## 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 ej - lg ei  = 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 = 2×10k/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 above-stated 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 mo1<< Mo2, the smaller mass (under the influence of gravitation forces) will move about the mass Mo2 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 , 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 .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 like-charges the resulting force in a projection to an interaction axis is equal

F = Fe - Fм,

F or heteronymic charges

F = – Fe + Fм.

But because ,

And ,

Then, having taken into account, that 1/ (eоmо) = c2 where eо and m0 – 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 like-charges repel. The mark (–) shows that heteronymic charges are attracted.

Excluding eо, 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 hо = 7,6957×10-37 j×s or the same hо 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,Dmc2 = Wcis the energy of charges connection, whereDm– defect of mass.

By definition, the defect of mass Dm = m – mo, where m and mo – 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 roe = ho / (moe c) – classical radius of a charge. As hо = moe crоe, the formula (1.7) can be presented in the following way: .                                              (1.10а)

Having designated r/roe = e, we receive: .                                                      (1.10)

If in the system under consideration which consists of two heteronymic charges, a positive charge has mass mo2 much greater than mass mo1 of a negative charge, then in the formula (1.10). E01 is an own energy of a negative charge which will start to go around mass mo2; and if the charge masses are equal, then Eо1 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

Jz = pj × r,                                                         (1.11)

where r – radius of a circular orbit; pj – a full impulse of mass mo1 relatively the axis of rotation.

The full impulse of mass mo1 in a circular orbit can be presented as the sum

pj = pe ± ps,                                                       (1.12)

where ре – impulse of mass m01 with delayed spin; ps –  impulse received by a charge when  spin disinhibition: during the process of spin disinhibition  the charge either increases the orbital speed, or reduces it. ps – 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 Jz = ho. Only quantums of electromagnetic radiation have such parameters (photon spin always equals 1);

2) if v = c, and i = 2, then Jz = ho/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 Ео= mo c2, we have .                                                     (1.16)

Uniting the formulas  (1.15) and (1.16) and taking into account that W = E – Ео  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= ho/Jzand vector basis αi, 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,Jz = hiis 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 =ho/hi. 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=hithen the equation (1.19) in operators of the quantum mechanics becomes .                (1.21)

If in this equation we accept hi = h, where h – a constant of Planca, and ai  = hо/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 hi 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 a2/k2 < 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 m01; φо – initial phase. For circular orbit S = 2pr. From here, hi/р = l, where D = l/2p, and р = h×k where р –  impulse of mass mо1. At the moment of time t = 0 initial phase φо = 0, i.e. for stationarity conditions we find, that ψ/ψ0 = = cos kS as kS = 2p (pr/h  where pr = Jz –  projection of the full moment of an impulse to the axis z, then ψ/ψ0 = cos 2p . The functionw = |ψ/ψ0|2 = cos2 2p , defines the density of probability to find out mass mo1on some distancerfrom the axisz. The density of probability, apparently, is maximal when the numbernj = Jz/ h1 accepts the whole and half-integer values.

In order to define the most probable distances it is necessary to connect e and nj. 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 nj, we define, that . Whenceit follows that .

Let's coordinate the received result to de Broglie wavelength. For a charge with the moment of impulse Jz = hо length of the De Broilia wave D = hо/p. Having substituted here hо= mo roe c and p according to the formulas (1.13), we receive .                                     (1.28)

Let's enter instead of b number nj into this formula according to the formulas (1.26). As a result of some simple transformations we have .                                                    (1.29)

But nj1/4= e, therefore l = roe e/ nj or l nj = roe e = r. From here it follows, that

l nj = 2p r.                                                        (1.30)

It is a well-known 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 = hi/р. From here, bearing in mind that hi = ho/a taking into consideration the formulas (1.13) and (1.26) we receive ,                                     (1.31)

i.e. we find, that or

l nj = 2p ref,                                                       (1.32)

where ref = r/ai – effective (characteristic) radius of a circular condition. At hierarchical transitions (from the structure of one level to the structure of other level) ai ® 1. Hence, in limiting transition ref = r, the formula (1.30). Besides from the formula (1.31) it follows that roe/ai = ro ef represents some new scale standard, and ro ef e = ref. 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 half-integer 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/ro ef = kg, we receive the equation

n2 - kg n – 1/4= 0.                                                  (1.34)

Its decision looks like .                                                (1.34a)

The greatest interest in this decision represent numbers kg 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 kg = l/ref, then kg = 2 for this harmonic.

Note: as kg corresponds to the number of a wave harmonic, quantizating de Broglie wavelength so under its physical content kg can accept only integer values. Having substituted kg = 2 in the formula (1.34a), we receive .

Whence we find, that n1 = 2,118034, n2 = – 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 n1 = 2, and n2 = 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 n1 = 2,118034 – 0,5 = 1,618034; n2 = – 0,618034. Being guided by these numbers, from  the formula l/ro ef = e/ n = kg we find, that e = kg n. Having substituted here kg = 2 and n = 1,618034, we receive e = r/ ref = 2 ×1,618, Whence r/1,618 = 2 ref or 0,618r = 2ro. But ref =l/2, therefore . This length is the side of a regular ten-square, incirculed in a circle of a condition and, apparently, it is measured by pieces, multipleref. 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 = ro ef, hence, 20 ro ef= rore= r/ ro 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 = kg n, we find, that n = 10, if kg = 2 and kg = 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: kg = 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 = kg N D1  n1.                                                       (1.35)

Where l1 – length wave of a link of a wave orbit, the radius of which is accepted for the identity element (l1 = 0,618034); n1 = 1,618034 – the moment of a in impulse appropriate to the length l1; N – the number of parts in length D1, 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 ten-square. Its perimeter SN = = N l1 = 10×0,618034 = 6,18034. Hence, the full moment of an impulse nN = SN×1,618034 = = 6,18034×1,618034 = 10. As a result e = kg 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 SN is in wave frequency rate to the perimeter of an initial orbit with perimeter S1 = 10l1 will be stable. For such orbits the number Nx = 10x. Hence, according to (1.35)

e = 2×10x,                                                        (1.36)

where in  general  x = k/m, where at m = 1, 2, 3 k = 0, 1, 2, 3.

If  the spin of mass mo1 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 n2 = – 0,118034 + 0,5 = 0,381966 =  (0,618034) 2. Having substituted n = (1,618034) 2 in the formula e = kg n, we receive e = kg (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 = kg (1,618) y,                                                   (1.37)

where y = k ¢/m ¢.

Ny = (1,618)y.

For the orbits with the number of the parts determined by the formula

NS= N y Nx = (1,618) y  10x.                                         (1.38)

Parameter e  from formulas (1.35)

e = 1,618 y kg 10x.                                                  (1.38à)

The formula (1.38à) shows, that energy of a charge connection (mass mo1) 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 8-th and 12-th 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 kg = … – 4; –3; –2; –1; 0; 1; 2; 3; 4; …

It is necessary to note, that at negative values kg  the spin is oppositely directed to orbital moment of the initial condition, and at positive kg – on the line of the orbital moment. Besides spin conditions are realized at odd kg, and non – spin – at even kg.

The greatest attention is paid to the decisions, where kg = 0, 1, 2 and 3. In particular:

for kg = 0       n1 = 1,118034,          n2 = – 0,118034;

for kg = 1       n1 = 1,618034,          n2 = – 0,618034;

for kg = 2       n1 = 1,118034,          n2 = – 0,118034;

for kg = 3       n1 = (1,618034) 2,     n2 = (0,618034) 2 etc.

Analyzing these conditions, we shall note the following. The condition for which kg = 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 kg = 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 kg = 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  kg = 0, when the spin is oriented towards the orbital moment it can be reduced to the condition at kg = 1, however, conditions with kg = 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 kg can accept values kg = 1, 2, 3. As for the higher kg, 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 8-th and 12-th harmonics and compare them with the periodogram analysis data.

So: 1) the formula (1.36.) e = kg×10 k/m. Its wave frequency rates look as follows: .

From here, having reduced kg and having taken the decimal logarithm, we  receive: .

2) The formula (1.37): e = kg (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 , see page 58-65 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  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 Ui – lg Uj = (0,21/m¢)×k¢, where at m ¢ = 1, 2, 3 … k ¢ = 0, 1, 2, 3 …. From here we find, that

Ui/Uj = (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 Ti = 6, 12, 18, 24 in full conformity with the periodogram (see fig. 2)  and the theoretical data of the formula (1.36). Hence, lg Ui – lg Uj = k/m, where at m = 1, 2, 3 … k = 0, 1, 2, 3 …. As a result we find, that

Ui/Uj = 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  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 х2,1981, 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 rкр = 5,582 . Since  ум/<y> > rcr, 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 [2-5] 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 jо = 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 x2 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 . The situation has cleared up with the occurrence of the method offered by the author of this article . 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 2-4. 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 Wo = 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 kg = 2.

 jо = 0,79
 jо = 0,69  `Resultant perioddiagram`
 jо = 0,81  Fig. 4. Periodograms with initial phases jо = 0,69; 0,79 и 0,81

and their total result

 jо = 0,80
 jо = 0,72  Resultant perioddiagram Fig. 5. Periodograms with initial phases  jо = 0,72; 0,80

and their total result Fig. 6. The total result of the periodograms with initial phases

jо = 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) № Wempir. (KeV) Wvalue calculated. (KeV) d, % k/m k¢/m¢ Wvalue 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) № Wempir. (KeV) Wvalue calculated. (KeV) d, % k/m k¢/m¢ Wvalue 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 р-resonant neutrons of a nucleus Formula (1.38a) Formula (1.37) № Wempir. (KeV) Wvalue calculated. (KeV) d, % k/m k¢/m¢ Wvalue 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 = 0,445 = 0,75

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Petrunenko V.V.

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