The physical nature of the decalogoriphmic periodicity phenomenon
The gist of this phenomenon is that in the distribution of structural objects of dynamically equilibrium systems on the given parameters the maxima forming a ranged number(line) of parameters can be observed, where the period of recurrence Т = lg e_{j}  lg e_{i} = k/m, and where at m = = 1, 2, 3, 4 … k = 0, 1, 2, 3, 4 …. As a result according to the empirical data it appears that maxima are distributed in the following way e = 2×10^{k/}^{m}. At m = 8 and 12 this formula covers the most part of the empirical data. However, later and more careful researches of statistical distributions by periodogram methods, especially of the periodograms, constructed by the method of interval coverings, show also maxima with periods Т = 0,10 and 0,21, dropping out of the abovestated formula. Thus period Т = 0,21 exists practically on all periodogramsand it is not possible to remove it by the selection of the initial periodogram phase. The experimental situation becomes a little obscure. The given formula turns into a more complex conglomerate of an unknown kind, however, decalogoriphmic periodicity is not cancelled by this. As a result it appears that such theory should be constructed that would describe the existing and the obscure situation.
Before we start to construct such a theory, let us define first of all the common phenomenon for all systems. The common phenomenon for all systems is the character of relative movement of the objects comprising the system. For example, when two megamass interact, one of which m_{o1}<< M_{o2}, the smaller mass (under the influence of gravitation forces) will move about the mass M_{o2} on a trajectory in the form of a circle or an ellipse. Such trajectories have received the name of orbits. The movement on the orbits (trajectories) in the megaworld can't be denied. In the theories describing the interaction of microparticles (charges) the concept of an orbit is absent, though the concept of orbital movement exists! In A. Zommerfeld's theory [9], developed for the system consisting of two charges (atom of hydrogen) it is shown, that movement of a charge on circular orbits is more preferable as the energy of connection for a circular orbit turns out to be the greatest. However, neither Zommerfeld, nor other scientistsdeveloped this theory any further. But science is developing. And here are some conclusions [10, c. 110]. " Now it is generally excepted in nuclear physics that a special role in occurrence of significant underpressures and condensations of nucleus levels plays quasiclassical quantization of movement on multivariate periodic orbits... These orbits are unwound and get tangled because of quantum fluctuations of system and only elementary orbits survive ".
In connection with this a question arises: what condition should these orbits satisfy, so that the condition of movement could be considered stationarity. The answer to this question can be found in work [11].It says, that from the experiments on particles dispersion it is found, that in all the investigated cases resonances arose under the condition of commensurability of the wave length of De Broilia with the geometrical sizes of system, irrespective of a nature of interaction.
Thus, the experiment shows, that irrespective from the nature of interaction the movement of the structural elements inside of a dynamically equilibrium system is carried out on wave orbits of the elementary type. Eventually, our problem is to find conditions at which the circular orbit will be stationarity, have the maximal durability and also to find in what ratio should be the parameters of these orbits among themselves within the limits of the wave relativistic quantum theory.
We shall start to solve the problem by analyzing of the elementary example of interaction, in particular we shall consider force with which two identical relativistic charges einteract, moving in parallel to each other with identical speed v relatively the laboratory system of readout.
The resulting force _{} of interaction of the charges moving in parallel consists of, [12, p. 198] two components: electric _{} and magnetic _{}. For the likecharges the resulting force in a projection to an interaction axis is equal
F = F_{e}  F_{м},
F or heteronymic charges
F = – F_{e} + F_{м}.
But because
_{},
And
_{},
Then, having taken into account, that 1/ (e_{о}m_{о}) = c^{2} where e_{о} and m_{0 }– dielectric and magnetic permeability of vacuum, and c – speed of photons in it. The sum equals:
_{}, (1.1)
where r – distance between charges. The mark (+) means, that the likecharges repel. The mark (–) shows that heteronymic charges are attracted.
Excluding e_{о}, this formula can be presented in the following way:
_{}, (1.2)
The size in brackets has dimension of mass. The physical sense of it is not clear, therefore at the beginning, having designated
_{},
where i – presumably, virial factor, we receive
_{}. (1.3)
On the other hand, if the right part of the formula (1.1) will be increased and divided by c and allocate in it the following size
_{}, (1.4)
where h_{о} = 7,6957×10^{37} j×s or the same h_{о} from the formula (1.2), for example
_{}, (1.5)
Then the resulting force
_{}. (1.6)
Comparing (1.3) and (1.6), we find
_{}, (1.7)
where i – virial factor.
But _{} is a module of the potential energy interaction. Hence,Dmc^{2} = W_{c}is the energy of charges connection, whereDm– defect of mass.
By definition, the defect of mass Dm = m – m_{o}, where m and m_{o} – masses of a charge in movement and in a condition of rest. But as _{}, then the energy of connection is
_{}, (1.8)
where E and Т – full and kinetic energy of a charge.
Resolving formulas (1.7) and (1.8) relatively to r, we receive, that the distance between the charges will be expressed by the formula
_{}, (1.9)
where r_{oe} = ho / (m_{oe} c) – classical radius of a charge. As h_{о} = m_{oe} cr_{о}_{e}, the formula (1.7) can be presented in the following way:
_{}. (1.10а)
Having designated r/r_{oe} = e, we receive:
_{}. (1.10)
If in the system under consideration which consists of two heteronymic charges, a positive charge has mass m_{o2} much greater than mass m_{o1} of a negative charge, then in the formula (1.10). E_{01} is an own energy of a negative charge which will start to go around mass m_{o2}; and if the charge masses are equal, then E_{о}_{1} is an own energy of the given mass because under the action of the attraction force both charges will come into movement relatively the common mass center with speed v. As a result, the system of charges gets the moment of an impulse. In relativistic dynamics there is no ready formula of the relativistic moment of an impulse. It needs to be found somehow.
In the elementary case of a plainly – circular movement relatively the axis z the moment of an impulse
J_{z} = p_{j} × r, (1.11)
where r – radius of a circular orbit; p_{j} – a full impulse of mass mo1 relatively the axis of rotation.
The full impulse of mass m_{o1} in a circular orbit can be presented as the sum
p_{j }= p_{e} ± p_{s}, (1.12)
where р_{е} – impulse of mass m_{01} with delayed spin; p_{s} – impulse received by a charge when spin disinhibition: during the process of spin disinhibition the charge either increases the orbital speed, or reduces it. p_{s} – is a small additive, but it can be observed by apparatus: spectral lines of radiation slightly fork.
As a whole the module of a full impulse
_{}, (1.13)
Having substituted formulas (1.9) and (1.13) in the formula (1.11), we receive, that the full moment of an impulse in a projection to the axis z
_{}. (1.14)
Thus:
1) if v = c, and i = 1, then J_{z} = h_{o}. Only quantums of electromagnetic radiation have such parameters (photon spin always equals 1);
2) if v = c, and i = 2, then J_{z} = h_{o}/2: the full moment of an impulse degenerates at the spin moment of a charge (spin of a charge is always equals ^{1}/_{2}). Such process takes place during the formation of charges by photons with the spin S = 1 in a strong electromagnetic field of a nucleus. As a result, the charges are born in pairs, each of which has a spin s = 1/2 in a projection to the axis z. Thus, the analysis of the formula (1.14) shows, that virial factor of a circular relativistic orbit i = 2 (strictly). It is an important result, but not final, therefore, we shall multiply the formulas (1.8) and (1.14). As a result we receive:
_{}. (1.15)
On the other hand, given that an impulse _{}, And Е_{о}= m_{o} c^{2}, we have
_{}. (1.16)
Uniting the formulas (1.15) and (1.16) and taking into account that W = E – Е_{о} we find the formula connecting the moment of an impulse and energy:
_{}. (1.17)
Let's allocate in this equality the first part, having accepted i = 1, and write it down in the following way:
_{}. (1.18)
But according to Zommerfeld, full energy E and full energy W of the system of charges are connected by formula E = W – U, and having substituted E in the formula (1.18), we receive the equation of a kind
_{}. (1.19)
In this equation vector _{} has the modulea= h_{o}/J_{z}and vector basis α_{i}, wherei= 1, 2, 3. The factora  is a constant of a thin structure and depends only on the charge nature, for example, for hydrogen atom electron_{}. Hence,J_{z} = h_{i}is the moment of an impulse of a chargeof an i version in its basic condition.
Basically the equation (1.19) can be reduced to the equation such as the Dirack equation. It is even necessary in order to use the already known decisions of the Dirack equation. For this purpose we shall proceed to the spherical system of coordinates.
In the spherical system of coordinates the operator is:
_{}. (1.20)
where _{} – vector, the module of which a =h_{o}/h_{i}. If the operator (1.20) is divided into the module of a vectorathen we receive:
_{},
where _{} – individual vector,k = ±(j +^{1}/_{2}),wherej– quantum number of the full moment of an impulse. But as h/a_{i }=h_{i}then the equation (1.19) in operators of the quantum mechanics becomes
_{}. (1.21)
If in this equation we accept h_{i} = h, where h – a constant of Planca, and a_{i }= h_{о}/h = = 7,6957^{37}/1,0545×10^{34} = 9,2976×10^{3} (1/a = 137,031) then we receive the Dirack equation for the hydrogen atom electron.
And in general, the equation (1.21) describes a charge energy spectrum of any mass if for this mass h_{i} and _{} are known.
In particular, for the hydrogen atom system, the full energy
_{}, (1.22)
Where N = 0, 1, 2 …. If N = 0, the conditions turn out to be the most simple (bifurcation of levels does not occur). According to Zommerfeld these conditions correspondwith the circular orbits. Having done some algebraic simplifications we find, that
_{}.
Because under the radical a^{2}/k^{2} < 1 (always), then the radical can be presented as sedate lines, in particular
_{}
_{},
where∑ О(a) – the line sum in square brackets. Then
_{}.
Comparing this result with the formula (1.10) or (1.10а) we come to a conclusion, that _{}, i.e.
_{}. (1.23)
Thus it appears, that the virial factor for a circular relativistic orbit equals two.
In order to find the reason for the circular orbit stationarity, we shall address to the analysis of decisions of the equation (1.21).
The decisions of this equation are the wave functions. When the negative charge with mass mo1 goes over the arch S of a radius r circle relatively a positive charge with mass mo2, its wave functions become:
_{}, (1.24)
where k = 2p/l – wave number; l – length of a wave of mass m_{01}; φ_{о} – initial phase. For circular orbit S = 2pr. From here, h_{i}/р = l, where D = l/2p, and р = h×k where р – impulse of mass m_{о}_{1}. At the moment of time t = 0 initial phase φ_{о} = 0, i.e. for stationarity conditions we find, that ψ/ψ_{0} = = cos kS as kS = 2p (pr/h where pr = J_{z} – projection of the full moment of an impulse to the axis z, then ψ/ψ_{0} = cos 2p _{}. The functionw = ψ/ψ_{0}^{2} = cos^{2} 2p_{}, defines the density of probability to find out mass m_{o1}on some distancerfrom the axisz. The density of probability, apparently, is maximal when the numbern_{j} = J_{z}/ h_{1} accepts the whole and halfinteger values.
In order to define the most probable distances it is necessary to connect e and n_{j}. For this purpose, resolving the formula of the moment of an impulse
_{} (1.25)
relatively b, we find, that
_{}. (1.26)
On the other hand, from the formula (1.9), having accepted i = 2, we define, that
_{}. (1.27)
Equating (1.26) and (1.27), and solving the received equality relatively n_{j}, we define, that _{}. Whenceit follows that _{}.
Let's coordinate the received result to de Broglie wavelength. For a charge with the moment of impulse J_{z }= h_{о} length of the De Broilia wave D = h_{о}/p. Having substituted here h_{о}= m_{o} r_{oe} c and p according to the formulas (1.13), we receive
_{}. (1.28)
Let's enter instead of b number n_{j} into this formula according to the formulas (1.26). As a result of some simple transformations we have
_{}. (1.29)
But n_{j} – ^{1}/_{4}= e, therefore l = r_{oe} e/ n_{j} or l n_{j} = r_{oe} e = r. From here it follows, that
l n_{j} = 2p r. (1.30)
It is a wellknown result used in the quantum mechanics for an evident illustration of the De Broilia waves utility.
If as the standard of measurement of the moment of an impulse in the system of the interacting charges we choose quantum of a charge action in its basic condition, and it can be received from the formula (1.14), then D = h_{i}/р. From here, bearing in mind that h_{i }= h_{o}/a taking into consideration the formulas (1.13) and (1.26) we receive
_{}, (1.31)
i.e. we find, that
or
l n_{j} = 2p r_{ef}, (1.32)
where r_{ef} = r/a_{i} – effective (characteristic) radius of a circular condition. At hierarchical transitions (from the structure of one level to the structure of other level) a_{i} ® 1. Hence, in limiting transition r_{ef} = r, the formula (1.30). Besides from the formula (1.31) it follows that r_{oe}/a_{i} = r_{o}_{ ef} represents some new scale standard, and r_{o}_{ ef} e = r_{ef}. As a result it turns out that formula (1.32) expresses a strict condition of a circular orbit stationarity: the circular orbit is steady in case the whole or halfinteger number of lengths of the De Broilia waves is stacked on its characteristic length. This condition is fair without any restrictions. However, it is necessary but insufficient. It is necessary to establish, how many of such l/2 should be stacked on length of a characteristic circle, so that the orbit was both stationarity and maximum strong. Let us present the formula (1.31) as equality
_{}. (1.33)
Having designated l/r_{o }_{ef} = k_{g}, we receive the equation
n^{2}  k_{g} n – 1/4= 0. (1.34)
Its decision looks like
_{}. (1.34a)
The greatest interest in this decision represent numbers k_{g} and n, characteristic for the basic harmonic of wave process. In the basic harmonic of a full wave cycle one loop occupies half of the wave's length. As k_{g} = l/r_{ef}, then k_{g }= 2 for this harmonic.
Note: as k_{g} corresponds to the number of a wave harmonic, quantizating de Broglie wavelength so under its physical content k_{g} can accept only integer values. Having substituted k_{g} = 2 in the formula (1.34a), we receive
_{}.
Whence we find, that n_{1} = 2,118034, n_{2} = – 0,118034. In the given situation the full moment of an impulse is equal to orbital (in projection to the axis z), but increased owing to the system relativizm on size 0,118034. In a non – relativistic case when n > >1/4, it follows from the equation (1.34), that n_{1} = 2, and n_{2} = 0. It should be noted that the spin of the rotating charge is perpendicular to the orbital moment in the given situation. Such orientation of a spin charge stabilizes a circular orbit relatively the axis z.
Let's turn the spin, having directed it to the side opposite to the orbital moment. Then we receive n_{1} = 2,118034 – 0,5 = 1,618034; n_{2} = – 0,618034. Being guided by these numbers, from the formula l/r_{o }_{ef} = e/ n = k_{g} we find, that e = k_{g }n. Having substituted here k_{g} = 2 and n = 1,618034, we receive e = r/ r_{o }_{ef} = 2 ×1,618, Whence r/1,618 = 2 r_{o }_{ef} or 0,618r = 2r_{o}. But r_{o }_{ef} =l/2, therefore _{}. This length is the side of a regular tensquare, incirculed in a circle of a condition and, apparently, it is measured by pieces, multipler_{o }_{ef}. It divides a circle into 10 equal parts, the length of each is equal to the length of the De Broilial. For this reason we find, that 10l= 2p r, i.e. 10D= r, or_{}. ButD/2 = r_{o }_{ef}, hence, 20 r_{o }_{ef}= rore= r/ r_{o }_{ef} = 20, i.e.e= 2×10¢.These are the parameters of the initial condition (limiting for the given system).
Comparing this number with the initial precondition, let us see the result. For this purpose , having substituted e = 20 in the initial formula e = k_{g }n, we find, that n = 10, if k_{g} = 2 and k_{g} = 12,36068, if n = 1,618034. As a result, there is discrepancy. Trying to eliminate the given mismatch, we shall present the last result as follows: k_{g} = 12,36068 = 2×10×0,618034. Then e = = 2×10×0,618034×1,618034 = 20. The received result represents a principle of the wave orbit organization which is carried out on the basis of the wave frequency rates of gold section. Here it is written down in the numerical variant, and in general it is possible to present it the formula
e = k_{g} N D_{1} n_{1}. (1.35)
Where l_{1} – length wave of a link of a wave orbit, the radius of which is accepted for the identity element (l_{1} = 0,618034); n_{1} = 1,618034 – the moment of a in impulse appropriate to the length l_{1}; N – the number of parts in length D_{1}, making the wave orbit which has the linear form at N = 1, and in other cases they get the form of the polygon incirculed in a circle of a condition, except the case when N = 2. The moment of an impulse is in direct proportion to the number of parts N, for example, the number of parts N = 10 for an orbit in the form of a tensquare. Its perimeter S_{N} = = N l_{1} = 10×0,618034 = 6,18034. Hence, the full moment of an impulse n_{N} = S_{N}×1,618034 = = 6,18034×1,618034 = 10. As a result e = k_{g }n = 2×10 = 20.
In such a way it is possible to form an infinite set of wave orbits. However, not all of them will be steady. Only those orbits the perimeter of which S_{N} is in wave frequency rate to the perimeter of an initial orbit with perimeter S_{1} = 10l_{1} will be stable. For such orbits the number N_{x} = 10^{x}. Hence, according to (1.35)
e = 2×10^{x}, (1.36)
where in general x = k/m, where at m = 1, 2, 3 k = 0, 1, 2, 3.
If the spin of mass m_{o1} is directed to the side of the orbital moment of an impulse of a system with n = 2,118034, then its full moment of an impulse n = 2,118034 + 0,5 = 2,618034 = (1,618034)^{2}, and n_{2} = – 0,118034 + 0,5 = 0,381966 = (0,618034) ^{2}. Having substituted n = (1,618034) ^{2} in the formula e = k_{g }n, we receive e = k_{g} (1,618034) ^{2}. In this formula the tendency of change of the impulse moment under the indicative law is visible. If this tendency is kept, then, in a more general case
e = k_{g }(1,618) ^{y}, (1.37)
where y = k ¢/m ¢.
Having substituted the received result in the formula (1.35), we receive
N_{y} = (1,618)^{y}.
For the orbits with the number of the parts determined by the formula
N_{S}= N _{y} N_{x} = (1,618) ^{y} 10^{x}. (1.38)
Parameter e from formulas (1.35)
e = 1,618 ^{y} k_{g} 10^{x}. (1.38à)
The formula (1.38à) shows, that energy of a charge connection (mass m_{o1}) in a system can change either due to the change of r, see the formula (1.36), or due to the change of n, see the formula (1.37), where at m ¢ = 1, 2, 3 … k ¢ = 0, 1, 2, 3 …. And if both the moment of an impulse n and distance r change, then the parameter of connection e is defined from the formula (1.38à).
The analysis of the static distributions of atom and nuclear conditions has shown, that they contain maxima of the formula (1.36), characteristic for the 8th and 12th harmonics. As for the formulas (1.37) and (1.38a), these should be checked out. However, before we proceed to this part of the question, let's take a look what is going to be if we increase the moment of an impulse n = 2,118034 and further in each 0,5 in both sides. As a result, we get the whole set of new decisions which can be described in the following way:
_{}, (1.40)
where k_{g }= … – 4; –3; –2; –1; 0; 1; 2; 3; 4; …
It is necessary to note, that at negative values k_{g} the spin is oppositely directed to orbital moment of the initial condition, and at positive k_{g} – on the line of the orbital moment. Besides spin conditions are realized at odd k_{g}, and non – spin – at even k_{g}.
The greatest attention is paid to the decisions, where k_{g} = 0, 1, 2 and 3. In particular:
for k_{g} = 0 n_{1} = 1,118034, n_{2} = – 0,118034;
for k_{g} = 1 n_{1} = 1,618034, n_{2} = – 0,618034;
for k_{g }= 2 n_{1 }= 1,118034, n_{2} = – 0,118034;
for k_{g} = 3 n_{1} = (1,618034) ^{2}, n_{2} = (0,618034) ^{2 }etc.
Analyzing these conditions, we shall note the following. The condition for which k_{g} = 1 is a condition of gold section in pure state. Such conditions are connected among themselves by the formula (1.36). The condition for which k_{g} = 3 is also a condition of gold section in pure state. Such conditions are connected among themselves by the formula (1.37). The condition for which k_{g} = 2, is a mixed type condition. Conditions of such a type are connected among themselves by the formula (1.38a). Here much depends on the external conditions in which the system exists. They can greatly influence on the spin orientation and the realization of conditions. As for the condition where k_{g} = 0, when the spin is oriented towards the orbital moment it can be reduced to the condition at k_{g} = 1, however, conditions with k_{g} = 0 are not realized, as in this case there is no secondary quantization, i.e. quantization of de Broglie wavelength, and there is no wave harmonics. Thus, the number k_{g} can accept values k_{g} = 1, 2, 3. As for the higher k_{g}, their conditions were not investigated.
For empirical check of theoretical conclusions, in particular formulas (1.36), (1.37) and (1.38à) we shall calculate first the expected wave frequency rates for the 8th and 12th harmonics and compare them with the periodogram analysis data.
So: 1) the formula (1.36.) e = k_{g}×10 ^{k/m}. Its wave frequency rates look as follows:
_{}.
From here, having reduced k_{g} and having taken the decimal logarithm, we receive:
_{}.
2) The formula (1.37): e = k_{g} (1,618)^{k }^{¢}^{/m }^{¢}. Its wave frequency rates look as follows:
_{}.
From here
_{}.
The results of the periods Т and Т¢ calculations under these formulas are submitted in table 1.
Table 1
Wave frequency rates of formulas (1.36) and (1.37) for m = 8 and m = 12
k = k ¢ 
The formula (1.36) 
The formula (1.37) 


Т 
Т ¢ 


m= 8 
m= 12 
m ¢ = 8 
m ¢ = 12 
1 
0,125 
0,08333 
0,026123 
0,01742 
2 
0,25 
0,16666 
0,05225 
0,03483 
3 
0,375 
0,25 
0,07837 
0,05225 
4 
0,5 
0,33333 
0,10448 
0,06966 
5 
0,625 
0,41666 
0,13062 
0,08708 
6 
0,75 
0,5 
0,15674 
0,10449 
7 
0,875 
0,58333 
0,18286 
0,12191 
8 
1,000 
0,66666 
0,20899 
0,13932 
9 
– 
0,75 

0,15674 
10 
– 
0,83333 

0,17415 
11 
– 
0,91666 

0,19157 
12 
– 
1,00000 

0,20899 
From the formula (1.38a):
_{},
i.e. its' periodogram consists of the periodogram sum formulas (1.36) and the formulas (1.37), within the limits of Т = 0 ¸ 0,25 on the average DТ = 0,03.
Taking into consideration all the facts mentioned above, we shall try to look for the wave frequency rates of gold section among nuclear conditions. As a basis of the researche we shall accept ionization potentials U given in the directory [13], see page 5865 where values of 952 potentials of all atoms are given. As far as their conditions are of great interest to us, so in order to select the most probable ones the further research will done in the following order.
First all data will be decalogarifmied. Then, using the received results, we shall construct the histogram, having accepted for this purpose an interval of accumulation 0,005. The histogram constructed with such a step appeared to be very long, therefore here it is represented in table 2 where the sums of the data, falling on each interval are given in lines, and the first number corresponds to lg U = 0,60, and for each two subsequent numbers the interval is 0,01.
Table 2
The histogram of the empirical data










10 






0,69 

20 


0,72 





30 


1 
1 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
1 
0 
3 
2 
1 
1 

2 
2 
1 
2 
1 
1 
6 
0 
4 
2 
0 
0 
1 
0 
1 
0 
1 
2 
3 
0 
1 
1 
0 
2 
2 
3 
1 
3 
0 
0 
5 

3 
1 
0 
2 
0 
1 
1 
0 
2 
0 
0 
2 
3 
0 
2 
1 
0 
0 
1 
1 
0 
1 
2 
0 
3 
3 
0 
2 
2 
2 
0 

4 
1 
0 
2 
1 
3 
1 
4 
2 
0 
0 
0 
1 
1 
0 
1 
0 
3 
0 
1 
1 
0 
1 
1 
2 
1 
2 
2 
0 
1 
1 
1,20 
5 
2 
2 
1 
2 
2 
1 
4 
0 
0 
0 
1 
3 
1 
1 
3 
3 
2 
1 
0 
1 
3 
2 
4 
0 
1 
2 
3 
1 
2 
0 

6 
0 
0 
1 
2 
1 
1 
1 
3 
1 
3 
3 
1 
0 
1 
0 
0 
3 
2 
1 
3 
3 
1 
2 
1 
3 
2 
1 
3 
2 
0 

7 
2 
1 
0 
4 
2 
2 
3 
1 
5 
2 
0 
1 
0 
3 
3 
3 
1 
0 
2 
2 
2 
1 
4 
0 
1 
1 
2 
2 
2 
1 

8 
2 
1 
8 
2 
2 
1 
3 
0 
3 
1 
2 
4 
3 
1 
3 
1 
1 
5 
4 
3 
0 
4 
2 
1 
3 
3 
1 
4 
5 
2 
1,80 
9 
1 
4 
5 
2 
0 
2 
5 
1 
0 
3 
3 
4 
4 
2 
0 
5 
2 
5 
2 
1 
7 
2 
3 
3 
3 
3 
0 
1 
3 
5 

10 
1 
4 
3 
1 
4 
2 
1 
5 
3 
5 
8 
0 
6 
1 
3 
2 
3 
2 
5 
4 
0 
3 
10 
1 
1 
13 
0 
2 
1 
5 

11 
2 
2 
8 
3 
0 
1 
4 
2 
3 
11 
2 
2 
9 
3 
0 
6 
1 
3 
10 
2 
8 
3 
2 
6 
2 
1 
5 
2 
6 
2 

12 
1 
3 
1 
5 
3 
4 
1 
3 
3 
1 
1 
1 
3 
3 
5 
2 
1 
2 
2 
1 
4 
0 
3 
1 
1 
2 
3 
2 
2 
4 
2,40 
13 
0 
2 
1 
2 
3 
2 
1 
1 
2 
0 
1 
2 
1 
1 
1 
1 
2 
1 
2 
1 
0 
1 
0 
3 
1 
2 
1 
1 
1 
2 

14 
1 
1 
0 
2 
1 
2 
2 
0 
2 
0 
4 
1 
2 
0 
0 
2 
1 
2 
0 
2 
1 
2 
1 
1 
0 
1 
3 
1 
2 
1 

15 
2 
1 
0 
2 
1 
0 
2 
1 
1 
2 
2 
1 
1 
0 
2 
0 
1 
1 
2 
1 
1 
0 
1 
2 
1 
1 
0 
2 
1 
1 

16 
1 
0 
1 
2 
0 
3 
0 
0 
0 
2 
0 
1 
1 
1 
0 
0 
1 
2 
1 
0 
2 
0 
1 
1 
1 
1 
0 
3 
0 
0 
3,0 
17 
2 
0 
1 
1 
0 
1 
0 
2 
2 
0 
1 
1 
1 
1 
1 
1 
1 
2 
0 
0 
2 
1 
1 
1 
0 
1 
3 
0 
1 
1 

18 
1 
1 
1 
1 
2 
1 
0 
2 
0 
2 
1 
0 
1 
3 
0 
2 
0 
2 
1 
1 
1 
2 
0 
2 
1 
1 
1 
0 
4 
0 

19 
1 
0 
2 
2 
0 
2 
0 
1 
0 
2 
0 
0 
3 
0 
1 
1 
0 
1 
1 
0 
0 
1 
0 
0 
0 
1 
0 
1 
0 
1 

20 
0 
0 
0 
1 
0 
0 
0 
1 
0 
1 
0 
1 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
1 
3,60 
21 
0 
0 
0 
1 
0 
0 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
1 
0 
0 

22 
1 
0 
0 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
1 

23 
0 
0 
1 
0 
0 
0 
1 
0 
0 
1 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
1 
0 
1 
0 
4,05 
The histogram given in the table is the initial base for getting helpful information. In order to extract it we shall apply one of the periodogram analysis method, in particular, the method of interval coverings developed by Petrunenko V.V. For details see work [14] or go to www.stavedu.ru section " Scientific associations (Cycles) ". The periodograms constructed by this method, one in the interval lg U = 0,69 ¸ 3,6, and the other  in the interval lg U = 0,72 ¸ 3,6 (take as datum 0,03) are presented on fig.1 and 2.
Fig. 1. The periodogram of Petrunenko Vasily for the interval
t = 0,69¸3,6; у = (<n> –2,166)×50
Fig. 2. The periodogram of Petrunenko Vasily for the interval
t = 0,72¸3,6; у = (<n> –2,19)×50
From fig.1 it is clear, that period Т = 0,21 can be divided successfully into two parts. As a result we receive Т = 10,5; 5,25. Each of these periods admits repetitions. For example, repeating the period Т = 5,25, we find Т = 5,25; 10,5; 15,75; 21; 26, which fully complies with the periodogram data and theoretical conclusions, see table 1. As a result we come to the conclusion, that lg U_{i} – lg U_{j} = (0,21/m¢)×k¢, where at m ¢ = 1, 2, 3 … k ¢ = 0, 1, 2, 3 …. From here we find, that
U_{i}/U_{j} = (1,618) ^{k}^{¢}^{/m}^{¢}. (1.41)
The formula (1.41) is an empirical analogue of the theoretical formula (1.37). The similar analysis (fig. 2) shows, that period Т = 0,24 can be also divided into parts. When dividing successfully we receive Т = 12, 6, 3. Repeating the period Т = 6, we receive T_{i} = 6, 12, 18, 24 in full conformity with the periodogram (see fig. 2) and the theoretical data of the formula (1.36). Hence, lg U_{i} – lg U_{j} = k/m, where at m = 1, 2, 3 … k = 0, 1, 2, 3 …. As a result we find, that
U_{i}/U_{j} = 10 ^{k/m}.(1.41)
Fig. 3. Shooster's periodogramfor the interval t = 0,72¸3,6
Fig. 3 presents the periodogram constructed by the method of Shooster [15] with the step DТ = 0,01 in the same data interval, as periodogram in fig. 2. The initial data for its construction are taken from the previous histogram at the double interval of accumulation. Comparing fig.2 and fig.3, we find, that their results do not contradict. However it is visible, that Shooster's periodogram has a trend (a line without breaks). The correlation analysis shows with 99,9% probability, that function у = 0,02878 х^{2,1981}, where х = 10×Т describes the trend. Subtracting thetrend, we receive a line with the average value <y> = 9,302. The relation of the maximal emission appropriate to period Т = 0,24 to average is equal 108,858/9,302 = 11,702.. The critical size of this relation for n = 25 on a significance value a = 0,01 is equal to r_{кр} = 5,582 [16]. Since у_{м}/<y> > r_{cr, }then the period T= 0,24 passes with probability of more than 99%, together with it passes periodogram conclusion, i.e. the formula (1.42) which fully correlates to the formula (1.36). Besides formula (1.36) proves to be true and on the nuclear level [25] with probability no worse than 95 %. As for the degree of the formula (1.41) significance, we shall construct some more periodogram, displacing each time the datum to the right on 0,01, in order to define the formula by the same method of interval coverings. Then, having selected the similar ones, we shall combine them. If at such addition regular maxima amplify, and irregular become weak, then the regular picture extracted from the statistical distribution can be trusted. The periods received by a deduction of initial phases of the regularly repeating periodograms can be relied. They are a good addition to the revealed law as in statistical distribution regular recurrence of periodograms is not a casual event.
Fig. 4 represents three periodograms with initial phases j_{о} = 0,69; 0,79; 0,81. They were selected as coincident. The periodogram which is the sum of first three is represented at the bottom of the fig. It shows maximum with periods Т = 0,05; 0,10; 0,15; 0,21; 0,25; 0,30; 0,07 and 0,17, coinciding with the formula (1.37) data. there is also a period Т = 0,03. Fig.5 shows a periodogram, which is the sum of the periodograms with initial phases 0,72 and 0,80. The maxima with periods Т = 0,04; 0,06; 0,08; 0,16; 0,20  0,21; 0,24 and 0,30 can be seen in it. Fig.6 shows the result of addition of three more periodograms. Formula (1.36) describes well the data of fig. 5 and fig. 6. The displacement of the periodograms (fig. 1 and fig. 2) on average on DТ = 0,03 is the empirical confirmation of the formula (1.38à). Thus, summing up this part of the work, it is possible to assert that the empirical data completely confirm the theoretical conclusions of formulas (1.36), (1.37), (1.38a). It appeared that resonances of the formula (1.36) alternate with resonances of the formula (1.37), and from time to time they are superimposed on them. Much work has been done to divide them (as searches were conducted at random). In particular, the application of method x^{2} to the common statistical distribution has not allowed to reveal its thin structure though and it was clear that resonances of the formula (1.36) are a little bit impaired, see [6]. The situation has cleared up with the occurrence of the method offered by the author of this article [14]. As a result there appeared an opportunity of concrete calculation of separate nucleus levels. Such calculations were done for nucleus _{},_{}And partly for_{}. The results of these calculations are submitted in tables 24. It should be noted that energy of nucleons levels in a nucleus and electrons in atom were calculated using one and the same formula
_{},
where W_{o} = 938,28 МeV.
The comparison of the empirical and estimated data shows that the formula (1,38à) gives the best results: the average deviation d from the empirical data makes ~0,5% whereas for the formula (1.37) <d> » 1%, it is necessary to note that the calculations were made at k_{g} = 2.




Fig. 4. Periodograms with initial phases j_{о} = 0,69; 0,79 и 0,81
and their total result



Fig. 5. Periodograms with initial phases j_{о} = 0,72; 0,80
and their total result
Fig. 6. The total result of the periodograms with initial phases
j_{о} = 0,70; 0,76 и 0,82
The good accord of the empirical and estimated data shows that the theory of the decalogoriphmic periodicity phenomenon is developed correctly.
It follows from the theory that decalogoriphmic periodicity is a universal phenomenon, in particular, it is common for all dynamically equilibrium microsystems. It is common because common principles for all microsystems of the organization of quantum conditions are at the basis of the theory. The quantum condition is especially steady in case it is organized within the framework of wave frequency rates of gold section. A prominent feature of such organization is the division of a characteristic circle of the initial length condition of the De Broilia wave of the rotating charge (mass) on 10 equal parts. The conditions organized in such a way appear to be spatially and energylike, i.e. they are in spatial and power frequency rates among themselves. The phenomenon of the decalogoriphmic periodicity reflects this frequency rate. In its turn the construction of conditions (wave orbits) gold section appeared to be possible because the interacting objects (in this case charges):
1. show wave properties;
2. their mass depends on the relative movement speed ;
3. the rotating charge has a spin.
In other words, the decalogoriphmic periodicity is a direct consequence of the charge system wave relativism.
.
Table 3.
Power levels of a nucleus _{}
Formula (1.38a) 
Formula (1.37) 

№ 
W_{empir. }(KeV) 
W_{value calculated. }(KeV) 
d, % 
k/m 
k¢/m¢ 
W_{value calculated. } (KeV) 
d, % 
k¢/m¢ 

1 
4518 
4507 
0,24 
13/8 
21/48 
4608 
1,99 
98/12 

2 
4276,7 
4286,2 
0,22 
13/8 
13/24 
4253 
0,55 
100/12 

3 
3908,2 
3908,5 
0,0077 
14/8 
1/8 
3925 
0,43 
68/8 102/12 

4 
3052,8 
3035,4 
0,57 
15/8 
1/16 
3085,9 
1,08 
72/8 108/12 

5 
2987,4 
3005 
0,59 
15/8 
1/12 
2964,6 
0,76 
109/12 

6 
2718,2 
2718,4 
0,0074 
15/8 
7/24 
2736,1 
0,66 
74/8 111/12 

7 
2560,8 
2559,7 
0,043 
15/8 
5/12 
2575,4 
0,61 
75/8 

8 
2230,8 
2208,7 
0,99 
16/8 
1/8 
2239 
0,37 
116/12 

9 
1637,7 
1623,5 
0,86 
17/8 
2/12 
1624,5 
0,81 
124/12 

10 
776,5 
776,7 
0,026 
19/8 
4/8 
772,8 
0,48 
95/8 



<0,355> 


<0,77> 

Electrons W (eV) (potentials of ionization) 





11 
195 
195,8 
0,41 
48/8 
3/8 
195/98 
0,50 
349/12 

12 
170 
170,7 
0,41 
49/8 
1/16 
170,3 
0,18 
235/8 

13 
145 
146,9 
1,31 
49/8 
2/8 
142,2 
1,93 
238/8 357/12 

14 
115 
117,0 
1,74 
50/8 
7/8 
116,4 
1,22 
362/12 

15 
94 
93,2 
0,85 
51/8 
1/8 
93,33 
0,71 
245/8 

16 
72 
71,3 
0,97 
78/12 
1/12 
71,92 
0,14 
374/12 

17 
59 
58,8 
0,34 
79/12 
1/12 
58,85 
0,17 
379/12 

18 
37,48 
37,77 
0,77 
81/12 
5/24 
37,86 
1,01 
260/8 390/12 

19 
16,908 
16,90 
0,047 
57/8 
1/12 
16,977 
0,41 
410/12 

20 
8,993 
9,038 
0,50 
59/8 
3/16 
8,937 
0,62 
284/8 420/12 




<0,73> 



<0,69> 


_{}.
Table 4.
Power levels of a nucleus _{}
Formula (1.38a) 
Formula (1.37) 

№ 
W_{empir}_{. }(KeV) 
W_{value calculated. } (KeV) 
d, % 
k/m 
k¢/m¢ 
W_{value calculated. }(KeV) 
d, % 
k¢/m¢ 

1 
5034,2 
5031,9 
0,046 
13/8 
5/24 
4993,1 
0,046 
64/8 96/12 

2 
4516,5 
4551,9 
0,78 
13/8 
5/12 
4427,2 
1,98 
66/8 99/12 

3 
4017,2 
4007,3 
0,25 
14/8 
1/12 
4085,9 
1,71 
– 101/12 

4 
3534,4 
3482,6 
1,46 
14/8 
3/8 
3480,4 
1,53 
70/8 105/12 

5 
3067,4 
3065,9 
0,049 
15/8 
1/24 
3085,9 
0,60 
72/8 – 

6 
2618,5 
2611,6 
0,26 
15/8 
3/8 
2628,6 
0,39 
– 112/12 

7 
2190,6 
2208,8 
0,83 
16/8 
1/8 
2151,0 
1,81 
78/8 117/12 

8 
1788 
1786,9 
0,062 
25/12 
2/12 
1715,9 
0,44 
81/8 

9 
1415,1 
1418,5 
0,24 
26/12 
3/12 
1411,8 
0,23 
85/8 – 

10 
1076,5 
1079,4 
0,27 
18/8 
5/12 
1087,8 
0,10 
– 134/12 

11 
775,7 
777,7 
0,26 
19/8 
4/8 
773,6 
0,27 
95/8 – 

12 
517,9 
523,8 
1,14 
21/8 
1/8 
507,8 
1,95 
102/8 153/12 

13 
307,21 
305,27 
0,64 
34/12 
3/12 
301,5 
1,86 
– 166/12 

14 
148,41 
147,49 
0,62 
38/12 
2/12 
146,5 
1,29 
– 184/12 

15 
44,91 
45,07 
0,36 
29/9 
7/16 
45,8 
1,98 
142/8 – 




<0,48> 



<1,08> 


Potentials of ionization (eV) 

16 
160 
160,7 
0,44 
48/8 
3/16 
160,38 
0,24 
236/8 354/12 

17 
140 
138,3 
1,2 
49/8 
4/8 
142,20 
1,57 
238/8 

18 
120 
120,5 
0,42 
50/8 
3/16 
118,72 
1,07 
241/8 

19 
104 
103,7 
0,29 
50/8 
4/8 
103,17 
0,80 
– 365/12 




<0,59> 



<0,92> 


Table 5.
Power levels S and рresonant neutrons of a nucleus
Formula (1.38a) 
Formula (1.37) 

№ 
W_{empir}_{. }(KeV) 
W_{value calculated. } (KeV) 
d, % 
k/m 
k¢/m¢ 
W_{value calculated. }(KeV) 
d, % 
k¢/m¢ 
1 
0,19 ± 0,02 
0,190 
0 
72/8 
7/16 
0,1924 
0,126 
348/8; 522/12 
2 
0,34 ± 0,07 
0,341 
0,89 
70/8 
5/12 
0,347 
0,26 
338/8 
3 
0,41 ± 0,04 
0,417 
1,7 
70/8 
0 
0,4076 
0,5854 
503/12 
4 
1,52 ± 0,03 
1,513 
0,46 
65/8 
5/16 
1,531 
0,72 
470/12 
5 
1,95 ± 0,04 
1,958 
0,41 
64/8 
3/8 
1,947 
0,154 
464/12 
6 
3,01 ± 0,03 
3,005 
0,17 
63/8 
1/12 
3,027 
0,56 
302/8; 453/12 
7 
3,53 ± 0,02 
3,483 
1,33 
62/8 
3/8 
3,553 
0,65 
449/12 
8 
4,58 ± 0,05 
4,552 
0,61 
61/8 
5/12 
4,612 
0,70 
295/8; 443/12 
9 
4,85 ± 0,04 
4,855 
0,10 
92/12 
1/12 
4,898 
0,99 
294/8 
10 
6,62 ± 0,04 
6,577 
0,65 
60/8 
2/8 
6,616 
0,060 
294/8; 433/12 
11 
6,71 ± 0,04 
6,710 
0 
90/12 
5/24 
6,750 
0,60 
280/8; 433/12 
12 
7,77 ± 0,05 
7,777 
0,090 
59/8 
4/8 
7,924 
1,98 
286/8 
13 
9,08 ± 0,03 
9,129 
0,54 
59/8 
2/12 
8,937 
1,57 
284/8; 476/12 
14 
9,79 ± 0,14 
9,849 
0,60 
88/12 
5/24 
9,684 
1,08 
424/12 
15 
11,73 ± 0,04 
11,696 
0,29 
58/8 
2/8 
11,834 
0,89 
419/12 
16 
12,30 ± 0,03 
12,287 
0,001 
87/12 
2/12 
12,318 
0,15 
418/12 
17 
13,14 ± 0,01 
13,191 
0,39 
58/8 
0 
13,347 
1,58 
416/12 
18 
14,34 ± 0,07 
14,394 
0,38 
57/8 
5/12 
14,461 
0,84 
276/8 


<d> = 0,445 

<d> = 0,75 
Literature
1. Бегжанов Р.Б., Беленький В.М., ЗалюбовскийИ.И., Кузнеченко А.В. Справочникпоядернойфизике. Книга 1. Ташкент: ФАН, 1989. 740 p. (_{} р.25).
2. Бегжанов Р.Б., Беленький В.М., Залюбовский И.И., Кузнеченко А.В. Справочник по ядерной физике. Книга 2. Ташкент: ФАН, 1989. 828p. (_{}р.526).
3. Анхименков В.П., Гагарский А.М., Голосовская С.П. и др. Исследование нарушения пространственной частности и интерференционных эффектов в угловых распределениях осколков деления _{} резонансными нейтронами.
4. Аллен К.У. Астрофизические величины. М.: Мир, 1977. 446 p.
Petrunenko V.V.
220015, Минск, Я. Мавра, д.32, кв, 28, email: This email address is being protected from spambots. You need JavaScript enabled to view it.