Global Journal of Science Frontier Research, A: Physics and Space Science, Volume 23 Issue 1

r = λ / 2π (28) r h = λ h / 2π = (3.958 x 10 -10 m) / 6.283 = 6.30 x 10 -11 m (29) r x = λ x / 2π = (2.856 x 10 -10 m) / 6.283 = 4.456 x 10 -11 m (30) resulting in the fine geometry of the electron mass-energy torus: E = m e c 2 =∫= λ e c 2 = p e W x = λ e W k W x = λ e (2πr h υ k ) (2πr x υ ce ) (31) The structure of the electron mass-energy, broken down this way into its basic constituents, presents itself as a flux of electric energy shaped as a torus due to the superimposed wavefunctions W k and W x . When the geometric mean of the two wavefunction radii is calculated, the result is the Bohr radius which compares favorably with the CODATA 2018 value of 0.529 x 10 -10 m: (r h r x ) 0.5 = [(6.30 x 10 -11 m) (4.456 x 10 -11 m)] 0.5 = 0.535 x 10 -10 m (32) The implication is that the Bohr radius is not really the radius of a spherical cavity, or of a quasi-spherical cloud where a point-mass may locate once the probability wave collapses, but merely the geometric mean of the two radii of the electron mass-energy torus. The latter has the "normal" volume given by [6]: V = 2π 2 r h r x 2 (33) III. C onclusions The proposed model is a much more satisfactory and rigorous description of the electron and explanation of the phenomenological nature of the Bohr radius. Rather than leading a nebulous existence as a supposed particle, wave, cloud, cavity, or “point particle in a cloud of probable locations”, it is best understood as a “precise toroidal volumetric flux structure” of electric energy resulting from two longitudinal waves being superimposed upon and trapped with each other, “that occupies the location of the entire ‘cloud’ [16].” The torus model also rejoins the problematics of "the ring electron" originally proposed in 1915 by Alfred Parson [17], and taken up during 1917-1921 by others, in particular by Arthur Compton [18, 19] and H. Stanley Allen [20]. However, unlike the ring electron model, the model proposed by the Correas does not invoke for the electron a vortical ring structure, but rather that of a closed torus composed by a large number of continuous rings. The two models share many of the features that overcome the present-day conceptualization and treatment of the electron. Loss of energy by radiation of the closed-loop mass-energy is done away simply by assuming rotation of a ring- shaped charge, and the virtually impossible complications raised by independent orbitals required to explain diamagnetic atoms melt away, given that it is the electron mass-energy itself that has a diamagnetic moment [6]. Likewise, paramagnetism and X-ray diffraction patterns are explained by the tilting of the toroidal ring, confirming the contention of the above three physicists that the elementary magnet is not the atom, but the electron itself. The magnetic susceptibility of paramagnetic substances flows directly from the electron retaining its magnetic properties at low temperatures. Asymmetric scatter of X-radiation is explained by the striking of distinct loops of the torus. Runge's rule that abstrusely related variable magnetic effects to the ratio e/m [20] is replaced by the elegant and eloquent result of equations 21, 25 and 26: W k = λ h υ k = p e / λ e =∫= e / m e (34) Well beyond what the ring electron model could provide, the Correa model of the electron torus addresses other features still. In effect, the number of toroidal loops is directly obtained by the relation that accounts for both the composite nature of the mass of the electron and the proportionality role played by the reciprocal of the fine structure constant – whose actual value is thereby revised [21] – in the topogeometry of the electron: No. loops = λ e / λ x = α -2 = 19,205.9 = (138.5853745) 2 (35) The very fact that the number of loops is very nearly an integer may suggest that it is simply 19,206 or, instead, that the fractional value actually indicates a kink in the toroidal structure (the shorter loop), one that may have phenomenologically induced existing physics into believing that the electron is a point-mass engaged in orbital 1 Year 2023 5 © 2023 Global Journals Global Journal of Science Frontier Research Volume XXIII Issue ersion I VI ( A ) The Toroidal Fine-Structure of the Electron