Periodic System of Atoms in Biquaternionic Representation

Private monochromatic solutions of the free-field equation of electro-gravimagnetic charges and currents are constructed in the differential algebra of biquaternions, which describe elementary particles as standing electro-gravimagnetic waves. The two classes of solutions of this biquaternionic wave equation have been investigated, generated by scalar potentials (pulsars) and vectorial potentials (spinors). Their asymptotic properties are considered, on the base of which they are classified into heavy (boson) and light (lepton) elementary particles. The biquaternion representation of the hydrogen atom is given. The periodic system of elements is produced, which is built on the principle of the musical structure of a simple gamma.


Introduction
In [1] [2] [3], the author developed a biquaternion model of the electrogravimagnetic field (EGM-field) and electro-gravimagnetic interactions. Its basis is made up of biquaternion representations of the generalized Maxwell and Dirac equations. The biquaternion representation of the Maxwell equations expresses the biquaternion of the mass-charge density and the EGM-current through the bigradient of the EGM-field tension. The biquaternionic representation of Dirac equations determines the transformation of the density of mass charges and currents under the influence of external EGM-fields. In particular, in the absence of external fields, it is the biquaternionic wave (biwave) equation for the free field of mass-charges and currents, which is a field analogue of first Newton's law (inertia law).

Inertia Law for Monochromatic Fields of Charges-Currents
The equation for the free field of charge-currents has the form of a homogeneous biwave equation [1] [2]: x t j x t ρ are the gravimagnetic charge density and current density; , ε µ are the constants of electric conductivity and magnetic permeability of vacuum, 1 c εµ = is speed of light, i is imaginary unit.
The action of the biquaternion differential operators − ∇ and + ∇ (mutual bigradients) is determined, according to the quaternion multiplication rule (see Appendix), by the formula grad rot The biquaternion of the energy-momentum of the F-field is given by Here the bar over the symbol means complex conjugation.
The scalar part W is the energy density of the F-field, and P is the analog of the generalized Poynting vector of the F-field, just as a generalized Poynting In this case, from the Equation (2) it follows that the biamplitudes satisfy the Helmholtz equation where the potentials for any function j φ that is integrable on a sphere of radius ω .

Biquaternions of Harmonic Elementary Particles
We consider particular solutions of the Helmholtz equation  It is natural to take these solutions for the construction of elementary particles, which can be called harmonic. Among them we select the ones generated by the scalar potential, which we call pulsars: and particles, generated by a vector potential, we call spinors: The latter are polarized in the direction of the coordinate axes, respectively, to the index 1, 2, 3 j = .

Biquaternions of Monochromatic Structures. Crystals
Using structural biquaternions of arbitrary form ( ) K x , on their basis, by the operation of biquaternion convolution where jlm ε is the Levi-Civita pseudo-tensor, it is possible to construct a variety of monochromatic fields of charge-currents: In (8) there are functional convolutions, which for integrable functions have the integral form: x k x y k x y y y y ρ ρ * = − ∫ Component convolutions for vectors are written similarly. By virtue of the differentiation property of convolution, the convolutions (9) are also solutions of the equations (1).
Formulas (9) allow us to construct various crystal lattices from harmonic elementary particles, if we take lattices as the structural biquaternion the different shifts of the δ-function, and other generalized functions.
We give here a simple example of an inhomogeneous rectangular lattice with Journal of Modern Physics more details, see [2]). And their frequency superpositions are generally immeasurable.

Elementary Spherical Harmonic Pulsars and Their Properties
Among the solutions of the Helmholtz Equations (7), only one is spherically symmetric [5]. It is Here The biamplitude of the corresponding pulsar is Whence follows Calculating the biquaternion of its energy-momentum It follows from (11)-(12) that the density of the mass-charge decreases as 1 r − by increasing r, and the oscillation energy decays even more rapidly, as 2 r − . It is interesting to investigate the asymptotic of these quantities as   The properties of spherical pulsars. In spherical harmonic pulsars at the center at 0 x = , the mass-charge density is equal to its oscillation frequency ω ; the density of the EGM-current is zero; the energy density is equal to

Elementary Spherical Harmonic Spinors and Their Properties
Consider a spinor polarized in direction 1 X : Whence follows  Properties of harmonic spherical spinors. For spherical harmonic spinors at the center (for 0 x = ), the mass-charge density is zero, the norm of the density of the EGM-current vector is ω , the energy density is 2 2 ω , the Poynting vector is equal to zero.
Thus, spherical harmonic spinors in terms of the density of the EGM-charge belong to light elementary particles-leptons.

Biquaternion Model of the Hydrogen Atom
Thus, we have shown that among the monochromatic solutions of the chargecurrent free field Equations (1), only harmonic spherical pulsars have a nonzero density at their center, which is not the case for harmonic spinors. This suggests that spherical harmonic pulsars can be used to construct a biquaternion model of atoms. Table 1 two musical systems are given ( simple gamma, [7]), which can be taken as a basis, in which the ratio frequency of tones is a rational number. For such tones (notes), there is the total period of oscillations, which is determined by the least common multiple for the period their harmonies, which makes it possible to harmoniously sound the accords from different notes. For each of them, in nature, there are substances that possess the described above properties. Which of them corresponds to Mendeleyev's periodic table? It should be the subject of a special study for specialists in area of physical chemistry, spectral properties of substances.
Perhaps among these three structures in Table 1 there is no such. But a similar musical scale should be, which contains the frequencies of these scales. The number of tones within an octave can be changed with growth octave numbers, but all the similar tones of the previous octave in it should be present, which explains the repeatability of chemical properties of substances in columns of periodic Mendeleyev's system, just as the musical sounds are harmonious for perception for octaves and chords composed of them. Journal of Modern Physics Here the frequency of oscillations of the atom For it, all the above formulas for a spherical harmonic pulsar are correct by the frequency of oscillations corresponding to it.

Conclusions
How many such natural octaves exist? Obviously, no less than the number of rows in the periodic system of Mendeleyev. Let us note that the twelve-tempered musical scale, now accepted in classical music, with twelve notes inside the octave, can not be taken, since the ratio of frequencies of consecutive tones in it is a number irrational 12 2 and the general period of oscillation for any set of tones in the octave does not exist. Complete harmonious sound in this system can not be achieved. This is well known to the orchestral musicians of strings and wind instruments, the sound of which is determined by the above described musical arrangements. As is known, with disproportionate oscillation frequencies, beats occur.
Such periodic systems can be constructed for elementary harmonic leptons (spinors and asymmetric pulsars), whose addition to atoms with the same vibration frequency apparently creates isotopes of these atoms. Moreover, the addition of the spinors is connected with the magnetization of matter. It is possible to construct many different isotopes with the same asymptotic density of the EGM charge. Which of them exist in nature is also a matter of special experimental research.
We also note that this description of atoms is based on the construction of