Phi in Chemistry and Physics

by Nick Seewald

1. Hydrogen Bond Distances

It has been found recently (in 2005, actually), that the golden ratio plays a significant role in atomic physics in that it governs what is known as the Bohr radius (the radius of atoms and ions that enables quantitative discussion of bond lengths between atoms and partial ionic character (D. Yu et al.). Here we will discuss it in a phenomenon known as hydrogen bonding, in which the partially positive hydrogen atom is attracted to an electronegative atom (one that strongly pulls electrons toward it).
In a triatomic O-H—O hydrogen bond, D. Yu et al. point out that the total bond strength (the sum of the strengths of the individual bonds, which can be measured thermodynamically or in other ways), is 1 (211). (Just to clarify, the two bonds in question are O-H and H–O.) Now notice that each of these bonds consist of a larger anionic (negatively-charged) radius (from O) and a smaller cationic radius (from H). So we see that the total distance D(H+) is equal to the sum of the smaller distances in both the strong bond (O-H) and weak bond (H—O):


Image from D. Yu et al. 212

The hydrogen radius changed when undergoing bonding, since the atom has become polarized in forming a hydrogen bond. Therefore, we can consider the following image from D. Yu et al. 212 (click to enlarge):


The presence of the golden ratio in hydrogen bonding.
Note that the ratio of bond distances is 0.618 (1/phi) for both the strong and weak bonds. The validity of these results are further discussed in D. Yu et al 213-214.

2. Quasicrystals

Some years ago it was shown that rapid solidification of an aluminum-manganese alloy can produce a solid that diffracts electrons like a crystal but has no lattice-based translational symmetry (as crystals do). “A quasiperiodic structure is generated using a pair of polyhedra which are the three-dimensional analogues of the two-dimensional Penrose tiles, which tile the plane aperiodically (for more information on Penrose tiles, see These polyhedra will produce a diffraction pattern when bombarded with gamma rays (this is a method of analyzing crystal structures) that suggests that the two polyhedra occur in the ratio of 1.618:1. The NGR (Nuclear Gamma-ray Resonance) spectrum of this crystal structure appears as follows (click to enlarge):


NGR Spectrum for crystallized Al-Mn alloy, Image from Swartzendruber et al. 1384.
The above spectrum displays two quadripole doublet peaks (this is fancy crystallographer-talk). The least-squares regression fit for these curves saw minimized standard deviations (which is statistically a good thing–the lower the standard deviation, the closer the data is to the mean) when the intensity ratio was constrained to be 1.618, the golden ratio. This suggests that phi is involved in determining the spacing between the atoms in the quasicrystalline lattice structure (Swartzendruber et al. 1384-1385).

3. Optics

It is possible to show that one-dimensional quasicrystals are self-similar with a self-similarity factor of phi (Grushina et al. 408). This, then, implies that optical devices constructed in the same manner as one-dimensional quasicrystals must also exhibit phi-based self-similarity. We see this in their diffraction patterns:


Diffraction of light by gratings, image from Grushina et al. 408.
Upon inspection, we notice that Fig. 1a (the Fibonacci grating) generates a fractal-like addition pattern (that is self-similar by definition) rather than anything periodic. Also, the ratios of the lengths of the arrows between peaks is the golden ratio. Also, the self-similar characteristics of these transmission spectra exhibit some amount of stability, as when 5% of the layers of the diffraction grating (through which light is passed) are replaced by layers with a different refractive index (which regulates the amount light bends in a certain material), the locations of the major energy gaps remains the same (see below, specifically Fig. 2b and Fig. 2c):


Relative stability of phi-based diffraction patterns, image from Grushina et al. 409.
“One of the main conclusions to be drawn from our calculations is that the self-similarity in the structures of diffraction gratings and multilayer Fibonacci systems related to the golden ratio is clearly seen in their optical characteristics. Also, the definite consistency in the behavior of optical parameters featuring a unique self-similarity factor is pointed out” (Grushina et al. 411). The above discussion serves to illuminate just another interesting scientific property of the golden ratio.

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