Longitudinal Degrees Of Freedom Research Articles

Anisotropic fluid dynamics in the early stage of relativistic heavy-ion collisions

A formalism for anisotropic fluid dynamics is proposed. It is designed to describe boost-invariant systems with anisotropic pressure. Such systems are expected to be produced at the early stages of relativistic heavy-ion collisions, when the timescales are too short to achieve equal thermalization of transverse and longitudinal degrees of freedom. The approach is based on the energy–momentum and entropy conservation laws, and may be regarded as a minimal extension of the boost-invariant standard relativistic hydrodynamics of the perfect fluid. We show how the formalism may be used to describe the isotropization of the system (the transition from the initial state with no longitudinal pressure to the final state with equal longitudinal and transverse pressure).

Longitudinal cooling of non-neutral plasma by energy exchange

The optimal values of Q and Deltaomega (Deltaomega identical withomega-Omega) for cooling a pure electron plasma with a microwave bath have been calculated. An electron plasma, which has no internal degree of freedom, cannot be cooled below the temperature of a heat bath. However, longitudinal cooling can be achieved by energy transfer from the poorly cooled longitudinal degree of freedom to the well-cooled (by synchrotron radiation) transverse degree of freedom. To do this, a microwave bath is introduced to the electron plasma. A microwave tuned to a frequency below the gyrofrequency forces electrons moving towards the microwave to absorb a microwave photon. The electrons move up one in Landau state and then lose their longitudinal momenta. In this process, the longitudinal temperature of the electron plasma decreases. On the basis that the perpendicular temperature is below the Landau temperature of the plasma, we set up two level transition equations and then derive a Fokker-Planck equation from them. With the aid of a finite element method (FEM) code for the equation, the cooling times for several values of the magnetic field, the microwave cavity (Q), and the relative detuning frequency from the gyrofrequency (Deltaomega) are calculated. Thus optimal values of the microwave cavity and the detuning frequency for longitudinal cooling of a strongly magnetized electron plasma with a microwave bath have been found. By applying these optimal values with an appropriate microwave intensity, the best cooling can be obtained. For an electron plasma magnetized to 10 T, the cooling time to the solid state is approximately two hours.

Tkachenko oscillations and the compressibility of a rotating Bose-Einstein condensate.

The elastic oscillations of the vortex lattice of a cold Bose gas (Tkachenko modes) are shown to play a crucial role in the saturation of the compressibility sum rule, as a consequence of the hybridization with the longitudinal degrees of freedom. The presence of the vortex lattice is responsible for a q2 behavior of the static structure factor at small wave vectors q, which implies the absence of long range order in 2D configurations at zero temperature. Sum rules are used to calculate the Tkachenko frequency in the presence of harmonic trapping. Results are derived in the Thomas-Fermi regime and compared with experiments as well as with previous theoretical estimates.

Electromagnetic Weibel instability in intense charged particle beams with large energy anisotropy

In plasmas with strongly anisotropic distribution functions, collective instabilities may develop if there is sufficient coupling between the transverse and longitudinal degrees of freedom. Our previous numerical and theoretical studies of intense charged particle beams with large temperature anisotropy [E. A. Startsev, R. C. Davidson, and H. Qin, Phys. Rev. ST Accel. Beams 6, 084401 (2003); Phys. Plasmas 9, 3138 (2002)] demonstrated that a fast, electrostatic, Harris-type instability develops, and saturates nonlinearly, for sufficiently large temperature anisotropy (T⊥b/T∥b≫1). The total distribution function after saturation, however, is still far from equipartitioned. In this paper the linearized Vlasov–Maxwell equations are used to investigate detailed properties of the transverse electromagnetic Weibel-type instability for a long charge bunch propagating through a cylindrical pipe of radius rw. The kinetic stability analysis is carried out for azimuthally symmetric perturbations about a two-temperature thermal equilibrium distribution in the smooth-focusing approximation. The most unstable modes are identified, and their eigenfrequencies, radial mode structure and instability thresholds are determined. The stability analysis shows that, although there is free energy available to drive the electromagnetic Weibel instability, the finite transverse geometry of the charged particle beam introduces a large threshold value for the temperature anisotropy [(T⊥b/T∥b)Weibel≫(T⊥b/T∥b)Harris] below which the instability is absent. Hence, unlike the case of an electrically neutral plasma, the Weibel instability is not expected to play as significant a role in the process of energy isotropization of intense unneutralized charged particle beams as the electrostatic Harris-type instability.

Analytical theory and nonlinearδfperturbative simulations of temperature anisotropy instability in intense charged particle beams

In plasmas with strongly anisotropic distribution functions $({T}_{\ensuremath{\parallel}b}/{T}_{\ensuremath{\perp}b}\ensuremath{\ll}1)$ a Harris-like collective instability may develop if there is sufficient coupling between the transverse and longitudinal degrees of freedom. Such anisotropies develop naturally in accelerators and may lead to a deterioration of beam quality. This paper extends previous numerical studies [E. A. Startsev, R. C. Davidson, and H. Qin, Phys. Plasmas 9, 3138 (2002)] of the stability properties of intense non-neutral charged particle beams with large temperature anisotropy $({T}_{\ensuremath{\perp}b}\ensuremath{\gg}{T}_{\ensuremath{\parallel}b})$ to allow for nonaxisymmetric perturbations with $\ensuremath{\partial}/\ensuremath{\partial}\ensuremath{\theta}\ensuremath{\ne}0$. The most unstable modes are identified, and their eigenfrequencies, radial mode structure, and nonlinear dynamics are determined. The simulation results clearly show that moderately intense beams with ${s}_{b}={\stackrel{^}{\ensuremath{\omega}}}_{pb}^{2}/2{\ensuremath{\gamma}}_{b}^{2}{\ensuremath{\omega}}_{\ensuremath{\beta}\ensuremath{\perp}}^{2}\ensuremath{\gtrsim}0.5$ are linearly unstable to short-wavelength perturbations with ${k}_{z}^{2}{r}_{b}^{2}\ensuremath{\gtrsim}1$, provided the ratio of longitudinal and transverse temperatures is smaller than some threshold value. Here, ${\stackrel{^}{\ensuremath{\omega}}}_{pb}^{2}=4\ensuremath{\pi}{\stackrel{^}{n}}_{b}{e}_{b}^{2}/{\ensuremath{\gamma}}_{b}{m}_{b}$ is the relativistic plasma frequency squared, and ${\ensuremath{\omega}}_{\ensuremath{\beta}\ensuremath{\perp}}$ is the betatron frequency associated with the applied smooth-focusing field. A theoretical model is developed based on the Vlasov-Maxwell equations which describes the essential features of the linear stages of instability. Both the simulations and the analytical theory predict that the dipole mode (azimuthal mode number $m=1$) is the most unstable mode. In the nonlinear stage, tails develop in the longitudinal momentum distribution function, and the kinetic instability saturates due to resonant wave-particle interactions.

Dispersive electron cooling experiments at the heavy ion storage ring TSR

At low relative velocities, the transverse cooling rates of fast stored ions by cold electrons are usually smaller than the longitudinal ones. By dispersive electron cooling, however, it is possible to transfer heat from the horizontal into the longitudinal degree of freedom, thereby increasing the horizontal cooling rate on the expense of the longitudinal rate. The cooling scheme requires a horizontal gradient of the longitudinal cooling force which can be achieved by displacing the electron beam relative to the ion beam, and a finite dispersion in the ion–electron interaction region. Operating the heavy ion test storage ring TSR at a moderate dispersion of D S =1.81 m , we investigated the dispersive electron cooling process and found good agreement with the expectations from fundamental considerations.

Topological twistons in crystalline polyethylene

We introduce an alternate model to describe twistons in crystalline polyethylene. The model couples torsional and longitudinal degrees of freedom and appears as an extension of a model that describes only the torsional motion. We find exact solutions that describe stable topological twistons, in good agreement with the torsional and longitudinal interactions in polyethylene.

Identification of gyroplane lateral/directional stability and control characteristics from flight test

This paper presents an analysis of test data recorded during flight trials of a gyroplane. This class of rotary-wing aircraft has found limited application in areas other than sport or recreational flying. However, the accident rate is such that a study of the configuration's stability and control characteristics is timely, and in addition substantive data are required for a new airworthiness and design standard that is under development. The paper complements previous work on the longitudinal degrees of freedom and, as a consequence, serves to consolidate the understanding of gyroplane stability and control. The identified derivatives are related to specific aspects of the layout of the gyroplane, and hence the influence of design on the static and dynamic behaviour is quantified. It is concluded that robust estimates of the lateral and directional stability and control derivatives have been identified. This analysis has focused on ‘high-speed’ flight, and the identified derivatives highlight benign and ‘conventional’ characteristics in this part of the flight envelope.

Non-critical light-cone string

A free string quantum model is constructed directly in the light-cone variables in the range of dimensions 1 < D < 25. The longitudinal degrees of freedom are described by an abstract Verma module with the central charge restricted by the requirement of closure of the non-linear realization of the Poincaré algebra. The spin content of the model is analyzed. In particular for D = 4 the explicit formulae for the character generating functions of the open and closed strings are given and the spin spectrum of the first twelve excited levels is calculated. It is shown that for the space-time dimension in the range 1 < D < 25 this light-cone quantum model is equivalent to the critical massive string and to the non-critical Nambu-Goto string.

Energy exchange between a test charge and an unstable electron beam

An investigation is made of the interaction between a test charge and an electron beam which is convectively unstable against plasma oscillations. An analysis is made of the negative-mass instability of the longitudinal motion. Expressions are derived for the reaction force exerted by the electron plasma polarized by the moving charge and for the diffusion tensor arising from electric field fluctuations. The approximation of uniform rectilinear motion of a test particle is used to estimate the systematic and diffusion components of the rates of change in its energy. It is shown that in this case, interaction with an unstable electron beam leads to acceleration of the particle in respect to the longitudinal degree of freedom (antifriction) and under certain conditions enhances the friction effect (cooling) in respect to the transverse degrees of freedom. An assessment is made of the possibility of obtaining an overall positive friction effect by using a special “sweeping” procedure whereby the longitudinal velocity of the electron beam is changed in a specific manner consistent with the cooling process.

Second quantization in a waveguide with variable cross section

The motion of a system consisting of noninteracting bosons in a waveguide with variable cross section is studied. These particles have transverse as well as longitudinal degrees of freedom, but only a finite number of transverse modes can propagate in the waveguide. While for a waveguide with constant cross section, the numbers of particles in a given state of longitudinal and transverse modes remain constant, in the case of a waveguide with variable cross section there is conversion between these modes, although the total number of particles is conserved. By considering the equations of motion for the annihilation (or creation) operator, it is shown that the boundaries act as an external force, and thus generate localized transverse modes in the waveguide.

A String Model for Hadron Interactions at High Energies

An iterative string cascade model is extended to the SPS energies, where high order terms in the fragmentation series are considered. The derived fragmentation series converges rapidly. Terms of second and third order have significant values only at SPS energies, while the zero and the first order terms are sufficient for the hadron interactions at a few GeV. The charged particle multiplicity and the relative cross section of the production of the resonance and the prompt hadrons are reproduced with fair agreement with experimental data. 2 > It was found that the perturbative series for the total cross section converges reasonably fast. On the other hand, the problem of the multi-particle production in pp collision is dealt by different models based on soft hadronic collisions, 3 > the 1/N expansion of the amplitude for processes in QCD and string type models, describing quark transi­ tions into hadrons. At high energies there exist diagrams of complicated topology which correspond to processes with the exchange of several Pomerons. 4 > The contri­ bution of such diagrams to the scattering amplitude increases rapidly with energy more than the one Pomeron exchange contribution. The properties of iterative cascade jets with emphasis on the longitudinal degrees of freedom are studied by Field and Feynman through a dynamical picture in space-time and energy-momentum space, based upon string dynamics. In this model a scheme is considered which is symmetric with respect to particle generation from the quark and antiquark ends in a two-jet system. The particle production is considered as a tunneling process in a constant force field. Because of the three-gluon coupling, the color flux lines will not spread out over all space, as the electromagnetic field lines do, but rather be con­ strained to a thin tube like region. Within this tube, new qij pairs can be created from the available field energy. Thus the original system breaks into smaller and smaller pieces, until only ordinary hadrons remain. In the field behind the original, outgoing quark Qo a new quark pair Q1 ii1 is produced, so that the Qo may join with ih to form a hadron Qo fh, leaving the Q1 unpaired. The production of another pair qd(2 will give a hadron Q1 fh, etc. To denote this flavor ordering of the hadrons, the concept of rank is introduced. 5 >' 6 > Q;-1 q; is the ith rank hadron. From these assump­ tions one may find the resulting particle spectra in a jet. A fragmentation function

A Study of the Efficiency of a Double-Action Worm Gear Set

A double-action worm gear set—so called because the worm has both rotational and longitudinal degrees of freedom—has been invented primarily for use in a motorized spacecraft hinge. This paper presents an analysis of the mechanical efficiency of the double-action worm gear set, together with preliminary test results. Results of analysis and test show that the double-action feature of the worm makes it possible to control the forward and backward driving efficiency of the gear set. Compared to a standard worm gear set, the double-action worm gear set has the advantages of the ability to absorb small applied displacements to the wormwheel shaft in an unlocked position, the ability to be non-backdrivable with a high degree of certainty when locked against an end stop and the ability to maintain a specific preload in a locked position.

A Lyapunov technique approach to electron cooling devices

An axial magnetic field of sufficient strength strength is imposed in the electron cooling devices on charged-particle beams, such that the mean cyclotron radius of electrons is much smaller than the average distance among electrons themselves. In this paper it is shown that this project condition implies an ordered evolution of system, without allowing thermal equilibrium between transversal and longitudinal degrees of freedom. This result has been obtained using a Lyapunov exponents technique in a simplified beam model.

On using the Freedman–Townsend model to generate massive vectors

The Freedman–Townsend model involves a gauge invariant coupling of an antisymmetric tensor field [Formula: see text] to a non-Abelian vector gauge field [Formula: see text] and an auxiliary gauge field [Formula: see text]. We analyze the dynamical degrees of freedom in this model and investigate unitarity by considering several four-point functions at the tree level. The model is found to describe a massive vector meson with the transverse degrees of freedom residing in [Formula: see text] and the longitudinal degree of freedom in [Formula: see text]. If an extra gauge-invariant term involving the square of the divergence of [Formula: see text] is added to the Lagrangian, then a scalar polarization appears in this field and [Formula: see text] is no longer simply auxiliary. This scalar mode serves to cancel the bad ultraviolet behaviour of the longitudinal mode, but only at the expense of having no lower bound to the energy spectrum of the theory. Furthermore, an examination of four-point tree level graphs indicates that if we consider elastic scattering of longitudinal modes, the amplitudes grows with the energy scale of the process, whether or not the scalar mode is present.

Gas-dynamic acceleration of ions in an inhomogeneous magnetic field

This paper evaluates the method of gas-dynamic acceleration of ions in an inhomogeneous magnetic field in two variants - a strongly scattering and nonscattering foil F. In the first variant the presence of the inhomogeneous magnetic field leads to an increase in ion current related to increase in the flux area in the acceleration region, while in the second the increase in current is insignificant, but nevertheless the efficiency of ion acceleration is increased, because all the energy of the oscillating electrons in the accelerated region will be included in a longitudinal degree of freedom. This fact leads to an increase in the rate of expansion of the plasma synthesized from ions and oscillating electrons. Because of this increase in the mean rate of plasma expansion the efficiency of acceleration increases also.

Defect concepts for vector spin glasses

The “order parameter” for describing low-temperature configurations of an m-vector spin glass is an O( m) orthogonal matrix R( x ), representing a rotation of one configuration relative to another, which is relatively slowly varying in space. The meaning of a “defect” description of the low-lying metastable states is clarified and the topologically stable defects and textures of R( x ) are reviewed. R( x ) is generalized to a more general ( m × m matrix) order parameter with longitudinal degrees of freedom, which in a hydrodynamic picture are associated with a longitudinal spin-stiffness constant and a spin-configuration “flattening” energy. It is shown that, in the most useful form of the Edwards-Anderson correlation function, the longitudinal fluctuations must be projected out. The relation of “defect” pictures to the phenomenological “two-level system” models and to experiments is discussed. Independent measures of the volume per defect can be derived from (i) the Edwards-Anderson correlation function, and (ii) the number of local minima in large systems. It is noted that in the limit m → ∞, (defect separation) ⪢ (core size) ⪢ (nearest-neighbor distance).

Aerodynamic Coupling between Lateral and Longitudinal Degrees of Freedom

An oscillatory technique for routine measurement of the direct, cross, and cross-coupling moment derivatives due to pitching and yawing has been developed and a series of comprehensive results at Mach numbers 0.7 and 0.25 and at angles of attack up to 40 deg has been obtained in a large wind tunnel. It was found that some of the dynamic cross-coupling derivatives, which at low angles of attack were all insignificant, at higher angles of attack could reach values that rendered the pertinent cross-coupling terms in the equations of motion comparable in magnitude to the well-established damping terms. This applied in particular to results obtained at Mach number 0.7. Large nonlinear variations with angle of attack were observed in many of the measured derivatives including the important damping-in-pitch and damping-in-yaw derivatives. It was concluded that it may be desirable to include certain dynamic cross-coupling derivatives in the equations of motion, and to consider all of these equations simultaneously, rather than in separate longitudinal and lateral groups.

Interacting-string picture of dual-resonance models

Dual-resonance models are analyzed by means of operators which act within the physical Hilbert space of positive-metric states. The basis of the method is to extend the relativistic-string picture of Goldstone, Goddard, Rebbi and Thorn to interacting particles. Functional methods are used, but their relation to the operator formalism is evident, and factorization is maintained. An expression is given for the N-point amplitude in terms of physical-particle operators. For the three-point function the Neumann functions which occur in this expression are evaluated, so that we have a formula for the on- and off-energy-shell vertex. We assume that the string has no longitudinal degrees of freedom, and our results are Lorentz invariant and dual only if d=26.

Conservative Trapping of Charged Particles in a Static Magnetic Field

A computer analysis is made of the trajectories of charged particles injected tangentially into the mid-plane of a physically reasonable static mirror field. Conditions necessary for the nonadiabatic transfer of particle energy from the plane perpendicular to the field to the direction parallel to the field are discussed. It is shown that this energy transfer can be qualitatively predicted from the Mathieu equation. Two exchange mechanisms are observed; when the requirements for parametric resonance are satisfied energy is transferred into the longitudinal degree of freedom, and when the particle exhibits the characteristic looping behavior energy may be transferred between the transverse and longitudinal degrees of freedom. The coupled nonlinear equations of motion for single particles are numerically integrated using the fourth-order Runge-Kutta method. Based on the results of the various computer experiments, an effective trapping time is obtained.