Dynamics of charged clusters in a self-consistent field
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C PROPERTIES OF SOLIDS
Dynamics of Charged Clusters in a Self-consistent Field A. S. Chikhachev State Research Center All-Russia Electrical Engineering Institute, Moscow, 111250 Russia e-mail: [email protected] Received May 30, 2006
Abstract—The problem of self-consistent description of dense clusters of charged particles characterized by time-dependent fields is considered. The classical and quantum problems are considered. The results can be applied to the study of clusters in laser fields, as well as to the study of the acceleration of multicharge ions and extreme states of matter. PACS numbers: 52.35.-g, 03.65.-w, 41.20.-q DOI: 10.1134/S1063776106110161
The study of the dynamics of dense charged clusters of particles interacting with self-consistent fields is a topical problem in view of the rise of new problems such as the behavior of clusters in laser fields [1], acceleration of heavy multicharge ions [2], and extreme states of matter (for example, a dust plasma [3]). Usually, the study of clusters is a quite transparent problem when there are integrals of motion of the particles. For time-independent problems, there exists the energy integral, which plays an important role in the study of steady-state systems. In the case of time-dependent problems, one may use the generalizations of the integral of energy for systems that explicitly depend on time; one of such generalizations is given by the well-known Courant–Snyder invariant. This invariant was used for studying a quantum-mechanical oscillator with variable frequency (see [4]). In addition, there are a number of classical problems that are primarily related to acceleration technology (see [5, 6]). A variable-scale technique has become quite popular for solving physical problems. For example, the introduction of a coordinate-dependent time scale allowed the authors of [7, 8] to reduce the Kepler problem to the oscillator problem. When solving a time-dependent Schrödinger equation, one frequently uses time-dependent coordinates in which the length scale depends on time. Note the papers [9–14]. In [9, 10], an exact solution was obtained by separation of variables, and expressions for the propagator of a time-dependent equation were found. In [11], Grosche studied a solution in the form of the integral along trajectories. In [12, 13], exact solutions to a time-dependent Schrödinger equation were investigated under the conditions when the separation of variables is impossible but the use of a variable scale simplifies the equation. In [14], Berry and Klein considered the eigensolutions of the Schrödinger equation
in time-dependent coordinates and a nonstationary classical ensemble of particles in statistical mechanics. In the present paper, we apply a generalization of the energy integral in time-dependent coordinates to describe the dynamics of spherically symmetric clusters that interact with their own field. 1. The expression m 2 md 2 2 I = ---- ξ ( t )r˙ – ---- ----- ( ξ ( t ) )r ⋅ r˙ 2 2 dt 2
(1)
2
r mr d 2 + V ⎛ ---------⎞ + --------- -------2 ( ξ (
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