In 1922, an experiment was carried out [1], which determined the presence of a spin in an electron. For our theory, this means that two more elements must be introduced into its model, which differ from the first two (1.2, 1.3) in an oppositely directed imaginary vector. If the parametric graph of an electron and a proton with an initial spin (now we will call it positive or plus) is a spiral twisting clockwise (Fig. 1-2), then an electron and a proton with a negative spin is a counterclockwise spiral. The models of these particles, therefore, will be as follows: \[\overline {A ^ e (x)} = {\overline {N ^ e} \over x} \exp [\, \mathbf {i} (- \pi x / \overline {N ^ e} + \pi / 4)] \tag {2.1} \] \[\overline {A ^ p (x)} = {\overline {N ^ p} \over x} \exp [\, \mathbf {i} (- \pi x / \overline {N ^ p} - 3 \pi / 4)] \tag {2.2} \] The upper line above the parameter will mean that the parameter itself belongs to a group of particles with negative spin. With the resulting model, you can do all the same manipulations as with the previous one, with a positive spin, and get the same results. For example, in the interaction of an electron and a proton with a negative spin, we will also find that the optimal energy orbits are in the wave nodes, and their numbers correspond to positive integers, as in (1.10).

The VEL interaction law, in general, looks like this: \[p = {h R \over x ^ 2} \bigg [\Phi_1 ^ e + \overline {\Phi_1 ^ e} + \Phi_1 ^ p + \overline {\Phi_1 ^ p} \bigg] \bigg [\Phi_2 ^ e + \overline {\Phi_2 ^ e} + \Phi_2 ^ p + \overline {\Phi_2 ^ p} \bigg] \tag {2.3} \] Here: \(\Phi_i ^ a \) is the wave function of particles, where \(a \) - indicates that the function belongs to the class of particles (electrons, protons), and \(i \) - - interaction group number. If a line is drawn above the wave function, it means that the spin of this group of particles is negative.

One of the special cases of the above general law of interaction is Coulomb's law [2]. We simply remove from the general expression those terms that do not participate in the interaction and obtain a solution for a specific option. Also, we assume that electrons and protons with different spins are involved in the process, and their average statistical value is the same: \[N ^ e = \overline {N ^ e}, \quad N ^ p = \overline {N ^ p} \]

Recall that at the moment we are using models of only two types of particles: an electron and a proton, therefore the law of gravitation, in this form, is applicable only for protium atoms. As the species diversity of particles expands, this law will also be supplemented. The form of this law completely coincides with (2.1), since all terms of this expression are used for it (Fig.5b): \[p = {h R \over x ^ 2} \bigg [\Phi_1 ^ e + \overline {\Phi_1 ^ e} + \Phi_1 ^ p + \overline {\Phi_1 ^ p} \bigg] \bigg [\Phi_2 ^ e + \overline {\Phi_2 ^ e} + \Phi_2 ^ p + \overline {\Phi_2 ^ p} \bigg] \tag {2.7} \]

Let's assign an electron to the first group of particles, and a proton from a protium atom (or any hydrogen-like atom) to the second. Then their interaction will be described as follows: \[p = {h R \over x ^ 2} \Phi_1 ^ e \Phi_2 ^ p \tag {2.9} \] In this formula, \(\beta \) is not taken into account due to its smallness. When all electrons occupy the optimal energy state and the corresponding orbits, then the distance \(x \) between the groups will take only integer values, which follows from formula (1.10) of the previous part. Let's denote these numbers as follows: \[n = x, \quad n = 1,2,3,4, ... \tag {2.10} \] Also, we understand that the number of electrons in an atom will be equal to the number of protons: \[Z = N ^ e = N ^ p \tag {2.11} \] Then the interaction formula becomes: \[p = \mathbf {i} {h R Z ^ 2 \over n ^ 2} \tag {2.12} \] The imaginary unit in front of the right-hand side of the expression shows that the momentum and energy of the particles are in the optimal state. Everything is correct. But what will happen if an electron, being in an excited state in an orbit with \(n = 2 \), goes to a smaller orbit, with \(n = 1 \)? The atom will emit a quantum of Planck energy, equal to \(h \nu \), where \(\nu \) is the frequency of this radiation. Let's deduce the same, but through the impulse and the length of the emitted wave \(\lambda \): \[{h \over \lambda} = p_1 - p_2 = \mathbf {i} h RZ ^ 2 \left ({1 \over n_1 ^ 2} - {1 \over n_2 ^ 2} \right) \tag {2.13} \] where \(n_1 \) and \(n_2 \) - will be the numbers of the final and initial orbits. We already know that being between the optimal orbits, the electron emits energy, which is what happens in our case. Consequently, the reactive energy is converted into active energy of the same amplitude, which means that the imaginary unit in the last expression can be removed. From this we obtain the Rydberg formula [3]: \[{1 \over \lambda} = RZ ^ 2 \left ({1 \over n_1 ^ 2} - {1 \over n_2 ^ 2} \right) \tag {2.14} \] This formula describes well the real spectra of hydrogen-like atoms [4].

The energy of an electron in orbit can be determined from the formula for the interaction of an electron and a proton (2.12), simply by multiplying it by the speed of light \(c\) \[E_n^e = p c = \mathbf{i} {h R Z^2 c \over n^2} \tag{2.15} \] which fully corresponds in absolute value to the Bohr energy of an electron [5]. The sign of the imaginary unit in front of the expression indicates that this energy is reactive. This advantage of our model immediately gives a physical meaning to this energy. In the standard model, it is made negative, which completely confuses university students :)