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PLANCK CONSTANT AND FINE STRUCTURE CONSTANT

By Charles Rhodes, P.Eng., Ph.D.

RADIATION AND MATTER:
When an electromagnetic wave passes a particle, atom or a molecule one of three things happen:
a) Energy is transferred from the particle, atom or molecule to the wave;
b) Energy is transferred from the wave to the particle, atom or molecule;
c) No energy transfer occurs.

The probability of (a) occurring is the probability of stimulated energy emission by the particle, atom or molecule.
The probability of (b) occurring is the probability of energy absorption by the particle, atom or molecule.

If an energy transfer does occur the amount of energy dE transferred is given by:
dE = h dF
where:
dE = one quantum of energy
dF = the electromagnetic wave frequency
= the change in particle, atom or molecule natural frequency
and
h = a proportionality factor known as the Planck Constant.

In interactions between matter and radiation energy is only transferred in quantized amounts where the magnitude of the transferred energy amount is proportional to the radiation frequency.
 

PLANCK CONSTANT:
Matter stores energy in electromagnetic configurations known as spheromaks. A spheromak is a complex closed current path that has associated with it both electic and magnetic field energy.

A photon is a quantum of radiant energy either emitted by or absorbed by a spheromak. The Planck Constant is the proportionality factor which relates the magnitude of the quantum of emitted or absorbed energy to the radiation frequency. The Planck Constant is actually a composite of other physical constants. This web page shows the origin of the Planck Constant.

The change in energy dE of a particle in proportion to the frequency dF of the absorbed or emitted radiation in accordance with:
dE = h dF
where:
h = the Planck Constant
has been observed in many physical experiments involving different assemblies of charged atomic particles.
 

Analysis of the Planck constant provides insight into the mechanism by which nature stores energy in rest mass and the reasons for quantum mechanical behavior.

Any physical measurement involves emission or absorption of radiant energy quanta by the system being examined. Hence there is always a potential error the equivalent of one energy quantum uncertainty in the measure of any physical parameter. This issue is known as quantum uncertainty.

In our local universe there is an overall tendency for energy carried by high frequency radiation to be absorbed by matter and to be re-emitted from that matter carried by lower frequency radiation. This tendancy determines the direction of evolution of most chemical and nuclear reactions.
 

PLANCK CONSTANT DERIVATION:
On this web page spheromak theory is used to derive the Planck Constant from first principles. In this derivation it is implicitly assumed that the energy contained in a particle's confined photon is constant. It is shown that the Planck constant h is in part a geometrical constant known as the Fine Structure constant and is in part a function of an electron charge quantum Q, the speed of light C and the permiability of free space Muo. Energy is quantized because the structure of a stable spheromak consists of integer numbers of poloidal and toroidal charge hose turns that form the spheromak wall.
 

SPHEROMAK OPERATION:
A spheromak's electric and magnetic field structure allows quantized charges to act as stable stores of electro-magnetic energy. The behavior of these spheromak based energy stores is governed by the well known laws of electricity and magnetism. This web page shows the mathematical relationship between spheromaks and quantum mechanics.

It is shown herein that the static field energy Ett of a quantum charge electro-magnetic spheromak at steady state in field free space is given by:
Ett = h Fh
where Fh is the characteristic natural frequency of the spheromak and h is a composite of other constants that together are generally referred to as the Planck constant. If radiation is absorbed or emitted:
dEtt = h dF
where dF is the radiation frequency. Note that the total energy of an atomic particle consists of the sum of its static field energy and its contained photon energy. Typically the contained photon energy is two orders of magnitude larger than the static field energy. However, in stable particles such as electrons and protons the contained photon energy is only released in particle/anti-particle interactions.
 

SPHEROMAK GEOMETRY:
The toroidal shape of a spheromak can be characterized by its inner radius Rc and its outer radius Rs. The ratio of Rs to Rc is defined by the spheromak shape parameter So where:
So^2 = (Rs / Rc).
 

RADIATION AND MATTER:
Atomic quantum charged particles have associated electro-magnetic spheromaks. Electro-magnetic spheromaks are stable energy states resulting from emission or absorption of radiation. During radiant energy emission and absorption total system energy and total system momentum are conserved. Charged particles and radiation, both have characteristic natural frequencies. During photon emission the emitting spheromak's natural frequency Fh decreases and the amount of radiant energy present increases. During photon absorption the absorbing spheromak's natural frequency Fh increases and the amount of radiant energy present decreases.

One of the most fundamental formulae in physics is:
dEtt = h dFh
where:
dEtt = change in spheromak energy;
dFh = change in spheromak natural frequency;
and
h = Planck constant.

This formula is the basis of quantum mechanics. Spheromaks form the static field structure of all electromagnetic particles with rest mass. Since spheromaks are the main sources and sinks of radiant energy, the spheromak properties in large measure determine the radiant energy properties.
 

Over time electromagnetic spheromaks in free space will absorb or emit energy until they reach their stable state.
At this stable state the value of (dEtt / dFh) for an electromagnetic spheromak is given by:
(dEtt / dFh) = h, where:
Fh = the natural frequency of the circulating quantum charge that forms an electromagnetic spheromak and dFh is the frequency of the radiation emitted of absorbed when a quantum of energy dEtt is exchanged.
 

ORIGIN OF THE PLANCK CONSTANT:
Historically the Planck Constant h was assumed to be a natural constant that related the energy Ep carried by a photon to the frequency Fp of that photon via the formula:
Ep = h Fp
However, that formula gave no insight as to the underlying mechanisms.

It is shown herein that a quantum static charge spheromak with energy Ett and frequency Fh changes energy in accordance with:
dEtt = h dFh

If a spheromak's static electromagnetic field energy Ett changes from Ea to Eb and the spheromak frequency Fh changes from Fa to Fb then:
dEtt = (Ea - Eb)
= h (Fa - Fb)
= h dFh

The constant h can be determined theoretically by calculation of:
h = dEtt / dFh
 

PLANCK CONSTANT DEFINITION:
Although the Planck Constant is normally defined in terms of photon properties the photon energy quantization is actually due to energy quantization within the electromagnetic spheromaks that absorb or emit the photons.

Issues in high precision experimental measurement of the Planck Constant include suppression of external electric and magnetic fields that can distort the spheromak geometry and allowance for recoil energy. These two issues make the experimentally measured value of the Planck Constant dependent on the method used for its experimental measurement. The analysis herein shows that at very high resolution the Planck constant is slightly dependent on the quantum state of the system. For example the quantum state of a free electron in a vacuum may differ slightly from the quantum state of a conduction electron in a metal. The Planck constant is normally quantified by measuring the frequency of a photon emitted or absorbed during a known step change in spheromak energy. The most accurate measurement of the Planck constant is done using an apparatus known as a Kibble balance.
 

RECOIL KINETIC ENERGY:
Due to conservation of linear momentum a small portion of a change in spheromak potential energy may be converted into emitting spheromak kinetic energy instead of into photon energy. The reverse is true on photon absorption. This situation can cause a small error in experimental measurement of the Planck Constant h. Similarly thermal kinetic energy can lead to small error in the measurement of the Planck Constant. Generally precise measurements are done at low temperatures to minimize the effects of thermal kinetic energy.
 

CHARGE STRINGS:
Our universe is composed of a large number of closed charge strings. Each closed charge string contains one quantum of net electric charge, approximately 1.602 X 10^-19 coulombs. The net charge is uniformly distributed along the charge string. The net charge current circulates around the charge string path at the speed of light C, approximately 3 X 10^8 m / s. In a stable charged particle at every point along the charge string the electric and magnetic forces are in balance. For an isolated charged particle in a vacuum that geometry is a spheromak. Hence isolated electrons and protons have a spheromak geometry.
 

PARAMETER DEFINITIONS:
A spheromak in free space has an axis of symmetry from which radii are measured:
Rc = minimum radius of inner spheromak wall;
Rs = maximum radius of outer spheromak wall;
Zs = distance of spheromak wall from the equatoral plane;
Zf = maximum value of |Zs|
2 Zf = spheromak overall height parallel to its main axis of symmetry;
Rf = spheromak wall radius at Z = Zf and at Z = - Zf;
A = 2 Zf / (Rs - Rc) = A spheromak geometrical parameter near unity;
Ro = A (Rs Rc)^0.5 = radius where the potential energy well is deepest;
So = [A Rs / Ro] = [Ro / A Rc] = spheromak shape parameter;
So^2 = (Rs / Rc);
Lh = spheromak charge hose path length;
Np = number of poloidal charge hose turns contained in Lh;
Nt = number of toroidal charge hose turns contained in Lh;
Nr = Np / Nt;
Lp = Pi (Rs + Rc) = charge hose poloidal turn length;
Lt = Kc Pi (Rs - Rc) = charge hose toroidal turn length;
Kc = (ellipse perimeter length) / (circle perimeter length);
Bpo = poloidal magnetic field strength at the center of the spheromak;
Uo = (Bpo^2 / 2 Mu) = maximum field energy density at the center of the spheromak along the main axis of spheromak symmetry;
 

SPHEROMAK ENERGY:
On the web page titled: SPHEROMAK ENERGY it is shown that the total static field energy Ett of a spheromak is given by:
Ett = [Uo Ro^3 Pi^2 / A^2] {4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
where:
Uo = energy density at spheromak center;
Pi = 3.14159265
 

On the web page titled: ELECTROMAGNETIC SPHEROMAK it is shown that:
Uo Ro^3 = [1 / 2 Epsilono][Qs / (4 Pi)]^2 [1 / Ro]
  = [Muo C^2 / 2][Qs / (4 Pi)]^2 [1 / Ro]

Hence the static field energy of a charged particle spheromak is given by:
Ett
= [Muo C^2 Qs^2 / 32 A^2] [1 / Ro] {4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}

 

ELECTROMAGNETIC SPHEROMAKS:
For the purpose of analysis of electromagnetic spheromaks it is helpful to remember that electromagnetic spheromaks arise from the electric current formed by net charge circulation at the speed of light around a closed spiral path defined by the spheromak walls. The term [1 / Ro] is replaced by the function F(Fh, So) so that the spheromak static field energy instead of being expressed in terms of spheromak radius is instead expressed in terms of spheromak natural frequency Fh and the spheromak shape parameters So and A.

The change from a radial dimension Ro to frequency Fh involves a proportionality constant known as the Fine Structure Constant.
 

RELATIONSHIP BETWEEN ENERGY AND FREQUENCY:
On its equatorial plane a spheromak has an inside radius Rc and an outside radius Rs. Let Np be the number of poloidal charge hose turns and let Nt be the number of toroidal charge hose turns. Then Pythagoras theorem gives the total charge hose length Lh as:
Lh = {Lp^2 + Lt^2}^0.5
= {[2 Pi Np (Rs + Rc) / 2]^2 + [2 Pi Nt (Rs - Rc) Kc / 2]^2}^0.5
where Kc is a function of A.

Since the spheromak net charge Qs circulates at the speed of light C the circulation frequency is:
Fh = (C / Lh)
where:
Lh = {[2 Pi Np (Rs + Rc) / 2]^2 + [2 Pi Nt (Rs - Rc) Kc / 2]^2}^0.5
= [Pi {[Np (Rs + Rc)]^2 + [Nt (Rs - Rc) Kc]^2}^0.5]
= [Pi Rc {[Np (So^2 + 1)]^2 + [Nt (So^2 - 1) Kc]^2}^0.5]
= [Pi Rc Z]
= [Pi Ro / (A So)] [Z]
where:
So^2 = (Rs / Rc)

Hence:
(1 / Ro)
= [Pi Z / (A So Lh)]
 
= [Pi / A][Z / So][Fh / C]
where:
Ro = (A Rs / So)
= (A Rc So)
= nominal spheromak radius

and
Z = {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2 Kc^2]}^0.5

Note that the quantity:
[Pi Fh Z / A C So] = (1 / Ro)
and hence is proportional to the spheromak's static field energy.

The ratio (Lh / Ro) is the ratio of the spheromak charge hose length to the spheromak nominal radius. This ratio is a geometric constant for stable spheromaks.

Lh = C / Fh
= Pi Ro (Z / A So)

Hence:
[Lh / Ro] = [Pi Z / A So]

Note that Z is in effect a vector magnitude where:
[Np (So^2 +1)] is the poloidal magnetic vector and [Nt (So^2 - 1] is the toroidal magnetic vector. The ratio [Lh / Ro] is a highly stable constant which is the same for all stable spheromaks. The length Lh contains an imbedded factor of Pi.

In order for (Lh / Ro) to be the same for all stable spheromaks:
Np, (Nt Kc), So, A
must all be the same for all stable spheromaks.
 

The static field energy of a spheromak can be expressed as:
Ett
= [Muo C^2 Qs^2 / 32 A^2] [1 / Ro] {4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [Muo C^2 Qs^2 / 32 A^2] [Pi Fh / A C][Z / So]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [Muo C^2 Qs^2 / 32 A^2] [Fh / C][Lh / Ro]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [Muo C Qs^2 / 32 A^2] [Fh][Lh / Ro]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}

 

PLANCK CONSTANT:
The total static field energy Ett of an electromagnetic spheromak can be expressed in the form:
Ett = [Muo C Qs^2 / 32 A^2] [Fh][Lh / Ro]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
which is of the form:
Ett = h Fh
where:
h = [Muo C Qs^2 / 32 A^2] [Lh / Ro]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}

where:
{Lh / Ro} = [Pi Z / A So]
= [Pi / A So]{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2 Kc^2]}^0.5

The parameter h is known as the Planck Constant.

In a stable spheromak So, [Lh / Ro] and A are constants.
 

STRUCTURE OF [Lh / Ro]:
Recall that:
[Lh / Ro] = [Pi / A So]{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2 Kc^2]}^0.5

Recall from the web page titled: ELECTROMAGNETIC SPHEROMAK that the spheromak boundary condition is:
Nr^2 + [(So^2 - 1) Kc / (So^2 + 1)]^2 = 4 A^4 / Pi^2
An issue that is very important with respect to this boundary condition is that Nr^2 and So^2 are both firm constants. They do not change. Hence:
dSo = 0
and
dNr ~ 0

This boundary condition gives the conditions:
Nr^2 = (Np / Nt)^2 = Fx [4 A^4 / Pi^2]
and
(So^2 - 1)^2 Kc^2 / (So^2 + 1)^2 = (1 - Fx)[4 A^4 / Pi^2]
where Fx is a constant in the range:
0 < Fx < 1

Thus:
[Lh / Ro]^2
= [Pi / A So]^2 {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2 Kc^2]}
 
= [Pi / A So]^2 {[Np^2 (So^2 + 1)^2]
+ [{Np^2 / Fx [4 A^4 / Pi^2]} (So^2 + 1)^2 (1 - Fx)[4 A^4 / Pi^2]]}
 
= [Pi / A So]^2 {[Np^2 (So^2 + 1)^2]
+ [{Np^2 / Fx} (So^2 + 1)^2 (1 - Fx)]}
 
= [Pi / A So]^2 [Np^2 (So^2 + 1)^2][1 + [(1 - Fx) / Fx]]
 
= [Pi / A So]^2 [Np^2 (So^2 + 1)^2][1 / Fx]
 
= [Pi / A So]^2 [Np^2 (So^2 + 1)^2][4 A^4 / Pi^2] / (Np / Nt)^2
 
= [1 / So]^2 [Nt^2 (So^2 + 1)^2][4 A^2]

Thus:
[Lh / Ro] = [2 A Nt (So^2 + 1) / So]
 

Note that Lh consists of two mathematically orthogonal terms, one proportional to Np and one proportional to Nt. However, the spheromak boundary condition links Np and Nt in a manner such that they lock together providing a stable value for:
[Lh / Ro].
 

FINE STRUCTURE CONSTANT Alpha:
It is convenient to reduce the complexity of Planck Constant analysis by making the substitution:
Muo C Q^2 = 2 h Alpha
or
h = Muo C Q^2 / 2 Alpha
where Alpha is a unitless quantity known as the Fine Structure Constant.

P>The published CODATA experimentally measured value of Alpha^-1 = 137.03599915 corresponding to:
h = 6.636070150 X 10^-34 J-s.

In this context Alpha^-1 is calculated from Kibble (Watt) balance measurements of h. The linked wiki web site indicates that there is no known way of calculating Alpha from first principles. However, the mathematical formalism developed herein provides a method of calculating the theoretical value Alpha.

(1 / Alpha) = 2 h / (Muo C Q^2)
 
= [2 / (Muo C Q^2)][Muo C Qs^2 / 32 A^2] [Lh / Ro]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [1 / 4 A^2] [Lh / Ro]{So [ So^2 - So + 1] / [(So^2 + 1)^2]}

Recall that:
[Lh / Ro] = [2 A Nt (So^2 + 1) / So]

Hence:
(1 / Alpha)
= [1 / 4 A^2] [Lh / Ro]{So [ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [1 / 4 A^2] [2 A Nt (So^2 + 1) / So] {So [ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [Nt / 2 A] {[ So^2 - So + 1] / [(So^2 + 1)]}
 
= [Nt / 2 A]{1 - [So / (So^2 + 1)]}

This equation indicates that at a stable value of (1 / Alpha):
Nt, So and A are all constant. Note that (1 / Alpha) is proportional to the spheromak static energy and has a characteristic value at a spheromak relative energy minimum.
 

[Lh / Ro] STABILITY:
A key issue in Fine Structure Constant analysis is to realize that the stability of (1 / Alpha) rests on the stability of [Lh / Ro]. Recall that:
[Lh / Ro] = [Pi / A So]{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2 Kc^2]}^0.5
or
[Lh / Ro]^2 = [Pi / A So]^2 {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2 Kc^2]}
 

At the state of [Lh / Ro} stability:
d{[Lh / Ro]^2} = 0
which implies that its derivative:
[Np (So^2 + 1) / A So] d[Np (So^2 + 1) / A So]
+ [Nt (So^2 - 1) Kc / A So] d[Nt (So^2 - 1) Kc / A So]
+ {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2 Kc^2]}[- 2 Pi^2 dA / A^3 So^2]
= 0
or
[Np (So^2 + 1) / So] {[dNp (So^2 + 1) / So] + Np d[(So^2 + 1) / So]
+ [Nt (So^2 - 1) Kc / So] {dNt[(So^2 - 1) Kc / So] + Nt d[[(So^2 - 1) Kc / So]} - [Lh / Ro]^2 [2 dA / A]
= 0
or
dNp Np [(So^2 + 1) / So]^2 + dNt Nt [(So^2 - 1) Kc / So]^2
+ Np^2 [(So^2 + 1) / So] d[(So^2 + 1) / So]
+ Nt^2 (So^2 - 1) Kc / So] d[(So^2 - 1) Kc / So]
- [Lh / Ro]^2 [2 dA / A]
= 0

or
dNp Np [(So^2 + 1) / So]^2 + dNt Nt [(So^2 - 1) Kc / So]^2
+ Np^2 [(So^2 + 1) / So] [(So 2 So - (So^2 + 1)) dSo / So^2]
+ Nt^2 (So^2 - 1) Kc / So] [(2 So^2 Kc dSo - (So^2 - 1) Kc dSo/ So^2]
- [Lh / Ro]^2 [2 dA / A]

However, since So = constant, dSo = 0

Hence:
dNp Np [(So^2 + 1) / So]^2 + dNt Nt [(So^2 - 1) Kc / So]^2
- [Lh / Ro]^2 [2 dA / A]
= 0

 

Hence:
(dNp/dN) Np [(So^2 + 1) / So]^2 + (dNt /dN) Nt [(So^2 - 1) Kc / So]^2
- [Lh / Ro]^2 [2 (dA / dN) / A]
= 0

or
(dNp/dN) Np [(So^2 + 1) / So]^2 + (dNt /dN) Nt [(So^2 - 1) Kc / So]^2
- [Lh / Ro]^2 [2 d[Ln(A)] / dN]
= 0

Note that Ln(A) is a very slowly varing function of A which itself is a weak function of So^2.

 

NO COMMON FACTORS:
A spheromak will collapse if Np and Nt share a common factor other than one. A family of potential solutions for (Np / Nt) should have the characteristic that Np and Nt share no common factors.

If (dA / dN) = 0:
the (Np / Nt) Solution Family is given by:
(Np / Nt) = N / [P - 2 N]
where:
P = a fixed prime number
N = an integer in the range
0 < 2 N < P
Np = N
dNp / dN = 1
Nt = [P - 2 (N + 1)]
dNt / dN = - 2
(dNp / dNt) = (-1 / 2)
or
2 dNp = - dNt

If dA / dN is small but non-zero the solution family remains the same except that the prime number P becomes a slowly varing function of N. Thus P only remains constant over a limited range of N. Any solution for P is only valid for N values close to the point of stability. At the N value corresponding to the point of stability:
dP / dN = 0,
but for significantly different values of N:
dP / dN is non-zero.

Thus at the N value corresponding to the point of stability:
dNt = - 2 dNp
but we cannot rely on:
(dNp / dNt) = (- 1 / 2)
ratio being true at significantly different values of N.
 

CONSTRAINT ON THE RANGE OF Np:
Nr = Np / (P - 2 Np)
and
Nr^2 + (B Kc)^2 = 4 A^4 / Pi^2

Hence:
Nr < 2 A^2 / Pi
or
Np / (P - 2 Np) < 2 A^2 / Pi
or
Np < (P - 2 Np) 2 A^2 / Pi
or
Np (1 + 4 A^2 / Pi) < P [2 A^2 / Pi]
or
Np < P [2 A^2 / Pi] / (1 + 4 A^2 / Pi) or
Np < P [2 A^2] / [Pi + (4 A^2)]
 

CONSTRAINT ON THE RANGE OF So:
Recall that:
Nr^2 + Kc^2 B^2 = 4 A^2 / Pi^2

Hence:
B Kc < 2 A^2 / Pi
or
B < 2 A^2 / Pi Kc
or
(So^2 - 1) / (So^2 + 1) < [2 A^2 / Pi Kc]
or
(So^2 - 1) < [2 A^2 / Pi Kc](So^2 + 1)
or
So^2 (1 - [2 A^2 / Pi Kc]) < (1 + [2 A^2 / Pi Kc])
or
So^2 < (1 + [2 A^2 / Pi Kc]) / (1 - [2 A^2 / Pi Kc])
or
So^2 < [(Pi Kc + 2 A^2) / (Pi Kc - 2 A^2)]

At A = 1.000, Kc = 1.000:
So^2 < [(Pi + 2) / (Pi - 2)]
or
So^2 < 5.14 / 1.14 = 4.508

So^2 = Rs / Rc
where:
Rs > Rc
Hence So^2 > 1

Hence:
1 < So^2 < 4.508
 

[Lh / Ro] STABILITY CONTINUED:
The d{Lh / Ro] = 0 requirement of:
dNp Np [(So^2 + 1) / So]^2 + dNt Nt [(So^2 - 1) Kc / So]^2
- [Lh / Ro]^2 [2 (dA / A)]
= 0
becomes:
dNp Np [(So^2 + 1) / So]^2 + (- 2 dNp) Nt [(So^2 - 1) Kc / So]^2
- [Lh / Ro]^2 [2 dNp (dA / dNp A)] = 0
or
dNp{ Np [(So^2 + 1) / So]^2 + (- 2 Nt) [(So^2 - 1) Kc / So]^2
- [Lh / Ro]^2 [2 (dA / dNp A)]} = 0

At the N value corresponding to [Lh / Ro] stability:
(dA / dNp) = 0
so this equation gives the important condition:
Np [(So^2 + 1) / So]^2 + (- 2 Nt) [(So^2 - 1) Kc / So]^2 = 0
or
[Np / Nt] = 2 [(So^2 - 1) Kc / So]^2 / [(So^2 + 1) / So]^2
= 2 [(So^2 - 1) Kc]^2 / [(So^2 + 1)]^2

Recall that the spheromak boundary condition is:
(Np / Nt)^2 + [(So^2 - 1) Kc / (So^2 + 1)]^2 = 4 A^4 / Pi^2

Combining these two condition equations at the point of stability gives:
(Np / Nt)^2 + (Np / 2 Nt) = 4 A^4 / Pi^2

or
Nr^2 + (Nr / 2) - 4 A^4 / Pi^2 = 0

This quadratic equation has the real physical solution:
Nr = {- (1 / 2) + [(1 / 4) + 4(1)(4 A^4 / Pi^2)]^0.5} / 2
= (- (1 / 4) + (1 / 4)[1 + 64 A^4 / Pi^2]^0.5

If A = 1.00000 the computed value of (Np / Nt) is:
(Np / Nt) = 0.4339479041
One of the challenges is determination of the A value at the point of stability.

THIS VALUE OF (Np / Nt) AND ITS DEPENDENCE ON A IS KEY TO UNDERSTANDING SPHEROMAKS. Initial calculations are done with A = 1.0000. However, we will discover that A > 1 which means that the real value of (Np / Nt) is larger than for A = 1.0000.

Note that as shown later if A = 1.0000 then:
P = 947
and
(Np / Nt) = (220 / 507)
= 0.4339250493

However, for A > 1.0000 Nt will decrease from 507 to 505, 503, 501, 499, 497, 495, 493, 491,.... and Np will increase from 220 to 221, 222, 223, 224, 225, 226, 227, 228 .... A further complication is that with the change in N the prime number P will also likely change.

(Np / Nt) = Np / [P - 2 Np]
 

FUNCTIONS OF Nr:
Recall that:
Nr = (Np / Nt)
= {(- 1 / 4) +(1 / 4) [1 + (64 A^4 / Pi^2)]^0.5}

At A = 1.00000, Np = 220, Nt = 507:
(Np / Nt) = 0.4339250493
and
(Np / Nt)^2 = 0.1882909484
 

FUNCTIONS OF B:
Define B by:
B = [(So^2 - 1) / (So^2 + 1)]

Recall that:
Nr^2 + [(So^2-1) Kc / (So^2+ 1)]^2 = 4 A^4 / Pi^2

Define:
B = (So^2 - 1) / (So^2 + 1)
or
Nr^2 + Kc^2 B^2 = 4 A^4 / Pi^2
or
B^2 = [(4 A^4 / Pi^2) - (Nr^2)] / Kc^2

At A = 1.000000, Kc = 1.00000, Np = 220, Nt = 507
B^2 = [0.4052847355 - 0.1882909484]
= 0.2169937871
and
B = 0.4658259193
and
1 - B^2 = 0.7830062129
and
(1 - B^2)^0.5 = 0.8848763828

dB / dN = 0
 

FUNCTIONS OF So:

Thus:
B = [(So^2 - 1) / (So^2 + 1)]
or
(So^2 - 1) = B (So^2 + 1)
or
So^2 (1 - B) = (1 + B)
or
So^2 = [(1 + B) / (1 - B)]
or
So^2 + 1 = 2 / (1 - B)
and
So = [(1 + B) / (1 - B)]^0.5

At A = 1.00000 So^2 = [(1 + B) / (1 - B)]
= 1.4658259193 / 0.5341740807
= 2.744097799

So / (So^2 + 1) = [(1 + B) / (1 - B)]^0.5 [(1 - B) / 2]
= {[(1 + B) (1 - B)]^0.5} / 2
= {[1 - B^2]^0.5} / 2
 

(So^2 - 1) = [(1 + B) / (1 - B)] - 1
= [(1 + B) - (1 - B)] / (1 - B)
= 2 B / (1 - B)

So / (So^2 - 1) = [(1 - B) / 2 B][(1 + B) / (1 - B)]^0.5
= [(1 + B)(1 - B)]^0.5 / 2 B
= [1 - B^2]^0.5 / 2 B
 

*************************************************************

INSTABILITY OF (1 / Alpha):
Recall that:
(1 / Alpha) = [Nt / 2 A]{1 - [So / (So^2 + 1)]}

Assume that (1 / Alpha) is stable with respect to N. Substituting this expression into the expression for:
d(1 / Alpha) / dN = 0
gives:
d(1 / Alpha) / dN = (1 / 2 A){1 - [(1 - B^2)^0.5 / 2]}[dNt / dN]
+ [Nt / 4 A] d{1 - [So / (So^2 + 1)]}
 
= (1 / 2 A){1 - [(1 - B^2)^0.5 / 2]}[dNt / dN]
 
= (1 / 2 A){1 - [(1 - B^2)^0.5 / 2]}[-2]
 
This expression has the fundamental problem that it is always negative. To provide stability it must go to zero. Hence we can conclude that (1 / Alpha) is unstable with respect to N and we must identify a different cause of the apparent stability of (1 / Alpha). That stability is caused by the stability of [Lh / Ro].
 

FIND A STABLE SOLUTION BASED ON d[Lh / Ro] = 0
Np = N
and
Nt = (P - 2 N)
where:
P = prime number
and N = integer in the range:
0 < 2 N < P

Note that for each of these quantum number (Np / Nt) pairs there is no common factor and that between adjacent number pairs the relationship:
dNt = - 2 dNp
applies.

Then:
(Np / Nt) = N / [P - 2 N]
and
dNp / dN = 1
and
dNt / dN = - 2
so that
dNt = - 2 dNp
and
dSo / dN = 0

Thus:
Np / Nt = [N / (P - 2 N)]
= {(- 1 / 4) +(1 / 4) [1 + (64 A^4 / Pi^2)]^0.5}

At A = 1.00000:
(Np / Nt) = 0.4339250493
= [N / (P - 2 N)]

Hence:
(P - 2 N)(0.4339250493) = N
or
P(0.0.4339250493) = N (1 + 2(0.4339250493))
= N (1.867850099)

Thus:

P = N [(1.867850099) / (0.4339250493)]
= N (4.304545455)

Hence if A = 1.000000 a valid value of P when divided by 4.304545455 will give an integer result.

Prime Number P Quotient
997 231.6156283
991 230.2217529
983 228.3632524
977 226.969377
971 225.5755016
967 224.6462513
953 221.3938754
947 220.0000000***
941 218.6061246
937 217.6768743
929 215.8183738
919 213.4952481
911 211.6367476

Note that at P = 947 the quotient 220.00000 resulting from:
(P / 4.304545455)
has zero remainder. Hence:
Np = 220
P = 947
Nt = 947 - 2(220) = 507

Hence:
Nr = 220 / 507 = 0.4339250493

Recall that:
So / (So^2 + 1) = {[1 - B^2]^0.5} / 2 At A = 1.00000
B^2 = [(4 A^4 / Pi^2) - (Nr^2)] / Kc^2
= [(4 / Pi^2) - (Nr^2)]
(1 - B^2)^0.5 = 0.8848763828
{1 - So / (So^2 + 1)}
= [1 - (0.8848763828 / 2)]
= 0.557561809

(1 / Alpha) = [Nt / 2 A][1 - [So / (So^2 + 1)]]
= [507 / 2 A] [0.557561809]
= 141.3419186 / A

We must study the origin of A to resolve its size.
 

SUMMARY:
Nr^2 + (Nr / 2) - 4 A^4 / Pi^2 = 0 and
Nr = N / [P - 2 N]
where:
Nr = (Np / Nt)
Kc = Function of A
So = [(1 + B) / (1 - B)]^0.5
B^2 = [4 A^4 / Pi^2 - Nr^2] [1 / Kc]^2
Nr^2 + Kc^2 B^2 = 4 A^4 / Pi^2

Nr = {(- 1 / 4) + (1 / 4) [1 + (64 A^4 / Pi^2)]^0.5}

Note that certain values of A are not permitted because they could cause a spheromak collapse due to Nr becoming a simple fraction. For example: If:
[1 + (64 A^4 / Pi^2)]^0.5 = 3.

In this case:
[1 + (64 A^4 / Pi^2)] = 9
or
(64 A^4 / Pi^2) = 8
or
A^4 = Pi^2 / 8
or
A^2 = Pi / 8^0.5 = 3.14159265 / 2.828427125
= 1.110720733
or
A = 1.053907365

In this case:
Nr = (-1 / 4) + (1 / 4)(3)
= (1 / 2)

The above are the fundamental equations of quantum mechanics that must be solved in any physical situation to find Nr, P, Np and Nt. With those values, if A is known one can solve for (1 / Alpha) which leads to the Planck constant.

The keys to finding these equations were to recognize that:
dSo / dN = 0
and to prevent spheromak collapse locally:
Nr = (Np / Nt) = N / [P - 2 N}

The solution was to recognize that subject to variations in A, So and Nr behave as constants. To solve a real problem we must find the P and N values corresponding to a particular Nr value.

To find the stable state for [Lh / Ro] we need to find A, calculate Kc, specify prime number P values and for each P value step through N values near:
N ~ (P / 4)
looking for an integer solution at which the measured Np / Nt value.
 

REVERSE ANALYSIS:
In this section we express the spheromak parameters in a manner that lends itself to decomposition of experimental data.

Recall that:
Nr^2 + Kc^2 B^2 = 4 A^4 / Pi^2

Hence:
Kc^2 B^2 = (4 A^4 / Pi^2) - (Nr^2)
or
B^2 = [(4 A^4 / Kc^2 Pi^2) - (Nr^2 / Kc^2)]

Recall that:
B = (So^2 - 1) / (So^2 + 1)

Hence:
(So^2 - 1) / (So^2 + 1) = [(4 A^4 / Kc^2 Pi^2) - (Nr^2 / Kc^2)]^0.5
= (1 / Kc)[(4 A^4 / Pi^2) - (Nr^2)]^0.5

Recall that:
Nr = Np / (P - 2 Np)

Hence:
(So^2 - 1) / (So^2 + 1) = [1 / Kc]{(4 A^4 / Pi^2) - [Np / Nt]^2 }^0.5

Hence:
(So^2 - 1) = (So^2 + 1) (1 / Kc){(4 A^4 / Pi^2) - [Np / Nt]^2}^0.5
or
So^2[1 - (1 / Kc){(4 A^4 / Pi^2) - [Np / Nt]^2 }]
= [1 + (1 / Kc){(4 A^4 / Pi^2) - [Np / Nt]^2}]
or
So^2 = [1 + (1 / Kc){(4 A^4 / Pi^2) - (Np^2 / Nt^2}]
/ [1 - (1 / Kc){(4 A^4 / Pi^2) - (Np^2 / Nt^2)}]

So^2 + 1 = 2 / [1 - (1 / Kc){(4 A^4 / Pi^2) - (Np^2 / Nt^2)}]

Hence:
So / (So^2 + 1)
= (1 / 2)[1 - (1 / Kc){(4 A^4 / Pi^2) - (Np^2 / Nt^2)}]^0.5
[1 + (1 / Kc){(4 A^4 / Pi^2) - (Np^2 / Nt^2)}]^0.5  
= (1 / 2)[1 - (1 / Kc)^2{(4 A^4 / Pi^2) - (Np^2 / Nt^2)}^2]^0.5
 

(1 / Alpha) = [Nt / 2 A] {1 - [So / (So^2 + 1)]}

Rearrange this equation as:
[Nt / 2 A] = (1 / Alpha) + [Nt / 2 A][So / (So^2 + 1)]
or
Nt = (2 A / Alpha) + [Nt So / (So^2 + 1)]
= (2 A / Alpha)
+ [(Nt / 2)[1 - (1 / Kc)^2{(4 A^4 / Pi^2) - (Np^2 / Nt^2)}^2]^0.5
 
= (2 A / Alpha)
+ [(Nt / 2 Kc)[Kc^2 - {(4 A^4 / Pi^2) - (Np^2 / Nt^2)}^2]^0.5
 
= (2 A / Alpha)
+ [(Nt / 2 Kc)[Kc^2 - {(4 A^4 / Pi^2)^2 + (Np^4 / Nt^4) - 8(A^4 / Pi^2)(Np^2 / Nt^2)}]^0.5
 
= (2 A / Alpha)
+ [(Nt / 2 Kc)[Kc^2 - (4 A^4 / Pi^2)^2 - (Np^4 / Nt^4) + 8(A^4 / Pi^2)(Np^2 / Nt^2)]^0.5
 

Thus we have the equation:
Nt = (2 A / Alpha)
+ [(Nt / 2 Kc)[Kc^2 - A^8 (2 / Pi)^4 - (Np / Nt)^4 + A^4 (8 / Pi^2) (Np / Nt)^2]^0.5
 

Further rearrange this equation to get:
[Nt - (2 A / Alpha)][2 Kc / Nt]
= [Kc^2 - A^8 (2 / Pi)^4 - (Np / Nt)^4 + A^4 (8 / Pi^2) (Np / Nt)^2]^0.5
or
Kc [2 - (4 A / Nt Alpha)] = [Kc^2 - A^8 (2 / Pi)^4 - (Np / Nt)^4 + A^4 (8 / Pi^2) (Np / Nt)^2]^0.5
or
Kc^2 [2 - (4 A / Nt Alpha)]^2 = [Kc^2 - A^8 (2 / Pi)^4 - (Np / Nt)^4 + A^4 (8 / Pi^2) (Np / Nt)^2]
or
Kc^2 [4 - A (16 / Nt Alpha) + A^2 (16 / Nt^2 Alpha^2)] = [Kc^2 - A^8 (2 / Pi)^4 - (Np / Nt)^4 + A^4 (8 / Pi^2) (Np / Nt)^2]
or
Kc^2 [3 - A (16 / Nt Alpha) + A^2 (16 / Nt^2 Alpha^2)] = [- A^8 (2 / Pi)^4 - (Np / Nt)^4 + A^4 (8 / Pi^2) (Np / Nt)^2]

The web page titled: THEORETICAL SPHEROMAK shows that for A close to unity:
Kc = {[4 (A + 1)^2 + (A - 1)^2] / [8 (A + 1)]}
= [(A + 1) / 2] + [(A - 1)^2 / 8 (A + 1)]

Thus:
Kc^2 = [(A + 1) / 2]^2 + 2 [(A + 1) / 2][(A - 1)^2 / 8 (A + 1)]
+ [(A - 1)^2 / 8 (A + 1)]^2
 
= [(A + 1) / 2]^2 + [(A - 1)^2 / 8] + [(A - 1)^2 / 8 (A + 1)]^2

 
= (A^2 / 4) + (2 A / 4) + (1 / 4) + (A^2 / 8) + (-2 A / 8) + (1 / 8)
+ [(A^2 / 16) + (- 2 A / 16) + (1 / 16)]^2
 
~ (3 A^2 / 8) + (A / 4) + (3 / 8)
=(3 A^2 + 2 A + 3) / 8

Thus the equation in A becomes:
[3 A^2 + 2 A + 3][3 - A (16 / Nt Alpha) + A^2 (16 / Nt^2 Alpha^2)]
= 8 [- A^8 (2 / Pi)^4 - (Np / Nt)^4 + A^4 (8 / Pi^2) (Np / Nt)^2]

Thus we can test prospective values of A, Np and Nt using this equation with knowledge that the experimentally measured value of (1 / Alpha) is given by:
(1 / Alpha) = 137.03599915

We need to next find the value of A via the following ellipse analysis.
 

ELLIPSE STABILITY:
On the web page titled: THEORETICAL SPHEROMAK we identified that the the spheromak toroidal path Lt is actually an ellipse in which:
A = (major axis parallel to axis of symmetry) / (minor axis parallel to spheromak equatorial plane).

The spheromak boundary condition gives:
Nr^2 + Kc^2 B^2 = 4 A^4 / Pi^2

For a stable spheromak:
dNr = 0
implying that:
d[(4 A^4 / Pi^2) - Kc^2 B^2] = 0
or
(4 A^3 / Pi^2) dA - Kc^2 2 B dB - B^2 2 Kc dKc = 0

However, for a stable spheromak:
dSo = 0 which implies that dB = 0.

Hence for a stable spheromak:
(4 A^3 / Pi^2) dA = B^2 2 Kc dKc

However, the web page titled: THEORETICAL SPHEROMAK shows that for |A - 1| << 1:
Kc = [(1 + A) / 2] {1 + (1 / 4)[(A - 1) / (A + 1)]^2
+ (1 / 64)[(A - 1) / (A + 1)]^4 + (1 / 256)[(A - 1) / (A + 1)]^6 +...}

The first two terms of this expression simplify to:
Kc = {[4 (A + 1)^2 + (A - 1)^2] / [8 (A + 1)]}

Differentiating this simplified expression gives:
dKc
= {[8 (A + 1)][8(A + 1)dA + 2(A - 1) dA] - [4 (A + 1)^2 + (A - 1)^2] 8 dA}
/ [8 (A + 1)]^2
 
= {[64 (A + 1)^2 + 16 (A^2 - 1)] - [32 (A + 1)^2 + 8 (A - 1)^2]} dA
/ [8 (A + 1)]^2
 
= {[32 (A + 1)^2 + 16 A^2 - 16] - [ + 8 (A^2 - 2 A + 1)]} dA
/ [8 (A + 1)]^2
 
= {[32 (A + 1)^2 + 8 A^2 + 16 A - 24]} dA
/ [8 (A + 1)]^2
 
= {[32 (A^2 + 2 A + 1) + 8 (A^2 + 2 A - 3)]} dA
/ [8 (A + 1)]^2
 
= {[40 A^2 + 80 A + 8)]} dA
/ [8 (A + 1)]^2
 
= {8 [5 A^2 + 10 A + 1)]} dA
/ [8 (A + 1)]^2
 
= {[5 A^2 + 10 A + 1)]} dA
/ [8(A + 1)^2]

 

Thus at spheromak stability:
(4 A^3 / Pi^2) dA = B^2 2 Kc dKc
becomes:
(4 A^3 / Pi^2) dA
= B^2 2 {[4 (A + 1)^2 + (A - 1)^2] / [8 (A + 1)]} {[5 A^2 + 10 A + 1)]} dA
/ [8(A + 1)^2]
or
(4 A^3 / Pi^2)
= B^2 2 {[4 (A + 1)^2 + (A - 1)^2] / [64 (A + 1)^3]} {[5(A + 1)^2]}
or
(A^3 / Pi^2)
= B^2 {[4 (A + 1)^2 + (A - 1)^2] / [128 (A + 1)]} {[5]}
or
B^2 = {[128 (A + 1)](A^3 / Pi^2)} / {5 [4 (A + 1)^2 + (A - 1)^2]}

Thus for A = 1.00000:
B^2 = {256 / Pi^2} / {80}
= [32 / 10 Pi^2]
or
B = (3.2)^0.5 / Pi
= 0.5694100354

.

So^2 = (1 + B) / (1 - B)
= 1.5694100354 / 0.4305899646
= 3.644790089
 

CALCULATING SPHEROMAK PARAMETERS BY ITTERATION:
The practical way of finding the exact values of the spheromak parameters is by itteration.

a) Start with A = 1.0000. Use the formula:
B^2 = {[128 (A + 1)](A^3 / Pi^2)} / {5 [4 (A + 1)^2 + (A - 1)^2]}

to find an interim value for B^2.

b) Using this same value of A calculate an interim value for Kc using the formula:
Kc = [(1 + A) / 2] {1 + (1 / 4)[(A - 1) / (A + 1)]^2
+ (1 / 64)[(A - 1) / (A + 1)]^4 + (1 / 256)[(A - 1) / (A + 1)]^6 +...}
Note that A = 1.00000 the initial interim value of Kc will be 1.00000.

c) Calculate an interim value for Nr using the same value of A and the formula:
Nr = (-1 / 4) + (1 / 4)[1 + 64 A^4 / Pi^2]^0.5

d) Then insert the interim values of Nr^2, Kc^2 and B^2 into the formula:
A^4 = (Pi^2 / 4)[Nr^2 + Kc^2 B^2]
to find a new interim value of A.

e) Use the new interim value of A to repeat step (a) and thus obtain a new interim value of B.

f) Use the same new interim value of A to repeat step (b) and thus obtain a new interim value for Kc.

g) Use the same new interim value of A to repeat step (c) and thus obtain a new interim value for Nr^2.

h) Use the new interim values of B, Kc and Nr to calculate a new interim value of A as in step (d)

i) Repeat this process as many times as necessary to make the calculated value of A to converge to a precise value at the spheromak point of stability. There will be a corresponding real number value for Nr.

j) Find the rational number equal to this Nr value real number using the formula:
(Np / Nt) = Np / (P - 2 Np)

k) Find the corresponding values of So^2 and So using the formula:
So^2 = (1 + B) / (1 - B)

l) Calculate (1 / Alpha) using the formula:
(1 / Alpha) = (Nt / 2 A){1 - [So / (So^2 + 1)]}
where:
So / (So^2 + 1) = [(1 - B^2)^0.5] / 2

m)Compare the calculated value of (1 / Alpha) to the experimentally measured value of 137.03599915 corresponding to:
h = 6.636070150 X 10^-34 J-s
Note that to obtain really accurate results more terms must be used in the power series expansions of Kc and dKc.

n) Thus we have developed a methodology for precise calculation of the Fine Structure Constant from first principles. It is necessary to use an itterative calculation methodology because of the complexity of the relationship between the perimeter length of an ellipse and the linear dimensions of an ellipse. Ideally the A value calculated herein can be confirmed via a field energy density analysis along the Z axis calculated using the same value of So as calculated herein.
 

******************************************

Prime numbers less than 1000 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997.
 

On the web page titled: ELECTROMAGNETIC SPHEROMAK we speculated that:
Np = 223
Nt = 495

However, we have a possible hint.
[(1 / Alpha)]^2 ~ [137]^2 + Pi^2

EXPLANATION:
Recall that:
Ett = [Muo C Qs^2 / 2][Lh /(4 Ro)] Fh {[So][ So^2 - So + 1] / [(So^2 + 1)^2]}
The spheromak total static field energy Ett is the product of two functions, a energy function:
[Muo C Qs^2 / 2][Lh / 4 Ro] Fh
which increases with decreasing Ro:
and an energy reducing spheromak shape parameter function:
S(So) = {So [So^2 - So + 1] / [(So^2 + 1)^2]}
which decreases as So increases from unity.

The shape parameter function S(So) has three important properties:
a) At So = 1.0 the S(So) function value is unity;
b) At So = 1.0 [dS(So) / dSo] = 0;
b) At So = 2.0 the S(So) function value is (24 / 25);
c) At So = 2.0 the ratio:
[dS(So) / dSo] / S(So) = (- 1 / 10)

To understand the stability of So we have to understand how the spheromak boundary condition sets So.

The boundary condition is:
Nr^2 + (B Kc)^2 = 4 A^4 / Pi^2
where:
B = (So^2 - 1) / (So^2 + 1)
and
Nr = (Np / Nt)
 

Z QUANTIZATION:
The Fine Structure constant stability comes from inherent So stability. Larger changes in spheromak energy occur via integer changes in Np and Nt which cause quantum changes in Z.

Spheromak energy changes causing a change in So occur in quantum jumps caused by integer changes in Np and/or Nt. Ideally the energy changes associated with an incrementation or decrementation in Np are smaller than the energy changes associated with an incrementation or decrementation in Nt. Thus a multi-electron atomic spheromak can exhibit energy shells where Np indicates the number of electrons in a shell and Nt indicates which shell is involved.

Note that the rest mass energy of a charged particle such as an electron or proton is much higher than the spheromak static field energy due to the presence of a confined photon.

FINE STRUCTURE CONSTANT SOLUTION:
EXPERIMENTAL
On first inspection solving the equation for the Fine Structure Constant appears quite difficult. However, we are aided by experimental data which indicates that:
(1 / Alpha)^2 ~ (137)^2 + Pi^2
or
(1 / Alpha) ~ [(137)^2 + Pi^2]^0.5
or
(1 / Alpha) = 137.035999
 

Define:
S(So) = {[So / 2] [So^2 - So + 1] / [(So^2 + 1)^2]}

Note that at So = 2 this equation simplifies and gives:
S(So) = 3 / 25
 

Note that:
(1 / Alpha) = [Lh / 2 Ro] [So / 2]{[ (So^2 - 1) + (2 - So)] / [(So^2 + 1)^2]}
= [Lh / 2 Ro] S(So)

Recall that:
Z = [Lh / Ro] [So / Pi]
giving:
(1 / Alpha)
= [(Pi / So) Z (1 / 2)] [So / 2]{[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [(Pi / 4) Z] {[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [(Pi / 4) Z] {[(So^2 - 1) + (2 - So)] / [(So^2 + 1)^2]}

 
= [(Pi / 2) Z / So][So / 2] {[(So^2 - 1) + (2 - So)] / [(So^2 + 1)^2]}

At So = (2 + dSo):
S(So) = (3 / 25) + {[dS(So) / dSo]|So = 2} dSo

[dS(So) / dSo] = {[(So^2 + 1)^2]{[1 / 2][So^2 - So + 1] + [So / 2][2 So - 1]}
- [So / 2] [So^2 - So + 1][2 (So^2 + 1) 2 So]}
/ [(So^2 + 1)^4]
 
= {[(So^2 + 1)]{[1 / 2][So^2 - So + 1] + [So / 2][2 So - 1]}
 

Thus we have a tentative solution to a real quantum mechanical problem obtained by a good guess supported by a precise experimental data. However, we need to extend the spheromak theory to be able to apply it systematically to more general quantum mechanical problems.

Experimental data indicates that:
(1 / Alpha) = 137.03599915

An important issue worthy of noting is that if So = 2.0000:
(1 / Alpha) = (Lh / 2 Ro) S(So)
= (Lh / 2 Ro)(3 / 25)
which suggests that for:
(1 / Alpha) = 137.03599915:
then:
[Lh / Pi Ro] = (50 / 3 Pi)(1 / Alpha)
= (50 / 3 Pi)(137.03599915) = 726.9985557

Remember that Lh contains a factor of Pi whereas Ro does not.
 

> **************************************************************************

FINE STRUCTURE CONSTANT ISSUE:
Note that (1 / Alpha) is a function of the spheromak shape parameters So^2 and A. Note the following:
1) The spheromak energy is proportional to (1 / Alpha). Hence the spheromak has maximum energy stability when a plot of (1 / Alpha) versus So is at a relative minimum.

2) Experimentally (1 / Alpha) is a stable constant indicating that at the spheromak operating point:
d(1 / Alpha) / dSo = 0
further confirming that the spheromak operates at a relative minimum in a plot of (1 / Alpha) versus So.

3) Thus quantification of the Fine Structure Constant reduces to quantification of (1 / Alpha) at its relative minimum with respect to So where:
d(1 / Alpha) / dSo = 0
 

Alpha^-1 is a geometric ratio measured to be:
Alphas^-1 = 137.03599915

Note that the measured value of Alpha is slightly dependent on the system quantum state that can vary depending on the system environment. For example, the quantum state of a free electron in a vacuum may differ from the quantum state of a free electron in a metal which may differ again from the quantum state of an electron in a superconductor. the quantum state is specified by two numbers, Np and Nt.
 

SOLUTION:

A preliminary BASIC program solution indicates that there is a broad relative minimum in [1 / (Alpha Nt)] located at So = 2.026. When So is precisely:
So = 2.02606822 the corresponding value of (1 / Alpha) is given by:
(1 / Alpha) = 137.035999

The parameter So is a function of the electromagnetic boundary condition which determines the electric field ratio parameter F.

A spheromak consists of Np poloidal turns and Nt toroidal turns. The numbers Np and Nt are both integers. The ratio:
Nr = (Np / Nt)
is a rational number.

The initial computer program suggests that:
Np = 223
and
Nt = 305

Note that So^2 is a real number involving Pi.

In order for a particle to be stable it must exactly conform to an equality between the rational number:
Nr^2 = (Np / Nt)^2
and a real number which is a function of Pi^2 and So^2 and the boundary condition.

Hence, since Np and Nt are whole numbers So^2 is quantized. Thus, the particle static field energy, which is a function of So^2, is also quantized.

Hence for an isolated charged particle the Planck constant is really just an indication of the energy difference between adjacent stable spheromak energy states.

When there are multiple charged particles involved the geometry becomes more complex but the underlying principle is the same. Stable solutions only exist at values of Nr^2 = (Np / Nt)^2 that correspond to a precise balance between the electric and magnetic forces along the entire length of the charge string path. These precise balance situations are a function of Pi and F, where F is set by environmental parameters. The numbers Np and Nt cannot have any common factors. Otherwise the spheromak would not be stable.  

Note that while a computer may be used to find an analytical solution for So remember that the equation for So is only approximately true due to quantization of Np and Nt. Thus the precise value of So that is actually adopted will depend on the quantum state [Np / Nt]. This quantum state is set by the electromagnetic boundary condition on the spheromak. For details on this boundary condition matter refer to the web page titled: ELECTROMAGNETIC SPHEROMAK

Note that for each of these quantum number pairs there is no common factor and that between adjacent number pairs the relationship:
dNt = - 2 dNp
applies.

Hence Alpha as a function of So can be accurately calculated provided that the functional dependence between Nr = Np / Nt and the environment can be determined.

Remember that Nr^2 is affected by the spheromak environment. Consider a pure silicon crystal. At some Nr^2 and So values there maybe no electron energy states within a certain energy range. Then an energy bandgap will occur.

Similarly the electron energy states in a metal are less than the energy of an electron in free space by the work function of the metal.

Note the relationship between Nr^2 which is a rational number and Pi^2 which is a real number. This relationship is only valid for integer values of Np and Nt and corresponding discrete values of So.
 

RECOIL KINETIC ENERGY ERROR:
Historically h was defined as:
h = Ep / Fp where:
Ep = photon energy
and
Fp = photon frequency

In circumstances where the charged particle recoil kinetic energy is negligibly small:
Ep ~ dEtt
and
Fp ~ dFh

In reality there is a small difference between Ep and dEtt due to the charged particle recoil kinetic energy caused by the momentum of the photon. This issue of recoil momentum becomes important in very high accuracy measurements of the Planck constant h.

The recoil kinetic energy error will depend on the energy Ep of the photons used and the particle mass which is usually either an electron or an atomic nucleus.

Consider an electron with rest mass Me. The rest potential energy of the electron is:
Etta = Me C^2.

The photon energy experimentally used for determination of h by non-magnetic methods is typically of the order of:
Ep = 1 eV.

For electrons: Etta = Me C^2 = 9.1 X 10^-31 kg X (3 X 10^8 m / s)^2 X 1 eV / 1.602 X 10^-19 J
= 51.12 X 10^4 eV

Hence if h is measured via photon emission from an ionized gas and if the term:
[1 +(Ep / 2 Etta)]
is assumed to be unity we can expect an error in the experimentally determined value of h of about:
(Ep / 2 Er) = 1 eV / [2 (51.12 X 10^4 eV)]
= 1 / (102.24 X 10^4)
~ 0.978 X 10^-6

Thus the discrepency between the theoretical value of h and the experimental value of h may in part be caused by failure to properly take into account recoil kinetic energy when a spheromak emits a photon during an atomic energy transition. If this error is uncorrected the experimentally measured value of h will be slightly larger than the theoretical value of h calculated herein.

Note that the h value calculated herein is actually the change in spheromak static potential energy Ett with respect to a change in spheromak natural frequency. The energy carried away by the photon will be slightly less than the decrease in spheromak potential energy due to the small increase in charged particle kinetic energy on emission of a photon. This increase in charged particle kinetic energy must occur to satisfy the law of conservation of linear momentum.

This issue is also known as the Mossbauer Effect and is experimentally demonstrable via use of the doppler effect and certain nuclear energy transitions.

For precise measurement of h it is important that the participating atoms be cold to minimize error due to thermal molecular motion.

Note that an experimental measurement of h will be more precise if the particle emitting or absorbing the photon has a larger rest mass. Then the recoil momentum results in less recoil energy. This issue likely improves the resolution of medical Magnetic Resonance Imaging (MRI) equipment that relies on photon absorption and emission by protons in water. However, in that application there is still disturbance of the external magnetic fields caused by the circulating electrons of the hydrogen and oxygen atoms.
 

RECOIL KINETIC ENERGY:
In crude experimental measurements it is generally assumed that:
(Esa - Esb) = Ep = photon energy
and
(Fsa - Fab) = Fp = photon frequency

However, the Planck constant is normally evaluated by measuring the frequency of the photon emitted or absorbed during a change in spheromak energy. Due to conservation of linear momentum a small portion of the change in spheromak potential energy is converted into spheromak recoil kinetic energy instead of into photon energy. The reverse is true on photon absorption. This situation causes a small error in experimental measurement of hs.

Thus the change in particle energy is:
(Esa - Esb) = Ep + dEk
where:
dEk = particle recoil kinetic energy

Experimental measurements of the Planck constant generally actually measure the parameter:
h = (Esa - Esb) / Fp
= (Ep + dEk) / Fp
= hs + (dEk / Fp)
 

When a spheromak gains or loses potential energy by absorption or emission of a photon the spheromak transitions from state "a" with potential energy Etta and natural frequency Fha to state "b" with potential energy Ettb and natural frequency Fhb. The change in spheromak potential energy is:
(Ettb - Etta) = hs (Fhb - Fha)

When a spheromak absorbs a photon with energy Ep it also absorbs that photon's linear momentum. From Einstein's famous special relativistic relationship:
E^2 = P^2 C^2 + Mo^2 C^4
the momentum Pp of a photon with no rest mass but with energy Ep is:
Pp = Ep / C
where:
C = speed of light.
 

PHOTON ABSORPTION:
If a spheromak at rest in field free space with initial spheromak potential energy Eao absorbs a photon with energy Ep to conserve momentum the spheromak with combined total energy:
Eb = (Eao + Ep)
also acquires the photon momentum Pp. Hence after photon absorption:
Eb^2 = (Eao + Ep)^2
= Pp^2 C^2 + Ebo^2
or
(Eao + Ep)^2 = Ep^2 + Ebo^2
where Ebo is the spheromak rest potential energy after absorption of the photon.

Hence:
(Eao + Ep)^2 = Ep^2 + Ebo^2
or
Eao^2 + 2 Eao Ep = Ebo^2
or
Ebo = [Eao^2 (1 + 2 Ep / Eao)]^0.5
= Eao (1 + 2 Ep / Eao)^0.5

Hence:
(Ebo - Eao) = Eao (1 + 2 Ep / Eao)^0.5 - Eao
= Eao [(1 + (2 Ep / Eao))^0.5 - 1]
~ Eao [1 + (Ep / Eao) - [(2 Ep / Eao)^2 / 8] - 1]
= Ep - (Ep^2 / 2 Eao)
= Ep [1 - (Ep / 2 Eao)]

Hence for photon absorption:
Ep = (Ebo - Eao) / [1 - (Ep / 2 Ea)]
 

PHOTON EMISSION:
If a spheromak at rest with initial potential energy Eao emits a photon with energy Ep to conserve momentum the spheromak with the new total energy (Ea - Ep) acquires the photon momentum Pp. Hence: (Eao - Ep)^2 = Pp^2 C^2 + Ebo^2
or
(Eao - Ep)^2 = Ep^2 + Ebo^2
where Ebo is the spheromak rest mass potential energy after emission of the photon.

Hence:
(Eao - Ep)^2 = Ep^2 + Ebo^2
or
Eao^2 - 2 Eao Ep = Ebo^2
or
Ebo = [Eao^2 (1 - 2 Ep / Eao)]^0.5
= Eao (1 - 2 Ep / Eao)^0.5

Hence:
(Eao - Ebo) = Eao - Eao (1 - 2 Ep / Eao)^0.5
= Eao [1 - (1 - 2 Ep / Eao)^0.5]
~ Eao [ 1 - (1 - (Ep / Eao) - (2 Ep / Eao)^2 / 8)]
= Eao [ (Ep / Eao) + (2 Ep / Eao)^2 / 8)]
= Ep + (Ep^2 / 2 Eao)
= Ep [1 + (Ep / 2 Eao)]

Hence for photon emission:
Ep = (Ea - Eb) / [1 + (Ep / 2 Ea)]
 

EXPERIMENTAL MEASUREMENT OF h:
The Planck constant h is usually defined by:
Ep = h Fp
or
h = Ep / Fp
where:
Ep = photon energy
and
Fp = photon frequency

If the experimental methodology involves measurement of the frequency of photons emitted by spheromaks at rest the formula that should be used for determining h is:
h = (Ea - Eb) / {Fp [1 + (Ep / 2 Ea)]}
Note that on emission of a photon the change in spheromak potential energy (Ea -Eb) is slightly greater than the photon energy Ep and on photon absorption by a spheromak at rest the change in spheromak potential energy is slightly less than the photon energy Ep. These issues are further complicated by thermal motion of the particles.
 

EXPERIMENTAL ERROR:
Some high resolution experimental measurements of h rely on spectroscopic measurement of the frequency of photons emitted by excited electrons. In such experiments lab personnel often incorrectly assume that the term:
[1 +/- (Ep / 2 Ea)] = 1

However, at resolutions in measurement of h with 5 or more significant figures that assumption may be wrong and the claimed experimentally measured values of h will consistently deviate from the precise theoretically calculated value of:
h = (Eb - Ea) / (Fb - Fa).
Hence, in high resolution experimental measurements of h it is necessary to account for the charged particle recoil kinetic energy on absorption or emission of a photon.

Using spheromak theory we can precisely calculate a theoretical value for:
h = (Eb - Ea) / (Fhb - Fha)
= dEtt / dFh

Note that the spheromak spacial energy density assumptions are only truly valid in field free space, which is often not the case during many practical high precision measurements of the Planck Constant. While the spheromak internal magnetic fields are large compared to an applied laboratory magnetic field, the system is not totally distortion free. Proximity of other particles can cause interfering fields that potentially affect the measurement.
 

THERMAL MOTION:
Note that if the spheromaks are in thermal motion there is broadening of the emission and absorption frequency bands which further complicates precision measurements.
 

CONFINED PHOTONS:

There is yet another confusing issue. The inertial mass of electrons and protons is typically three orders of magnitude higher than the mass equivalent of the static electric and magnetic field energies in the spheromak. It appears that this extra mass energy is carried by a photon which is confined by the spheromak walls.

These confined photons each contain energy Ec given by:
Ec = h Fc

However, Ec and Fc are independent of small rapid changes in the spheromak static field parameters. Thus normally when the spheromak absorbs or emits a photon there is little or no change in the confined photon energy. The confined photon energy represents most of a particle's rest mass and is very stable. Generally the confined photon energy is only emitted on a particle/anti-particle interaction.
 

ASSUMED VALUE FOR PLANCK CONSTANT:
Under the proposed new SI units the value of the Planck Constant h is fixed at:
h = 6.62607015 X 10^-34 J-s
= 6.62607015 10-34 m^2 kg / s.

The reason for giving h this new value is to redefine a kilogram. However, redefining a kilogram in this manner forces new precise definitions of other parameters such as the quantum charge Q, permiability of free space Muo and permittivity of free space Epsilono based on the value of the Fine Structure constant.
 

ORIGIN OF PLANCK CONSTANT:
The parameter hs is a function of:
Muos = [2 Alphas h / Qs^2 C] = permiability of free space;
Alphas = fine structure constant;
C = speed of light in a vacuum;
Qs = 1.602176634 X 10^-19 C = proton charge;
Pi = (circumference / diameter) of a circle
= 3.141592653589793
Pi^2 = 9.869604401

The definition of the fine structure constant Alpha is:
Muo C Qs^2 = 2 h Alpha

Hence:

(Muo C Qs^2 / 4 Pi) = [Alpha h / 2 Pi]

However, there is a complication. Alpha is not constant. Alpha is a weak function of the spheromak parameter So. Alpha can only be treated as a constant in circumstances where So is reliably constant. In reality So is constant because [Lh / Ro] seeks a low energy relative minimum.

To understand the relationship of spheromak parameters to the Planck constant it is necessary to derive a closed form expression for the total electric and magnetic static field energy of a spheromak.
 

As shown on the web page titled ELECTROMAGNETIC SPHEROMAK the peak magnetic field strength Bpo at the center of a spheromak can be expressed as:
Bpo = [(Muos C Qs) / (4 Pi Ro^2)]

or as:
Bpo = I [(Muos Qs C) / (2 Pi^2 Rc^2)] {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5}
= I (Muo Qs C) / (2 Pi^2 Ro^2)(Ro / Rc)^2 {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5}
= I (Muo Qs C / (2 Pi^2 Ro^2) So^2 {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5}
where:
I = Integral from Z = 1 to Z = So^2 of:
Z^3 dZ Nr / ([(So^2 - Z)(Z - 1)]^0.5 {[Nr Z]^2 + [(So^2 - 1) / 2]^2}^0.5 [So^2 Z - So^2 + Z]^1.5)

where:
So^2 = (Rs / Rc)
and
Nr = (Np / Nt)
where:
Np = integer number of poloidal magnetic field generation turns
and
Nt = integer number of toroidal magnetic field generation turns.

In order to determine the spheromak operating point for each value of So^2 find the corresponding value of Nr^2 using the common boundary condition formula:
Nr^2
= {(8 / Pi^2) - [(So^2 - 1) / (So^2 + 1)]^2} / {1 - (16 / [Pi (So^2 - 1)]^2)}

which formula is derived on the web page titled: ELECTROMAGNETIC SPHEROMAK and then do a numerical integration to determine I.
 

To find the exact low energy point we need to find the Nr^2, So^2 combination that gives the spheromak its lowest total energy Ett while maintaining Np and Nt as integers.

The exact value of Nr^2 corresponding to a particular value of So is given by:

Nr^2 = [+ {8} - {[Pi (So^2 - 1) / (So^2 + 1)]^2}] / {[Pi^2] - [4 / (So^2 - 1)]^2}
 

The common boundary condition can then be used to find the precise value of So when the spheromak is in its operating state. This value of So can be used to determine the Planck Constant h which is:
h = dEtt / dFh

******************************************************* FIX FOLLOWING THEORY TO INCLUDE PARAMETER A

In this formula at steady state So spontaneously adopts the value that minimizes Ett while satisfying the required quantization of So. As shown by the following graph of the So dependent term of Ett vs So the operating value of So is:

Note that in plotting this graph Nr^2 is itself a complex function of So.

In the expression for the Planck constant:
Pi = 3.141592653589793
and
Pi^2 = 9.869604401

Hence:
Ett = Efs {1 - [(So - 1)^2 / (So^2 + 1)]^2}
or
Ett = [(Mu C Qs^2) / (4 Pi)] [Pi^2 / 8] [Fh Nt]
[(So^2 - 1) / So] [(8 {So^4 + 2 So^2 - 1} / {(So^2 - 1)^2 (Pi^2) - (16)})^0.5]
[1 - {(So - 1)^2 / (So^2 + 1)}^2]

Ett is a function of Fh and So. Hence:
dEtt = (dEtt / dFh) dFh + (dEtt / dSo) dSo

At steady state Fh is constant so:
dFh = 0
and
Nr = (Np / Nt)
adjusts so that:
(dEtt / dSo) ~ 0

This operating point is a spheromak field energy minimum.

Plot:
Ett / {[Muo C Qs^2 / 4 Pi] [Pi^2 / 8] Fh Nt}
=[(So^2 - 1) / So] [(8 {So^4 + 2 So^2 - 1} / {(So^2 - 1)^2 (Pi^2) - (16)})^0.5]
[1 - {(So - 1)^2 / (So^2 + 1)}^2]
versus So to find the value of So that minimizes Ett at constant Fh. At that relative minimum:
(dEtt / dSo) = 0.

Ett / {[Muo C Qs^2 / 4 Pi] [Pi^2 / 8] Fh Nt}
=[(So^2 - 1) / So] [(8 {So^4 + 2 So^2 - 1} / {(So^2 - 1)^2 (Pi^2) - (16)})^0.5]
[1 - {(So - 1)^2 / (So^2 + 1)}^2]
= 2.2882

Hence at that So value:
Ett = [Muo C Qs^2 / 4 Pi] [Pi^2 / 8] Fh Nt [2.2882]

FIX TO INCLUDE PARAMETER A.  

At this spheromak minimum energy operating state a small change in So causes no change in spheromak energy. At this stable operating state the spheromak energy Ett is directly proportional to the spheromak frequency Fh.

The proportionality constant h between spheromak energy Ett and frequency Fh is defined by:
h = [(Muo C Qs^2) / (4 Pi)] [Pi^2 / 8] Nt [2.2882]
and is known as the Planck Constant.

It is convenient to define the unitless Fine Structure Constant Alpha by:
[Muo C Qs^2] = 2 Alpha h

Then substitution in the above equation gives:
h = [(2 Alpha h) /(4 Pi)] [Pi^2 / 8] Nt [2.2882]
or
(1 / Alpha) = [(2) /(4 Pi)] [Pi^2 / 8] Nt [2.2882]
= [Pi / 16] Nt [2.2882]

According to this derivation:
Nt = (1 / Alpha) / {[Pi / 16][2.2882]}
= 305.00769
If this value is correct the field parameter A must be quite large.

Since by definition:
h = [Muo C Qs^2] / ( 2 Alpha)
the unitless constant Alpha can be determined from the Planck Constant.

Experimental measurements of the Planck Constant for stable particles indicate that:
Alpha ~ 137.03
which suggests that Nt is likely 303, 304 or 305.

SUMMARY:
Ett = [Muo C Qs^2 / 4 Pi] [Pi^2 / 8] Fh Nt [2.2882]
is the approximate low energy stable state of a charged particle spheromak.

When a spheromak is at its stable low energy state a small change in spheromak field energy Ett is almost entirely due to a corresponding small change in spheromak frequency Fh. At this low energy state the constant of proportionality between spheromak energy and spheromak frequency is:
dEtt / dFh = h
= [(Muo C Qs^2) / (4 Pi)] [Pi^2 / 8] Nt [2.2882]
= 6.626 X 10^-34 J-s
 

FINE STRUCTURE CONSTANT Alpha:
The fine structure constant Alpha is defined by:
Muo C Q^2 = 2 h Alpha
or
Muos C Q^2 = 2 h Alphas

EVALUATION OF Alpha:
Recall that h is given by:
h = [(Muo C Qs^2) / (4 Pi)] [Pi^2 / 8] Nt
X {1 - [(So -1)^2 / (So^2 + 1)] + [2 So (So - 1)^2 / (So^2 + 1)^2]}
X {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

Evaluation of the terms of hs gives:
[(Muo C Qs^2) / (4 Pi)] [Pi^2 / 8] Nt
= [Alpha h / 2 Pi] [Pi^2 / 8] Nt
= [Alphas h Pi / 16] Nt
 

At (1 / Alpha) = 137.035999:
So = 2.02606822
and
So^2 = 4.104937443
and
Nr = (Np / Nt)
= (223 / 303) = 0.7359735974

and
Nr^2 ~ 0.541657136

{1 - [(So -1)^2 / (So^2 + 1)] + [2 So (So - 1)^2 / (So^2 + 1)^2]}
= {1 - [(1.02606822)^2 / (5.104937443)] + 2 (2.02606822) [(1.02606822)^2 / (5.104937443)^2]}
= {1 - 0.2062348469 + 0.1637026404}
= 0.9574677935

{[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]
= {[0.541657136 (5.104937443)^2] + [(3.104937443)]^2}^0.5 / [2.02606822]
= {[14.1157942] + [9.640636525]}^0.5 / [2.02606822]
= 4.874056906 / [2.02606822]
= 2.405672651

Hence:
hs = [(Muos C Qs^2) / (4 Pi)] [Pi^2 / 8] Nt
X {1 - [(So -1)^2 / (So^2 + 1) + [2 So (So - 1)^2 / (So^2 + 1)^2] ]}
X {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

= [Alpha h Pi / 16] Nt [0.9574677935] [2.405672651]
or
Alpha^-1 = [Pi / 16] Nt [0.9574677935] [2.405672651]
= [3.1415926535 / 16] (303) [0.9574677935] [2.405672651]
= 137.0355425

By comparison the CODATA recommended Alpha^-1 value obtained using a Kibble balance is:
Alpha^-1 = 137.03599915

The discrepency is:
(137.03599915 - 137.0355425) / 137.03599915
= 3.3 X 10^-6
 

CHECK THE FOLLOWING ACCURACY - HAS LIKELY IMPROVED

Note that there is agreement to within 3.3 parts per million between the value of Alphas^-1 calculated herein and the CODATA recommended value of Alpha^-1 based on experimental measurements. This error is likely due to improper treatment of the parameter A.
 

It is necessary to examine exactly how h is experimentally measured with a Kibble balance to understand the discrepency sources. A possible cause of experimental to theoretical discrepency is that the spheromaks related to the Josephson junctions used with Kibble (Watt) balances are not isolated in free space.
 

CONCLUSION:
The spheromak model of a charged particle provides a means of calculating the Fine Structure Constant Alpha and hence the Planck constant h in terms of Pi, Muo, Q and C. In highly accurate experimental measurements of Alpha and h it is necessary to take into account elliptical spheromak shape distortion and the charged particle recoil kinetic energy.

Note that for a spheromak at steady state conditions in field free space Alpha is independent of the charged particle spheromak nominal radius Ro and hence is also independent of the charged particle static field energy Ett.
 

This web page last updated May 18, 2019.

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