# Fine-structure constant (spiral)

Atomic energy levels emerge from fine structure constant alpha hyperbolic spiral

A specific form of hyperbolic spiral (limited to 2 revolutions) can be used to map electron co-ordinates during ionisation (in the H atom). At set intervals the spiral angles cancel and reduce to integer values (360°=4, 360+120°=9, 360+180°=16, 360+216°=25 ... 720°=∞). If we set radius r = Bohr radius and map the electron transit, we find these radius correspond to the principal quantum number n^2 and so can be used to calculate the orbital transition frequencies. As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are based on pi, we can ask if quantization of the atom is a geometrical function of this spiral.

${\displaystyle \varphi =(2)\pi ,\;4r}$ (360°)
${\displaystyle \varphi =(4/3)\pi ,\;9r}$ (240°)
${\displaystyle \varphi =(1)\pi ,\;16r}$ (180°)
${\displaystyle \varphi =(4/5)\pi ,\;25r}$ (144°)
${\displaystyle \varphi =(2/3)\pi ,\;36r}$ (120°)

A hyperbolic spiral is a type of spiral with a pitch angle that increases with distance from its center. As this curve widens (radius r increases), it approaches an asymptotic line (the y-axis) with the limit set by a scaling factor a (as r approaches infinity, the y axis approaches a).

### Principal quantum number n

The H atom has 1 proton and 1 electron orbiting the proton, the electron can be found at fixed radius (Bohr radius) from the proton (nucleus), these radius represent permitted energy levels (orbitals) at which the electron may orbit the proton and so are described as quantum levels. Electron transition (to higher energy levels) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to lower energy levels.

The principal quantum number n denotes the energy level for each orbital. As n increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as n increases, the electron is further from the nucleus). Accounting for two states of spin, each n (electron shell) can accommodate up to 2n2 electrons (Bohr model). As these orbitals are fixed according to integer n, the atom can be said to be quantized.

### Alpha spiral

In its simplest form, a fine structure constant spiral (or alpha spiral) is a specific hyperbolic spiral that appears in electron transitions between atomic orbitals in the Hydrogen atom.

It can be represented in Cartesian coordinates by

${\displaystyle x=a^{2}{\frac {cos(\varphi )}{\varphi ^{2}}},\;y=a^{2}{\frac {sin(\varphi )}{\varphi ^{2}}},\;0<\varphi <4\pi }$

This spiral has only 2 revolutions approaching 720° (${\displaystyle 4\pi }$) as the radius approaches infinity. If we set start radius r = 1, then at given angles radius r will have integer values (the angle components cancel).

${\displaystyle \varphi =(2)\pi ,\;r=4}$ (360°)
${\displaystyle \varphi =(4/3)\pi ,\;r=9}$ (240°)
${\displaystyle \varphi =(1)\pi ,\;r=16}$ (180°)
${\displaystyle \varphi =(4/5)\pi ,\;r=25}$ (144°)
${\displaystyle \varphi =(2/3)\pi ,\;r=36}$ (120°)

### Bohr model

The basic (alpha) radius for each n level uses the fine structure constant α whereby;

${\displaystyle r_{orbital}=2\alpha n^{2}}$

Such that at n = 1, the start radius r = 2α. We can map the electron orbit around the orbital as a series of steps with the duration of each step the frequency of the electron + proton wavelengths (${\displaystyle \lambda _{p}+\lambda _{e}}$). The steps are defined according to angle β;

${\displaystyle \beta ={\frac {1}{r_{orbital}{\sqrt {r_{orbital}}}{\sqrt {2\alpha }}}}}$

At specific n levels;

${\displaystyle \beta ={\frac {1}{4\alpha ^{2}n^{3}}}}$

This gives a length travelled per step as the inverse of the radius

${\displaystyle l_{orbital}={\frac {1}{2\alpha n}}}$
${\displaystyle v_{orbital}={\frac {1}{2\alpha n}}}$

The number of steps (orbital period) for 1 orbit of the electron then becomes

${\displaystyle t_{orbital}={\frac {2\pi r_{orbital}}{v_{orbital}}}=2\pi 2\alpha 2\alpha n^{3}}$

A base (reference) orbital (n=1)

${\displaystyle t_{ref}=2\pi 4\alpha ^{2}}$

### Photon orbital model

The electron can jump between n levels via the absorption or emission of a photon.

In the photon-orbital model[1], the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that ${\displaystyle orbital\;radius+photon=zero}$ (cancel). The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius). Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role.

The photon is actually 2 photons as per the Rydberg formula (denoted initial and final).

${\displaystyle \lambda _{photon}=R.({\frac {1}{n_{i}^{2}}}-{\frac {1}{n_{f}^{2}}})={\frac {R}{n_{i}^{2}}}-{\frac {R}{n_{f}^{2}}}}$
${\displaystyle \lambda _{photon}=(+\lambda _{i})-(+\lambda _{f})}$

The wavelength of the (${\displaystyle \lambda _{i}}$) photon corresponds to the wavelength of the orbital radius. The (+${\displaystyle \lambda _{i}}$) will then delete the orbital radius as described above (orbital + photon = zero), however the (-${\displaystyle \lambda _{f}}$), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+${\displaystyle \lambda _{i}}$) photon wavelength, the orbital radius does not change as the 2 photons cancel each other;

${\displaystyle r_{orbital}=r_{orbital}+(\lambda _{i}-\lambda _{f})}$

However, the (${\displaystyle \lambda _{f}}$) has the longer wavelength, and so after the (${\displaystyle \lambda _{i}}$) photon has been absorbed, and for the remaining duration of this (${\displaystyle \lambda _{f}}$) photon wavelength, the orbital radius will be extended until the (${\displaystyle \lambda _{f}}$) is also absorbed. As the orbital radius increases, the orbital rotation angle β will conversely decrease, and as the velocity of orbital rotation depends on β, the velocity will adjust accordingly.

For example, the electron is at the n = 1 orbital. To jump from an initial ${\displaystyle n_{i}=1}$ orbital to a final ${\displaystyle n_{f}=2}$ orbital, first the (${\displaystyle \lambda _{i}}$) photon is absorbed (${\displaystyle \lambda _{i}+\lambda _{orbital}=zero}$ which corresponds to 1 complete n = 1 orbit by the electron, the orbital phase), then the remaining (${\displaystyle \lambda _{f}}$) photon continues until it too is absorbed (the transition phase).

${\displaystyle \lambda _{i}=1t_{ref}}$
${\displaystyle \lambda _{f}=4t_{ref}}$ (n = 2)

After ${\displaystyle t_{ref}}$ steps, the (${\displaystyle \lambda _{i}}$) photon is absorbed, but the (${\displaystyle \lambda _{f}}$) photon still has a ${\displaystyle \lambda _{f}=(4-1)t_{ref}}$ wavelength remaining until it too is absorbed (during the transition phase).

Instead of a discrete jump between energy levels by the electron, absorption/emission takes place in steps, each step corresponds to a unit of ${\displaystyle r_{incr}}$;

${\displaystyle r_{incr}=-{\frac {1}{2\pi 2\alpha }}}$

As ${\displaystyle r_{incr}}$ has a minus value, the (${\displaystyle \lambda _{i}}$) photon will shrink the orbital radius accordingly, per step

${\displaystyle r_{orbital}=r_{orbital}+r_{incr}}$

Conversely, because of its minus term, the (${\displaystyle \lambda _{f}}$) photon will extend the orbital radius accordingly;

${\displaystyle r_{orbital}=r_{orbital}-r_{incr}}$

Starting with ${\displaystyle \varphi =0,\;r=2\alpha }$ (n=1), for each step during transition (anticlockwise in the diagram);

${\displaystyle \varphi =\varphi +\beta }$

As ${\displaystyle \beta }$ is proportional to the radius, as the radius increases the value of ${\displaystyle \beta }$ will reduce correspondingly (likewise reducing the orbital velocity). At discrete angles (${\displaystyle 0<\varphi <4\pi }$), the alpha spiral returns an integer value component and so we can use the spiral to calculate when the electron reaches each n level (here ni=1).;

${\displaystyle {\frac {{n_{f}}^{2}}{{n_{f}}^{2}-{n_{i}}^{2}}}={\frac {4\pi }{\varphi }}}$
${\displaystyle \varphi =(2)\pi ,\;r=4*(2\alpha )}$ (360°)
${\displaystyle \varphi =(8/3)\pi ,\;r=9*(2\alpha )}$ (360+120°)
${\displaystyle \varphi =(3)\pi ,\;r=16*(2\alpha )}$ (360+180°)
${\displaystyle \varphi =(16/5)\pi ,\;r=25*(2\alpha )}$ (360+216°)
${\displaystyle \varphi =(10/3)\pi ,\;r=36*(2\alpha )}$ (360+240°)

The transition frequency is a combination of the orbital phase and the transition phase.

${\displaystyle {\frac {{n_{f}}^{2}-{n_{i}}^{2}}{t_{orbital}+t_{transition}}}}$

The model assumes orbits also follow along a timeline z-axis, for example, at n=1 (note: the orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius);

${\displaystyle t_{orbital}=t_{ref}{\sqrt {1-{\frac {1}{(v_{orbital})^{2}}}}}}$

The basic formula for transition in the H atom becomes;

${\displaystyle f{(n_{i}\;to\;n_{f})}=({\frac {2c}{\lambda _{e}+\lambda _{p}}})({\frac {{n_{f}}^{2}-{n_{i}}^{2}}{t_{orbital}+t_{transition}}})}$

### Rydberg atom

In the H atom, the electron orbits the proton, however although the proton is heavier than the electron, a barycenter of the orbit would lie between the electron and the proton. If we consider a theoretical atom in which the nucleus is heavy enough that the barycenter resides within the nucleus, and the vicinity of the electron does not create distortions (a Rydberg atom), then we would obtain the following transition frequencies (r = Bohr radius);

${\displaystyle f{(n_{i}\;to\;n_{f})}=R(8\pi \alpha ^{2}{\frac {{n_{f}}^{2}-{n_{i}}^{2}}{t_{orbital}+t_{transition}}})}$

(nf = 2) f = R/1.33332787190 (~4/3), radius/(2α) = 4r; ${\displaystyle \varphi }$ = 360.0°

(nf = 3) f = R/1.12499717212 (~9/8), radius/(2α) = 9r; ${\displaystyle \varphi }$ = 480.0°

(nf = 4) f = R/1.06666482457 (~16/15), radius/(2α) = 16r; ${\displaystyle \varphi }$ = 540.0°

(nf = 5) f = R/1.04166532842 (~25/24), radius/(2α) = 25r; ${\displaystyle \varphi }$ = 576.0°

(nf = 6) f = R/1.02857038862 (~36/35), radius/(2α) = 36r; ${\displaystyle \varphi }$ = 600.0°

This compares with the calculated values for the H atom

(nf = 2) f = R/1.33404705279

(nf = 3) f = R/1.12560072732

(nf = 4) f = R/1.06723598132

(nf = 5) f = R/1.04222259604

(nf = 6) f = R/1.02912037991

### Notes

The simulation program used for the animation was was adapted from a gravity simulator that uses orbital pairs at the Planck scale [2]. Instead of Planck length increments, to reduce computation the program was modified to use increments in electron wavelengths, the simulation calculating the spiral radius only in terms of the fine structure constant with the wavelengths added separately.

I have used α ${\displaystyle \sim }$ 137 as this the commonly recognized value, however officially this is the inverse alpha α ${\displaystyle ^{-1}\sim }$ 137.

w:Richard Feynman, one of the originators and early developers of the theory of w:quantum electrodynamics (QED), referred to the fine-structure constant in these terms: