The chaotic atom model via a fractal approximation of motion

October 18, 2017 | Penulis: Zoltan Borsos | Kategori: IDE
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The chaotic atom model via a fractal approximation of motion

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Phys. Scr. 84 (2011) 045017 (16pp)


The chaotic atom model via a fractal approximation of motion M Agop1,2 , P Nica1,2 , S Gurlui1,3 , C Focsa1 , D Magop3 and Z Borsos4 1

Laboratoire de Physique des Lasers, Atomes et Molécules (UMR 8523), Université des Sciences et Technologies de Lille, Villeneuve d’Ascq cedex 59655, France 2 Department of Physics, Technical ‘Gh. Asachi’ University, Iasi 700050, Romania 3 Faculty of Physics, ‘Al. I. Cuza’ University, Blvd Carol I no. 11, Iasi 700506, Romania 4 Department of Physics, ‘Oil and Gas’ University, Blvd Bucuresti no. 39, Ploiesti 100680, Romania E-mail: [email protected]

Received 22 June 2011 Accepted for publication 31 August 2011 Published 27 September 2011 Online at Abstract A new model of the atom is built based on a complete and detailed nonlinear dynamics analysis (complete time series, Poincaré sections, complete phase space, Lyapunov exponents, bifurcation diagrams and fractal analysis), through the correlation of the chaotic–stochastic model with a fractal one. Some specific mechanisms that ensure the atom functionality are proposed: gun, chaotic gun and multi-gun effects for the excited states (the classical analogue of quantum absorption) and the fractalization of the trajectories for the stationary states (a natural way of introducing the quantification). PACS numbers: 05.45.Df, 47.53.+n, 03.65.−w (Some figures in this article are in colour only in the electronic version.)

nonlinear system of equations that describes this physical phenomenon, one obtains a Bohr image of an atom. Such (quantum-transition) jumps and their duration and physical mechanism have never been explained by the quantum theory of atoms. One thus offers, through a multi-gun effect, a physical explanation of the quantum model of the absorption of energy by the atom. Nevertheless, the chaotic–stochastic model of the atom from [12–15] does not offer an accurate approach (from the nonlinear dynamics point of view) to the complex phenomena that result in the atom functionality (e.g. the description of the stationary states is missing). Therefore, in the present paper, based on a complete and detailed nonlinear dynamics analysis of the chaotic–stochastic model of the atom from [12–14] in correlation with the fractal one from [15, 16], a new model of the atom is built. In this way, some specific mechanisms that are able to explain the excited and stationary states of the atom are emphasized. This paper is structured as follows. Using a nonlinear dynamics analysis of a particle–field nonlinear interaction (i.e. by means of the complete time series, Poincaré sections, complete phase space, Lyapunov exponents, bifurcation diagrams and fractal analysis), the excited states of the atom are presented in section 2; taking into account

1. Introduction The theoretical description of microphysical systems is realized by means of the wave mechanics of Schrödinger [1], the matrix mechanics of Heisenberg [2] or the path-integral mechanics of Feynman [3]. Another approach is the hydrodynamic formulation of quantum mechanics (the idea of ‘subquantum medium’—due to Madelung [4], de Broglie [5, 6], Takabayasi [7] and Bohm [8, 9]). All these models, although equivalent [10, 11], define, for example, only the stationary states of the atom. The transitory states of the atom remain to be explained, although there are some mechanisms in the framework of quantum mechanisms that give us insight into them. A new approach in the study of these transitory states was recently established through the chaotic–stochastic model of the atom [12–15]. This model implies the idea that a ‘magnetized’ charged particle in interaction with ‘resonant’ photons operates from one energy level to another higher one by a stochastic acceleration effect (the gun effect according to [12–15]) and suggests that such effects may represent a phenomenological physical mechanism that explains how an electron jumps to higher atomic orbits when it absorbs resonant photons. If one increases the number of iterations of the corresponding 0031-8949/11/045017+16$33.00

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© 2011 The Royal Swedish Academy of Sciences

Phys. Scr. 84 (2011) 045017

M Agop et al

the non-differentiability of the charged particle trajectories induced by the chaotic behavior of the same particle–field nonlinear interaction, the stationary states of the atom are obtained in section 3.

(ii) A stationary uniform magnetic field with the components B0 = (0, 0, B0 ). Then, using the dimensionless variables [12–14]

2. Excited states of the atom through nonlinear dynamics analysis

T = kct, X = kx, P =

2.1. The nonlinear movement equations and numerical solutions

∇ · Bb = 0, ∇ × Eb = −

1 q Bb (t) 1 q E x,y (t) = b , εx,y (T ) = , kc m 0 kc m 0 c s γ ωc ωP 1 nbq 2 ωB = = γ c ,  P = = , B = kc kc kc kc m 0 ε Bb (T ) =



1 ∂Eb ∇ × Bb = µjb + 2 . (4) c ∂t (ii) Relativistic equation of motion for a single particle of the beam: dr p = , dt m0γ

dp q 2R = qEb + × (Bb +B0 ). dt m0γ

(5a, b)

In the above relations, Eb and Bb are the electric and magnetic field intensities generated by the beam, B0 is the constant uniform magnetic field, R = p/q is the momentum per unit charge, jb is the current density of the beam and p is the relativistic particle momentum: p = m 0 γ v, v2 γ = 1 − ⊥2 c



In our application, we consider the following. (i) An electromagnetic field with a harmonic space dependence of the form:    Eb (x, t) =
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