heating is that energy can also be deposited directly in the ion component.

On infrared problems in muon decay

Abstract Radiation corrections to μ-decay are calculated in the context of the coherent states formulation of the asymptotic condition by using dimensional regularization techniques. The theoretical explanation of radiation damping phenomena at the end-point of the decay spectrum is also clarified.

Nonlinear Damping on an Adiabatic Plasma Species

The phenomenon of nonlinear Landau damping is investigated for damping on an adiabatic species. It is shown that the cancellation which leads to the (κλD)4 dependence for damping on a hydrodynamic species no longer occurs; rather, the four-wave resonance term (driven spectra term) of the governing matrix element dominates the resonant three-wave term (induced spectra term), leading to the conjecture that, for a plasma with an adiabatic constituent, nonlinear Landau damping on that constituent may well dominate the nonlinear mode coupling processes. A detailed investigation of such damping is conducted for the case of two equidense zero temperature ion beams streaming through a hot, adiabatic electron plasma. A nonlinear Landau damping instability on the electrons is found which cannot be limited in the usual quasilinear sense. The instability serves to heat the electrons driving the system to a linear ion-ion streaming instability.

Landau damping and the water-bag model

It is shown how the phenomenon of Landau damping arises in solutions of the water-bag model for the Vlasov equation.

A Simplified Method for Calculation of Strain Dependence of Internal Friction in Transverse Vibrations

Damping phenomena in higher modes of free-free transverse vibration are analysed and tire effects of the distribution of the vibrational strain on the apparent internal friction are treated quantitatively. The functional relation of the observed internal friction with the maximum strain amplitude or the maximum displacement amplitude is different when different mode of vibration is used. A simplified method for finding the true strain amplitude dependence of the internal friction is presented where the average strain amplitude is used as an experimental variable. It is suggested that the amplitude-dependent part of the internal friction can be dependent on the frequency of the vibration.

Hydrogen maser frequency shifts due to coherently excited Δm F =±1 transitions betweenF=1 levels of the atomic hydrogen ground state

Hydrogen maser frequency shifts, caused by the multiple quantum transition nonlinearities of a resonant multiple frequency excitation of the atomic hydrogen four level ground state system have been investigated. The oscillation characteristics of hydrogen maser operation with simultaneously excited, low frequencyΔm F =±1 transitions between theF=1 states of the atomic hydrogen ground state have been analysed theoretically and explicit formulas for hydrogen maser frequency shifts and amplitude response have been derived for arbitrary maser oscillation amplitude and a small signal approximation for theΔm F =±1 “Zeeman” transitions. The comparison with experimentally observed hydrogen maser frequency shifts was specialized to small magnetic fields, for which the difference between the resonance frequencies of the two low frequency,Δm F =±1 Zeeman transitions is small compared to the linewidth. Special emphasis was placed on the evaluation of frequency pulling effects for a Zeeman transition excitation at off-resonance conditions. For this case the theoretical formulation of frequency pulling effects becomes insensitive against simplifying assumptions about the radiation damping phenomena and a particular good agreement between experiment and theory can therefore be expected. Experimental conditions have been specified, for which the uncertainty of hydrogen maser frequency due to Zeeman transition induced frequency shifts does not restrict the present frequency stability of a hydrogen maser frequency standard.

Stochastic Acceleration

Charged particles moving under the influence of randomly time-varying electromagnetic fields may be expected to experience a net acceleration. This process is analyzed for the special case of nonrelativistic motion in a static uniform magnetic field and time-varying electric field. Acceleration parallel and transverse to the magnetic field are considered separately. In the weak-field approximation, the motion may be described by a Fokker-Planck equation. The coefficients of this equation are expressible in terms of the correlation functions for the electric fields. In certain cases, the coefficients may be expressed in terms of the energy spectrum of the field. The Fokker-Planck equation derived for motion along the magnetic field is closely related to an equation of the quasilinear theory of plasma instability. One may also show that the equation is closely related to the phenomenon of Landau damping. Longitudinal acceleration is effected by waves with phase velocities slightly greater than the particle velocity. A similar statement is true for transverse acceleration, except that the "resonant" waves are in addition shifted by the particle gyrofrequency. In the absence of any other effects (such as "loading" of the accelerating field by the accelerated particles), the transverse energy distribution tends to a Maxwellian form with a temperature which increases linearly with time. The same is true for longitudinal acceleration if the spectrum of the electric field is flat over the range of phase velocities of interest. The equations related to transverse acceleration show that the high-energy electrons observed in the transition region outside the earth's magnetosphere may have been accelerated by quite weak random or quasirandom electric fields.

Dynamic Pulse Hysteresis of Magnetic Devices

Dynamic pulse hysteresis is defined as the instantaneous hysteresis of a magnetic device during the risetime of an applied pulse field. A method of measuring the dynamic pulse hysteresis of magnetic devices has been developed. The pulse hysteresis differs from the quasistatic hysteresis in its ability to display magnetic damping phenomena. By applying sampling technique the flux sensitivity of this method is extended to 10−4 Mx. The theoretical pulse hysteresis obtained by computer solution of Gilbert's equation is compared with the measured pulse hysteresis of a 10 000-Å uniaxial Permalloy film at 7-nsec risetime. From the comparison, the intrinsic damping constant of the film is determined to be 8×108 sec−1. The measured pulse hysteresis of a ferrite core in a time-limited partial switching mode is also presented.

Anomalous plasma diffusion in ion cyclotron resonance

We have carried out experimental investigations of the increased diffusion of a low-temperature plasma across the lines of force of a constant homogeneous magnetic field under conditions of high-frequency heating for a frequency approaching ion cyclotron frequency. The plasma was created in a glass tube by a pulse discharge with oscillating electrons in hydrogen at a pressure range of 5 × 10−4–5 × 10−2 torr. Initial plasma density exceeded 1.7 × 1013 cm−3. A high-frequency electric field with a frequency of 7.45 × 106 Hz and an axial period of 23 cm was excited in the plasma by an alternating current flowing along a spiral surrounding the discharge tube. The spiral and the capacitors connected to it formed an artificial line that was fed from the pulse generator with self-excitation. In the region of magnetic fields where there occurs a strong absorption of highfrequency energy by the plasma caused by ion cyclotron resonance time of plasma decay decreases sharply if the amplitude of the high-frequency current in the spiral exceeds some critical value depending on neutral gas pressure. Here ion diffusion across the magnetic field lines increases and chaotic oscillations arise in the plasma. It has been ascertained in experiments that for neutral gas pressures exceeding 10−2 torr the ratio between the critical current value and the pressure remains constant. On these grounds the assumption was made that anomalous diffusion arises when the ion drift velocity under the effect of an alternating azimuthal electric field exceeds a critical value. This is possibly associated with the appearance of “flute” instability. The possibility for flute instability to arise is apparently confirmed also by this fact: as is shown by evaluation, the velocity gained by the ion between two collisions in an azimuthal electric field for a current approaching a critical current equals in order of magnitude thermal ion velocity. We note that the investigated phenomenon of anomalous plasma diffusion in ion cyclotron resonance can be the reason for increased velocity of plasma decay in all devices in which the phenomenon of strong damping of an ion cyclotron wave in the region of ion cyclotron resonance is used for plasma heating and where considerable high-frequency powers are introduced into the plasma.

Plasma Oscillations: II. Kinetic theory of waves in plasmas

This work is the second part (and chapter VI) of a three-part series reviewing the subject of plasma oscillations. Part I (Nuclear Fusion 1 (1960) 3) was devoted to the macroscopic theory of plasmas in general, and plasma waves in particular, in both collision-free and collision-dominated plasmas, with and without magnetic fields, in some cases with boundaries and spatial gradients. The present article is concerned mostly with the kinetic theory of collision-free plasmas in the absence of magnetic fields, boundaries and spatial gradients and with neglect of the coupling with the radiation field. It is this area that has been most extensively expounded and that can be adapted to a heuristic theory of the non-equilibrium statistical mechanics of plasmas.The plan of the work is as follows: First the small longitudinal oscillations of an infinite homogeneous collisionless plasma are considered. Clearly the model requires that the frequency of collisions, Coulomb or other, be much smaller than the characteristic frequencies and growth rates of the system. The general theory is applied to an investigation of the stability of such systems and the polarization of its environment by a test charge. The latter provides the basic data for a heuristic theory of the non-equilibrium statistical mechanics of plasmas, which yields, for instance, a derivation of the Fokker-Planck equation with polarization corrections and a treatment of fluctuation phenomena in plasmas. This last is applied to the theory of the experimentally interesting phenomenon of the scattering of light by plasmas.The phenomenon of Landau damping has been treated both by the Laplace-transform technique and the normal-mode technique, and the equivalence of both methods has been shown. A special class of finite-amplitude electrostatic waves has been treated, and its relation to the normal modes of the linearized equations has been given. Also plasma oscillations of a large number of electron beams are considered, and the limit of an infinite number of beams is shown to yield the results of a plasma with a continuous velocity distribution.Part III of this series will deal with the effects of magnetic fields, relativity and coupling with the radiation field on the kinetic and non-equilibrium statistical-mechanical theories.

WAVES IN WARM QUIESCENT PLASMAS

In this work the comparison is made between the analyses of waves in a plasma which use perturbation theory, the Cartesian tensor or spherical harmonic expansion, and the integrated velocity moment equations. When the wave phase velocities are much greater than the average particle thermal velocities and when collision frequency is independent of velocity, then to first order in thermal transport terms, the three approaches give the same answer for plane waves, except for the phenomenon of Landau damping. Neglecting such damping and remembering the restriction on phase velocity, the use of integrated moment equations can then be justified in attacking non-planar wave problems.First-order relativistic effects are also considered.

Gas Law and Conductivity of a Collision-Free Plasma

The moment method is used to derive a form of two-component magneto-gas dynamics for a collision-free plasma. Closure of the moment equations is achieved by ignoring variations of fourth moments of the peculiar velocities for each component. This provides a ``fully adiabatic gas law'' which represents a generalization of the single or double adiabatic laws in that it predicts the gyrations of the pressure tensor, as well as the principal pressures. The currents which small perturbing electric fields cause to flow in each species in accordance with its own adiabatic gas dynamics are calculated. A complex conductivity tensor thus is deduced. This tensor is compared with that resulting from rigorous kinetic theory (without closure), for the case of a uniform plasma. It is found to be identical with the ``warm plasma approximation'' which takes into account temperature to first order. Hence a two-component fully adiabatic theory describes supersonic phenomena adequately but misses (altogether) the phenomenon of Landau damping. It could serve to provide pessimistic stability tests for nonuniform (confined) plasmas.