The Physical nature of the decalogoriphmic
Periodicity phenomenon
Part 1. Microsystems
The phenomenon of decalogoriphmic periodicity was found out first in
distributions of trial bodies of the Solar system (at the end of 1974) on
parameter lg e, where e = r/r_{og} – r – distance between
interacting objects, r_{og }– gravitational radius of the central mass
[1]. Then in distributions of sections of radiating capture of neutrons by a
nucleus atom [2] on size lg W, where W – kinetic energy of neutrons, and also
in distributions lg W_{c}, where W_{c} – energy of a nucleon
connection in a nucleus of atom [35], electron in atom [6], then in
conglomerations of galaxies [7] and, finally, in distributions of quasars, i.e. in the Metagalaxy [8].
Apparently, the spectrum of the phenomenon demonstration is very wide. This
phenomenon is universal.
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 periodograms and
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 scientists
developed 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 e interact, 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, where Dm – 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}
c r_{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 module a = h_{o}/J_{z}_{
}and vector basis α_{i}, where i = 1, 2, 3. The factor a  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 charge of 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 vector a then
we receive:
_{},
where _{} – individual vector, k = ± (j +^{1}/_{2}), where j – 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 correspond with 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
function w =  ψ/ψ_{0} ^{2}
= cos^{2} 2p _{},
defines the density of probability to find out mass m_{o1} on some distance r from the axis z. The density of probability, apparently, is
maximal when the number n_{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 _{}. Whence it 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
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, multiple r_{o }_{ef}. It divides a circle into 10 equal
parts, the length of each is equal to the length
of the De Broilia l. For this reason we find, that 10l = 2p r, i.e. 10D = r, or _{}.
But D/2
= r_{o }_{ef}, hence, 20
r_{o }_{ef} = r or e = 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


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
periodogram for 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 the trend,
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., Залюбовский И.И., Кузнеченко А.В. Справочник по ядерной физике. 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:
bed_nasty@inbox.ru