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) is used to map electron co-ordinates during ionisation in the H atom. At set intervals the spiral angle reduces to an integer value n² (n = 2, 3, 4, ... ∞). This n², although a function of the spiral angle, also corresponds to the principal quantum number n and so can be used to calculate the transition frequencies for each n. As these spiral angles can be used to derive the n quantum levels, we can ask if quantization of the atom is a geometrical function of this spiral.

Hyperbolic spiral

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).

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

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

This spiral has only 2 revolutions approaching 720° ($4\pi$) 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).

$\varphi =(2)\pi ,\;r=4$ (360°)

$\varphi =(4/3)\pi ,\;r=9$ (240°)

$\varphi =(1)\pi ,\;r=16$ (180°)

$\varphi =(4/5)\pi ,\;r=25$ (144°)

$\varphi =(2/3)\pi ,\;r=36$ (120°)

Electron at different n level orbitals

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.

Bohr model

The base radius for each n level uses the fine structure constant α whereby;

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

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 1 electron wavelength ($\lambda _{e}$). The steps are defined according to angle β;

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

electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference

At specific n levels;

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

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

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

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

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

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

A base (reference) orbital (n=1)

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

Physical orbital model

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

In the photon-orbital model, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that $orbital\;radius+photon=zero$ (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.

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

$\lambda _{photon}=(+\lambda _{i})-(+\lambda _{f})$

The wavelength of the ($\lambda _{i}$) photon corresponds to the wavelength of the orbital radius. The (+$\lambda _{i}$) will then delete the orbital radius as described above (orbital + photon = zero), however the (-$\lambda _{f}$), 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 (+$\lambda _{i}$) photon wavelength, the orbital radius does not change as the 2 photons cancel each other;

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

However, the ($\lambda _{f}$) has the longer wavelength, and so after the ($\lambda _{i}$) photon has been absorbed, and for the remaining duration of this ($\lambda _{f}$) photon wavelength, the orbital radius will be extended until the ($\lambda _{f}$) 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 $n_{i}=1$ orbital to a final $n_{f}=2$ orbital, first the ($\lambda _{i}$) photon is absorbed ($\lambda _{i}+\lambda _{orbital}=zero$ which corresponds to 1 complete n = 1 orbit by the electron, the orbital phase), then the remaining ($\lambda _{f}$) photon continues until it too is absorbed (the transition phase).

$\lambda _{i}=1t_{ref}$

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

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

orbital transition during orbital rotation

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

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

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

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

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

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

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

$\varphi =\varphi +\beta$

As $\beta$ is proportional to the radius, as the radius increases the value of $\beta$ will reduce correspondingly (likewise reducing the orbital velocity). At discrete angles ($0<\varphi <4\pi$), the alpha spiral returns an integer value component;

$\varphi =(2)\pi ,\;r=4*(2\alpha )$ (360°)

$\varphi =(8/3)\pi ,\;r=9*(2\alpha )$ (360+120°)

$\varphi =(3)\pi ,\;r=16*(2\alpha )$ (360+180°)

$\varphi =(16/5)\pi ,\;r=25*(2\alpha )$ (360+216°)

$\varphi =(10/3)\pi ,\;r=36*(2\alpha )$ (360+240°)

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

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

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);

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

H atom

The formula for transition (including the wavelengths of the proton and electron);

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

Bohr radius during ionization, as the H atom electron reaches each n level, it follows 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)

We can use the spiral to calculate when the electron reaches each n level (here n_i=1).

${\frac {{n_{f}}^{2}}{{n_{f}}^{2}-{n_{i}}^{2}}}={\frac {4\pi }{\varphi }}$

We then take an electron that is being ionized, and calculate the transition frequency f as it reaches each integer radius r on the spiral (as shown in the animation).

Literature:

H(1s-2s) = 2466 061 413 187.035 kHz

H(1s-3s) = 2922 743 278 665.79 kHz

H(1s-4s) = 3082 581 563 822.63 kHz

Solving for $f{(n_{i}\;to\;n_{f})},\;n_{i}=1$:

r = 4, f = 2466 048 524 808.881 kHz (n_f = 2)

r = 9, f = 2922 719 094 056.415 kHz (n_f = 3)

r = 16, f = 3082 553 353 068.955 kHz (n_f = 4)

r = 25, f = 3156 533 237 582.609 kHz (n_f = 5)

r = 36, f = 3196 719 581 863.051 kHz (n_f = 6)

r = 49, f = 3220 950 816 709.533 kHz (n_f = 7)

r = 64, f = 3236 677 704 164.894 kHz (n_f = 8)

See also: Quantum_gravity_(Planck) § Atomic_orbitals

Notes

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

The simulation was adapted from a gravity simulation program (which derives gravitational orbits from the sum of individual particle-particle orbital pairs at the Planck scale) by adding a transition between orbitals. The program runs an integer loop, incrementing with each additional unit of $r_{incr}$, and so for convenience a value for alpha was chosen to give an integer value for $3t_{ref}$. The program can be further modified for greater precision, however the purpose of the simulation discussed here was to demonstrate the spiral principle.

$\alpha ={\sqrt {\frac {472129.66667}{8\pi }}}$ = 137.059996197

The latest experimental value, published by the end of 2020, gives

1/α = 137.035999206(11),

with a relative accuracy of 8.1×10⁻¹¹
, which has a significant discrepancy from the previous experimental value.

The 2022 w:CODATA recommended value $\alpha ^{-1}$ = 137.035999177(21).

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:

“

There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.)

Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by humans. You might say the "hand of God" wrote that number, and "we don't know how He pushed His pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out – without putting it in secretly!

”

External links

• Gravity at the Planck scale
• Alpha and the Planck units as geometrical objects
• Simulation hypothesis modelling at the Planck scale using geometrical objects