We analyze the obtained solutions of the non-linear Shroedinger equation for spherically and axially symmetrical electrons density oscillations in plasma. The conditions of the oscillations process existence are examined. It is shown that in the center or on the axis of symmetry of the systems the static density of electrons enhances. This process results in the increasing of density and pressure of the ion gas. We suggest that this mechanism could occur in nature as rare phenomenon called the fireball and could be used in carrying out the research concerning controlled fusion. The description of the experiments, carried out for the purpose to generate long-lived spherical plasma structures, is presented.
I reply here to the comment of Dr Shmatov on my recent work and demonstrate the invalidity of his criticism of the classical physics description of the formation of bound states of electrons participating in spherically symmetric oscillations of plasma.
We study spherically symmetric oscillations of electrons in plasma in the frame of classical electrodynamics. Firstly, we analyze the electromagnetic potentials for the system of radially oscillating charged particles. Secondly, we consider both free and forced spherically symmetric oscillations of electrons. Finally, we discuss the interaction between radially oscillating electrons through the exchange of ion acoustic waves. It is obtained that the effective potential of this interaction can be attractive and can transcend the Debye-Huckel potential. We suggest that oscillating electrons can …
In this paper, we obtain the dispersion relation for the lower-hybrid oscillations/waves in a cold magnetized electron-positron-ion (e-p-i) plasma. It is found that the frequency of the lower-hybrid oscillations in an e-p-i plasma is approximate $1/\sqrt{3}$ times the frequency of lower-hybrid oscillations in an electron-ion (e-i) plasma, provided the plasma density is high, and the equilibrium densities of the positron and the ion are balanced. The present work may have some relevance for laboratory/astrophysical e-p-i plasmas.
We study low energy quantum oscillations of electron gas in plasma. It is shown that two electrons participating in these oscillations acquire additional negative energy when they interact by means of a virtual plasmon. The additional energy leads to the formation a Cooper pair and possible existence of the superconducting phase in the system. We suggest that this mechanism supports slowly damping oscillations of electrons without any energy supply. Basing on our model we put forward the hypothesis the superconductivity can occur in a low energy ball lightning.
Nonlinear oscillations within a plasma slab are analyzed. Two types of solutions are found, depending on the initial value of the electron density. The first represents regular oscillations within the plasma slab, while the second gives rise to explosive growth at the slab centre or at the edges. The results are discussed.
A new model for the joint neutrino flavor and plasma oscillations is introduced, in terms of the dynamics of the neutrino flavor polarization vector in a plasma background. Fundamental solutions are found for both time-invariant and time-dependent media, considering slow and fast variations of the electron plasma density. The model is shown to be described by a generalized Hamiltonian formalism. In the case of a broad spectrum of electron plasma waves, a statistical approach indicate the shift of both equilibrium value and frequency oscillation of flavor coherence, due to the existence of a tu…
In this work, we suggest an easy-to-code higher-order finite volume semi-discrete scheme to analyze the nonlinear behavior of the electron-plasma oscillations by solving electron fluid equations numerically. The present method employs a fourth-order accurate centrally weighted essentially nonoscillatory reconstruction (CWENO4) polynomial for estimating the numerical flux at the grid-cell interfaces, and a fourth-order Runge-Kutta method for the time integration. The numerical implementation is validated by reproducing earlier results for both non-dissipative and dissipative cold plasmas. The s…
We construct the model of a long lived plasma structure based on spherically symmetric oscillations of electrons in plasma. Oscillations of electrons are studied in frames of both classical and quantum approaches. We obtain the density profile of electrons and the dispersion relations for these oscillations. The differences between classical and quantum approaches are discussed. Then we study the interaction between electrons participating in spherically symmetric oscillations. We find that this interaction can be attractive and electrons can form bound states. The applications of the obtained…
We develop a method for investigating the relationship between the shape of a 1-particle distribution and non-linear electrostatic oscillations in a collisionless plasma, incorporating transverse thermal motion. A general expression is found for the maximum sustainable electric field, and is evaluated for a particular highly anisotropic distribution.
We consider the collective neutrino plasma interactions, and study the electron plasma instabilities produced by a nearly mono-energetic neutrino beam in a plasma. We describe the mutual influence of neutrino flavor oscillations and electron plasma waves. We show that the neutrino flavor oscillations are not only perturbed by electron plasmas waves, but also contribute to the dispersion relation and the growth rates of neutrino beam instabilities.
A classical model of radiation reaction for the betatron oscillation of an electron in a plasma wakefield accelerator is presented. The maximum energy of the electron due to the longitudinal radiation reaction is found, and the betatron oscillation damping due to both the longitudinal and transverse radiation reaction effects is analyzed. Both theoretical and numerical solutions are shown with good agreements. The regime that the quantum radiation takes effect is also discussed. This model is important for designing future plasma based super accelerators or colliders.
A first-principle study of the collective oscillation spectrum of a strongly correlated one-component plasma in a strong magnetic field is presented. The spectrum consists of six fundamental modes which are found to be in good agreement with results from the Quasi-Localized Charge approximation. At high frequencies additional modes are observed that include Bernstein-type oscillations and their transverse counterparts which are of importance for the high-frequency optical and transport properties of these plasmas.
Nonlinear plasma oscillations in an arbitrary mass ratio cold plasma have been studied using 1-D particle-in-cell simulation. In contrast to earlier work for infinitely massive ion plasmas it has been found that the oscillations phase mix away at any amplitude and that the rate at which phase mixing occurs, depends on the mass ratio ($Δ= m_{-}/m_{+}$) and the amplitude. A perturbation theoretic calculation carried upto third order predicts that the normalized phase mixing time $ω_{p-} t_{mix}$ depends on the amplitude $A$ and the mass ratio $Δ$ as $\sim [(A^{2}/24)(Δ/\sqrt{1 + Δ})]^{-1/3}$. We…
We use cold plasma theory to calculate the response of an ultracold neutral plasma to an applied rf field. The free oscillation of the system has a continuous spectrum and an associated damped quasimode. We show that this quasimode dominates the driven response. We use this model to simulate plasma oscillations in an expanding ultracold neutral plasma, providing insights into the assumptions used to interpret experimental data [Phys. Rev. Lett. 85, 318 (2000)].
Previously developed method for finding asymptotic solutions of Vlasov equations using two-dimensional (in coordinate x and time t) Laplace transform is applied to low-collision electron-ion plasmas. Taking into account Coulomb collisions in the limit m_e << m_i, \bar{v_i} << \bar{v_e}, and T_e m_e << T_i m_i, results in the expression for longitudinal high-frequency plasma-electron oscillation/wave decrement essentially depending on oscillation frequency ω. This expression is quite different from the used one for low-frequency plasma sound, which can be derived using expansion in asymptotical…
An alternative explanation to the mechanism behind self-sustained oscillations of ions in direct current (DC) glow discharges is provided. Such description is distinguished from the one provided by the fluid models, where oscillations are attributed to the positive feedback mechanism associated with photoionization of particles and photoemission of electrons from the cathode. Here, oscillations arise as consequence of interaction between an ion and the surface charges induced by it at the bounding electrodes. Such mechanism provides an elegant explanation to why self-sustained oscillations occ…
We report the observation of plasma oscillations in an ultracold neutral plasma. With this collective mode we probe the electron density distribution and study the expansion of the plasma as a function of time. For classical plasma conditions, i.e. weak Coulomb coupling, the expansion is dominated by the pressure of the electron gas and is described by a hydrodynamic model. Discrepancies between the model and observations at low temperature and high density may be due to strong coupling of the electrons.
The generation of electron surface oscillations in overdense plasmas irradiated at normal incidence by an intense laser pulse is investigated. Two-dimensional (2D) particle-in-cell simulations show a transition from a planar, electrostatic oscillation at $2ω$, with $ω$ the laser frequency, to a 2D electromagnetic oscillation at frequency $ω$ and wavevector $k>ω/c$. A new electron parametric instability, involving the decay of a 1D electrostatic oscillation into two surface waves, is introduced to explain the basic features of the 2D oscillations. This effect leads to the rippling of the plasma…
Methods of plasma heating by electromagnetic waves, interaction of electro-magnetic waves with magnetoactive plasma, plasma turbulence and transport processes are investigated in the trap of the linear plasma device OMT2. An ultrahigh frequency (UHF) contactless method is used to fill the open magnetic trap by plasma. We propose a new method of open magnetic trap filling with plasma whereby plasma is injected in the trap along the magnetic field from an independent stationary UHF source. The source is located outside the trap and plasma formation in it takes place in the strongly nonuniform ma…
We present a derivation of the dispersion relation for electrostatic oscillations (ESOs) in a zero temperature quantum plasma. In the latter, degenerate electrons are governed by the Wigner equation, while non-degenerate ions follow the classical fluid equations. The Poisson equation determines the electrostatic wave potential. We consider parameters ranging from semiconductor plasmas to metallic plasmas and electron densities of compressed matter such as in laser-compression schemes and dense astrophysical objects. Due to the wave diffraction caused by overlapping electron wave function due t…
The self-excitation of plasma series resonance (PSR) oscillations is a prominent feature in the current of low pressure capacitive radio frequency (RF) discharges. This resonance leads to high frequency oscillations of the charge in the sheaths and enhances electron heating. Up to now, the phenomenon has only been observed in asymmetric discharges. There, the nonlinearity in the voltage balance, which is necessary for the self-excitation of resonance oscillations with frequencies above the applied frequencies, is caused predominantly by the quadratic contribution to the charge-voltage relation…
The boundary problem about behavior (oscillations) of the electronic plasmas with arbitrary degree of degeneration of electronic gas in half-space with specular boundary conditions is analytically solved. The kinetic equation of Vlasov--Boltzmann with integral of collisions of type BGK (Bhatnagar, Gross, Krook) and Maxwell equation for electric field are applied. Distribution function for electrons and electric field in plasma in the form of expansion under eigen solutions of the initial system of equations are received. Coefficients of these expansions are found by means of the boundary condi…
The boundary problem about behaviour (oscillations) of the electronic plasmas with arbitrary degree of degeneration of electronic gas in half-space with diffusion boundary conditions is analytically solved. The kinetic equation of Vlasov - Boltzmann with integral of collisions of type BGK (Bhatnagar, Gross, Krook) and Maxwell equation for electric field are applied. Distribution function for electrons and electric field in plasma in the form of expansion under eigen solutions of the initial system of equations are received. Coefficients of these expansions are found by means of the boundary co…
Three dimensional rotatory modes of oscillations in a one-dimensional chain of rodlike charged particles or dust grains in a plasma are investigated. The dispersion characteristics of the modes are analyzed. The stability of different equilibrium orientations of the rods, phase transitions between the different equilibria, and a critical dependence on the relative strength of the confining potential are analyzed. The relations of these processes with liquid crystals, nanotubing, and plasma coating are discussed.
Previously developed method for finding asymptotic solutions of Vlasov equations using two-dimensional (in coordinate x and time t) Laplace transform is here applied to consider transversal oscillations and waves in low-collision quasi-neutral (n_i \simeq n_e) Maxwellian electron-ion plasmas. We obtain two branches of electron waves: the ubiquitous one of high-frequency and high-velocity oscillations and the unusual low-velocity one. Taking into account Coulomb collisions in the limit m_e << m_i, \bar{v_i} << \bar{v_e}, and T_e m_e << T_i m_i results in expressions for transversal plasma-elect…
It is shown that a Friedel type oscillations accompany skin effect in degenerate plasma of a metal. It was learnt earlier that Friedel oscillations take place under charge screening in quantum plasma. However the nature of Friedel oscillations is not in the quantum character of the plasma, but in the features of degenerate Fermi distribution, namely, in its sharp transformation into zero directly just the other side of the Fermi surface. This circumstance leads to the Frieldel-type oscillations under anomalous skin effect.
Micron-size charged particles can be easily levitated in low-density plasma environments. At low pressures, suspended particles have been observed to spontaneously oscillate around an equilibrium position. In systems of many particles, these oscillations can catalyze a variety of nonequilibrium, collective behaviors. Here, we report spontaneous oscillations of single particles that remain stable for minutes with striking regularity in amplitude and frequency. The oscillation amplitude can also exceed 1 cm, nearly an order of magnitude larger than previously observed. Using an integrated experi…
Transport properties of the ionic component of dense plasmas are investigated on the basis of effective potentials taking into account the presence of an external alternating electrical (laser) field. The latter generates single particle oscillations of electrons and significantly changes the screening of the test charge in plasma. Therefore, transport coefficients become non-monotonic function of the oscillation frequency. The results for the diffusion and viscosity coefficients are presented. These transport coefficients are computed using the generalized Coulomb logarithm within the Chapman…
A method for solving model nonlinear equations describing plasma oscillations in the presence of viscosity and resistivity is given. By first going to the Lagrangian variables and then transforming the space variable conveniently, the solution in parametric form is obtained. It involves simple elementary functions. Our solution includes all known exact solutions for an ideal cold plasma and a large class of new ones for a more realistic plasma. A new nonlinear effect is found of splitting of the largest density maximum, with a saddle point between the peaks so obtained. The method may sometime…
The spatio-temporal evolution of nonlinear oscillations in multi-species plasma is revisited to provide more insight into the physics of phase mixing by constructing two sets of nonlinear solutions up to the second order. The first solution exhibits perfect oscillations in the linear regime and phase mixing appears only nonlinearly in the second order as a response to the ponderomotive forces. This response can be both direct and indirect. The indirect contribution of the ponderomotive forces appears through self-consistently generated low frequency fields. Furthermore, the direct and indirect…
Plasma dipole oscillation (PDO) depicted as harmonic motion of a spatially localized block of electrons has, until now, been hypothetical. In practice, the plasma oscillation occurs always as a part of a plasma wave. Studies on radiation burst from plasmas have focused only on coupling of the plasma wave and electromagnetic wave. Here we show that a very-high-field PDO can be generated by the electrons trapped in a moving train of potential wells. The electrons riding on the potential train coherently construct a local dipole moment by charge separation. The subsequent PDO is found to persist …
Five-moment approximation in hydrodynamics includes not only continuity equation and momentum balance equation, but also energy equation. Set of hydrodynamic equations is presented for system of non-relativistic quantum particles with Coulomb interaction. This set of equations is linearized, and dispersion relation for Langmuir waves in quasi-neutral system of electrons and immobile ions is obtained.
In the present paper the linearized problem of half-space plasma oscillations in external longitudinal alternating electric field is solved analytically. Specular - accommodative boundary conditions of electron reflection from the plasma boundary are considered. Coefficients of continuous and discrete spectra of the problem are found, and electron distribution function on the plasma boundary and electric field are expressed in explicit form.
We evidence the existence of plasma oscillations of electrons-positron pairs created by the vacuum polarization in an uniform electric field with E < Ec. Our general treatment, encompassing also the traditional, well studied case of E > Ec, shows the existence in both cases of a maximum Lorentz factor acquired by electrons and positrons and allows determination of the a maximal length of oscillation. We quantitatively estimate how plasma oscillations reduce the rate of pair creation and increase the time scale of the pair production. These results are particularly relevant in view of the exper…
A two-fluid model developed originally to describe wave oscillations in the vacuum arc centrifuge, a cylindrical, rapidly rotating, low temperature and confined plasma column, is applied to interpret plasma oscillations in a RF generated linear magnetised plasma (WOMBAT), with similar density and field strength. Compared to typical centrifuge plasmas, WOMBAT plasmas have slower normalised rotation frequency, lower temperature and lower axial velocity. Despite these differences, the two-fluid model provides a consistent description of the WOMBAT plasma configuration and yields qualitative agree…
We demonstrated experimentally canard induced mixed mode oscillations (MMO) in an excitable glow discharge plasma, and the results are validated through numerical solution of the FitzHugh Nagumo (FHN) model. When glow discharge plasma is perturbed by applying a magnetic field, it shows mixed mode oscillatory activity, i.e., quasiperiodic small oscillations interposed with large bounded limit cycles oscillations. The initial quasiperiodic oscillations were observed to change into large amplitude limit cycle oscillations with magnetic field, and the number of these oscillation increases with inc…
The linearized problem of plasma oscillations in layer (particularly, in thin films) in external longitudinal alternating electric field is solved analytically. Specular - accommodative boundary conditions of electron reflection from the plasma boundary are considered. Coefficients of continuous and discrete spectra of the problem are found, and electron distribution function on the plasma boundary and electric field are expressed in explicit form. Absorption of energy of electric field in layer is calculated.
We study the spatio-temporal evolution of the nonlinear electrostatic oscillations in a cold magnetized electron-positron (e-p) plasma using both analytics and simulations. Using a perturbative method we demonstrate that the nonlinear solutions change significantly when a pure electrostatic mode is excited at the linear level instead of a mixed upper-hybrid and zero-frequency mode that is considered in a recent study. The pure electrostatic oscillations undergo phase mixing nonlinearly. However, the presence of the magnetic field significantly delays the phase-mixing compared to that observed …
The results of experimental investigations of electron ejection from gas-discharge nonneutral electron plasma at the formation and radial oscillations of vortex structure have been presented. The electrons are injected from the vortex structure and the adjacent region of electron sheath in the form of pulses the duration and periodicity of which are determined by the processes of evolution and dynamics of this structure. The possible mechanisms of pulse ejection of electrons are considered. The influence of electron ejection on other processes in discharge electron sheath is analyzed.
The dispersion relation of longitudinal electrostatic oscillations in a relativistic plasma is studied in the context of the nonextensive statistics formalism proposed by Tsallis [C. Tsallis, J. Stat. Phys. {\bf 52}, 479 (1988)], where nonextensivity is characterized by a parameter $q$ in Tsallis's entropy. $q=1$ corresponds to the usual Boltzmann-Gibbs, extensive statistics formalism. In the nonrelativistic regime, normalizability of the equilibrium distribution function implies that $-1\leq q\leq\infty$. We show that in the relativistic regime much tighter constraints must be satisfied, name…
We study the frequency of the plasma oscillations of electron-positron pairs created by the vacuum polarization in an uniform electric field with strength E in the range 0.2 Ec < E < 10 Ec. Following the approach adopted in [1] we work out one second order ordinary differential equation for a variable related to the velocity from which we can recover the classical plasma oscillation equation when E -> 0. Thereby, we focus our attention on its evolution in time studying how this oscillation frequency approaches the plasma frequency. The time-scale needed to approach to the plasma frequency and …
The self-excitation of Plasma Series Resonance (PSR) oscillations plays an important role in the electron heating dynamics in Capacitively Coupled Radio Frequency (CCRF) plasmas. In a combined approach of PIC/MCC simulations and a theoretical model based on an equivalent circuit, we investigate the self-excitation of PSR oscillations and their effect on the electron heating in geometrically symmetric CCRF plasmas driven by multiple consecutive harmonics. The discharge symmetry is controlled via the Electrical Asymmetry Effect, i.e. by varying the total number of harmonics and tuning the phase …
Research of influence of collisions on Friedel oscillations in quantum degenerate collisional plasma (T=0) is carried out for the first time. It is shown that presence of collisions in plasma leads to exponential decreasing of amplitude and phase shift of the Friedel oscillations. In linear approximation the phase shift is equal to the half of quantity inverse to product of Fermi's wave number by free length path of electrons. The correct expression for longitudinal dielectric permeability of the quantum collisional plasma found by the authors (see arxiv:1001.3937 [math-ph] 22 Jan 2010) is use…
The influence of the shape of a sample on the type of uniform dipole collective electrons oscillations is discussed. In samples of a bulk shape uniform bulk dipole oscillation (Langmuir oscillation) cannot exist. It exists in samples of a thin slab shape only. As uniform bulk dipole oscillations cannot penetrate ellipsoidal samples of conductive material, they exist in the surface layer of a sample only (Mie oscillations). Frequencies of Mie oscillations are alculated for a sample of the shape of an arbitrary ellipsoid.
An experimental investigation on the periodic and chaotic oscillations in a reflex plasma source in presence of magnetic field is presented. The experiment is conducted in a reflex plasma source, consisting of two cathodes and a ring anode. A penning type DC glow discharge in an uniform axial magnetic field is initiated in the background of argon gas. The current-voltage characteristics near the breakdown voltage show a hysteresis with two distinct discharge current regimes. The effect of magnetic field on the discharge current and floating potential oscillations is studied when the discharge …
We study the dispersion properties of surface plasmon oscillations in a semi-bounded semiconductor plasma with the effects of the Coulomb exchange force associated with the spin polarization of electrons and holes as well as the effects of the Fermi degenerate pressure and the quantum Bohm potential. Starting from a quantum hydrodynamic (QHD) model coupled to the Poisson equation, we derive the general dispersion relation for surface plasma waves. Previous results in this context are recovered. The dispersion properties of the surface waves are analyzed in some particular cases of interest and…
In the present paper the linearized problem of plasma oscillations in slab (particularly, thin films) in external longitudinal alternating electric field is solved analytically. Specular boundary conditions of electron reflection from the plasma boundary are considered. Coefficients of continuous and discrete spectra of the problem are found, and electron distribution function on the plasma boundary and electric field are expressed in explicit form. Absorption of energy of electric field in slab is calculated.
Restoring force acts on the electronic cloud of the outer electrons of a neutral or charged impurity atom when it is shifted relative to the inner charged core. Because of this the dipole oscillation arises, which influences considerably the dispersion law of the plasma oscillation in bulk donor semiconductors. Assuming that only one transition of the outer electron from the ground state to the first excited state is essential, the dispersion law is calculated. It is shown that calculated dispersion law consists of two separate branches, one of them originates from the regular plasma oscillati…
The theory of nonlinear dipole plasma oscillations generated in a metal spheroidal nanoparticle by a laser-wave field has been developed. Approximate (to within the cubic term) analytic expressions for the nanoparticle dipole moment have been obtained in the case where the laser field is oriented along the spheroid rotation axis.
Nonlinear axisymmetric cylindrical plasma oscillations in magnetized collisionless plasmas are a model for the electron fluid collapse on the axis behind an ultrashort relativisically intense laser pulse exciting a plasma wake wave. We present an analytical description of the strongly nonlinear oscillations showing that the magnetic field prevents closing of the cavity formed behind the laser pulse. This effect is demonstrated with 3D PIC simulations of the laser-plasma interaction. An analysis of the betatron oscillations of fast electrons in the presence of the magnetic field reveals a chara…
It is well known that oscillations at the electron plasma frequency may appear due to instability of the plasma sheath near a positively biased electrode immersed in plasma. This instability is caused by transit-time effects when electrons, collected by this electrode, pass through the sheath. Such oscillations appear as low-power short spikes due to additional ionization of a neutral gas in the electrode vicinity. Herein we present first results obtained when the additional ionization was eliminated. We succeeded to prolong the oscillations during the whole time a positive bias was applied to…
The effect of short wavelength quantum electrodynamic (QED) correction on plasma-wave propagation is investigated. The effect on plasma oscillations and on electromagnetic waves in an unmagnetized as well as a magnetized plasma is investigated. The effects of the short wavelength QED corrections are most significant for plasma oscillations and for extraordinary modes. In particular, the QED correction allow plasma oscillations to propagate, and the extra-ordinary mode looses its stop band. The significance of our results is discussed.
Phase mixing of relativistic large amplitude nonlinear plasma wave in presence of a time independent space periodic ion density profile has been investigated. Inhomogeneous ion along with the relativistic variation of electron mass make the characteristic frequency of the wave to acquire a space dependency and thereby it breaks at arbitrarily small amplitude due to phase mixing. An approximate space time dependent solution is obtained in the weakly relativistic limit by Bogoliuboff and Kryloff method of averaging. We find that the change in the ion density perturbation and also the relativisti…
The effect of spin of particles in the propagation of plasma waves is studied using a semi-classical kinetic theory for a magnetized plasma. We focus in the simple damping effects for the electrostatic wave modes besides Landau damping. Without taking into account more quantum effects than spin contribution to Vlasov's equation, we show that spin produces a new damping or instability which is proportional to the zeroth order magnetization of the system. This correction depends on the electromagnetic part of the wave which is coupled with the spin vector.
We analyze the role of collisional effects on the coupling between ion-acoustic waves and neutrino flavor oscillations, discussing its relevance for plasma instabilities in extreme plasma environments like in type II supernovae, where intense neutrino bursts exist. Electrons (leptons) are coupled to the electron-neutrino fluid through the weak Fermi force, but the electron-neutrinos are allowed to convert to other neutrino flavors and vice-versa. Due to the typically slow frequency of neutrino flavor oscillations, many orders smaller than e.g. the plasma frequency, an effective energy transfer…
Betatron oscillation is a commonly known phenomenon in laser or beam driven plasma wakefield accelerators. In the conventional model, the plasma wake provides a linear focusing force to a relativistic electron, and the electron oscillates in one transverse direction with the betatron frequency proportional to $1/\sqrtγ$, where $γ$ is the Lorentz factor of the electron. In this work, we extend this model to three-dimensional by considering the oscillation in two transverse and one longitudinal directions. The long-term equations, with motion in the betatron time scale averaged out, are obtained…
