The Golden Parameter

The four different periodic tables account for the observed elemental diversity and provide compelling evidence that the properties of atomic matter are intimately related to the local properties of space-time, conditioned by the golden parameter r = l/ f . The appearance of r in the geometrical description of the very small (atomic nuclei) and the very large (spiral galaxies) emphasizes its universal importance and implies the symmetry relationship of self-similarity between all states of matter. This property is vividly illustrated by the formulation of r as a continued fraction  

The golden parameter is perhaps best known from botanical phyllotaxis and from its relationship to five-fold symmetry, as evidenced by trigonometric formulae such as  

These angles are all fractions of 37t/5 = 108°, characteristic of a regular pentagon. 

The geometrical principles that underlie interactions inside atomic nuclei must also prevail in molecular space and the golden parameter is likely to surface again in problems of chemical bonding. The advantage is that once 

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The only parameter that has been fixed in the above three sequential stages is the HRAT. We can subsequently update the H RAT by performing a one-dimensional search using the golden section search method, which is shown as the outside loop in Figure 8.20. 

Fig. 4. The temperature dependence of the rate of non-radiative transitions y for some (given) values of the interaction parameter w parameters co0 and cr0 are the same as in Fig. 1(a). The sharp peaks result from the divergence of the resolvent in equation (31) for w > 10 at some positive co > o>m- F°r comparison, the golden rule result is also presented (the thick line below).

Recognition of space-time curvature as the decisive parameter that regulates nuclear stability as a function of the ratio, Z/N, with unity and the golden mean, r, as its upper and lower limits, leads to a consistent model for nucleogenesis, based on the addition of -par tides in an equilibrium chain reaction. This model is also consistent with the limitations imposed by the number spiral. 

The natural appearance of nuclear magic numbers, and the golden-ratio limitation on nuclear distribution, indicate the development of an excess surface layer of protons, which correlates well with periodic variation of nuclear spin, and which may be an important parameter in the understanding of superconductivity. 

Molecular structure and shape are related to orbital angular momentum and chemical change is shown to be dictated by the quantum potential. The empirical parameters used in computer simulations such as molecular mechanics and dynamics are shown to derive in a fundamental way from the relationship between covalence and the golden ratio. 

These estimates bracket the NASA-JPL and lUPAC recommendations of 6.5x10 and 7.7 x 10 cm molecule s [9,60]. It is therefore possible fo reconcile fhe thermochemistry proposed here with the observed lO + NO2 recombination kinetics while employing reasonable input parameters for the unimolecular model. Nevertheless it must be stressed, as emphasized earlier [16], that there is considerable uncertainty in some of the input parameters to an RRKM analysis, especially the Frot term. It is of interest to compare the present kinetic calculations with the Multiwell [61] Master Equation calculations on this system by Golden [16]. He used a Morse potential to locate the centrifugal maximum, and from the bond extension Frot 2.1 is derived, about 1/7 of fhaf used here. On the other hand, the higher Eo value yields a density of sfafes larger by a facfor of 6, and fhese two factors largely cancel. 

Equations (12.55), sometime referred to as multiphonon transition rates for reasons that become clear below, are explicit expressions for the golden-rule transitions rates between two levels coupled to a boson field in the shifted parallel harmonic potential surfaces model. The rates are seen to depend on the level spacing 21, the normal mode spectrum mo,, the normal mode shift parameters Ao-, the temperature (through the boson populations ) and the nonadiabatic coupling... 

Equation 14) and BJ theory with the Golden rule single sum FCWD and the with the double sum FCWD of Equation (19). Fig. 10 shows how small the effect of using cow as a simple multiplier in Equation (17) is compared to using it in the exponential terms, as in Equation (19). It will also be noted that for small cav values the BJ theory predicted Vab exceeds that produced by the adiabatic limit using the parameters quoted. 

Figure 11-9. The cluster with 222 symmetry of four VPl coat proteins taken from the human rhinovirus is enclosed in molecular forms with vertices at points of an integral tetragonal lattice with lattice parameters a = ra( /9, b = c = tuq/S, in which Uq is the icosahedral lattice parameter and t the golden ratio (adapted from [27], courtesy lUCr)...

Figure 9.2. Renormalized quantum energy gap law, that is, the activation energy —AP versus the free energy change of reaction Ap° — E IE vs. AG/ , in our notation), for H O with immersed donor and acceptor molecules of radii 3.5 A (point chaiges in spherical cavities) at different separation, as compared against the classical Marcus law. The parameter AP was evaluated from the Golden Rule formula by the method of steepest descents. The corresponding simulated relaxation spectrum is shown in Figure 9.3. (Reproduced from [41c] with permission. Copyright (1997) by the American Institute of Physics.)...

The Golden Rule rests on the assumption of weak electronic coupling. Weak, first of all, means nonadiabatic, that is, the following condition for the Massey parameter J should be fulfilled [298] ... 

The main building blocks of the proposed new model are the relationship between geometry, numbers and space the theory of relativity and the periodicity of atomic matter. Taken together, these considerations indicate a cosmic symmetry that defines a harmonious holistic system that embraces all objects from the subatomic to extragalactic scales. The common geometrical factor is the ubiquitous golden parameter, r = 0.61803... 

As recently shown (Boeyens, 2009) the Bode -Titius law, which hints at some harmonious regular organization of planetary motion in the solar system, is dictated by a more general self-similar symmetry that applies from subatomic systems to galactic spirals. The common parameter is the golden ratio, r = 0.61803. Any such cosmic symmetry should be dictated by a successful cosmological model. 

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