Nuclide periodicity

In the excess plot the isotopes of a given element occur on a straight line through the points (NX,A) = (0,2Z) and (1,3Z). In the ratio plot these isotopes lie on circular segments between the coordinates (r, A) = (1,2Z) and (1/2, 3Z). The straight lines and circles intersect along a straight line 

The negative quantity in brackets is an irrational number known as the golden ratio, t = 0.61803. The solution Z = — iV(1.61803...) = — hlV defines = 1/t. The field of nuclide stability, as defined on both plots, converges to a point on the line r = Nx = r at A 267, the maxinum possible mass number for nuclides, stable against /1-type decay. By definition, this maximum, 

The heaviest non-radioactive nuclide, Bi, is interpreted as the maximum, on this scale, in the gravitational field of the solar system. Jq/No is postulated to be the heaviest stable nuclide anywhere in the cosmos. 

The nuclides of each modular group are spread along 11 festoons that terminate in a radioactive nuclide at each end. Nuclides on the high-ratio side decay by positron emission or electron capture those on the low-ratio side by /3-emission. All nuclides with A 209 decay by cr-emission. The 81 naturally stable elements have, on average, 3 isotopes each. The predicted 100 elements on the cosmic scale, by the same reckoning, correspond to 300 isotopes. 

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In Figure 5.6 we arrange the natural numbers along a spiral with a pitch of 24. All prime numbers, except for 2 and 3, occur on eight radial lines as p = 6n 1. By mapping the natural elements to these radial lines the periodicity of 16 = 2 x 8 is accounted for at the same time as the nuclide periodicity of 24. This arrangement is known as Plichta s prime-number cross. It has the remarkable property that the sum of all numbers over any complete cycle is given by... 

Figure 5.4 signifies more than elemental or nuclide periodicity. It summarizes the appearance of ponderable matter in all modifications throughout the universe. Following the extended hemlines from top left at Z/N = 1.04 — bottom left at 0 —> top right at Z/N = 1.04 bottom right at 0, and back to top left, the involuted closed path, which is traced out, is mapped to the non-orientable surface of a Mobius band in Figure 5.7. The two sides of the double cover are interpreted to represent both matter and antimatter.

The prominent role of the golden ratio that conditions the observed periodic table of the elements hints at a general self-similarity between atomic and celestial structures. By exploiting this similarity the Bode -Titius law is shown to be based on the same number theory as nuclide periodicity. All planets, moons and rings in the solar system obey the same rules of com-mensurability and move on quantized orbits like those assumed in planetary... 

To show that the periodic table of the elements is a subset of the more general nuclide periodicity, the data of Fig. 5 are replotted on axes of Z/N vs Z in Fig. 7. 

Fig. 7 The periodic distribution of stable nuclides as a function of atomic number. Open circles represent odd mass numbers and filled circles the two even mass-number series. The hemlines that define the nuclide periodicity of 24 are no longer parallel to the Z/N axis, and their points of intersection with the lines at Z/N = r and 1 are of special importance in the definition of elemental periodicity as a subset of the nuclide periodic function...

The characteristic values of Z/N = x and of 0.58 for observed and wave-mechanical periodicities are the limits of converging Fibonacci fractions around 3/5. The segmentation of the table into groups of 2 and 8 and of periods 2,8,18,32 summarizes the observed periodicity as a subset of nuclide periodicity. The sublevel structure, despite formal resemblance to the wave-mechanical H solution, emerges from number theory without reference to atomic structure. 

Fig. 1 Proton surface excess, x = Z — xN, as a function of mass number. Nuclide periodicity predicts maximal surface spin to occur in the regions as marked, in general agreement with the measured spin and elemental superconductivity of odd mass number nuclides...

In all cases where the golden section or the golden spiral correlates with chemical phenomena, convergence to some singularity is observed. The most striking example, shown in Fig. 4, occurs as the composition of stable nuclides, measured as Z/N, converges to the golden ratio as Z 102. At the same time, the hem lines, which define nuclide periodicity of 24, map out the observed periodic table of the elements at Z / A = t. ... 

Thermal Res. int. Main nuclides Period Decay mode... 

Our present views on the electronic structure of atoms are based on a variety of experimental results and theoretical models which are fully discussed in many elementary texts. In summary, an atom comprises a central, massive, positively charged nucleus surrounded by a more tenuous envelope of negative electrons. The nucleus is composed of neutrons ( n) and protons ([p, i.e. H ) of approximately equal mass tightly bound by the force field of mesons. The number of protons (2) is called the atomic number and this, together with the number of neutrons (A ), gives the atomic mass number of the nuclide (A = N + Z). An element consists of atoms all of which have the same number of protons (2) and this number determines the position of the element in the periodic table (H. G. J. Moseley, 191.3). Isotopes of an element all have the same value of 2 but differ in the number of neutrons in their nuclei. The charge on the electron (e ) is equal in size but opposite in sign to that of the proton and the ratio of their masses is 1/1836.1527. 

Charge number and mass number must be conserved in each reaction. Thus, each a particle decreases the nuclear charge by two units and the mass number by four units. Similarly, each P emission increases the nuclear charge by one unit but leaves the mass number unchanged. Consult a periodic table to identify the elemental S3Tnbol of each product nuclide. 

Figure 3. Parent daughter disequilibrium will return to equilibrium over a known time scale related to the half-life of the daughter nuclide. To return to within 5% of an activity ratio of 1 requires a time period equal to five times the half-life of the daughter nuclide. Because of the wide variety of half-lives within the U-decay-series, these systems can be used to constrain the time scales of processes from single years up to 1 Ma.

The previous section showed that if the decay chain remains undisturbed for a period of approximately 6 times the longest half-lived intermediate nuclide then the chain will be in a state of secular equilibrium (i.e., equal activities for all the nuclides). The key to the utility of the U-series is that several natural processes are capable of disrupting this state of equilibrium. 

Decay of the nuclide itself. The conceptually simplest approach is to take a known quantity of the nuclide of interest, P, and repeatedly measure it over a sufficiently long period. The observed decrease in activity with time provides the half-life to an acceptable precision and it was this technique that was originally used to establish the concept of half-lives (Rutherford 1900). Most early attempts to assess half lives, such as that for " Th depicted on the front cover of this volume, followed this method (Rutherford and Soddy 1903). This approach may use measurement of either the activity of P, or the number of atoms of P, although the former is more commonly used. Care must be taken that the nuclide is sufficiently pure so that, for instance, no parent of P is admixed allowing continued production of P during the experiment. The technique is obviously limited to those nuclides with sufficiently short half-lives that decay can readily be measured in a realistic timeframe. In practice, the longest-lived isotopes which can be assessed in this way have half-lives of a few decades (e.g., °Pb Merritt et al. 1957). 

The global average production rate of any nuclide, Q(t), at any time, t, will be primarily dependent on the cosmic ray intensity, I(t). If the intensity varies sinusoidally with a period T(u) = 2n/T), Q(t) will also vary sinusoidally. The standing crop of a nuclide in the sea water column for a production function, Q(t) = Q (1 + a cos tot), a being the amplitude, can be deduced to be 0... 

The amplitude attenuation factor, 1 + (tu/Ai)2 for nuclides satisfying relation [14], for various values of Ax and T are presented in figure 9. It is obvious from figure 9 that the attenuation is minimal when Ai > u>, i.e., when the removal residence time of the nuclide from sea water is less than the period in the variation of cosmic ray intensity. 

The pairs of elements that are out of order based on their atomic masses are presented here, together with their atomic numbers. The periodic table lists elements in order of increasing atomic number, not increasing atomic mass. For one of these pairs there is a further explanation. Most of the Ar in the atmosphere is thought to result from the radioactive decay of 40 K, a nuclide of that once was more plentiful than it is now. 

Only a few relevant points about the atomic structures are summarized in the following. Table 4.1 collects basic data about the fundamental physical constants of the atomic constituents. Neutrons (Jn) and protons (ip), tightly bound in the nucleus, have nearly equal masses. The number of protons, that is the atomic number (Z), defines the electric charge of the nucleus. The number of neutrons (N), together with that of protons (A = N + Z) represents the atomic mass number of the species (of the nuclide). An element consists of all the atoms having the same value of Z, that is, the same position in the Periodic Table (Moseley 1913). The different isotopes of an element have the same value of Z but differ in the number of neutrons in their nuclei and therefore in their atomic masses. In a neutral atom the electronic envelope contains Z electrons. The charge of an electron (e ) is equal in size but of opposite sign to that of a proton (the mass ratio, mfmp) is about 1/1836.1527). 

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