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166 RUGGERO MARIA SANTILLI Appendix A Aringazin’s Studies on Toroidal Orbits of the Hydrogen Atom under an External Magnetic Field In the main text of this book we have presented the theoretical and experimental foundations of the new chemical species of magnecules which is centrally dependent on individual atoms acquiring a generally toroidal configuration of the orbits of at least the peripheral electrons when exposed to sufficiently intense external magnetic fields, as originally proposed by Santilli [1] and reviewed in the main text of this book. In this Appendix we outline the studies by Aringazin [8] on the Schrödinger equation of the hydrogen atom under a strong, external, static and uniform magnetic field which studies have confirmed the toroidal configuration of the electron orbits so crucial for the existence of the new chemical species of magnecules. It should be stressed that when considered at orbital distances (i.e., of the order of 10 −8 cm), atoms and molecules near the electric arc of hadronic reactors (Section 4), and in the plasma region, are exposed to a strong magnetic field, whose intensity may be high enough to cause the needed magnetic polarization (see Fig. 9.D). A weak, external, static, and uniform magnetic field B causes an anomalous Zeeman splitting of the energy levels of the hydrogen atom, with ignorably small effects on the electron charge distribution. In the case of a more intense magnetic field which is strong enough to cause decoupling of a spin-orbital interaction (in atoms), eB/2mc > ∆E jj ′ ≃ 10 −3 eV, i.e., for B ≃ 10 5 Gauss, a normal Zeeman effect is observed, again, with ignorably small deformation of the electron orbits. More particularly, in the case of a weak external magnetic field B, one can ignore the quadratic term in the field B because its contribution is small in comparison with that of the other terms in Schrödinger equation, so that the linear approximation in the field B can be used. In such a linear approximation, the wave function of electron remains unperturbed, with the only effect being the well known Zeeman splitting of the energy levels of the H atom. In both
THE NEW FUELS WITH MAGNECULAR STRUCTURE 167 Zeeman effects, the interaction energy of the electron with the magnetic field is assumed to be much smaller than the binding energy of the hydrogen atom, eB/2mc ≪ me 4 /2 2 = 13.6 eV, i.e., the intensity of the magnetic field is much smaller than some characteristic value, B ≪ B 0 = 2.4 · 10 9 Gauss = 240000 Tesla (recall that 1 Tesla = 10 4 Gauss). Thus, the action of a weak magnetic field can be treated as a small perturbation of the hydrogen atom. In the case of a very strong magnetic field, B ≫ B 0 , the quadratic term in the field B makes a great contribution and cannot be ignored. Calculations show that, in this case, a considerable deformation of the electron charge distribution in the hydrogen atom occurs. More specifically, under the influence of a very strong external magnetic field a magnetic confinement takes place, i.e., in the plane perpendicular to the direction of magnetic field (see Fig. 9.D), the electron dynamics is determined mainly by the action of the magnetic field, while the Coulomb interaction of the electron with the nucleus can be viewed as a small perturbation. This adiabatic approximation allows one to separate variables in the associated Schrödinger equation [9]. At the same time, in the direction of the magnetic field the motion of electron is governed both by the magnetic field and the Coulomb interaction of the electron with the nucleus. The highest intensities of magnetic fields maintained macroscopically at large distances in modern magnet laboratories are of the order of 10 5 −10 6 Gauss (∼ 50 Tesla), i.e., they are much below B 0 = 2.4 · 10 9 Gauss (∼ 10 5 Tesla). Extremely intense external magnetic fields, B ≥ B c = B 0 /α 2 = 4.4 · 10 13 Gauss, correspond to the interaction energy of the order of the mass of electron, mc 2 = 0.5 MeV, where α = e 2 /c is the fine structure constant. In this case, despite the fact that the extremely strong magnetic field does characterize a stable vacuum in respect to creation of electron-positron pairs, one should account for relativistic and quantum electrodynamics (QED) effects, and invoke Dirac or Bethe-Salpeter equation. These contributions are of interest in astrophysics, for example, in studying the atmosphere of neutron stars and white dwarfs which are characterized by B ≃ 10 9 . . . 10 13 Gauss. Aringazin [8] has focused his studies on magnetic fields with intensities of the order of 2.4·10 10 ≤ B ≤ 2.4·10 13 Gauss, at which value nonrelativistic studies via the Schrödinger equation can be used to a very good accuracy, and the adiabatic approximations can be made. Relativistic and QED effects (loop contributions), as well as effects related to finite mass, size, and magnetic moment of the nucleus, and the finite electromag-
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Contents Preface 1 THE INCREASINGLY
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HADRONIC MATHEMATICS, MECHANICS AND
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