In a previous paper the theory of the toroidal spiral field was employed to derive equations for the stationary orbits of particles that belong to both the atomic nucleus and electrons orbiting the nucleus. This theory offered new relativistic relationships for both the mass and electrical charges, leading to the existence inside the nucleus of particles with not only opposite electrical charges in comparison with the electrons, but also with the masses of opposite sign. In this paper, this theory was applied to create a model of the atom that consists of positive and negative muons, positrons and electrons. The two methods that are used to determine the stationary orbits of particles are: (1) quantizing the particle angular momentum, and (2) quantizing the particle relative electrical charge. The existence of particles with the charges 1/3 and 2/3 is explained by the relativistic dependence of the electrical charges on the particle velocity and also by the quantum character of these charges. Creation of the particles and their distribution in the orbits follows the proposed rule of duality and the rule of stability. The theory describes the processes of creation, excitation and annihilation of the positron-electron pairs inside the atoms. In addition, it explains the nature of the self-sustaining movement of particles inside the atom and illustrates the similarity between the propagation of the positron--electron pair inside the atom and the propagation of electromagnetic waves in a free space. It gives a novel interpretation of the physical meaning of Maxwell's 'displacement current' and demonstrates that electromagnetic fields are intimately associated with electrical charges.