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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:
A photon is a quantum of radiant energy either emitted byr absorbed by a charged particle spheromak. The Planck Constant is the proportionality factor which relates the magnitude of the quantum of energy to the radiation frequency. The Planck Constant is actually a frequently reoccurring 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. This behaviour is due to formation of particle spheromaks.
 

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 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 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 individual quantized charges to act as stable stores of electro-magnetic energy. The behavior of these spheromak based energy stores is governed by the 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 emited:
dEtt = h dF
where dF is the radiation frequency.
 

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 spontaneously attempt to reach stable energy states by emission or absorption of radiation. During radiant energy absorption and emission 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 to or from which 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 energy Ett changes from Ea to Eb and frequency Fh changes from Fa to Fb then:
dEtt = (Ea - Eb)
= h (Fa - Fb)
= h dFh

Note that (Ea - Eb) is a change in spheromak static electromagnetic field energy.

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 precise definition of the Planck constant and 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.
 

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 electric charge, approximately 1.602 X 10^-19 coulombs. The charge is uniformly distributed along the charge string and 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 adopt a spheromak geometry.
 

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

where:
Uo = energy density at spheromak center;
Ro = (So Rc) = (Rs / So) = geometric spheromak radius;
So^2 = (Rs / Rc) ~ 4 where So is the spheromak shape parameter;
Rs = radius from spheromak axis of symmetry to furthest spheromak wall;
Rc = radius from spheromak axis of symmetry to nearest spheromak wall;
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] [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 electric current that circulates at the speed of light around a closed spiral path within the spheromak wall. The term [1 / Ro] is replaced by the function F(Fh, So) where 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 parameter So.

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:
A spheromak has an inside radius Rc and an outside radius Rs. Let Np be the number of poloidal current path turns and let Nt be the number of toroidal current path turns. Then Pythagoras theorem gives the total charge string length Lh as:
Lh = {[2 Pi Np (Rs + Rc) / 2]^2 + [2 Pi Nt (Rs - Rc) / 2]^2}^0.5

Since the current 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) / 2]^2}^0.5
= [Pi {[Np (Rs + Rc)]^2 + [Nt (Rs - Rc)]^2}^0.5]
= [Pi Rc {[Np (So^2 + 1)]^2 + [Nt (So^2 - 1)]^2}^0.5]
= [Pi Ro / So] {[Np (So^2 + 1)]^2 + [Nt (So^2 - 1)]^2}^0.5

Hence:
(1 / Ro)
= [Pi Fh / C] [{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5 / So]
 
= [Pi Fh / C] [Z / So]
where:
Ro = Rs / So = Rc So = nominal spheromak radius
and
Z = {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5

Note that:
[Pi Fh Z / 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 / So)

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

The ratio (Lh / Ro) behaves as a highly stable constant suitable for relating energy and mass in the metric system to distance and time. The length Lh contains an imbedded factor of Pi.

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

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

 
= [Muo C Qs^2 / 32] [Fh][Pi Z / So]
{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 / 2] [(Pi / 4) Fh] [Z]
{[ So^2 - So + 1] / [(So^2 + 1)^2]}
which is in the form:
Ett = h Fh
where:
h = [Muo C Qs^2 / 2] [(Pi / 4)] [Z]
{[ So^2 - So + 1] / [(So^2 + 1)^2]}

In a stable spheromak So and Z are constant. The parameter h is known as the Planck Constant.
 

FINE STRUCTURE CONSTANT:
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.

Then:
(1 / Alpha) = 2 h / (Muo C Q^2)
 
= [2 / (Muo C Q^2)][Muo C Qs^2 / 2][(Pi / 4)][Z]
{[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [(Pi / 4)][Z]{[ So^2 - So + 1] / [(So^2 + 1)^2]}

It has been experimentally shown that the Fine Structure Constant Alpha has the value given by:
(1 / Alpha) = 137.03599915
corresponding to:
h = 6.636070150 X 10^-34 J-s
 

Thus:
(1 / Alpha) = [(Pi / 4)][Z]{[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [(Pi / 4) {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5]
{[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [(Pi / 4)][Lh / Ro] [So / Pi]{[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [Lh / Ro] [So / 4]{[ So^2 - So + 1] / [(So^2 + 1)^2]}
 

MEANING OF (1 / Alpha):
For So ~ 2 we have:
(1 / Alpha) ~ (Lh / Ro) [1 / 2] [3] / [25]
or
(Lh / Ro) ~ (50 / 3)(137.035999)
= 2283.933317
or
(Lh / Pi Ro) = 726.9985549
which suggests that with a small correction in So possibly:
(Lh / Pi Ro) = 727

Recall that:
(Lh / Pi Ro) = [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5 / So
~ [Np^2 25 + Nt^2 9]^0.5 / 2
or
1454 ~ [Np^2 25 + Nt^2 9]^0.5
or
(1454)^2 ~ Np^2 25 + Nt^2 9

Hence: Np < 291 and Nt < 485. We can use a computer to check all possible Np, Nt value combinations.

(1 / Alpha) is an indication of the numbers of poloidal and toroidal charge hose turns required for the changes in magnetic field energy density to balance the changes in electric field energy density at the spheromak walls. As shown on the web page titled: ELECTROMAGNETIC SPHEROMAK So ~ 2.0
 

Recall that:
(Lh / Pi Ro) = [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5 / So

(Lh / Pi Ro) has maximum stability when:
[Np^2 (So^2 + 1)^2 = Nt^2 (So^2 - 1)^2
or
Np / Nt = (So^2 - 1) / (So^2 + 1)

At So = 2:
Np / Nt = 3 / 5

Recall that: (1454)^2 ~ Np^2 25 + Nt^2 9 = Np^2 (25 + (5 / 3)^2 9) = Np^2 (50) = Np^2 (7.0710678)^2 Np = 205.6266523 Nt = (5 / 3) Np
= 342.7110872 (1 / Alpha) = [(Pi / 4) {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5]
{[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
[(Pi / 4) {[42,282.32014 (25)] + [117,450.8893 (9)]}^0.5]
{[3] / [25]}
 
[3 Pi / 100][1,057,058.003 + 1,057,058.004]^0.5
 
[3 Pi / 100][1454.000002]
 
= 137.0362716
which is close to the desired result.

This analysis indicates that:
205 <= Np <= 206
and that:
342 <= Nt <= 343

Np = 205, Nt = 343 appears to be a good candidate.

We must do precise calculations around these values to determine the actual integer values of Np and Nt and the corresponding offset in So from So = 2.0000.
 

WE MUST BE CAREFUL BECAUSE THE SPHEROMAK BOUNDARY CONDITION AT R = Rc, H = 0 MAY REQUIRE THAT Np BE LARGER WITH RESPECT TO Nt. If the electric fields at R = Rc, H = 0 are in balance then the poloidal and toroidal magnetic fields must be in balance.
 

STABILITY OF (1 / Alpha):
It is important to understand the source of the Fine Structure constant stability.

(1 / Alpha)
= [Pi / 4] {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)^2]}

In an ideal world there would be a Np, Nt, So combination that makes:
d(1 / Alpha) / dSo = 0
as well as giving ( 1 / Alpha) its required magnitude. However, that is not the case. In reality (1 / Alpha) consists of two major compoonents, both of which have positive derivatives with respect to So. Hence (1 / Alpha) relies on the stability of So resulting from the spheromak boundary condition.

(1 / Alpha) = [Pi / 4] {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [Pi / 4] {[Np^2] + [Nt^2 (So^2 - 1)^2 / (So^2 + 1)^2]}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)]}
 
= [Pi / 4] Nt {[Nr^2] + [(So^2 - 1)^2 / (So^2 + 1)^2]}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)]}

Define R by:
R = (So^2 - 1) / (So^2 + 1)

Hence:
(1 / Alpha) = [Pi / 4] Nt {Nr^2 + R^2}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)]}

The spheromak boundary condition gives:
[Nr^2 + R^2] = 1 / [(Pi / 2)^2 - (F / R)^2]
where:
0.5 < F < 1.0
as a result of the spheromak's electric field configuration.

Hence:
(1 / Alpha) = [Pi / 4] Nt {Nr^2 + R^2}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)]}
 
= [Pi / 4] Nt {1 / [(Pi / 2)^2 - (F / R)^2]}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)]}
 
= [Nt / 2] {(Pi / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)]}

 

SOLUTION GUESS:
We will guess a solution of So ~ 2.0. We are aided in this guess by the fact that even at So ~ 2.0 the central poloidal magnetic field strength is only barely sufficient to maintain a spheromak. Our strategy is to show that at So ~ 2.0 a spheromak has all the experimentally observed and theoretically projected properties and is consistent with the computed electric field boundary condition.
 

SOLUTION STRATEGY:
The equation for (1 / Alpha) is highly non-linear. We will attempt to solve it by:
a) Guessing reasonable values of So = Soa and F = Fa that result in an approximate value of (1 / Alpha);
b) Using the calculus of variations to find improved values So = Sob and F = Fb;
c) Using a computer to calculate Fc from Sob and comparing that result to Fb. It may be necessary to itterate the entire process again using Fc in place of Fa.
 

Guess:
Soa = 2.0

Hence:
Ra = (Soa^2 - 1) / (Soa^2 + 1)
= 3 / 5

Then:
[ Soa^2 - Soa + 1] / [(Soa^2 + 1)] = 3 / 5

Guess:
{(Pi / 2)^2 / [(Pi / 2)^2 - (Fa / Ra)^2]}^0.5 = (5 / 3)

Hence:
{(Pi / 2)^2 / [(Pi / 2)^2 - (Fa / Ra)^2]} = 25 / 9
or
9 (Pi / 2)^2 = 25 [(Pi / 2)^2 - (Fa / Ra)^2]
or
(Fa / Ra)^2 = (16 / 25)(Pi / 2)^2
or
(Fa / Ra) = (4 / 5)(Pi / 2)
or
Fa = Ra (2 / 5) Pi
= (3 / 5)(2 / 5) Pi
= 0.24 Pi
= 0.7539
which is a reasonable mid-range value for F.

Nra^2 = {1 / [(Pi / 2)^2 - (Fa / Ra)^2]} - Ra^2
 
= {1 / [(Pi / 2)^2 - (16 / 25)(Pi / 2)^2]} - Ra^2
= {1 / [(9 / 25)(Pi / 2)^2]} - Ra^2
= {1 / [(9 / 25)(Pi / 2)^2]} - (9 / 25)
= {100 / 9 Pi^2} - (9 / 25)
= 1.1257900932 - 0.36
= 0.7657909319

Then:
(1 / Alpha)a = [Nta / 2] {(Pi / 2)^2 / [(Pi / 2)^2 - (Fa / Ra)^2]}^0.5
{[ Soa^2 - Soa + 1] / [(Soa^2 + 1)]}
 
= [Nta / 2]{5 / 3][3 / 5]
 
= (Nta / 2)
 
= 137

Hence: Nta = 274

At So = 2.0:
(1 / Alpha)a = (Pi / 4)[Npa^2 (Soa^2 + 1)^2 + Nta^2 (Soa^2 - 1)^2]
[(Soa^2 - Soa + 1) / (Soa^2 + 1)^2]
or
137 = (Pi / 4)[Npa^2 25 + Nta^2 9]^0.5 [(3) / 25]
or
[25 (137)(4) / Pi (3)] = [Npa^2 25 + (274)^2 9]^0.5
or
25 Npa^2 = [25 (137)(4) / Pi (3)]^2 - (274)^2 (9)
= 2,112,997.001 - 675,684
= 1,437,313.001

Npa = 239.7759789 ~ 240

Note that:
[25 (137)(4) / Pi (3)]^2 = 2,112,997.001

This rational number approximation of Pi^2 is of fundamental importance in quantum mechanics.

Pi^2 = [25 (137)(4) / (3)]^2 / 2,112,997 = (3.14159265)^2

(1 / Alpha)b - (1 / Alpha)a = (137.03599915 - 137.00000000)
=0.03599915

This difference might be a result of:
Fb > Fa
or
Sob < Soa
or
some combination of these two inequalities.

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

CONSIDER THE PARAMETER [Lh / Ro]:
[Lh / Ro] = (Pi Z / So)
 
= [Pi {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5 / So]
 
= [Pi Nt {[Nr^2 + R^2]}^0.5 / So](So^2 + 1)
 
= [Pi Nt {1 / [(Pi / 2)^2 - (F / R)^2]}^0.5 [(So^2 + 1) / So]
 
= [2 Nt {(Pi / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5 [(So^2 + 1) / So]
 
= {(Pi / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5 [2 Nt (So^2 + 1) / So]

Hence:
[Lh / Ro]a = (5 / 3)[2 Nta (So^2 + 1) / So]
 
= (10 / 3)(274)(5 / 2)
 
= 2283.333333
which indicates that at Soa = 2.0:
3 [Lh / Ro]a = 6850

A physical origin of the factor of 3 is not obvious.
 

On the web page titled: ELECTROMAGNETIC SPHEROMAKS we found that at So ~ 1.0 [Lh / Ro]a = 2 Pi Np

Recall that:
(1 / Alpha) = [Lh / Ro] [So / 4]{[ So^2 - So + 1] / [(So^2 + 1)^2]}
= [2 Pi Np] [1 / 4]{[1] / [4]}
= (Pi / 8) Np = 137.03599915

Hence:
Np = 8 (137.03599915) / Pi
= 348.9593067
~ 349
which is prime.

Hence it appears that So ~ 1 and Np ~ 349 is a likely viable solution needing further investigation. The corresponding value of [Lh / Ro] is:
[Lh / Ro = 2 Pi Np
= 2192.832

VARIATION IN (1 / Alpha):
Recall that:
(1 / Alpha) = [Nt / 2] {(Pi / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)]}

The quantity:
[(1 / Alpha)b - (1 / Alpha)a] is too small to be accounted for by an incrementation in Nt. Hence we can presume that:
Nta = 274
is correct and concentrate on variations in F and So.

Let us initially assume that:
[(1 / Alpha)b - (1 / Alpha)a] is entirely due to Fb > Fa. Then:
 
d(1 / Alpha) = [Nta / 2] d{(Pi / 2)^2 / [(Pi / 2)^2 - (Fa / Ra)^2]}^0.5
{[ Soa^2 - Soa + 1] / [(Soa^2 + 1)]}
 
= [Nta / 2]{- (Pi / 2)^2 (1 / 2)[- 2 (Fa / Ra)(1 / Ra) dFa]}
{[ Soa^2 - Soa + 1] / [(Soa^2 + 1)]}
/ [(Pi / 2)^2 - (Fa / Ra)^2]^1.5

Recall that:
(Fa / Ra) = (2 / 5) Pi
and
Ra = (3 / 5)

Hence if the entire error in (1 / Alpha)a is due to error in Fa then:
d(1 / Alpha) = [Nta / 2]{- (Pi / 2)^2 (1 / 2)[- 2 (Fa / Ra)(1 / Ra) dFa]}
{[ Soa^2 - Soa + 1] / [(Soa^2 + 1)]}
/ [(Pi / 2)^2 - (Fa / Ra)^2]^1.5
 
= [Nta / 2]{- (Pi / 2)^2 (1 / 2)[- 2 (2 Pi / 5)(5 / 3) dFa]}
{[ 4 - 2 + 1] / [(4 + 1)]}
/ [(Pi / 2)^2 - (2 Pi / 5)^2]^1.5
 
= [Nta / 2]{(Pi / 2)^2 [(2 Pi / 3) dFa]}
{3 / 5}
/ [(Pi / 2)^2 - (4 Pi^2 / 25)]^1.5
 
= [Nta / 2]{(1 / 2)^2 [(2 / 3) dFa]}{3 / 5} / [(1 / 2)^2 - (4 / 25)]^1.5
 
= [Nta / 2]{(1 / 6) dFa}{3 / 5} / [(25 / 100) - (16 / 100)]^1.5
 
= [Nta / 2]{(1 / 6) dFa}{3 / 5} / [(9 / 100)]^1.5
 
= [Nta / 2]{(1 / 10) dFa}(1000) / 27
 
= [Nta / 2]{(100 / 27) dFa}
 
= 1.85185185 Nta dFa
 
= 507.4074074 dFa

Thus (1 / Alpha) is very sensitive to even a small change in F. A tiny increase in F will account for the increase in (1 / Alpha). In order to pursue this avenue further we must find an exact solution for F as a function of So. It appears that a key issue in stability of (1 / Alpha) is stability of the ratio (F / R) at:
(F / R) ~ (2 Pi / 5)
 

IS AN EXACT SOLUTION AT So = 2.0 POSSIBLE?
To resolve this question we can study the equation:
(1 / Alpha) = [Pi / 4] {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)^2]}

If So = 2.0 is an exact solution there will exist integer Np and Nt values that precisely satisfy the equation:
(1 / Alpha) = [Pi / 4] {[Np^2 25] + [Nt^2 9]}^0.5 {[3 / 25]} = 137.03599915
or
[Np^2 25] + [Nt^2 9]}^0.5 = (137.03599915)(25 / 3)(4 / Pi)
or
[Np^2 25] + [Nt^2 9] = [(137.03599915)(25 / 3)(4 / Pi)]^2

RHS = 2,114,107.599 ~ 1454^2 = 2,114,116

At Np = 240, Nt = 274:
LHS = [1,440,000] + [675,684] = 2,115,684

Thus Soa = 2.0 does not give a precise solution.
 

FINDING THE PRECISE VALUE OF So:
Sob = Soa + dSo
= 2 + dSo
and
(1 / Alpha)
= [Pi / 4] {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5
{[ So^2 - So + 1] / [(So^2 + 1)^2]}

At So = 2.0:
(1 / Alpha)a
= [Pi / 4] {[Np^2 (25)] + [Nt^2 (9)]}^0.5 {3 / 25}

 
= [Pi / 4] {[240^2 (25)] + [274^2 (9)]}^0.5 {3 / 25}
 
= [Pi / 4] {1,440,000] + [675,684]}^0.5 {3 / 25}  
= [Pi / 4] {2,115,684}^0.5 {3 / 25}  
= 137.0870806

d(1 / Alpha) / dSo
= [Pi / 4] {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5
d{[ So^2 - So + 1] / [(So^2 + 1)^2]} / dSo
+ {[ So^2 - So + 1] / [(So^2 + 1)^2]}
[Pi / 4] d{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5 / dSo

Z = [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5

At So = 2.0:
[Pi / 4] Z = [Pi / 4] {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5
 
= [Pi / 4] {[Np^2 (25)] + [Nt^2 (9)]}^0.5
 

dZ / dSo = (1 / 2 Z) {Np^2 2 (So^2 + 1) 2 So + Nt^2 2 (So^2 - 1) 2 So}
 
= {Np^2 2 (So^2 + 1) 2 So + Nt^2 2 (So^2 - 1) 2 So}
/ 2 [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5
 
= 2 So {Np^2 (So^2 + 1) + Nt^2 (So^2 - 1)}
/ [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5

 

At So = 2.0:
dZ / dSo = 2 (2) {Np^2 (5) + Nt^2 (3)}
/ [Np^2 (25) + Nt^2 (9)]^0.5
 
= {Np^2 (20) + Nt^2 (12)}
/ [Np^2 (25) + Nt^2 (9)]^0.5

Thus:
[Pi / 4] [dZ / dSo]
= [Pi / 4]{Np^2 (20) + Nt^2 (12)}
/ [Np^2 (25) + Nt^2 (9)]^0.5

 

At So = 2.0:
{[ So^2 - So + 1] / [(So^2 + 1)^2]}
= [3 / 25]

d{[So^2 - So + 1] / [(So^2 + 1)^2]} / dSo = {(So^2 + 1)^2 (2 So - 1) - [So^2 - So + 1] 2 (So^2 + 1) 2 So} /(So^2 + 1)^4
 
= {(So^2 + 1) (2 So - 1) - [So^2 - So + 1] 2 2 So} /(So^2 + 1)^3
 
= {2 So^3 - So^2 + 2 So - 1 - 4 So^3 + 4 So^2 - 4 So} / (So^2 + 1)^3
 
= {- 2 So^3 + 3 So^2 - 2 So - 1} / (So^2 + 1)^3

At So = 2:
d{[So^2 - So + 1] / [(So^2 + 1)^2]} / dSo
 
= {- 2 So^3 + 3 So^2 - 2 So - 1} / (So^2 + 1)^3
 
= {- 16 + 12 - 4 - 1} / 125
 
= [-9 / 125]

Thus at So = 2:
d(1 / Alpha) / dSo
= [Pi / 4] {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5
d{[ So^2 - So + 1] / [(So^2 + 1)^2]} / dSo
+ {[ So^2 - So + 1] / [(So^2 + 1)^2]}
[Pi / 4] d{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5 / dSo  
= [Pi / 4] {[Np^2 (25)] + [Nt^2 (9)]}^0.5 [-9 / 125]
+ [3 / 25] [Pi / 4]{Np^2 (20) + Nt^2 (12)} / [Np^2 (25) + Nt^2 (9)]^0.5  
= [Pi / 4] {2,115,684}^0.5 [- 9 / 125]
+ [Pi / 4] [3 / 25] {240^2 (20) + 274^2 (12)} / [2,115,684]^0.5
 
= [Pi / 4] {-104.7268154 + [3 / 25][1,152,000 + 900,912] / 1454.539102}  
= [Pi / 4] {64.63916428}  
= d(1 / Alpha) / dSo d(1 / Alpha) = 137.03599915 - 137.0870806
= - .05108145 dSo = d(1 / Alpha) / {[Pi / 4] {64.63916428}}
= - .05108145 (4) / {Pi 64.63916428}
= - 0.0010061845

Thus at Np = 240, Nt = 274
So = 2.0 - 0.0010061845
= 1.998993816

An advantage of this method of So determination is that it is independent of F.

WE NEED TO CROSS CHECK THIS So VALUE.

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

At So = 2.0:
[Lh / Ro] = [Pi / 2]{Np^2 (25) + Nt^2 (9)}^0.5
 
= [Pi / 2]{1454.539102}

d[Lh / Ro] / dSo = [Pi / So^2] [So (dZ / dSo) - Z]

Z = [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5

dZ / dSo = (1 / 2 Z)[Np^2 2 (So^2 + 1) 2 So + Nt^2 2 (So^2 - 1) 2 So]

Hence:
d[Lh / Ro] / dSo = [Pi / So^2][So (dZ / dSo) - Z]
 
= [Pi / So^2](4 So^2 / 2 Z)[Np^2 (So^2 + 1) + Nt^2 (So^2 - 1)]
- [Pi Z / So^2]
 
= (2 Pi / Z)[Np^2 (So^2 + 1) + Nt^2 (So^2 - 1)]
- [Pi Z / So^2]
 
= (1 / Z So^2) {(2 Pi So^2)[Np^2 (So^2 + 1) + Nt^2 (So^2 - 1)]
- [Pi Z^2]}
 
= (1 / Z So^2) {(Pi)[Np^2 (2 So^4 + 2 So^2) + Nt^2 (2 So^^4 - 2 So^2)]
- [Pi][Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2}
 
= (1 / Z So^2) {(Pi)[Np^2 (So^4 - 1) + Nt^2 (So^4 - 1)]}
which is non-zero at So = 2

Hence this derivative should be taken into account in the calculation of (Lh / Ro).

At So = 2:
d[Lh / Ro] / dSo
= {(Pi)[Np^2 (So^4 - 1) + Nt^2 (So^4 - 1)] / Z So^2
 
= {(Pi / So^2)(So^4 - 1)[Np^2 + Nt^2]
/ [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5
 
= {(Pi / 4)(15)[Np^2 + Nt^2] / [Np^2 (25) + Nt^2 (9)]^0.5
 
= {(Pi / 4)(15)[57,600 + 75,076] / [57,600 (25) + 75,076 (9)]^0.5
 
= {(Pi / 4)(15)[132,676] / [1,440,000 + 675,684]^0.5
 
= Pi (497,535) / 1454.539102
 
= Pi (342.0568064)

Thus at So = (2 - 0.0010061845):
(Lh / Ro) = [(Lh / Ro)|So = 2] + {d[Lh / Ro] / dSo} dSo
 
= [Pi / 2](1454.539102) + Pi (342.0568064)(- 0.0010061845)  
= Pi [727.269551 + .3441722567]
 
= Pi [727.6137233]
 
= 2285.865925

INVESTIGATE d(1 / Alpha) / dSo:

Recall that the static field energy of a spheromak is:
Ett = [Muo C Qs^2 / 2][Lh /(4 Ro)] Fh {[So][ So^2 - So + 1] / [(So^2 + 1)^2]}
 

The change in energy with respect to a change in So is:
dEtt / dSo.

Note that h is of the form:
h = [Muo C Qs^2 / 2][Lh / 4 Ro] S(So)
where:
S(So) = {[So][ So^2 - So + 1] / [(So^2 + 1)^2]}
 

The change in h with respect to a change in So is:
dh / dSo = [Muo C Qs^2 / 2][Lh / 4 Ro]d[S(So)] / dSo
+ [Muo C Qs^2 / 2] S(So) d[Lh / 4 Ro] / dSo
or
[dh / dSo]
=[Muo C Qs^2 / 8] {[Lh / Ro]d[S(So)] / dSo + S(So) d[Lh / Ro] / dSo}

d[S(So)] / dSo
= d{So [ So^2 - So + 1] / [(So^2 + 1)^2]} / dSo
= {(So^2 + 1)^2 [So (2 So - 1) + ( So^2 - So + 1)]
- So [ So^2 - So + 1] 2 (So^2 + 1) 2 So}
/ [(So^2 + 1)^4]
 
= {(So^2 + 1) [So (2 So - 1) + ( So^2 - So + 1)]
- 4 So^2 [ So^2 - So + 1]}
/ [(So^2 + 1)^3]
 
= [(So^2 + 1)[3 So^2 - 2 So + 1] - [4 So^4 - 4 So^3 + 4 So^2] / [(So^2 + 1)^3]
or
= [3 So^4 - 2 So^3 + So^2 + 3 So^2 - 2 So + 1 - 4 So^4 + 4 So^3 - 4 So^2] / [(So^2 + 1)^3]
 
= [- So^4 + 2 So^3 - 2 So + 1] / [(So^2 + 1)^3]

Note that at So ~ 1:
dh / dSo = 0

Recall that:
[dh / dSo]
=[Muo C Qs^2 / 8]
{[Lh / Ro]d[S(So)] / dSo + S(So) d[Lh / Ro] / dSo}
 
=[Muo C Qs^2 / 8]
{[Pi Z / So] d[S(So)] / dSo + S(So)(2 Pi / Z)[Np^2 (So^2 + 1) + Nt^2 (So^2 - 1)] - {S(So) [Pi Z / So^2]}}

WE MUST NUMERICALLY EVALUATE THIS DERIVATIVE AT So = 2

Examine:
{[Pi Z / So] d[S(So)] / dSo} - {S(So) [Pi Z / So^2]}
 
= [Pi Z / So]{d[S(So)] / dSo - S(So)/ So}
 
= [Pi Z / So]{[- So^4 + 2 So^3 - 2 So + 1] / [(So^2 + 1)^3]
- [ So^2 - So + 1] / [(So^2 + 1)^2]}

Examine:
S(So)(2 Pi / Z)[Np^2 (So^2 + 1) + Nt^2 (So^2 - 1)]
 
= {So [ So^2 - So + 1] / [(So^2 + 1)^2]}(2 Pi / Z)[Np^2 (So^2 + 1) + Nt^2 (So^2 - 1)]
 
= {2 Pi So [So^2 - So + 1] / [(So^2 + 1)^2]}[Np^2 (So^2 + 1) + Nt^2 (So^2 - 1)]
/ [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5

 
= {2 Pi 2 [3] / [25]}[Np^2 (5) + Nt^2 (3)]
/ [Np^2 25 + Nt^2 9]^0.5

THUS FIND THE CHANGE IN ALPHA WITH RESPECT TO A CHANGE IN So. Calculate d(1 / Alpha) / dSo Find new So value. Find new (Lh / Ro) value.

The Fine Structure constant and hence the Planck Constant rely on the stability of So. Note that the spheromak energy is most stable against pertubations in So when: [- So^4 + 2 So^3 - 2 So + 1] = 0
or when So ~ 1.6
So is set by the spheromak boundary condition.
 

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 + R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]
where:
R = (So^2 - 1) / (So^2 + 1)
and
Nr = (Np / Nt)
and
0.5 < F < 1.0

Rearranging the boundary condition formula gives:
[(Pi / 2)^2 - (F / R)^2] [Nr^2 + R^2] = 1 or [(Pi R / 2)^2 - (F)^2] [Nr^2 + R^2] = R^2

However:
Nr^2 = K R^2
where at So = 2:
K = constant of the order of unity yet to be determined. Hence:
[(Pi R / 2)^2 - (F)^2] [K R^2 + R^2] = R^2
or
[(Pi R / 2)^2 - (F)^2] = [1 / (K + 1)]

Recall that:
R = (So^2 - 1) / (So^2 + 1)
or
(So^2 - 1) = R (So^2 + 1)
or
So^2 {1 - R} = {1 + R}
or
So^2 = (1 + R) / (1 - R)

At So^2 = 4:
R = 3 / 5
R^2 = 9 / 25
= 0.36
F = 0.753982236

[1 / (K + 1)] = [(Pi R / 2)^2 - (F)^2]
= (Pi / 2)^2 (9 / 25) - (.753982236)^2
= 0.88826 - 0.568489
= 0.31977

1 = 0.31977 (K + 1)
or
K = (1 - 0.31977) / 0.31977
= 2.1272

Thus if So = 2.0 then:
Nr^2 = 2.1272 R^2

Consider the function:
F(So) = [Muo C Qs^2 / 2][Lh / 4 Ro] The derivative of the product:
F(So) S(So)
with respect to So is given by:
d[F(So) S(So)] = F(So) {d[S(So)] / dSo} + S(So) {d[F(So)] / dSo}

Ideally for this function product to be stable at So = 2.0 the derivative of this function product with respect to So should be zero. Hence ideally at
So = 2.0:
and {F(So) (- 12 / 125)} + (24 / 25){d[F(So)] / dSo} = 0
or
(24 / 25){d[F(So)] / dSo} = {F(So) (12 / 125)}
or
{d[F(So)] / dSo} / {F(So)} = (25 / 24)(12 / 125)
= (1 / 10)
= 1 / 10

Thus to realize ideal function product stability at So = 2.0 :
{d[F(So)] / dSo} / {F(So)} = (1 / 10)

The essence of the Fine Structure constant is that it is supposedly constant, so if the spheromak operates at So = 2.0 the F(So) term should conform to this criteria.

Recall that:
F(So) = {[Muo C Qs^2 / 2][Lh / 4 Ro]}

dF(So) / dSo = 0

Ideally the parameters of this function would be such that {d[F(So)] / dSo} / {F(So)} = (1 / 10)
so that the Planck Constant and the Fine Structure Constant act as constants. However, that is definitely not the case.

Z QUANTIZATION:
Instead the Fine Structure constant stability comes from inherent So stability which in turn relies on the F stability. Larger changes in spheromak energy occur via integer changes in Np and Nt which cause quantum changes in Z.

In conclusion a spheromak's energy is stable because the electric field parameter F sets So^2. Since Np and Nt are integers if due to an external influence F changes causing a change in So the the spheromak's energy responds 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 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 the surface 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.

An important issue worthy of noting is that:
3 Lh / Ro = 6850

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

SPHEROMAK INTEGER REQUIREMENT:
Recall that:
(1 / Alpha) = [Lh / 2 Ro] [So / 2]{[ So^2 - So + 1] / [(So^2 + 1)^2]}

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

We will assume that the 137 is an integer resulting from the relationship:
3 Lh / Ro = 6850

Recall that:
Z = [Lh / Ro] [So / Pi]
= [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5

Thus:
Z = [6850 / 3] [So / Pi]
= [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5
where:
So = 1.999708036
and
Np = integer
and
Nt = integer

We can rewrite this equation as:
6850 = (3 Pi / So)[Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5
and use a computer to explore all the possible Np and Nt integer values with So = 1.999708036

The computer work is minimized by recognizing that:
(6850)^2 = (9 Pi^2 / So^2)[Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]
~ (9 Pi^2 / 4)[Np^2 (25) + Nt^2 (9)]

At Nt = 0 the maximum value of Np is given by:
Np^2 < [(6850)^2 (4)] / [25 (9 Pi^2)] or
Np < [(6850) (2)] / [5 (3 Pi)] or
Np < 291

Similarly, at Np = 0 the maximum value of Nt is given by:
Nt^2 < [(6850)^2 (4)] / [9^2 Pi^2]
or
Nt < [(6850) (2)] / [9 Pi]
or
Nt < 485

Thus there are less than 1000 possible Np, Nt choices, one of which will give the least error in the equation:
(6850)^2 = (9 Pi^2 / So^2)[Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]
at
So = 1.999708036

Z = (So / Pi)(Lh / Ro)
= [6850 / 3] [So / Pi]

dZ / dSo = (1 / Pi)(Lh / Ro)
= [6850 / 3 Pi]

Hence at So = 1.999708036:
dZ / dSo = (1 / Pi)(Lh / Ro)
= [6850 / 3 Pi]

Hence:
Z = [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5
or
dZ / dSo = (1 / 2 Z) {Np^2 2 (So^2 + 1) 2 So + Nt^2 2 (So^2 - 1) 2 So}
 
= {Np^2 2 (So^2 + 1) 2 So + Nt^2 2 (So^2 - 1) 2 So}
/ 2 [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5
 
= 2 So {Np^2 (So^2 + 1) + Nt^2 (So^2 - 1)}
/ [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5
 
~ 2 (2) {Np^2 (5) + Nt^2 (3)}
/ [Np^2 25 + Nt^2 9]^0.5
 
= {Np^2 (20) + Nt^2 (12)}
/ [Np^2 25 + Nt^2 9]^0.5 = [6850 / 3 Pi]

CONTINUE FROM HERE

Nt must be an integer. Hence:
Nt = 2 (137) = 274

There is a remote possibility that:
Nt = 1 (137) = 137
or
Nt = 3 (137) = 411
so we must eliminate those possibilities.

We can now match theory and experiment.

At So = 2 the equation for (1 / Alpha) simplifies as follows:
(1 / Alpha) = (Pi / 4) [{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5
[(So^^2 - So + 1) / (So^2 + 1)^2]
 
(Pi / 4) Nt [{[Np^2 / Nt^2] + [(So^2 - 1)^2 / (So^2 + 1)^2]}^0.5
[(So^^2 - So + 1) / (So^2 + 1)]
 
= [Pi / 4] Nt [{[Nr^2] + [R^2]}^0.5]
{[(So^2 - 1)+ (2 - So)] / [(So^2 + 1)]}
 

At So = 2:
(1 / Alpha) = [Pi / 4] Nt [{[Nr^2] + [R^2]}^0.5]
{(So^2 - 1) / (So^2 + 1)}
 
= [Pi / 4] Nt [{[Nr^2] + [R^2]}^0.5] R
 
= [Nt / 2][(Pi R) / 2][{[Nr^2] + [R^2]}^0.5]
where because Nt is an integer:
[(Pi R) / 2][{[Nr^2] + [R^2]}^0.5] = 1

or
[(Pi R) / 2]^2 [{[Nr^2] + [R^2]}] = 1

The spheromak boundary condition gives:
Nr^2 + R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]

Combining these two equations gives:
[(Pi R) / 2]^2 / [(Pi / 2)^2 - (F / R)^2] = 1
 

DETERMINING F:
Rearranging this equation gives:
[(Pi R) / 2]^2 = [(Pi / 2)^2 - (F / R)^2]}
or
(F / R)^2 = [(Pi / 2)^2] - [(Pi R) / 2]^2
or
F^2 = R^2 {[(Pi / 2)^2] - [(Pi R) / 2]^2}
= [R Pi / 2]^2 [1 - R^2]

Thus:
F = [R Pi / 2] [1 - R^2]^0.5

The definition of R gives:
R = (So^2 - 1) / (So^2 + 1)
which at So = 2 gives:
R = 3 / 5

Thus:
F = [R Pi / 2] [1 - R^2]^0.5
= [3 Pi / 10][1 - (9 / 25)]^0.5
= [3 Pi / 10][4 / 5]
= 0.24 Pi
= 0.753982236

We know from the definition of F that:
0.5 < F < 1.0
so this result is consistent.
 

DETERMINING Nr:
Recall that:
[(Pi R) / 2]^2 {[Nr^2] + [R^2]} = 1
or
{[Nr^2] + [R^2]} = [4 / (Pi R)^2]
or
Nr^2 = [4 / (Pi R)^2] - [R^2]
or
Nr = {[4 / (Pi R)^2] - [R^2]}^0.5
= {[4(25) / Pi^2 9] - [9 / 25]}^0.5
= {0.7657909319}^0.5
= 0.8750948131

Np = 0.8750948131 X 274
= 239.7759788

We anticipate Np = 240 in the exact solution.

We can repeat the entire above calculation for the cases of Nt = 137 and Nt = 411 and we find that those choices result in out of range F values. Thus we have confidence that:
Nt = 274 is correct. Moreover the F value can be checked by an electric field integration.

At Nt = 137:
[(Pi R) / 4][{[Nr^2] + [R^2]}^0.5] = 1 or
[(Pi R) / 4]^2[{[Nr^2] + [R^2]}] = 1
or
[(Pi R) / 4]^2 / [(Pi / 2)^2 - (F / R)^2] = 1
or
(Pi R) / 4]^2 = [(Pi / 2)^2] - [(F / R)^2]
or
(F / R)^2 = [(Pi / 2)^2] - [(Pi R) / 4]^2
or
F^2 = [(Pi R / 2)^2] - [(Pi R^2) / 4]^2
= [(Pi R / 2)^2][1 - (R^2 / 4)]

Hence:
F = (Pi R / 2)[1 - (R^2 / 4)]^0.5
= Pi (3 / 10)[1 - (9 / 100)]^0.5
= Pi (3 / 10)[.9539392014]
= 0.8990
which is likely too large to be a real F value.
 

At Nt = 411:
[(Pi R) 3 / 4][{[Nr^2] + [R^2]}^0.5] = 1 or
[(Pi R)3 / 4]^2[{[Nr^2] + [R^2]}] = 1
or
[(Pi R) 3 / 4]^2 / [(Pi / 2)^2 - (F / R)^2] = 1
or
(Pi R) 3 / 4]^2 = [(Pi / 2)^2] - [(F / R)^2]
or
(F / R)^2 = [(Pi / 2)^2] - [(Pi R) 3 / 4]^2
or
F^2 = [(Pi R / 2)^2] - [(Pi R^2) 3 / 4]^2
= [(Pi R / 2)^2][1 - (R^2 9 / 4)]

Hence:
F = (Pi R / 2)[1 - (R^2 9 / 4)]^0.5
= Pi (3 / 10)[1 - (81 / 100)]^0.5
= Pi (3 / 10)[0.435889]
= 0.4108
which is too small to be a real F value.

Thus subject to checking the F value with an electric field integration we are quite certain that Nt = 274.
 

We believe that (3 Lh / Ro) is an integer. If that belief is correct then at So = 2:
(1 / Alpha) = [Lh / Ro] [So / 4]{[ So^2 - So + 1] / [(So^2 + 1)^2]}
= [Lh / 2 Ro][So / 2]{[ So^2 - So + 1] / [(So^2 + 1)^2]} = [Lh / 2 Ro][2 / 2][3 / 25]
= [Lh / Ro][3 / 50]

Hence:
[3 Lh / Ro] = 3 [50 / 3][137] = 6850 = 2 X 5^2 X 137
where 137 is prime.
 

Note that (Lh / Ro) is a geometrical constant that applies to all simple spheromaks, regardless of their energy content.

Recall that: Lh / Ro = Pi [{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5 / So] Thus:
3 Lh / Ro = 3 Pi [{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5 / So]
= 6850
or
9 Pi^2 [{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]} / So^2 = (6850)^2
where Np and Nt are integers. Hence:
[{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]} / So^2 = [6850 / 3 Pi]^2
This formula is likely to have general quantum mechanical application. or
2^0.5 No / So = 6850 / 3 Pi
or
No = (6850 So) / (3 Pi 2^0.5)
= 1027.861129 No^2 = Np Nt (So^4 - 1) So = 1.999708036 So^4 - 1 = 14.9906592 Np Nt = 1056498.501 / 14.9906592
= 70477.1209 = (265.475)^2 240 X 274 = 65760 WE NEED TO RESOLVE THIS DISCREPENCY

Note that in reality:
F determines R which determines So which determines Nr which sets (1 / Alpha).

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

THE OVERLOOKED ISSUE:
For over a century it has been asserted that Alpha and h are constant. However, for that assertion to be true the following analysis must hold.

At its operating frequency Fh an electro-magnetic spheromak spontaneously reconfigures itself so as to minimize its total static field energy. Thus, subject to quantum limitations So, Np and Nt will spontaneously adopt a minimum energy configuration consistent with maintenance of Fh. Maintenance of Fh implies that dFh = 0 and dLh = 0.

Recall that:
(1 / Alpha) = [Pi / 4] [{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5]
{[ So^2 - So + 1] / [(So^2 + 1)^2]}

The inverse Fine Structure constant (1 / Alpha) is in the form:
(1 / Alpha) = [Fa Fb]
and at the spheromak operating point:
Fa dFb + Fb dFa = 0
where:
Fa = [Pi / 4] [{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5]
and
Fb = {[ So^2 - So + 1] / [(So^2 + 1)^2]}

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

Then:
Fa = [Pi / 4] [{[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}^0.5]
= [Pi / 4] Z

dFa = [Pi / 4][1 / 2 Z]
{d[Np^2 (So^2 + 1)^2] + d[Nt^2 (So^2 - 1)^2]]}
= [Pi / 4][1 / 2 Z]
{Np^2 2 (So^2 + 1) 2 So dSo + Nt^2 2 (So^2 - 1) 2 So dSo
+ 2 Np dNp (So^2+ 1)^2 + 2 Nt dNt (So^2 - 1)^2}

Recall that:
Fb = [ So^2 - So + 1] / [(So^2 + 1)^2]}

Differentiating Fb gives:
d[Fb]
= {[(So^2 + 1)^2][(2 So - 1)][dSo]
- (So^2 - So + 1) [2 (So^2 + 1) 2 So][dSo]}
/ [(So^2 + 1)^4]
 
= {[(So^2 + 1)][(2 So - 1)][dSo]
- (So^2 - So + 1) [4 So][dSo]}
/ [(So^2 + 1)^3]
 
= {[2 So^3 - So^2 + 2 So - 1][dSo ]
- [4 So3 - 4 So^2 + 4 So][dSo]}
/ [(So^2 + 1)^3]
 
= [- 2 So^3 + 3 So^2 - 2 So - 1][dSo]}
/ [(So^2 + 1)^3]
 

Thus:
dFb = [- 2 So^3 + 3 So^2 - 2 So - 1][dSo] / [(So^2 + 1)^3]

At the stable spheromak operating point:
Fa dFb + Fb dFa = 0

Substituting for each term gives:
[Pi / 4] [Z] [- 2 So^3 + 3 So^2 - 2 So - 1][dSo] / [(So^2 + 1)^3]
+ {[ So^2 - So + 1] / [(So^2 + 1)^2]}[Pi / 4][1 / 2 Z]
{Np^2 2 (So^2 + 1) 2 So dSo + Nt^2 2 (So^2 - 1) 2 So dSo
+ 2 Np dNp (So^2+ 1)^2 + 2 Nt dNt (So^2 - 1)^2}
= 0

FIX

To achieve spheromak stability two equilibrium equations must operate simultaneously:
[2 Np dNp (So^2 + 1)^2] + [2 Nt dNt (So^2 - 1)^2]} = 0
and
[Pi / 4] [Z] [- 2 So^3 + 3 So^3 - 2 So - 1][dSo] / [(So^2 + 1)^3]
+ {[ So^2 - So + 1] / [(So^2 + 1)^2]}[Pi / 4][1 / 2 Z]
{[Np^2 2 (So^2 + 1) 2 So dSo] + [Nt^2 2 (So^2 - 1) 2 So dSo]
= 0

Apply the important constraint condition that for a stable spheromak:
M dNp = - dNt
where M is a small integer, likely 3.

This is a critical constraint equation. It basically says that provided that suitable Np and Nt states are available the integer ratio (Np / Nt) quantizes the available So values according to the equation:
(Np / Nt) = M (So^2 - 1)^2 / (So^2 + 1)^2
= M R^2
where:
R = (So^2 - 1) / (So^2 + 1)

Then:
[2 Np dNp (So^2 + 1)^2] + [2 Nt dNt (So^2 - 1)^2]} = 0
becomes:
[2 Np (- dNt / M) (So^2 + 1)^2] + [2 Nt dNt (So^2 - 1)^2]} = 0
or
(Np / M)(So^2 + 1)^2 = Nt (So^2 - 1)^2]}
or
Np / Nt = M (So^2 - 1)^2 / (So^2 + 1)^2

Now consider the other equilibrium equation:
[Pi / 4] [Z] [- 2 So^3 + 3 So^3 - 2 So - 1][dSo] / [(So^2 + 1)^3]
+ {[ So^2 - So + 1] / [(So^2 + 1)^2]}[Pi / 4][1 / 2 Z]
{[Np^2 2 (So^2 + 1) 2 So dSo] + [Nt^2 2 (So^2 - 1) 2 So dSo]
= 0

Multiply through by [8 Z (So^2 + 1)^2 / Pi] to get:
[2 Z^2] [- 2 So^3 + 3 So^3 - 2 So - 1][dSo] / [(So^2 + 1)]
+ {[ So^2 - So + 1]} {[Np^2 (So^2 + 1)] + [Nt^2 (So^2 - 1)]}[4 So dSo]
= 0

Use the previously derived condition:
Np / Nt = M (So^2 - 1)^2 / (So^2 + 1)^2
to get:
Np^2 = {Nt^2 M^2 (So^2 - 1)^4 / (So^2 + 1)^4}
= Nt^2 M^2 R^4

Then:
Z^2 = {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}
 
= {[{Nt^2 M^2 (So^2 - 1)^4 / (So^2 + 1)^4} (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}
 
= {[{Nt^2 M^2 (So^2 - 1)^4 / (So^2 + 1)^2}] + [Nt^2 (So^2 - 1)^2]}
 
= {[{M^2 R^2 Nt^2 (So^2 - 1)^2}] + [Nt^2 (So^2 - 1)^2]}
 
Z^2 = {[(M^2 R^2) + 1] [Nt^2 (So^2 - 1)^2]}

Using similar substitutions:
[Np^2 (So^2 + 1)] + [Nt^2 (So^2 - 1)]
 
= [{Nt^2 M^2 (So^2 - 1)^4 / (So^2 + 1)^4}(So^2 + 1)] + [Nt^2 (So^2 - 1)]
 
= [{Nt^2 M^2 (So^2 - 1)^4 / (So^2 + 1)^3}] + [Nt^2 (So^2 - 1)]
 
= [{Nt^2 M^2 R^3 (So^2 - 1)}] + [Nt^2 (So^2 - 1)]
 
= [(M^2 R^3) + 1] [Nt^2 (So^2 - 1)]

Recall the equilibrium equation:
[Z^2] [- 2 So^3 + 3 So^3 - 2 So - 1][dSo] / [(So^2 + 1)]
+ {[ So^2 - So + 1]} {[Np^2 (So^2 + 1)] + [Nt^2 (So^2 - 1)]}[2 So dSo]
= 0

Substitution into this equation gives:
[(M^2 R^2) + 1] [Nt^2 (So^2 - 1)^2][- 2 So^3 + 3 So^3 - 2 So - 1][dSo] / [(So^2 + 1)]
+ {[ So^2 - So + 1]} [(M^2 R^3) + 1] [Nt^2 (So^2 - 1)][2 So dSo]
= 0

Cancel out Nt^2 (So^2 - 1) dSo to get:
[(M^2 R^2) + 1] [(So^2 - 1)][- 2 So^3 + 3 So^3 - 2 So - 1][1 / [(So^2 + 1)]
+ {[ So^2 - So + 1]} [(M^2 R^3) + 1] [2 So]
= 0

or
[(M^2 R^3) + R] [- 2 So^3 + 3 So^3 - 2 So - 1]
+ {[ So^2 - So + 1]} [(M^2 R^3) + 1] [2 So]
= 0

Collecting terms gives:
[M^2 R^3][- 2 So^3 + 3 So^3 - 2 So - 1 + 2 So^3 - 2 So^2 + 2 So]
+ [R][- 2 So^3 + 3 So^3 - 2 So - 1]
+ [2 So^3 - 2 So^2 + 2 So]
= 0
or
[M^2 R^3][3 So^3 - 2 So^2 - 1]
+ [R][So^3 - 2 So - 1]
+ [2 So^3 - 2 So^2 + 2 So]
= 0

This is the equation for analytical determination of So without consideration of the spheromak boundary condition. Absent the boundary condition this equation has no real solution because the LHS is always positive. Hence the behaviour of spheromaks is dominated by the boundary condition.
 

The other stability equation is:
Np / Nt = M (So^2 - 1)^2 / (So^2 + 1)^2

Recall that:
Z^2 = {[Np^2 (So^2 + 1)^2] + [Nt^2 (So^2 - 1)^2]}

The form of the equation for Z indicates that a spheromak is most stable when:
Np^2 (So^2 + 1)^2 ~ Nt^2 (So^2 - 1)^2

or substituting for Np:
M^2 (So^2 - 1)^4 Nt^2 / (So^2 + 1)^2 ~ Nt^2 (So^2 - 1)^2
or
M^2 (So^2 - 1)^2 / (So^2 + 1)^2 ~ 1
or
M ~ (So^2 + 1) / (So^2 - 1)

Clearly:
M = 1 is impossible
M = 2 gives: So^2 = 3
M = 3 gives: So^2 = 2
M = 4 gives: So^2 = (5 / 3)
M = 5 gives: So^2 = (3 / 2)
etc.

At small So values the spheromak is less energy stable. The most stable spheromaks will have M = 2.
 

FINE STRUCTURE CONSTANT ISSUES:
Note that (1 / Alpha) is a function of the spheromak shape parameter So. Note the following:
1) The spheromak energy is proportional to (1 / Alpha). Hence the spheromak has maximum energy stability when (1 / Alpha) is at a relative minimum.

2) Experimentally (1 / Alpha) is a nearly 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
 

EVALUATION OF (1 / ALPHA):
Note that in the analysis earlier on this web page:
R = (So^2 - 1) / (So^2 + 1)

Hence: (1 / Alpha)^2 = (Pi / 4)^2 Z^2 {(So^2 - So + 1)^2 / (So^2 + 1)^4}
 
= (Pi / 4)^2 [Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]
{(So^2 - So + 1)^2 / (So^2 + 1)^4}
 
= (Pi / 4)^2 [Np^2 + Nt^2 R^2] {(So^2 - So + 1)^2 / (So^2 + 1)^2}
 
= (Pi / 4)^2 [Nr^2 + R^2] Nt^2 {(So^2 - So + 1)^2 / (So^2 + 1)^2}

On the web page titled: ELECTROMAGNETIC SPHEROMAK the boundary condition is found to be:
Nr^2 + R^2 = {1 / [(Pi / 2)^2 - (F / R)^2]}

Hence:
(1 / Alpha)^2
= (Pi Nt / 4)^2 {1 / [(Pi / 2)^2 - (F / R)^2]}{(So^2 - So + 1)^2 / (So^2 +1)^2}

Hence:
(1 / Alpha)
= (Pi Nt / 4) {1 / [(Pi / 2)^2 - (F / R)^2]^0.5} {(So^2 - So + 1) / (So^2 + 1)}
= (Nt / 2){(Pi / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5 {(So^2 - So + 1) / (So^2 + 1)}
= (Nt / 2){1 / [1 - (2 F / Pi R)^2]}^0.5 {(So^2 - So + 1) / (So^2 + 1)}

P>This is the equation for analytical determination of So. Unambiguous interpretation of this equation requires independent quantification of the parameter F. Remember that the actual value of So adopted will depend on the nearest available quantum state.
 

From the web page titled:
ELECTROMAGNETIC SPHEROMAK the dependence of F on So is given by:
F^2 = {[(Pi / 2)^2] [Nr^2 + R^2] - 1} / {1 + [(Nr / R)^2]}

This relationship indicates that for So = 2.026 and M = 2:
FIX F^2 = 0.506967
and
F = 0.712016
corresponding to:
R = 0.6082235
and
So = 2.02606822

However, there is still uncertainty about the value of So and hence all the other numbers. Resolution of this uncertainty requires independent calculation of the parameter F.

Approximate numerical evaluation for:
F = 0.7071 = (1 / 2)^0.5 gives:
(1 / Alpha)
= (Nt / 2){1 / [1 - (2 F / Pi R)^2]}^0.5 {(So^2 - So + 1) / (So^2 + 1)}
= (305 / 2) {1 / [1 - [(2)(.7071067)(5.1046) / (Pi 3.1046)]^2]}^0.5
{4.1046 - 2.026 + 1) / (5.1046)
= (152.5) {1 / [1 - 0.54782566]}^0.5 (3.074) / (5.1)
= (152.5) (1.4871) {3.074) / (5.1046)
= 136.569

We need more exact input data to improve this calculation.

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. The most common quantum state is: Np/Nt = 240 / 274. However, the theoretically possible Np/Nt states include but are not limited to:
___________________

Note that in a particular quantum state there may be complications if the integers Np and Nt have common factors other than one. FIX

Note that we derived the expression:
Np / Nt = M (So^2 - 1)^2 / (So^2 + 1)^2 which for M = 2 at So = 2.026 gives:
Nr = Np / Nt = 2 (3.1047 / 5.1047)^2
= 0.7398

At Np = 222, Nt = 305, Nr = 0.7278

At Np = 223, Nt = 303, Nr = 0.7359

At Np = 224, Nt = 301, Nr = 0.7442

Thus there is good agreement between theory and experiment at So = 2.026

Hence at that operating point we can use experimantal data to calculate F as opposed to calculating F from first principles.

Provided that either the boundary condition or the operating point stability requirement leads to So = 1.9989 we have a tentative solution to the Fine Structure constant calculation.

INTEGER ISSUES:
Recall that:
Nr^2 = Np^2 / Nt^2
where Np and Nt are integers with no common factors. Thus to find Np and Nt it is necessary to test both the Np and Nt values for integer and factor compliance. This is not a huge task because we know that:
(1 / Alpha) ~ 137
which constrains the maximum size of the Np and Nt integer values to less than about 600.

We know that with the simple boundary condition:
(1 / Alpha Nt)^2 = (Pi / 4)^2 [(So^2 - So + 1)^2 / (So^2 + 1)^8] {2 (So^2 + 1)^2 - 4}
/ {- [4 / [(So^2 - 1)^2]] + [Pi^2 / 4]}

We know that:
(1 / Alpha) ~ 137

Hence we can estimate Nt using the equation:
Nt|estimate = (1 / Alpha) / (1 / (Alpha Nt))
= 137 /(1 / (Alpha Nt))

Then we can estimate Np using the equation:
Np|estimate = Nr Nt|estimate
where to calculate Nr we first calculate:
[Z^2 / Nt^2] = {2 - [4 / (So^2 + 1)^2]}
/ {+ [Pi^2 / 4][1 / (So^2 + 1)^2] - [4 / [(So^2 + 1)^2 (So^2 - 1)^2]]} and then calculate Nr^2 using the equation:
Nr^2 = {Z^2 / [Nt^2 (So^2 + 1)^2]} - {[(So^2 - 1)^2 / (So^2 + 1)^2]}
and the calculate Nr using the equation: Nr = [Nr^2]^0.5

These estimates in combination with a list of prime numbers lead to only a few Np, Nt combinations that need to be fully tested.

Prime numbers less than 774 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,

Nr = Np / Nt
where Np and Nt are both integers with no common factors and usually either Np or Nt is prime.
 

SOLUTION:
Examination of the list of prime numbers shows isolated primes at 293 and 211. These primes would lead to a stable solution if So was high enough at about 2.16.

However, the actual operating So value is superficially seems to be about 2.026 which causes the corresponding Np and Nt values to be 223 and 303. Note that 223 is prime.

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 spheromak structure has Np poloidal turns and Nt toroidal turns. Spheromaks have an energy minimum that potentially permits Np / Nt values of 226 / 297, 225 / 299, 224 / 301, 223 / 303, 222 / 305 and 221 / 307. These Np, Nt pairs each have the mathematical property that they do not share any common factors, which is one of the criteria required for the existence of the Planck constant. These integer pairs and the spheromak mathematical model precisely predict the experimentally measured Planck constant h, which is fundamental to quantum mechanics. Notice that when So is constant between adjacent number pairs:
dNt = - M dNp

An isolated plasma spheromak has a ratio of outside radius Rs to inside radius Rc of:
(Rs / Rc) = So^2 ~ 4.1

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 F^2.

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.  

INTEGER CONSIDERATIONS:
This arrangement confines the spheromak to a few Np/Nt state pairs each of which has no common factors other than one.

Consider for example the state:
Np / Nt = 223 / 303
In this case Np and Nt do not share common factors. Adjacent states are:
Np / Nt = 222 / 305, and Np / Nt = 224 / 301.

We must allow for the possibility that Np increments or decrements by one without changing Nt. Hence a more general rule is that an Np integer should not share any common factors with either its normal corresponding Nt integer or either of the two immediately adjacent Nt integers.

Consider factors of the following contemplated Np states:
221 = 13 X 17
222 = 2 X 11^2, 3 X 74
223 = prime
224 = 2^5 x 7
225 = 5^2 X 3^2
Thus we need to check if any of the contemplated Nt states have factors of 2, 3, 5, 7, 11, 13, 17.

Consider factors of the following contemplated Nt states:
309 = 3 X 103 307 = prime 305 = 5 X 61 303 = 3 X 101 301 = 7 X 43 299 = 13 X 23 297 = 3^3 X 11 295 = 5 X 59

Consider a spheromak that normally operates at:
Np / Nt = 223 / 303 or Np / Nt = 222 / 305.

The numbers 223 and 303 have no common factors. The numbers 222 and 305 have no common factors. Hence superficially these numbers appear compatible. However, suppose that the spheromak is slightly disturbed and Np increments or decrements by one without changing Nt. If the spheromak operating at:
Np / Nt = 223 / 303
is disturbed it might move into:
Np / Nt = 222 / 303 or Np / Nt = 224 / 303.
alternatively it might shift from operation at:BR> Np / Nt = 222 / 305
into:
Np / Nt = 221 / 305 or Np / Nt = 223 / 305.
Hence we need to examine if 221 and 223 are compatible with 305 and we need to examine if 222 and 224 are compatible with 303.

There is a potential problem because both 222 and 303 share the common factor 3. The problem may not be serious if normal spheromak operation is at:
Np / Nt = 222 / 305

I used a computer to find an analytical solution for So at:
So = 2.02606822
corresponding to:
Np = 223, Nt = 303 and (1 / Alpha) = 137.03599929

With the same value of So there are nearby solutions at:
Np = 222, Nt = 305 and (1 / Alpha) = 137.039963385
and at:
Np = 224, Nt = 301 and (1 / Alpha) = 137.03609541155

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 matter refer to the web page titled: ELECTROMAGNETIC SPHEROMAK

Computer analysis of experimental data exhibiting:
(1 / Alpha) = 137.035999
shows that for isolated charged particles with So = 2.026 Np and Nt follow the following table:
   
NpNt
226297
225299
224301
223303
222305
221307

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.

At the spheromak operating point of So ~ 2.026 the experimentally measured Planck and Fine Structure constants are generated and experimentally observed energy quantization occurs.

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.

The web page titled:
ELECTROMAGNETIC SPHEROMAK for isolated quantum charged particles there is a boundary condition of the form:
Nr^2 + R^2 = {1 / [(Pi^2 / 4) - (F / R)^2]}
where:
R = (So^2 - 1) / (So^2 + 1)
and F is a real number in the range:
0 < F < 1
related to the electric field distribution in the central core of the spheromak. Experimental measurements indicate that:
F ~ 0.7
and
F^2 ~ 0.5

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.

Since experimental data shows that at (1 / Alpha) = 137.035999:BR> So^2 = [2.02606822]^2 = 4.104952432
this relationship can be solved to find:
Nr^2 = {2 - [(So^2 - 1)]^2 [Pi^2 / 4][1 / (So^2 + 1)^2]}
/{[Pi^2 / 4] - [4 / [(So^2 - 1)^2]]}
 
= {2 - [3.104952432]^2 [Pi^2 / 4][ 1 / 5.104952432]^2} / {[Pi^2 / 4] - [4 / [(3.104952432)^2]]}
= {2 - 0.912780295} / {2.467401095 - 0.4149063571}

= 1.087219705 / 2.052494738

= 0.5297064518

Hence:
Nr = [0.5297064518]^0.5
= 0.7278093513

This compares to:
(Np / Nt) = 222 / 305 = 0.7278688525
which can be met by a very tiny correction in So and hence (1 / Alpha).  

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)]
 

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:
The spheromaks of real charged particles also contain confined photons. 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 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.
 

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 charged 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:
hs = (Eb - Ea) / (Fb - Fa).
Hence, in high resolution experimental measurements of hs 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 values for:
hs = (Eb - Ea) / (Fhb - Fha)
= dEtt / dFh

Note that the spheromak spacial energy density assumptions are only truly valid in field free space, which is 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 fields that potentially affect the measurement.

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 is carried by a photon which is confined by the spheromak walls.
 

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:

(Muos C Qs^2 / 4 Pi) = [Alphas 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 considered constant in circumstances where So is reliably constant.

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.
 

PARAMETER DEFINITIONS:
Define for a spheromak in free space:
Rc = minimum radius of inner spheromak wall;
Rs = maximum radius of outer spheromak wall;
Ro = (Rs Rc)^0.5 = radius where the potential energy well is deepest;
So = [Rs / Ro] = [Ro / Rc] = spheromak shape parameter;
So^2 = (Rs / Rc);
Hs = distance of spheromak wall from the equatoral plane;
Hf = maximum value of |Hs|
2 Hf = spheromak overall height along its main axis of symmetry;
Lh = charge hose length;
Np = number of poloidal charge path turns contained in Lh;
Nt = number of toroidal charge path turns contained in Lh;
Nr = Np / Nt;
Rf = spheromak wall radius at H = Hf and H = -Hf;
Lp = Pi (Rs + Rc) = wall tangential poloidal turn length;
Lt = Pi (Rs - Rc) = wall tangential toroidal turn length;
Bpo = poloidal magnetic field strength at the center of the spheromak;
Uo = Upo = (Bpo^2 / 2 Mu) = maximum field energy density at the center of the spheromak on the main symmetry axis;
 

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)]

As shown on the web page titled ELECTROMAGNETIC SPHEROMAK the peak magnetic field strength Bpo at the center of a spheromak can also be expressed as:
Bpo = I [(Muos Qs C) / (2 Pi^2 Rc^2)] {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5}
= I (Muos Qs C) / (2 Pi^2 Ro^2)(Ro / Rc)^2 {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5}
= I (Muos 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)}
FIX
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:
FIX
Nr^2 = [+ {8} - {[Pi (So^2 - 1) / (So^2 + 1)]^2}] / {[Pi^2] - [4 / (So^2 - 1)]^2}
 
For the approximate value of So of:
So = 2.026
Nr^2 = [+ {8} - {[Pi (3.104676) / (5.104676)]^2}] / {[Pi^2] - [4 / (3.104676)]^2}
 
= [+ {8} - {3.650860312}] / {[9.869587728] - [1.659920979]}
 
= [4.349139688] / {8.209666749}
 
=0.5297583716

Hence the approximate values are:
So = 2.026
So^2 = 4.104676
Nr^2 = 0.5297583716
Nr = 0.7278450189
Ett = [Mu C Qa^2 / 4 Pi] [Pi^2 / 8] Fh Nt [2.2882]
 

The approximate value of Nr points to the integer values:
Np = 222
and
Nt = 305
hich give the exact values:
Nr = (Np / Nt)
= 0.7278688525
and
Nr^2 = (Np / Nt)^2
= 0.5297930664

The alternative is the integer values:
Np = 223
and
Nt = 303
which give the exact values:
Nr = (Np / Nt)
= 0.73359735974
and
Nr^2 = (Np / Nt)^2
= 0.541657136

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

Note that 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

Note that 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.
 

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:
So ~ 2.026

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

In the expression for the Planck constant:
So = 2.02606822
and
So^2 = 4.104937443
and
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.

This plot shows that there is a broad relative minimum at:
So = 2.026
in spheromak energy Ett spanning the range:
2.025 < So < 2.027
or
4.100625 < So^2 < 4.108729
where:
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

Typically So = 2.026 but depending on the exact circumstances there are multiple stable So values in the range 2.025 to 2.027

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

 

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]

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

The published CODATA experimentally measured value of Alpha^-1 = 137.03599915. 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 Alphas assuming that electrons behave as spheromaks.
 

EVALUATION OF Alphas:
Recall that hs is given by:
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]

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

At (1 / Alpha) = 137.035999:
So = 2.02606822
and
So^2 = 4.104937443
and
Nr = (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]

= [Alphas hs Pi / 16] Nt [0.9574677935] [2.405672651]
or
Alphas^-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

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. We are uncertain as to the implicit assumptions that are embedded in the experimentally measured Alpha^-1 value chosen by CODATA. For example, this CODATA value may only be strictly valid in the presence of certain values of external magnetic fields caused by proximity of other particles.
 

These values of Muo and Muos are in agreement to an accuracy of 3.3 parts in 1,000,000. 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.
 

RECOIL KINETIC ENERGY ERROR:
The recoil kinetic energy error will depend on the energy Ep of the photons used and the particle mass which is usually an electron.

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 hs 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 hs calculated herein. Note that the hs value calculated herein is actually the change in spheromak static potential energy Ett with respect to a change in 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.

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.
 

CONCLUSION:
The spheromak model of a charged particle provides a means of calculating the fine structure constant Alphas and hence the Planck constant hs in terms of Pi, Muo, Q and C. In highly accurate experimental measurements of Alpha and h it is necessary to take into account spheromak shape distortion due to external fields, confined photon response to the measurement signal and the charged particle recoil kinetic energy. These error sources cause the spheromak to adopt slightly different quantum states which slightly change the measured value of the fine Structure constant Alpha.

Note that for a spheromak at steady state conditions in field free space Alphas 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 September 16, 2018.

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