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The logarithmic spiral of beta-stable odd isotopes Isayev R.
Логарифмическая спираль бета-стабильных нечетных изотопов
Исаев Р. Ш.
Исаев Рафаэль Шахбаз оглу /Isayev Rafael — бакалавр, кафедра общей и неорганической химии, химический факультет, Бакинский государственный университет, г. Баку, Азербайджанская Республика
Abstract: this research is a mathematical description of the line of beta-stability isotopes with use of a parametrical form of the equation of a logarithmic spiral. It gives the chance to prove Praut's hypothesis that hydrogen is primary matter of which by a nucleosynthesis atoms of all other elements were formed. Conducted the analogy is the result of this study with the processes which are developing in nature by the properties of the logarithmic spiral.
Аннотация: данное исследование является математическим описанием линии бета-стабильности изотопов с использованием параметрической формы уравнения логарифмической спирали. Это дает возможность обосновать гипотезу Праута о том, что водород является первичной материей, из которой путём нуклеосинтеза образовались атомы всех других элементов. Проведена аналогия результата данного исследования с процессами, которые развиваются в природе по свойствам логарифмической спирали.
Keywords: logarithmic spiral, odd isotopes, deuteron-cluster structure, binding neutron.
Ключевые слова: логарифмическая спираль, нечетные изотопы, дейтрон-кластерная структура,
связывающий нейтрон.
DOI: 10.20861/2304-2338-2016-55-002
Introduction
This research is a continuation of "Statistical analysis of stable and long-lived isotopes using deuteron cluster" [1] .It presents the Hydrogen model of atomic nucleus in which nucleus consists of deuterium nucleus and definite numbers of neutrons which bond deuterons in a unified structure. The specialty of this model is that the hydrogen is considered as a primary matter and atoms of other elements are formed from it by nucleosynthesis.
The idea of hydrogen being primary matter for all elements was proposed at the beginning of 19thcentury. In his evaluation John Dalton assumed that atomic mass of hydrogen is equal to 1, andtherefore atomic masses of all elements in his table are integers. After study of Dalton's table of atomicmasses in 1815-1816 years, William Prout came to a conclusion that all elements ultimately consist ofhydrogen and atomic masses differ as they consist of different number of hydrogen atoms. This standpoint is known as Prout hypothesis [2, 3]. Deviations from integrality were considered by Pratt as measuring errors. Further the most precise determinations of atomic masses did not confirm this position and Pratt's ideas were not been developed.
But Josef Mattauch in 1934, states that if two adjacent elements on the periodic table have isotopes of the same mass number, one of these isotopes must be radioactive. Two nuclides that have the same mass number (isobars) can both be stable only if their atomic numbers differ by more than one [4]. A Mattauch isobar rule nevertheless indicates a continuous sequence in the structure of atomic nuclei. In order to develop this idea in the Hydrogen model the atomic mass of hydrogen was taken as 2 [1] which corresponds to the mass of its isotope deuterium.
Analysis
If the weight of any isotope divided into deuteron clusters which are quantitatively equal to the charge of the nucleus, it gets to deuteron-cluster structure with a certain number of neutrons in the residue, which is supposed to link these clusters into a single structure [1].
Table 1. The deuteron - cluster structure of odd stable and long-lived isotopes
1 p 1D H
2 3D 3D+1n Li 5D 5D+1n B 7D 7D+1n N 9D+1n F
3 11D+1n Na 13D+1n Al 15D+1n P 17D+1n 17D+3n Cl
4 19D+1n 19D+3n K 21D+3n Sc 23D+5n V 25D+5n Mn 27D+5n Co 29D+5n 29D+7n Cu 31D+7n 31D+9n Ga 33D+9n As 35D+9n 35D+11n Br
5 37D+11n Rb 39D+11n Y 41D+11n Nb 43D+11n 43D+12n 43D+13n Tc 45D+13n Rh 47D+13n 47D+15n Ag 49D+15n 49D+17n In 51D+19n 51D+21n Sb 53D+21 I
6 55D+23n Cs 57D+25n La 59D+23n Pr 61D+23n 61D+24n 61D+25n Pm 63D+25n 63D+27n Eu 65D+29n Tb 67D+31n Ho 69D+31n Tm 71D+33n Lu
73D+35n Ta 75D+35n 75D+37n Re 77D+37n 77D+39n Ir 79D+39n Au 81D+41n 81D+43n Tl 83D+43n Bi 85D+39n 85D+40n 85D+41n At
7 87D+49n Fr 89D+49n Ac 91D+49n Pa 93D+51n Np 95D+51n 95D+53n Am 97D+53n Bk 99D+55 99D+57 Es 101D+57 Md 103D+53 103D+55 Lr
105D+63 Db 107D+65 Bh 109D+65 109D+67 Mt 111D+67 Rg 113D+69 113D+71 Uut 115D+71 Uup 117D+67 117D+69 Uus
For elements with nuclear charge 99-117 deuteron-cluster structure is calculated based on the periodicity of cluster structures in atomic nuclei [1] by adding 14 neutrons, to the deuteron-cluster structure of downstream elements, which binding the clusters.
Using mathematical expressions (1.0) and (2.0) we construct a logarithmic spiral:
= rcost = aeM cosí , (i o) y(t) = t sini = sin t, (2.0)
Where t - the angle of deviation from the zero point, r - radius vector of the point. The e being the base of natural logarithms, and a and b being arbitrary positive real constants.
Table 2. a - factor responsible for the distance between the coils, b - factor responsible for the density of coils of spiral
a 0.05
b 0.25
Binding neutron Angle (t) x y Isotopes
0 -0.1396263 0.04781488 -0.006719943 2H
0 -0.1396263 0.04781488 -0.006719943 6Li
1 0 0.05 0 7Li
0 -0.1396263 0.04781488 -0.006719943 10B
1 0 0.05 0 11B
0 -0.1396263 0.04781488 -0.006719943 14N
1 0 0.05 0 15N
1 0 0.05 0 19F
1 0 0.05 0 23Na
1 0 0.05 0 27Al
1 0 0.05 0 31P
1 0 0.05 0 35CI
3 0.27925268 0.05153842 0.014778405 37CI
1 0 0.05 0 39K
3 0.27925268 0.05153842 0.014778405 41K
3 0.27925268 0.05153842 0.014778405 45Sc
5 0.55850536 0.04875615 0.030466226 51V
5 0.55850536 0.04875615 0.030466226 55Mn
5 0.55850536 0.04875615 0.030466226 59Co
5 0.55850536 0.04875615 0.030466226 63Cu
7 0.83775804 0.04125146 0.045814388 65Cu
7 0.83775804 0.04125146 0.045814388 69Ga
9 1.11701072 0.02897943 0.05941664 71Ga
9 1.11701072 0.02897943 0.05941664 75As
9 1.11701072 0.02897943 0.05941664 79Br
11 1.3962634 0.01230942 0.069810193 81Br
11 1.3962634 0.01230942 0.069810193 85Rb
11 1.3962634 0.01230942 0.069810193 89Y
11 1.3962634 0.01230942 0.069810193 93Nb
Tc
13 1.67551608 -0.0079455 0.075596415 103Rh
13 1.67551608 -0.0079455 0.075596415 107Ag
15 1.95476876 -0.0305339 0.075573961 109Ag
15 1.95476876 -0.0305339 0.075573961 113In
17 2.23402144 -0.0538106 0.068874418 115In
19 2.51327412 -0.0758233 0.055088882 121Sb
21 2.7925268 -0.0944388 0.034372921 123Sb
21 2.7925268 -0.0944388 0.034372921 127I
23 3.07177948 -0.1075041 0.007517419 133Cs
23 3.07177948 -0.1075041 0.007517419 139La
23 3.07177948 -0.1075041 0.007517419 141Pr
Pm
25 3.35103216 -0.1130337 -0.024026063 151Eu
27 3.63028484 -0.1094103 -0.058174474 153Eu
29 3.90953752 -0.0955821 -0.092302597 159Tb
31 4.1887902 -0.0712413 -0.123393634 165Ho
31 4.1887902 -0.0712413 -0.123393634 169Tm
33 4.46804289 -0.0369621 -0.148246932 175Lu
35 4.74729557 0.00571769 -0.16373308 181Ta
35 4.74729557 0.00571769 -0.16373308 185Re
37 5.02654825 0.05428788 -0.167080925 187Re
37 5.02654825 0.05428788 -0.167080925 191Ir
39 5.30580093 0.10534203 -0.156175978 193Ir
39 5.30580093 0.10534203 -0.156175978 197Au
41 5.58505361 0.15474387 -0.129845526 203Tl
43 5.86430629 0.19788329 -0.088103319 205Tl
42 5.72467995 0.17739432 -0.110848275 206Pb
43 5.86430629 0.19788329 -0.088103319 209Bi
At
49 6.70206433 0.24398749 0.108630228 223Fr
49 6.70206433 0.24398749 0.108630228 227Ac
49 6.70206433 0.24398749 0.108630228 231Pa
51 6.98131701 0.21938697 0.184087527 237Np
51 6.98131701 0.21938697 0.184087527 241Am
53 7.26056969 0.17172677 0.254595402 243Am
53 7.26056969 0.17172677 0.254595402 247Bk
54 7.40019603 0.1394049 0.285822403 238U
55 7.53982237 0.10176025 0.313185857 253Es
57 7.81907505 0.01232351 0.352899181 255Es
57 7.81907505 0.01232351 0.352899181 259Md
Lr
63 8.65683309 -0.3131899 0.30244401 273Db
65 8.93608577 -0.4122191 0.219180786 279Bh
65 8.93608577 -0.4122191 0.219180786 283Mt
67 9.21533845 -0.4896853 0.104085821 285Mt
67 9.21533845 -0.4896853 0.104085821 289Rg
69 9.49459113 -0.5355166 -0.037446968 295Uut
70 9.63421747 -0.5437462 -0.115576832 298Fl
71 9.77384381 -0.5409255 -0.196880769 297Uut
71 9.77384381 -0.5409255 -0.196880769 301Uup
Binding neutrons of odd isotopes were taken for the making a logarithmic spiral, which are shown in Table i .After entering the values from Table 3 on the two-dimensional coordinate system, we obtain a logarithmic spiral, which is shown in Figure 2. It should be noted that the binding neutron from the deuteron-cluster structure of technetium, promethium, astatine and lawrencimn were not used in the making of the spiral.
Y
Mil
X \ D b
\ / 238U
/
/ / \ \
/ \ \ Fr
/ / \ 0.1 Tc
X N \
/ 1 / x \ \ \
6 -0.! 5 -0. 15 -0 4 -0. ¡5 -0 3 -0. >5 -0 2 -0. L5 r 1 -0. 15 [ Li' 5 0 L\ 5 0 2 o.; 5 0.
\ \ Bi
298F 1 V ! 06Pb -A
Uup
Fig. 2. The logarithmic spiral of odd isotopes actually describes the line of beta stability
As it is known technetium and promethium have no stable isotopes and they come before lead in the periodic table of elements, and astatine is the next odd element after isotope 209Bi, which is considered until some time [5] the hardest of naturally occurring element between stable isotopes. It is noteworthy that the value of technetium, promethium, astatine and lawrencium intersect abscissa and ordinate of logarithmic spiral. Hydrogen model shows that for doubly magic isotope 298Fl [6] the clusters neutrons ratio is equal to the value of the golden ratio.
298Fl = 114Deuteron cluster + 70 binding neutron = 114 / 70 = 1.6285... Golden ratio = 1.6180...
Fig. 3. The atmospheric depression which has arisen on the southwest coast of Iceland on September 4, 2003
(satellite picture)
Fig. 4. Examples of logarithmic spirals occurring in nature: in plants, spiral galaxies, in atmospheric air flow, etc.
Conclusions
The size of logarithmic helix turns increases gradually, but their shape remains constant. The increase of radius by circle length unit is constant. As a result of this feature the logarithmic helix appears in definite growing forms such as shell of mollusks, sunflower cap, spirals of cyclones and galaxies. Analysis of connection neutrons of odd elements deuteron-cluster structures also shows the regularity of growth and repeatability of atom features. In fact, logarithmic helix describes the power hole in which the energy transforms into mass (matter). The center of cyclones or galaxies in the place where the above mentioned power hole is collated with the general energy background. Energy entering into the power hole gradually transforms into more bound matter form, but as it goes to the center of the helix it reaches the maximum of bonding and then the reversed process of transformation of mass (matter) into energy takes place.
Figures 3 and 4 show examples in which the processes occur in nature on a logarithmic spiral. Having drawn an
analogy with the Figure 2, it can be concluded that the physical processes that are mathematically described by a
logarithmic spiral in nature are the same for the micro- and macrocosm. Such a physical process is the periodicity.
Periodic law is universal for the Universe.
References
1. Isayev R. Statistical analysis of stable and long-lived isotopes using deuteron cluster // Problems of modern science and education, 2015. Vol. 10 (52), p. 10-16.
2. William Prout (1815). On the relation between the specific gravities of bodies in their gaseous state and the weights of their atoms. Annals of Philosophy, 6: 321-330.
3. William Prout (1816). Correction of a mistake in the essay on the relation between the specific gravities of bodies in their gaseous state and the weights of their atoms. AnnalsofPhilosophy, 7: 111-113.
4. Thyssen Pieter; Binnemans Koen; (2011). Handbook on the Physics and Chemistry of Rare Earths. Elsevier. Vol. 41, p. 66.
5. Pierre de Marcillac, Noël Coron, Gérard Dambier, Jacques Leblanc, and Jean-Pierre Moalic (April 2003). «Experimental detection of a-particles from the radioactive decay of natural bismuth» // Nature 422 (6934), p. 876-878.
6. Fricke Burkhard. "Superheavy elements: a prediction of their chemical and physical properties". Recent Impact of Physics on Inorganic Chemistry 21, 1975, P. 93.
