📝 Abstract

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 results to the theory of natural plasmoids are considered.

💡 Analysis

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 results to the theory of natural plasmoids are considered.

📄 Content

Theoretical Studies of Long Lived Plasma  Structures Maxim Dvornikov*†

*Departamento de Física y Centro-Científico-Tecnológico de Valparaíso, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile; †N. V. Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radowave Propagation (IZMIRAN), 142190, Troitsk, Moscow region, Russia E-mail: maxim.dvornikov@usm.cl

Abstract 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 results to the theory of natural plasmoids are considered.

1. Introduction  The construction of a theoretical model for a stable spherically symmetric plasma structure is still a puzzle for the plasma physics [1]. The attempts to build a model of a spherical plasmoid with help of the classical Boltzmann and Gibbs statistics result in the formidable difficulties [2]. Moreover the static cluster of charged particles maintained by its own electromagnetic forces is unlikely to be stable since plasma has a tendency to expand at the absence of any external pressure or additional attractive forces like gravity etc. (see Ref. [3]). We can imagine a spherical plasma structure in the form of a spherically symmetric Langmuir wave [4]. Such a model of a spherical plasmoid has many advantages compared to the static electric charge distribution models. First our model does not require any additional external forces to provide the stability of the system. Second, a charged particles system making the spherically symmetric motion does not loose energy for radiation. Thus, such a spherical plasmoid will be dynamically stable compared to any other unstructured plasma formations since the latter will loose their energy and recombine with the time scale of several milliseconds. Third, in frames of our model one can point out an internal energy source which would compensate the inevitable energy losses and provide the long life-time of a plasmoid. In this paper we summarize our previous works on the theory of spherically symmetric plasma structures. In Sec. 2 we study spherically symmetric oscillations of electron gas in plasma in frames of both classical and quantum approaches. We obtain the expression for the density of electrons in the explicit form and the dispersion relations for these oscillations. The discrepancy between classical and quantum cases are discussed. In Sec. 3 we discuss the possible formation of a bound state of electrons participating in spherically symmetric oscillations due to the exchange of ion acoustic waves. Finally in Sec. 4 we discuss our results.
2. A model of a spherical plasmoid based  on electron gas oscillations in plasma  In this section we will show that one can describe spherically symmetric oscillations of electrons in plasma within both classical and quantum approaches. We will obtain the time dependent density profile of electrons and the dispersion relation for these oscillations. First we consider spherically symmetric oscillations in frames of the classical electrodynamics. We can suggest that only electrons participate in oscillations of plasma since the mobility of ions is low. At the absence of collisions and other forms of dissipation the system of the hydrodynamic equations for the description of the evolution of the electrons velocity v , the electric field E and plasma pressure p can be presented in the following way [5]: ext ( ) 0, 1 ( ) , 4 ( ) 4 ( , ), (1) e e e e i n n t e p t m mn e n n t π πρ ∂

• ∇⋅ = ∂ ∂

⋅∇ = − − ∇ ∂ ∇⋅ = − − + v v v v E E r

where en is the number density of electrons, 0 in n

is the constant number density of ions, m is the mass on an electron, and 0 e > is the proton charge. In Eq. (1) we include the possible external source ext ( , )t ρ r as well as take into account that a spherically symmetric system does not have a magnetic field (see Ref. [6]). If we consider small perturbations of the electrons density 0 0 en n n n −

 and linearize Eq. (1), we get a single differential equation for the electrons density perturbation [6], 2 2 2 0 ext 2 0 4 1 , (2) e en n p n n t m n m π ω ρ ∂ ∂ ⎛ ⎞ + − ∇

⎜ ⎟ ∂ ∂ ⎝ ⎠

where 2 0 4 / e n e m ω π

is the plasma frequency for electrons and 0 ( / ) p n ∂ ∂ is the derivative taken at 0 en n

, which depends on the equation of state of electrons gas in plasma. At the absence of the external source ext ( , )t ρ r the spherically symmetr

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