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PLANCK CONSTANT AND FINE STRUCTURE CONSTANT

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

EXPERIMENTAL OBSERVATION:
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.

On Earth there is an overall tendency for incident energy carried by solar radiation to be absorbed by matter. That energy is later re-emitted from that matter carried by lower frequency infrared radiation which is characteristic of the surface temperature of the matter. This tendancy determines the direction of evolution of many chemical reactions.
 

REST MASS ENERGY:
Matter stores energy in electromagnetic field energy configurations known as spheromaks. A spheromak has a closed current path with net charge that has associated with it both electic and magnetic field energies. The length Lh of this current path involves many toroidal and poloidal turns and hence is long compared to the nominal spheromak radius Ro.
 

PHOTON ENERGY:
A photon is a quantum of radiant energy dE either emitted by or absorbed by a spheromak. To change energy a spheromak absorbs or emits photons of energy dE and frequency dFh.
 

PLANCK CONSTANT:
In interactions between matter and radiation, energy is only transferred in quantized amounts where the magnitude of the quantum of transferred energy is proportional to the emitted or absorbed photon frequency. This photon frequency Fp is equal to a change dFh in inherent particle frequency Fh.

The quantum of energy dE transferred between rest mass and the electromagnetic radiation with frequency dFh is set via the relationship:
h = dE / dFh
where:
h = a proportionality factor known as the Planck Constant.

This quantum of electromagnetic radiation is known as a photon.
 

HISTORICAL 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 by itself gave no insight as to the underlying mechanisms.
 

RADIATION AND MATTER:
Atomic quantum charged particles have associated electro-magnetic spheromaks. Electro-magnetic spheromaks are stable energy states. These stable states are reached by 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 increases. During photon absorption the absorbing spheromak's natural frequency Fh increases and the amount of radiant energy decreases.

Over time electromagnetic spheromaks in free space will absorb or emit energy until they reach their stable steady states.
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 net charge that forms an electromagnetic spheromak and dFh is the frequency of a radiation emitted or absorbed.

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

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

PLANCK CONSTANT DERIVATION:
The Planck Constant is actually a composite of other physical constants. On this web page spheromak theory is used to derive the Planck Constant from first principles. 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 stable integer numbers of poloidal and toroidal current path turns that form the spheromak wall. The parameter h is constant because the spheromak energy Ett and the spheromak frequency Fh are both exactly inversely proportional to the nominal spheromak radius Ro, so the ratio:
(spheromak energy) / (spheromak frequency)
is constant independent of the spheromak size parameter Ro.

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 = (C / Lh)
is the natural frequency of the spheromak and h is a composite of other constants that together are generally referred to as the Planck constant. If radiant energy dEtt is absorbed or emitted by the spheromak:
dEtt = h dFh
where:
dFh = Fp
is the radiation photon frequency. Note that the radiation frequency is lower then the superficially apparent static energy frequency by the factor:
[Lh / 2 Pi Ro]:
where Lh = spheromak circulating current path length.
 

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

SPHEROMAK GEOMETRY:
The quasi-toroidal shape of a spheromak can be geometrically characterized by its inner radius Rc, its outer radius Rs and its height 2 Zf parallel to its main axis of symmetry. The ratio of Rs to Rc is defined by the spheromak shape parameter So where:
So^2 = (Rs / Rc).
The spheromak height 2 Zf is given by:
2 Zf = A (Rs - Rc)
where A is a constant in the range 1 < A < 2.
 

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SPHEROMAK GEOMETRIC PARAMETERS:
Define:
So^2 = (Rs / Rc)
and
Ro^2 = A^2 Rs Rc
giving:
(Rs / Ro) = (So / A)
and
(Rc / Ro) = (1 / So A)
where So is a spheromak geometrical shape parameter where:
So^2 = (Spheromak outside radius) / (spheromak inside radius)
and
A / B = (Ellipse major diameter parallel to main axis of symmetry)
/ (Ellipse minor diameter in the spheromak eqitorial plane).

The ratio (Lh / 2 Pi Ro) is the ratio of the spheromak current path length Lh to the spheromak nominal circumference (2 Pi Ro). The ratio [Lh / (2 Pi Ro)] is a highly stable geometrical constant which is the same for all stable spheromaks, independent of nominal spheromak radius Ro. The length Lh contains embedded factors of Pi and [1 / A] as well as the embedded constant Kc. The constant:
[Lh A / 2 Pi Ro]
has the advantage that its value is independent of Pi and A.

Note that there is an inherent assumption that Lt is the length of the perimeter of an ellipse.

In order for [Lh A / (2 Pi Ro)] to be the same for all stable spheromaks:
the parameters Np, Nt, So, and Kc
must all be the same for all stable spheromaks. Note that Kc is an ellipse parameter that can be calculated from the ellipse major axis A and minor axes B via a power series.

However, Kc can also be calculated from a spheromak existence condition. The two different Kc values will only be in agreement when the values of Np and Nt are correct. Thus Np and Nt can be found via computer itteration.
 

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SPHEROMAK STABLE STATE:
Over time electromagnetic spheromaks in free space will absorb or emit radiation photons until they reach their stable steady 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 net charge that forms an electromagnetic spheromak and dFh is the frequency of a radiation photon emitted or absorbed.
 

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

Although the Planck Constant is normally defined in terms of photon properties the apparent photon energy quantization is largely due to the properties of 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 slightly 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 is 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 motion of the emitting or absorbing particle 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 particle thermal motion.
 

CHARGE STRING CURRENT PATH:
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 which flows along the string (current path) at the speed of light C, approximately 3 X 10^8 m / s. The net charge is uniformly distributed along the charge string. In a stable charged particle at every point along the current path 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.
 

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SPHEROMAK ANALYSIS STRATEGY:
The Planck Constant and its contained Fine Structure Constant naturally arise from spheromak analysis. A spheromak has Np poloidal current path windings and Nt toroidal current path windings. The numbers Np and Nt are always positive integers and are related to a prime number P by the formula:
P = 2 Np + Nt.

It is necessary to find P and Nt.

The analysis strategy is to develop two functions, each involving the spheromak constant [Lh A / 2 Pi Ro]. These two functions are combined to find the A value corresponding to guessed P and Nt values. The guessed P and Nt values can also be use to calculate a Kc value, which value will only be correct when the guessed P and Nt values are correct. However, the A value can be used to calculate another Kc value referred to herein as Kc(new). These two different Kc values will only be in agreement when the prime integer P and integer Nt both match physical reality.

Thus it is necessary to check a range of candidate prime number P values and toroidal turn Nt values to find which P, Nt pair gives the correct value of Kc. Due to the non-invertable nature of some spheromak mathematical functions it is impractical to find P by other means. This problem is solvable because both P and Nt are integers. Experimental data gives some guidance as to the approximate value of Nt.
 

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PARAMETER DEFINITIONS:
A spheromak in free space has a toroidal shape with an axis of symmetry from which the radii Rc, Rs and Rf are measured:
Rc = minimum radius of inner spheromak wall;
Rs = maximum radius of outer spheromak wall;
Zs = distance of any point on the spheromak wall from the spheromak equatorial plane;
Zf = maximum value of |Zs|
2 Zf = spheromak overall height parallel to its axis of symmetry;
Rf = spheromak wall radius at Z = Zf and at Z = - Zf;
A = 2 Zf / (Rs - Rc) = A spheromak geometrical parameter usually slightly greater than unity;
A = (ellipse major axis) / (ellipse minor axis);
B = 1.000 = Z axis coefficient;
Ro = A (Rs Rc)^0.5 = nominal spheromak radius;
So = [A Rs / Ro] = [Ro / A Rc] = spheromak shape parameter;
So^2 = (Rs / Rc);
Lh = spheromak current path length;
Np = integer number of poloidal current path turns contained in Lh;
Nt = integer number of quasi-toroidal current path turns contained in Lh;
Nr = Np / Nt;
Lp = Pi (Rs + Rc) = average current path poloidal turn length;
Lt = Kc Pi (Rs - Rc) = charge path quasi-toroidal turn length;
Kc = (ellipse perimeter length) / (contained circle perimeter length);
Bpo = poloidal magnetic field strength at the center of the spheromak;
Uo = (Bpo^2 / 2 Mu) = magnetic field energy density at the center of the spheromak;
 

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SPHEROMAK ENERGY:
On the web page titled: SPHEROMAK ENERGY it is shown that if the field energy density outside the spheromak wall is:
U = Uo {Ro^2 / [Ro^2 + (A R)^2 + (B Z)^2]}^2
where:
Uo = field energy density at spheromak center;
and if the field energy density inside the spheromak wall is:
U = Uto [Ro^2 / R^2]
and if the two field energy densities are equal at the spheromak wall, then the total static field energy Ett of a spheromak is given by:
Ett = [(Uo Ro^3) Pi^2 / (A^2 B)] {4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
where:
Pi = 3.14159265
and
So^2 = (Rs / Rc)
 

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SPHEROMAK ENERGY VERSUS NOMINAL RADIUS Ro:
Recall from the web page titled: Spheromak Energy that:
dEtt
= d(Uo Ro^3) [Pi^2 / (A^2 B)] {4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}

There is potential confusion because Uo has buried within itself a
(1 / Ro^4) term arising from the boundary condition that in the far field the energy density must match the electric field energy density of a quantum charge.

In the far field: U = Uo {Ro^2 / [Ro^2 + (A R)^2 + (B Z)^2]}^2
= Uo {Ro^2 / X^2}^2

For a quantum charge Q the electric field E along the Z axis is:
E = Q / (4 Pi Epsilono Z^2)
and the electric field energy density is:
Ue = (Epsilono / 2) E^2
= (Epsilono / 2)[Q / (4 Pi Epsilono Z^2)]^2
= [Q^2 / (32 Epsilono Pi^2)][1 / Z^4]

In the far field along the Z axis:
U = Uo {Ro^2 / [Ro^2 + (A R)^2 + (B Z)^2]}^2
= Uo {Ro^2 / [Ro^2 + (B Z)^2]}^2
= Uo {Ro^2 / [(B Z)^2]}^2
and since:
Ue = U
hence:
[Q^2 / (32 Epsilono Pi^2)][1 / Z^4] = Uo {Ro^2 / [(B Z)^2]}^2
[Uo Ro^4 / B^4] = [Q^2 / (32 Epsilono Pi^2)]
or
Uo = [Q^2 B^4 / (32 Ro^4 Epsilono Pi^2)]

Hence:
Uo Ro^3 = [Q^2 B^4 / (32 Ro Epsilono Pi^2)]
and
d[Uo Ro^3] = - [Q^2 B^4 / (32 Ro^2 Epsilono Pi^2)] dRo

Thus:
dEtt
= d(Uo Ro^3) [Pi^2 / (A^2 B)] {4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
= - [Q^2 B^4 / (32 Ro^2 Epsilono Pi^2)] dRo [Pi^2 / (A^2 B)]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}

= - [Q^2 B^4 Muo C^2 / (32 Ro^2 Pi^2)] dRo [Pi^2 / (A^2 B)]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}

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

Note that the change in spheromak energy is proportional to (- 1 / Ro^2) dRo.
 

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SPHEROMAK FREQUENCY Fh:
Each spheromak has a characteristic current path length Lh, where (Lh / 2 Pi Ro) is a constant common to all spheromaks. Thus:
Lh = (Lh / Ro) Ro
and:
Lh Fh = C
or
Fh = C / Lh
= [C /[(Lh / Ro)}[1 / Ro]

Hence:
dFh = [C /(Lh / Ro)] d[1 / Ro]dRo = [C /(Lh / Ro)] [- 1 / Ro^2] dRo = {[2 Pi Ro / Lh][C / 2 Pi][-1 / Ro^2]} dRo
 

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PLANCK CONSTANT:
Thus the change in static field energy Ett of a spheromak with respect to the change in frequency Fh can be expressed in the form:
h = dEtt / dFh = [Muo C^2 Qs^2 B^4 / (32 A^2 B)] [- 1 / Ro^2] dRo
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
/ {[2 Pi Ro / Lh][C / 2 Pi][-1 / Ro^2] dRo}
 
= [Muo C Qs^2 B^4 / (32 A^2 B)]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
/ [2 Pi Ro / Lh][1 / 2 Pi]
 
= [Lh / 2 Pi Ro][Muo C Qs^2 B^4 2 Pi / (32 A^2 B)]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [ Lh / 2 Pi Ro][Muo C Qs^2] [B^3 Pi / (16 A^2)]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}

 
= h which is of the form:
dEtt / dFh = h
where:
h = [Muo C Qs^2 Pi B^3 / 16 A^2][Lh / (2 Pi Ro)]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}

The parameter h is known as the Planck Constant. It is constant to the extent that the spheromak relative geometrical shape parameters:
[Lh / (2 Pi Ro)] and So are both constant. Note that both of these parameters are independent of the spheromak size parameter Ro.
 

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FINE STRUCTURE CONSTANT:
The Fine Structure Constant Alpha is defined by:

[1 / Alpha] = 2 h / (Muo C Q^2)
 
= [2 / (Muo C Q^2)](Lh A / 2 Pi Ro) [Muo C Q^2 Pi B^3 / (16 A^3)]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [Pi B^3 / 8 A^3](Lh A / 2 Pi Ro) {4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= (Lh A / 2 Pi Ro) {[(Pi / 8) (B^3 / A^3)][4 So] [So^2 - So + 1] / [(So^2 + 1)^2]}
 
= (Lh A / 2 Pi Ro) {(Pi / 2)(B / A)^3)][So / (So^2 + 1)] [1 - (So / (So^2 + 1))]}
= [1 / Alpha]
where Alpha is the Fine Structure constant which is a combination of spheromak parameters:
 

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SPHEROMAK INNER WALL BOUNDARY CONDITION LEADS TO PARAMETER A:
The web page titled: Theoretical Spheromak shows that a spheromak inner wall boundary condition gives:
Uto = [Uo / A^2] [So^2 /(So^2 + 1)^2]
where Uto = toroidal magnetic field energy density at R = Ro.

>NOW ASSUME THAT THE ENERGY DENSITY INSIDE THE SPHEROMAK WALL IS ENTIRELY CAUSED BY THE TOROIDAL MAGNETIC FIELD

Recall that from electromagnetic field theory:
Uto = [Bto^2 / 2 Muo]
= {[Muo Nt Ih / 2 Pi Ro]^2 / 2 Muo}
= [Muo / 8] [Nt^2 Ih^2 / (Pi^2 Ro^2)]
= [Muo / 8] [Nt^2 Q^2 C^2 / (Lh^2 Pi^2 Ro^2)]

Recall that in the far field the electric field boundary condition along the Z axis gives:
Uo = [B^4 Muo C^2 Qs^2 / (32 Pi^2)] / Ro^4

Thus substituting for both Uto and Uo in the equation:
Uto = [Uo / A^2] [So^2 / (So^2 + 1)^2]
gives: [Muo / 8] [Nt^2 Q^2 C^2 / Lh^2 Pi^2 Ro^2]
= [B^4 Muo C^2 Qs^2 / (32 Pi^2 Ro^4)] [1 / A^2][So^2 /(So^2 + 1)^2]
or cancelling terms:
[Nt^2 / Lh^2]
= [ 1 / (4 Ro^2)] [B^4 / A^2][So^2 /(So^2 + 1)^2]
or rearranging:
A^2 = B^4 (Lh / 2 Ro)^2 (1 / Nt)^2 [So^2 /(So^2 + 1)^2]
or
A^4 = B^4 (Lh A / 2 Pi Ro)^2 (Pi / Nt)^2 [So^2 /(So^2 + 1)^2]
or
A^2 = B^2 (Lh A / 2 Pi Ro) (Pi / Nt)[So / (So^2 + 1)]
or
[Nt / Pi] = (B / A)^2 (Lh A / 2 Pi Ro)[So / (So^2 + 1)]
 

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SIMPLIFICATION OF EXPRESSION FOR [1 / Alpha]:
Substitution of Parameter A into the equation for the Fine Structure Constant gives:
[1 / Alpha]
= (Lh A / 2 Pi Ro) {(Pi / 2)(B / A)^3)][So / (So^2 + 1)] [1 - (So / (So^2 + 1))]}
  = [Nt / Pi][Pi / 2] [B / A] [1 - (So / (So^2 + 1))]
 

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

Thus calculation of the Fine Structure Constant requires determination of Nt, (A / B) and So.
 

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UPPER LIMIT ON [1 / Alpha]:
The function:
[So / (So^2 + 1)] [1 - (So / (So^2 + 1))]
provides an indication of the value of So at the maximum possible value of [1 / Alpha]. At this function relative maximum:
d{[So / (So^2 + 1)] [1 - (So / (So^2 + 1))]} / dSo = 0

Let X = [So / (So^2 + 1)] [So / (So^2 + 1)] [1 - (So / (So^2 + 1))] = [X (1 - X)] d[X (1 - X)] = X(-1) + (1 - X)
= 1 - 2 X
= 0

Hence:
X = (1 / 2) = [So / (So^2 + 1)]
or
So^2 + 1 = 2 So
or
So^2 - 2 So + 1 = 0
or
So = 1
 

Thus there is an upper limit on [1 / Alpha] of:
[1 / Alpha]|max = (Lh A / 2 Pi Ro) {[Pi / (2 A^3 B)][So / (So^2 + 1)] [1 - (So / (So^2 + 1))]}
= (Lh A / 2 Pi Ro) {[Pi / (2 A^3 B)][1 / 4]
 

In a real spheromak:
So ~ 2
X = So /(So^2 + 1) ~ 2 / 5
1 - X ~ 3 / 5
X (1- X) ~ (2 / 5) (3 / 5)
~ 6 / 25

Note that [1 / Alpha] is only weakly dependent on So.
 

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NO COMMON FACTORS:
A spheromak will collapse if Np and Nt share a common factor other than one. The spheromak turns must not overlap or intersect. For the same reason Np cannot be an integer multiple of Nt. Hence Np cannot equal Nt except at:
Np = Nt = 1
For example:
Np = 2 Nt
is prohibited.
 

Prime number theory together with the requirement for no common factors in Np and Nt gives two formulae for sets of number pairs Np and Nt that share no common factors other than one. Those formula are:
Family A:
P = 2 Nt + Np
and
Family B:
P = 2 Np + Nt
where:
P = prime number.

If Family A was involved [dNt / dNp] would be -(1 / 2) instead of - 2. The consequence would be that Np would be much smaller with respect to Nt causing the magnetic field at R = 0, Z = 0 to be insufficient to meet the Uo energy density requirement at R = 0, Z = 0. Hence Family A is excluded from spheromaks by this magnetic field strength requirement. Only Family B corresponds to real spheromaks.

For Family B:
P = 2 Np + Nt
or
Nt = (P - 2 Np) which gives:
dNt = - 2 dNp
which restricts real spheromaks to Family B.

P = 2 Np + Nt

= 2 (Np / Nt) Nt + Nt
= [2 (Np / Nt) + 1] Nt

or
(P / Nt) = [2 (Np / Nt) + 1]

Recall that:
Nt = P - 2 Np
which gives:
(Np / Nt) = Np / (P - 2 Np)
= [(P - Nt) / 2] / [P - 2((P - Nt) / 2)]
= [(P - Nt) / 2] / [P - ((P - Nt))]
= [(P - Nt) / 2 Nt]

(Np / Nt) values occur in the range:
2 / [2 (P - 2)] < [(P - Nt) / 2 Nt] < [(P - 1) / 2]

In summary, Np and Nt conform to the equation:
P = 2 Np + Nt
 

There is nothing obvious in the analytical mathematics of spheromaks that sets Nt. However, experimentally the Fine Structure constant has a firm value indicating that something other than analytical mathematics sets the Nt value.
 

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PRIME NUMBERS FOR SPHEROMAK STABILITY ENHANCEMENT:
The prime numbers P provide spheromak stability as long as the spheromak complies with the equation:
P = 2 Np + Nt

However, a stable spheromak needs to be tolerant of ousid disturbances that cause minor fluctuations in Np and Nt. In normal spheromak operation:
dNt = - 2 dNp However, for good spheromak stability:
If dNp = 0, then:
dNt = -2, -1, +1, +2
should not cause spheromak collapse and if dNt = 0, then:
dNp = -1, +1
should not cause spheromak collapse.

The issue is that the prime number P will not change but an external disturbance can potentially cause either Np or Nt to temporarily deviate from its nominal value. A stable spheromak should not collapse under such a condition.

This tolerance can be partially achieved if Nt is located between 2 prime numbers (Nt + 2) and at (Nt - 2). Similarly Np may be disturbance tolerant if there are primes at Np + 1 and Np - 1. Sometimes by a fluke the adjacent Np or Nt values share no common factors that can cause spheromak collapse.

The situation that must be carefully investigated is what happens if Nt becomes even. In the aforementioned example the spheromak must not collapse if Nt briefly becomes (Nt + 1) or (Nt - 1). To avoid this problem the normal value of Np must be odd. If Np is enen and Nt becomes even then both Np and Nt will share the common factor 2 which may lead to spheromak collapse.
 

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FIND THE LOWER LIMIT OF Kc:
Recall that:
Kc^2 = 2 (So^4 + 1) / (So^2 - 1)^2

Rearranging this equation gives:
(So^2 - 1)^2 Kc^2 = 2 (So^4 + 1)
or
(So^4 - 2 So^2 + 1) Kc^2 = 2 (So^4 + 1)
or
So^4 (Kc^2 - 2) - 2 Kc^2 So^2 + (Kc^2 - 2) = 0

So^2 = {2 Kc^2 + / - [4 Kc^4 - 4 (Kc^2 - 2)^2]^0.5} / [2 (Kc^2 - 2)]
= {2 Kc^2 + / - [4 Kc^4 - 4 (Kc^4 - 4 Kc^2 + 4)]^0.5} / [2 (Kc^2 - 2)]
= {2 Kc^2 + / - [ 16 Kc^2 - 16)]^0.5} / [2 (Kc^2 - 2)]
= {2 Kc^2 + / - 4 [ Kc^2 - 1)]^0.5} / [2 (Kc^2 - 2)]

Clearly:
Kc^2 > 2

At Kc^2 = 3: So^2 = {6 + 4 [2^0.5]} / 2
= 5.8284

At Kc^2 = 4:
So^2 = {8 + 4 [3^0.5]} / 4
= 3.732

At Kc^2 = 5:
So^2 = {10 + 4 [2]} / 6
= 3.000

Thus the assumed spheromak operating conditions are:
3 < So^2 < 5.8284
or
3 < Kc^2 < 5
 

FIND THE RANGE OF A:
Recall that:
(Np / Nt)^3 = 2 [So^2 + (1 / So^2)]^2

At So^2 = 3:
(Np / Nt)^3 = 2 [So^2 + (1 / So^2)]^2
= 2 [3 + (1 / 3)]^2
= 2 [100 / 9]
= 200 / 9
= 22.222

Hence:
Np / Nt = [22.222]^(1 / 3)
= 2.8114

Hence:
P = 2 Np + Nt
= 2 (2.8144 Nt) + Nt
= 6.6228 Nt

At So^2 = 5:
(Np / Nt)^3 = 2 [So^2 + (1 / So^2)]^2
= 2 [5 + (1 / 5)]^2
= 2 [26 / 5]^2
= 54.08

Hence:
Np / Nt = [54.08]^(1 / 3)
= 3.7816

Hence:
P = 2 Np + Nt
= 2 (3.7816 Nt) + Nt
= 8.5632 Nt

Thus for the assumed spheromakoperating conditions P lies in the range:
6.6228 Nt < P < 8.5632 Nt
where a midrange value is about:
P ~ 7.6 Nt

Thus in searching for suitable primes we choose a candidate Nt value and look for suitable prime P values near:
P ~ 7.6 Nt

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

FINDING THE REQUIRED PRIME VALUES:
The analytical mathematics determines the spheromak [Np / Nt] ratio but it is the location of suitable primes that actually fix Nt, P and Np.

1) Start with a prime number table. Based on experimental data we think that:
Nt < 930
and
P < 8000

2) Identify strings of primes that are each less than 500. The desired Nt value likely lies in one of these prime number strings.

3)Choose a Nt value to test.

4) For that Nt value list the P values that lie in range:
6.6 Nt < P < 8.6 Nt

5) For each such P value calculate:
Np = (P - Nt) / 2

6) If Np is even that P value is unsuitable for use with the chosen Nt value.

7) Do Np and (Nt + 1) have any common factors. If so that P value is unsuitable for use with the chosen Nt value.

8) Do Np and (Nt - 1) have any common factors? If so that P value is unsuitable for use with the chosen Nt value.

9) Do Np and (Nt + 2) have any common factors? If so that P value is unsuitable for use with the chosen Nt value.

10) Do Np and (Nt - 2) have any common factors? If so that P value is unsuitable for use with the chosen Nt value.

11) Note that it is advantageous for (Nt + 2) and/or (Nt - 2) to be prime to automatically satisfy #9 and/or #10.

12) Note that it is highly advantageous for Np to be prime so that tests #7,#8, #9 and #10 are all automatically satisfied.

13) Do (Np + 1) and Nt share any common factors? If so that P value is unsuitable for use with the chosen Nt value.

14) Do (Np - 1) and Nt share any common factors? If so that P value is unsuitable for use with the chosen Nt value.

15) Note that it is advantageous for Nt to be prime so that #12 and #13 are automatically satisfied.

16) A highly desireable arrangement is for Nt, P and Np to all be prime so that tests #6, #7, #8, #9, #10, #13 and #14 are all satisfied. Thus before any other work is done it is helpful to scan all the available primes looking for three primes that satisfy:
P = 2 Np + Nt

If such a prime number combination exists it may be unique and if so is likely the basis of spheromak existence.

The amount of scanning is reduced by the conditions that:
2 Nt < Np < 3.8 Nt
 

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

POTENTIAL SOLUTIONS FOR Nt:
The equation:
[1 / Alpha] = [Nt / (2 A B)][1 - (So / (So^2 + 1))]
allows us to make a reasonable guess as to Nt based on experimental measurements of [1 / Alpha] and suitable prime number availability. We can assume that:
[1 / Alpha] ~ 137
Nt = 461 or 463
So ~ 2.0 (from plama spheromak photographs) [1 - (So / (So^2 + 1))] ~ 0.6 A = 461 (0.6)/ 2(237) = 1.00948 B = 1.0

However, there is a problem. A typical value for A is about 2.35. To keep the other parameters constant we must increase Nt from 461 to about: 2.35 (461) = 1083

There exists a string of suitable prime numbers at potential Nt prime values of 877, 881, 883, 887. Any choice of Np must be compatible with spheromak non-collapse at Nt values of 879 and 885.

There are three other possible Nt prime value strings. They are:
853, 857, 859, 863 and 1277, 1279, 1283 and 1301, 1303, 1307.

Thus including the in between Nt values there are up to 20 Nt values in prime value strings to be examined.

The alternative is that we find a set of three prime numbers for P, Np, Nt where:
P = 2 Np + Nt.
These particular primes may be isolated so the entire set of primes less than 10,000 needs to be examined to find a 3 primes that meet this criteria.

Note that for spheromak existence the spheromak core magnetic field strength must be sufficient to balance the electric field. Hence:
Np > 2 Nt
 

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

SPHEROMAK CURRENT PATH LENGTH Lh:
Electromagnetic spheromaks arise from the electric current formed by distributed net charge Qs circulating at the speed of light C around the closed spiral path of length Lh which defines the spheromak wall. On the equatorial plane measured from the main axis of symmetry the spheromak inside radius is Rc and the spheromak outside radius is Rs.

Let Np be the integer number of poloidal currrent path turns in Lh and let Nt be the integer number of toroidal current path turns in Lh.

The spheromak wall contains Nt quasi-toroidal turns equally spaced around 2 Pi radians in angle Theta about the main spheromak axis of symmetry.
Each purely toroidal winding turn has length:
2 Pi (Rs - Rc) Kc / 2 = Pi (Rs - Rc) Kc
so the purely toroidal spheromak winding length is:
Nt Pi (Rs - Rc) Kc

Note that for a round spheromak cross section toroid Kc = 1. If for an elliptical cross section spheromak:
A > 1
then:
Kc > 1

The spheromak wall contains Np poloidal turns which are equally spaced around the ellipse perimeter. The average purely poloidal turn length is:
2 Pi (Rs + Rc) / 2 = Pi (Rs + Rc)
and the purely poloidal winding length is:
Np Pi (Rs + Rc)

In one spheromak cycle period the poloidal angle advances Np (2 Pi) radians. In the same spheromak cycle period the toroidal angle advances Nt (2 Pi) radians.
Hence:
(poloidal angle advance) / (toroidal angle advance) = Np / Nt

While a current point moves radially outward from Rc to Rs the toroidal angle advance is Pi radians and the toroidal travel is Lt / 2. The corresponding distance along the equatorial outer circumference is:
Pi (Np / Nt) Rs.
Thus Pythagoras theorm gives the current point travel distance along the winding for the toroidal half turn as:
[(Lt / 2)^2 + ( Pi Np Rs / Nt)^2]^0.5

While the current point moves radially inward from Rs to Rc the toroidal angle advance is Pi radians and the toroidal point travel is Lt / 2. The corresponding distance along the equatorial inner circumference is:
Pi (Np / Nt) Rc.
Thus Pythagoras theorm gives the current point travel distance along this winding toroidal half turn as:
[(Lt / 2)^2 + ( Pi Np Rc / Nt)^2]^0.5

Thus the total winding length Lh is:
Lh = Nt [(Lt / 2)^2 + (Pi Np Rs / Nt)^2]^0.5
+ Nt [(Lt / 2)^2 + ( Pi Np Rc / Nt)^2]^0.5
 
= [(Nt Lt / 2)^2 + (Pi Np Rs)^2]^0.5
+ [(Nt Lt / 2)^2 + (Pi Np Rc)^2]^0.5

Recall that:
Lt = [Kc Pi (Rs - Rc)]
and from spheromak geometry:
Rs = Ro So / A
and
Rc = Ro / A So

Thus:
Lh = {(Nt Lt / 2)^2 + (Pi Np Rs)^2}^0.5
+ {(Nt Lt / 2)^2 + (Pi Np Rc)^2}^0.5
 
= {(Nt [Kc Pi (Rs - Rc)] / 2)^2 + (Pi Np Rs)^2}^0.5
+ {(Nt [Kc Pi (Rs - Rc)] / 2)^2 + (Pi Np Rc)^2}^0.5
 
= {[Nt Kc Pi (Rs - Rc) / 2]^2 + (Pi Np Rs)^2}^0.5
+ {[Nt Kc Pi (Rs - Rc) / 2]^2 + (Pi Np Rc)^2}^0.5
 
= {[Nt Kc Pi ((Ro So / 2 A) - (Ro / 2 A So))]^2 + (Pi Np (Ro So / A))^2}^0.5
+ {[Nt Kc Pi ((Ro So / 2 A) - (Ro / 2 A So))]^2 + (Pi Np (Ro / A So))^2}^0.5
 
= [Pi Ro / A]{[Nt Kc ((So / 2) - (1 / 2 So))]^2 + [Np So]^2}^0.5
+[Pi Ro / A] {[Nt Kc ((So / 2) - (1 / 2 So))]^2 + [(Np / So)]^2}^0.5
 
= [Pi Ro / 2 So A] {[Nt Kc ((So^2) - (1))]^2 + [Np (2 So^2)]^2}^0.5
+[Pi Ro / 2 So A] {[Nt Kc ((So^2) - (1))]^2 + [2 Np]^2}^0.5
 
= [Pi Ro / 2 So A] {[Nt Kc (So^2 - 1)]^2 + [Np (2 So^2)]^2}^0.5
+[Pi Ro / 2 So A] {[Nt Kc (So^2 - 1)]^2 + [2 Np]^2}^0.5
 
= [Pi Ro Nt / 2 So A] {[Kc (So^2 - 1)]^2 + [(Np / Nt) (2 So^2)]^2}^0.5
+[Pi Ro Nt / 2 So A] {[Kc (So^2 - 1)]^2 + ([Np / Nt][2])^2}^0.5

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

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

= [Lh A / 2 Pi Ro]
=La + Lb
where:
La = Nt {[Kc (So^2 - 1) / 4 So]^2 + (Np / Nt)^2 [So / 2]^2}^0.5
and
Lb = Nt {[Kc (So^2 - 1)/ 4 So]^2 + (Np / Nt)^2 [1 / (2 So)]^2}^0.5

This equation is the result of spheromak geometric analysis.
 

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

SPHEROMAK [Lh A / 2 Pi Ro] STABILITY:
Recall that:
[Lh A / 2 Pi Ro]
= Nt {[Kc (So^2 - 1) / 4 So]^2 + (Np / Nt)^2 [So / 2]^2}^0.5
+ Nt {[Kc (So^2 - 1)/ 4 So]^2 + (Np / Nt)^2 [1 / (2 So)]^2}^0.5

 
= {Nt^2 [Kc (So^2 - 1) / 4 So]^2 + (Np)^2 [So / 2]^2}^0.5
+ {Nt^2 [Kc (So^2 - 1)/ 4 So]^2 + (Np)^2 [1 / (2 So)]^2}^0.5
 
= (La + Lb)
where:
La = {Nt^2 [Kc (So^2 - 1) / 4 So]^2 + (Np)^2 [So / 2]^2}^0.5
and
Lb = {Nt^2 [Kc (So^2 - 1)/ 4 So]^2 + (Np)^2 [1 / (2 So)]^2}^0.5

For a stable spheromak:
d[Lh A / 2 Pi Ro] = 0
implying that:
dLa + dLb = 0
or
d[Lh A / 2 Pi Ro] = {1 / 2 La} {2 Nt dNt [Kc (So^2 - 1) / 4 So]^2 + 2 Np dNp [So / 2]^2}
+ {1 / 2 Lb} {2 Nt dNt [Kc (So^2 - 1) / 4 So]^2 + 2 Np dNp [1 / (2 So)]^2}
= 0

Recall that a spheromak is subject to the winding constraint that:
P = 2 Np + Nt
and since P = constant:
2 dNp + dNt = 0
or
dNt = - 2 dNp

Make substitution:
dNt = - 2 dNp
to get:
d[Lh A / 2 Pi Ro]
= {1 / 2 La} {2 Nt (- 2 dNp) [Kc (So^2 - 1) / 4 So]^2 + 2 Np dNp (So / 2)^2}
+ {1 / 2 Lb} {2 Nt (-2 dNp) [Kc (So^2 - 1) / 4 So]^2 + 2 Np dNp [1 / (2 So)]^2}
= 0

or
d[Lh A / 2 Pi Ro]
= {1 / 2 La} {- 4 Nt [Kc (So^2 - 1) / 4 So]^2 + Np (So^2 / 2)}
+ {1 / 2 Lb} {-4 Nt [Kc (So^2 - 1) / 4 So]^2 + Np (1 / 2 So^2)}
= 0

which sets the relationship between Np and Nt at the spheromak operating point.

Bring terms to a common denominator to get:
Lb {- 4 Nt [Kc (So^2 - 1) / 4 So]^2 + Np (So^2 / 2)}
+ La {-4 Nt [Kc (So^2 - 1) / 4 So]^2 + Np (1 / 2 So^2)}
= 0

or
Lb {- 4 Nt [Kc (So^2 - 1) / 4 So]^2 + Np (So^2 / 2)}
= La {+ 4 Nt [Kc (So^2 - 1) / 4 So]^2 - Np (1 / 2 So^2)}

Multipy through by 16 So^2 to get:
Lb [8 Np So^4 - 4 Nt [Kc (So^2 - 1)]^2]
= La [4 Nt [Kc (So^2 - 1)]^2 - 8 Np]
or squaring both sides:
Lb^2 [8 Np So^4 - 4 Nt [Kc (So^2 - 1)]^2]^2
= La^2 [4 Nt [Kc (So^2 - 1)]^2 - 8 Np]^2<

Rearrangiing gives:
La^2 / Lb^2
= [8 Np So^4 - 4 Nt [Kc (So^2 - 1)]^2]^2
/ [4 Nt [Kc (So^2 - 1)]^2 - 8 Np]^2

However, from the geometric expressions:
La = {Nt^2 [Kc (So^2 - 1) / 4 So]^2 + (Np)^2 [So / 2]^2}^0.5
and
Lb = {Nt^2 [Kc (So^2 - 1)/ 4 So]^2 + (Np)^2 [(1 / 2 So)]^2}^0.5

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

Equating the two expressions for (La^2 / Lb^2} gives:
[8 Np So^4 - {4 Nt [Kc (So^2 - 1)]^2}]^2
/ [{4 Nt [Kc (So^2 - 1)]^2} - 8 Np]^2
= {[Nt^2][Kc (So^2 - 1)]^2 + (Np^2)(2 So^4)}
/ {[Nt^2][Kc (So^2 - 1)]^2 + (Np^2)(2)}

or
[- {4 Nt [Kc (So^2 - 1)]^2} + 8 Np So^4]^2
/ [{4 Nt [Kc (So^2 - 1)]^2} - 8 Np]^2
= {[Nt^2][Kc (So^2 - 1)]^2 + (Np^2)(2 So^4)}
/ {[Nt^2][Kc (So^2 - 1)]^2 + (Np^2)(2)}

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

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

Let X = {Np / [Nt Kc (So^2 - 1)]}
Then:
[- 1 + (Nt / Np)X^2 (2) So^4]^2
/ [{1 - (Nt / Np) X^2(2)]^2
= {1 + X^2 (2 So^4)}
/ {1 + X^2 (2)}

Hence:
{1 + X^2 (2)}[- 1 + (Nt / Np) X^2 (2) So^4]^2
= {1 + X^2 (2 So^4)}[1 - (Nt / Np) X^2 (2)]^2

Expand these terms: {1 + X^2 (2)}{1 - 2 (Nt / Np) X^2 (2) So^4 + [(Nt / Np) X^2 (2) So^4]^2}
= {1 + X^2 (2 So^4)}{1 - 2 (Nt / Np) X^2 (2) + [(Nt / Np) X^2 (2)]^2 }

Further expansion gives:
[1 - 2 (Nt / Np) X^2 (2) So^4 + (Nt / Np)^2 X^4 (4) So^8
+ 2 X^2 - 8 (Nt / Np) X^4 So^4 + 8 (Nt / Np)^2 X^6 So^8
= [1 - 2 (Nt / Np) X^2 (2) + (Nt / Np)^2 X^4 (4)] ]
+ X^2 (2 So^4) - 8 (Nt/ Np) X^4 So^4 + (Nt / Np)^2 X^6 (8) (So^4)

Collecting terms gives:
+ 8 (Nt / Np)^2 X^6 So^8
- 8 (Nt / Np)^2 X^6 So^4
+ 4 (Nt / Np)^2 X^4 So^8
- 4 (Nt / Np)^2 X^4
- 8 (Nt / Np) X^4 So^4
+ 8 (Nt/ Np) X^4 So^4
- 4 (Nt / Np) X^2 So^4
+ 4 (Nt / Np) X^2
+ 2 X^2 - X^2 (2 So^4)
+ 1 - 1
= 0

Simplify:
+ 8 (Nt / Np)^2 X^6 So^8 - 8 (Nt / Np)^2 X^6 (So^4)
- 4 (Nt / Np)^2 X^4 [1 - So^8]
- 4 (Nt /Np) X^2 (So^4 - 1)
+ 2 X^2 (1 - So^4) = 0 or
+ 4 (Nt / Np)^2 X^6 (So^8 - So^4)
+ 2 (Nt / Np)^2 X^4 [So^8 - 1]
- 2 (Nt /Np) X^2 (So^4 - 1)
- X^2 (So^4 - 1) = 0

or
4 (Nt / Np)^2 X^4 So^4 (So^4 - 1)
+ 2 (Nt / Np)^2 X^2 [So^8 - 1]
-2 (Nt /Np) (So^4 - 1)
- (So^4 - 1)
= 0

Divide through by (So^4 - 1) to get:
4 (Nt / Np)^2 X^4 So^4
+ 2 (Nt / Np)^2 X^2 [So^4 + 1]
-2 (Nt /Np)
- 1
= 0

or
+ 4 (Np / Nt)^2 {1 / [Kc (So^2 - 1)^4]} (So^4)
+ 2 {1 / [Kc (So^2 - 1)]^2} [So^4 + 1]
- 2 (Nt /Np)
- 1
= 0

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

or
4 (Np / Nt)^3 {So^4 / [Kc (So^2 - 1)]^4}
+ 2 (Np / Nt) {(So^4 + 1) / [Kc (So^2 - 1)]^2}
- (Np / Nt)
- 2
= 0

or
(Np / Nt)^3 {4 So^4 / [Kc (So^2 - 1)]^4}
+ (Np / Nt) {2 {(So^4 + 1) / [Kc (So^2 - 1)]^2} - 1}
= 2

Note that this equation has two entirely different solutions for (Np / Nt). First solution:
If Kc ~ 1.0 then:
(Np / Nt) ~ 2 / {2 {(So^4 + 1) / [Kc (So^2 - 1)]^2} - 1}
~ 2
which is inconsistent with spheromak existence.

The second solution is consistent with spheromaks.
 

SPHEROMAK SOLUTION:
If the spheromak existence condition:
2 {(So^4 + 1) / [Kc (So^2 - 1)]^2} = 1
is true, giving:
Kc^2 = 2 {(So^4 + 1) / [(So^2 - 1)]^2}
in which case Kc > (2)^0.5.
Then:
(Np / Nt)^3 = 2 / {4 So^4 / [Kc (So^2 - 1)]^4}
= [Kc (So^2 - 1)]^4 / 2 So^4
= 4 (So^4 + 1)^2 / 2 So^4
= 2 [(So^4 + 1) / So^2] [(So^4 + 1) / So^2]
= 2 [So^2 + (1 / So^2)][So^2 + (1 / So^2)]
= 2 [So^2 + (1 / So^2)]^2

This equation says that the spheromak solution:
can exist if:
(Np / Nt)^3 = 2 [So^2 + (1 / So^2)]^2
can exist if:
Kc^2 = 2 {(So^4 + 1) / [(So^2 - 1)]^2}

This equation in effect prescribes the necessary spheromak shape for the existence of spheromak solutions.

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

FINDING So^4:
This equationis quadratic in So^4.
Let:
Nr = (Np / Nt)
Then: Nr^3 = 2 (So^4 + 1)^2 / So^4
or
So^4 Nr^3 = 2 (So^8 + 2 So^4 + 1)
or
2 So^8 + (4 - Nr^3) So^4 + 2 = 0
giving:
So^4 = {-(4 - Nr^3) + / - [(4 - Nr^3)^2 - 4(2)(2)]^0.5} / 4
= {(Nr^3 - 4) + / - [(Nr^3 - 4)^2 - 16]^0.5} / 4

For:
Nr^3 < 8
or for:
Nr < 2
there is no real solution.

At:
Nr^3 = 8
or at
Nr = 2
So^4 = 1

At:
So^4 = 16 Nr^3 = 36.125:

Spheromaks typically operate at about:
So^2 ~ 4.2
corresponding to:
Nr^3 = 37
or
Nr = 3.33
 

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

FINDING So^2:
Let:
Np = (Npr)^2
and let:
Nt = (Ntr)^2

Recall that:
(Np / Nt)^3 = 2 [So^2 + (1 / So^2)]^2

Hence:
(Npr^6 / Ntr^6) = 2 [So^2 + (1 / So^2)]^2
or
(Npr^3 / Ntr^3) = 2^0.5 [So^2 + (1 / So^2)]
or
Npr^3 = 2^0.5 Ntr^3 [So^2 + (1 / So^2)]
or
So^2 Npr^3 = 2^0.5 Ntr^3 So^4 + 2^0.5 Ntr^3
or
2^0.5 Ntr^3 So^4 - Npr^3 So^2 + 2^0.5 Ntr^3 = 0

This quadratic equation has solutions:
So^2
= {Npr^3 + / - [Npr^6 - 4 (2^0.5 Ntr^3)(2^0.5 Ntr^3)]^0.5} / [2 (2^0.5 Ntr^3)]
 
= {Npr^3 + / - [Npr^6 - 8 Ntr^6]^0.5} / [2 (2^0.5 Ntr^3)]
= {Npr^3 + / - Npr^3 [1 - 8 (Ntr / Npr)^6]^0.5} / [2 (2^0.5 Ntr^3)]
= [Npr^3 / Ntr^3]{1 + / - [1 - 8 (Ntr / Npr)^6]^0.5} / [2 2^0.5]
= [Np / Nt]^(3 / 2) {1 + / - [1 - 8 (Nt / Np)^3]^0.5} / [2 2^0.5]
= [Np / 2 Nt]^(3 / 2) {1 + / - [1 - (2 Nt / Np)^3]^0.5}
= [Np / 2 Nt]^(3 / 2) + / - {[Np / 2 Nt]^3 - 1}^0.5

For real physical solutions:
So^2 = {[Np / 2 Nt]^3}^0.5 + {[Np / 2 Nt]^3 - 1}^0.5

Note that:
(Np / 2 Nt) > 1

Typically:
4 < [Np / 2 Nt]^3 < 5
which gives:
(2 + 3^0.5) < So^2 < (5^0.5 + 2)
 

At So^2 = 4:
(Np / Nt)^3 = 2 [So^2 + (1 / So^2)]^2 = 2 [4 + (1 / 4)]^2
= 2 [17 / 4]^2
= 289 / 8

Thus:
(Np / Nt) = (1 / 2) 289^(1 / 3) = (1 / 2)(2.57128 1591) = 1.285640795

Recall that:
P = 2 Np + Nt
or
P / 2 Nt = (Np / Nt) + (1 / 2)
or
(P / 4 Nt ) - (1 / 4) = (Np / 2 Nt)
or
(Np / 2 Nt) = (1 / 4)[(P / Nt) - 1]
= (1 / 4)[(P - Nt) / Nt]
or
(Np / 2 Nt)^3 = (1 / 64) [P - Nt]^3 / Nt^3

Thus:
So^2 = {[Np / 2 Nt]^3}^0.5 + {[Np / 2 Nt]^3 - 1}^0.5
= {(1 / 64) [P - Nt]^3 / Nt^3}^0.5 + {[(1 / 64) [P - Nt]^3 / Nt^3] - 1}^0.5
= [1 / 8] {{[P - Nt]^3 / Nt^3}^0.5 + {[[P - Nt]^3 / Nt^3] - 64}^0.5}
 

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

THE RANGE OF P:
Recall that due to Np and Nt having no common factors:
P = 2 Np + Nt
or
(P / Nt) = 2 (Np / Nt) + 1
or
[(P / Nt) - 1] / 2 = Nr

Thus:
Nr^3 = [(P / Nt) - 1]^3 / 8

Recall that for real spheromak solutions to exist:
Nr^3 > 8. Hence:
[(P / Nt) - 1}^3 > 64
or
[(P / Nt) - 1} > 4
or
(P / Nt) > 5

Typically:
Nr^3 ~ 16
so that:
[(P / Nt) - 1]^3 ~ 128
or
[(P / Nt) - 1] ~ 5.0396
or
P / Nt ~ 6.0396

Hence to solve practical problems start by chosing P values in the range:
5 Nt < P < 8 Nt

Beware that experimental evidence suggests that Nt ~ 1000 so P can easily be as large as 8000.
 

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FINDING Kc:
Recall that:
(Np / Nt)^3 {4 So^4 / [Kc (So^2 - 1)]^4}
+ (Np / Nt) {2 {(So^4 + 1) / [Kc (So^2 - 1)]^2} - 1}
- 2
= 0

A condition for spheromak existence is:
{2 {(So^4 + 1) / [Kc (So^2 - 1)]^2} - 1} = 0
or
Kc^2 = 2 {(So^4 + 1) / [(So^2 - 1)]^2}

Typical Kc values:
At So^2 = 5, Kc^2 = 52 / 16 = 3.25
At So^2 = 4, Kc^2 = 34 / 9 = 3.7777
At So^2 = 3, Kc^2 = 20 / 4 = 5.000
 

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FINDING So^2 FROM Kc:
Kc^2 = 2 {(So^4 + 1) / [(So^2 - 1)]^2}

or
[Kc (So^2 - 1)]^2 = 2 (So^4 + 1)
or
Kc^2 [So^4 - 2 So^2 + 1] = 2 (So^4 + 1)
or
(Kc^2 - 2) So^4 - 2 Kc^2 So^2 + (Kc^2 - 2) = 0

This equation has the quadratic solution:
So^2 = {2 Kc^2 + / - [4 Kc^4 - 4(Kc^2 - 2)^2]^0.5} / 2(Kc^2 - 2)
= {2 Kc^2 + / - [4 Kc^4 - 4(Kc^4 - 4 Kc^2 + 4)]^0.5} / 2(Kc^2 - 2)
= {2 Kc^2 + / - [16 Kc^2 - 16)]^0.5} / 2 (Kc^2 - 2)
= {2 Kc^2 + / - 2 [4 Kc^2 - 4)]^0.5} / 2 (Kc^2 - 2)
= {Kc^2 + / - [4 Kc^2 - 4)]^0.5} / (Kc^2 - 2)

For real solutions:
So^2 = {Kc^2 + [4 Kc^2 - 4)]^0.5} / (Kc^2 - 2)

Note that:
Kc^2 > 2

Note that the calculated value of So^2 goes to infinity as Kc^2 approaches 2 and goes to unity at large values of Kc.

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

TYPICAL [Np / Nt] VALUE:
When the spheromak existence condition is satisfied:
(Np / Nt)^3 {4 So^4 / [Kc (So^2 - 1)]^4} = 2
or
(Np / Nt)^3 {4 So^4 / 4 (So^4 + 1)^2} = 2
or
(Np / Nt)^3 = 2 (So^4 + 1)^2 / So^4
= [2 (So^4 + 1)^2 / So^4]
= [2 (So^2 + (1 / So^2))^2]

At So^2 = 4:
(Np / Nt)^3 = [2 (17 / 4)^2]
= 289 / 8
giving:
2 (Np / Nt) = (289)^(1 / 3)
= 6.611489018
or
(Np / Nt) = 3.305744509

Each prime number P yields a family of (Np / Nt) integer pairs. The spheromak equations chose a specific Nt value from this family by changing So. The value of So is in effect chosen by Kc, which is proportional to the ellipse perimeter length. Hence the elliptical cross section of the spheromak enables the stable spheromak's existence.
 

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FINDING PARAMETER [A / B]:
Recall that from the inner wall boundary condition the equation for A is:
[Nt / Pi] = (B / A)^2 (Lh A / 2 Pi Ro)[So / (So^2 + 1)]
 
or
(Lh A / 2 Pi Ro) = [Nt / Pi] [A / B]^2 [(So^2 + 1) / So]

Spheromak geometry gives:
[Lh A / 2 Pi Ro] = + Nt {[Kc (So^2 - 1) / 4 So]^2 + (Np / Nt)^2 [So / 2]^2}^0.5
+ Nt {[Kc (So^2 - 1)/ 4 So]^2 + (Np / Nt)^2 [1 / (2 So)]^2}^0.5

Equating these two expressions for
[Lh A / 2 Pi Ro]
gives:
[Nt / Pi][A / B]^2 [(So^2 + 1) / So]
= + Nt {[Kc (So^2 - 1) / 4 So]^2 + (Np / Nt)^2 [So / 2]^2}^0.5
+ Nt {[Kc (So^2 - 1)/ 4 So]^2 + (Np / Nt)^2 [1 / (2 So)]^2}^0.5

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

Recall that for spheromaks:
Kc^2 = 2 {(So^4 + 1) / [(So^2 - 1)]^2}
or
Kc^2 (So^2 - 1)^2 = 2 (So^4 + 1)

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

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

For typical values of So^2 = 4 and (Np / Nt]^2 = 10:
[A / B]^2
= [Pi / 5] {[34 / 16] + 10 (4)}^0.5
+ [Pi / 5] {[34 / 16] + 10 (1 / 4)}^0.5
 
= [Pi / 5] {42.125}^0.5
+ [Pi / 5] {4.625}^0.5
 
= [Pi / 5] [6.4904 + 2.1506] = 5.4293

Hence a typical value for [A / B] is:
[A / B] = 5.4293^0.5
= 2.33

Thus [A / B] is an easily computed function of [Np / Nt] or So^2.

The equation for [[A / B] is essential for solution of spheromak problems. Note that Np and Nt are integers.

Note that the quantity:
[A / B]^2 [(So^2 + 1) / So]
can in principle be confirmed via the magnetic integral.
 

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

QUANTIFICATION OF So^4:
Recall that for spheromaks:
(Np / Nt)^3 = {2 [So^2 + (1 / So^2)]^2}
= 2 [So^4 + 2 + 1 / So^4]
or
2 So^4 + [4 - (Np / Nt)^3] + 2 / So^4 = 0
or
2 So^8 + [4 - (Np / Nt)^3]So^4 + 2 = 0
giving:
So^4 = {- [4 - (Np / Nt)^3] + / - [[4 - (Np / Nt)^3]^2 - 4 (2)(2)]^0.5} / 4
= {[(Np / Nt)^3 - 4] + [[(Np / Nt)^3 - 4]^2 - 16]^0.5} / 4
which has real solutions for:
(Np / Nt)^3 > 8
or for
(Np / Nt)^2 > 4
or for
(Np / Nt) > 2

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

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

EVALUATION OF [Lh A / 2 Pi Ro]:
Recall from the spheromak geometry that:
[Lh A / 2 Pi Ro]
= + Nt {[Kc (So^2 - 1) / 4 So]^2 + (Np / Nt)^2 [So / 2]^2}^0.5
+ Nt {[Kc (So^2 - 1)/ 4 So]^2 + (Np / Nt)^2 [1 / (2 So)]^2}^0.5

=La + Lb
where:
La = Nt {[Kc (So^2 - 1) / 4 So]^2 + (Np / Nt)^2 [So / 2]^2}^0.5
and
Lb = Nt {[Kc (So^2 - 1)/ 4 So]^2 + (Np / Nt)^2 [1 / (2 So)]^2}^0.5

This equation was the result of spheromak geometric analysis.
 

or
[Lh A / 2 Pi Ro Nt] = + {[Kc^2 (So^2 - 1)^2 / 16 So^2] + (Np / Nt)^2 [So^2 / 4]}^0.5
+ {[Kc^2 (So^2 - 1)^2 / 16 So^2] + (Np / Nt)^2 [1 / (4 So^2)]}^0.5

Recall that the spheromak existence condition gives:
{2 {(So^4 + 1) / [Kc (So^2 - 1)]^2} - 1} = 0
or
Kc^2 (So^2 - 1)^2 = 2 (So^4 + 1)

Hence:
[Lh A / 2 Pi Ro Nt]
= + {[(So^4 + 1) / 8 So^2] + (Np / Nt)^2 [So^2 / 4]}^0.5
+ {[(So^4 + 1) / 8 So^2] + (Np / Nt)^2 [1 / (4 So^2)]}^0.5
or
[Lh A / 2 Pi Ro Nt] = + {[1 / 8][(So^2 + (1 / So^2)] + (Np / Nt)^2 [So^2 / 4]}^0.5
+ {[1 / 8][(So^2 + (1 / So^2)] + (Np / Nt)^2 [1 / (4 So^2)]}^0.5

 

SPHEROMAK SOLUTION:

Recall that for a spheromak:
So^4 = {[(Np /Nt)^3 - 4] + [[(Np / Nt)^3 - 4]^2 - 16]^0.5} / 4
which has real solutions for:
(Np / Nt)^3 > 8
for
(Np / Nt) > 2

Thus for any (Np / Nt) > 2 value there is a corresponding computed real number value for the spheromak parameters [A / B], Kc, and So^2. Note that we do not use the formula:
Kc^2 = 2 [(So^4 + 1) / (So^2 - 1)^2]
because the inverse of this formula is used to compute a new So^2 value from the Kc^2 value obtained from A and B from the using ellipse geometry. The spheromak parameters are considered correct when they converge after several program iterations.

However, to find the spheromak related quantities:
[Lh A /2 Pi Ro] and [1 / Alpha]
it is necessary to first determine Nt.
 

EVALUATION OF Np AND Nt:
Recall that:
P = 2 Np + Nt
and the ratio:
Nr = (Np / Nt)
is accurately known as a real number from analytical calculations. Hence:
P = Nt [2 (Np / Nt) + 1]
= Nt [2 Nr + 1]
is an integer.

The practical way to find Nt is to first find a precise real number solution for the ratio:
Nr = (Np / Nt)
and then increment through the range of possible Nt integer values looking first for the quantity (Nt Nr) to be an integer and then checking against a table of prime numbers to see if:
P = Nt[2 Nr + 1] is a prime number.
 

ELLIPSE MATHEMATICS:
The general case of an ellipse described by:
(Y^2 / a^2) + (Z^2 / b^2) = 1
where:
a = ellipse minor radius
b = ellipse major radius
mathematicians have shown that the ellipse perimeter length Lt is given by the power series:
Lt = Pi (a + b) Kh
= Pi (a + b) [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]

where:
h = (a - b)^2 / (a + b)^2

For a spheromak with an elliptical cross section:
Zs = (A / B)[(Rs - R)(R- Rc)]^0.5
Zf = (A / B)[(Rs - (Rs + Rc) / 2)((Rs + Rc) / 2 - Rc)]^0.5
= (A / B)[(Rs - Rc) / 2)((Rs - Rc) / 2)]^0.5
= (A / B)[(Rs - Rc) / 2]

Hence:
b = (A / B) a
or
b B = a A

and
(b / a) = (A / B)
where if b > a then (A / B) > 1.0

Recall that:
a = (Rs - Rc) / 2

Hence:
(a + b) = [(Rs - Rc) / 2] [1 + (A / B)]
and
(b - a) = [(Rs - Rc) / 2] [(A / B) - 1]
and
h = (a - b)^2 / (a + b)^2
= [((A / B) - 1)]^2 / [(A / B) + 1]^2
= [A - B]^2 / [A + B]^2

Recall that:
Kh = [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]

Note that Kh is always greater than unity.
Then:
Lt = Pi (a + b) Kh
= Pi [(Rs - Rc) / 2] [1 + (A / B)] Kh
= 2 Pi [(Rs - Rc) / 2] [1 + (A / B)] [Kh / 2]

Define the lumped constant Kc by:
Kc = [1 + (A / B)] [Kh / 2]
= {Perimeter length of ellipse with a = (Rs - Rc) / 2}
/{Perimeter length of circle with radius a = (Rs - Rc) / 2}

Thus:
Lt = Pi (Rs - Rc) Kc

Note that if b > a then A > 1 and since Kh > 1 thus Kc > 1.
 

CALCULATION OF A AND B:
Recall that from the far field boundary condition:
2 A^2 + B^2 = 3 or
2 A^2 + A^2 (B / A)^2 = 3
or
A^2 [2 +(B / A)^2] = 3
or
A = {3 / [2 +(B / A)^2]}^0.5
and
B = A (B / A)
= (B / A){3 / [2 +(B / A)^2]}^0.5
 

CALCULATION SEQUENCE:
1) Start by assuming an initial value of:
So^2 = 4

2) Use the formula:
(Np / Nt)^3 = {2 [So^2 + (1 / So^2)]^2}
to calculate the corresponding real number value of (Np / Nt)^3
eg (Np / Nt)^3 = 36.125

3) Use the formula:
(Np / Nt) = [(Np / Nt)^3]^(1 / 3)
to calculate the corresponding real number value of (Np / Nt).
eg. (Np / Nt) = 3.305744509

4) Use the formula:
(Np / Nt)^2 = [(Np / Nt)]^2
to calculate the real number value of (Np / Nt)^2
eg (Np / Nt)^2 = 10.000

P>5) Calculate the corresponding real number value of [A / B] using the formula:
[A / B]^2
= +[Pi / (So^2 + 1)] {[2 (So^4 + 1) / (4)^2] + (Np / Nt)^2 [So^2 / 2]^2}^0.5
+ [Pi / (So^2 + 1)] {[2 (So^4 + 1) / (4)^2] + (Np / Nt)^2 [1 / (2)]^2)^0.5

eg [A / B] = 2.33

6) Calculate the corresponding real number value of A using the formula:
A = {3 / [2 +(B / A)^2]}^0.5
eg A = ______

6) Calculate the corresponding real number value of B using the formula:
B = (B / A) {3 / [2 +(B / A)^2]}^0.5
eg B = ______

7) Calculate ellipse parameter h using the equation:
h = [A - B]^2 / [A + B]^2

8) Calculate ellipse parameter Kh using the equation:
Kh = [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]

9) Calculate ellipse parameter Kc using the equation:
Kc = [1 + (A / B)] [Kh / 2]

10) Calculate So^2 using the formula:
So^2 = {Kc^2 + [4 Kc^2 - 4)]^0.5} / (Kc^2 - 2)

11) Note that this is a feedback equation that potentially may have too much gain. Remember that the old Kc value was given by: Kc^2 = 2 (So^4 - 1) / (So^2 - 1)^2

To obtain certain convergence it may be necessary to reduce the amount of feedback per program cycle.

12) Loop back to step #1.

13) Run the program until there is very good real number convergence for all the spheromak parameters.

14) After all the spheromak parameters have converged as real numbers find the corresponding P and Nt integer values using the formula:
P = Nt [2 Nr + 1]
and a table of prime numbers.

15)Check that P, Np and Nt are all prime or failint that check that Np and Nt share no common factors.

16) Record all the calculated parameters.

17) The correct parameter values are used to accurately calculate Lh A / 2 Pi Ro
and [1 / Alpha].

18) Compare the theoretical value of [1 / Alpha] to the experimentally measured value of[1 / Alpha].

Thus we have developed a methodology for precise calculation of the Fine Structure Constant from first principles. This same methodology can potentially be applied to other quantum mechanical problems. It is necessary to use a computer based successive approximation calculation because of the mathematical 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 So value as calculated herein.
 

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

FINE STRUCTURE CONSTANT ISSUES:
Note that (1 / Alpha) is a function of the spheromak shape parameters So^2 and [A / B]. 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.
 

NUMERICAL SOLUTION:
A preliminary BASIC program solution indicated 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

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.

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 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 = (Np / Nt) that correspond to a precise balance between the electric and magnetic forces along the entire length of the current path. 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. States are stable when both Np and Nt are prime numbers.

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 is affected by the spheromak environment. Consider a pure silicon crystal. At some Nr 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 which is a rational number and Pi which is a real number. This relationship is only valid for integer values of Np and Nt and corresponding discrete values of So.
 

FIRST SOLUTION DETAIL:
Recall that:
(Np / Nt)^3 {4 So^4 / [Kc (So^2 - 1)]^4}
+ (Np / Nt) {2 {(So^4 + 1) / [Kc (So^2 - 1)]^2} - 1}
- 2
= 0

Now let:
X = (Np / Nt).
Then this equation takes the form:
A X^3 + B X + C = 0
where:
A = 4 {So^4 / [Kc (So^2 - 1)]}^4
and
B = + 2 {(So^4 + 1) / [Kc (So^2 - 1)]}^2- 1
and
C = - 2

This third order equation can be expressed in the form:
A (X - a) (X - b) (X - c) = 0

where a, b , c are potential real solutions.

To find the values of a, b, c expand this equation as follows:
A (X - a) (X^2 - (b + c) X + bc) = 0
A [X^3 - (b + c) X^2 + b c X - a X^2 + a(b + c) X - abc] = 0
A [X^3 - (a + b + c) X^2 + (bc + ab + ac) X - abc] = 0

Comparing terms gives:

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

(a + b + c) = 0

B = + 2 {So^4 / [Kc (So^2 - 1)]}^2
+ 2 {1 / [Kc (So^2 - 1)]}^2
- 1
= A {(bc + ab + ac)

C = - 2
= A [-a b c]

Hence:
C / A = [- a b c] = - 2
or
a b c = 2

Recall that:
a + b + c = 0

Try potential solution:
a = -1, c = -1, b = 2

A [X^3 - (a + b + c) X^2 + (bc + ab + ac) X - abc] = 0
or
A [X^3 + (-2 -2 +1) X - 2] = 0
or
A [X^3 - 3 X - 2] = 0

Solutions to this equation are:
X = 2
or
Np / Nt = 2

and X = -1
or
Np = - Nt

The equation:
Np^3 - 3 Np Nt^2 - 2 Nt^3 = 0
or
(Np / Nt)^3 - 3 (Np / Nt) - 2 = 0
has solutions:
(Np / Nt) = 2
and
(Np / Nt) = -1

Neither of these two solutions leads to a spheromak.
 

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

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

DETERMINATION OF Nt:
Up to this point we have the quantity {(Lh A) / (2 Pi Ro Nt)} but we do not yet have a means of evaluating Nt separate from {(Lh A) / (2 PiRo)}. However, we can evaluate:
Nr = (Np / Nt)
and we can find Nt using formula:
Np = Nt Nr,
where Np is an integer. We can potentially obtain Np from the magnetic field strength at the center of the spheromak.
 

RELATING [So / (So^2 + 1)] TO THE MAGNETIC INTEGRAL:
On the web page titled: ELECTROMAGNETIC SPHEROMAK it is shown that:
[Np / Nt] = A (So^2 + 1) / {So [INTEGRAL]}
where [INTEGRAL] is a non-invertable function of So and A
where:
[INTEGRAL] = Integral from X = Xc to X = Xs of:
{(X^2) / {X^2 + (A / B)^2 [Xs - X) (X - Xc)]}^1.5}
{4 / [Kh (1 + (A / B))(Xs - Xc)]}
dX {[(Xs - X) (X - Xc)] + (A / 2 B)^2 [Xs + Xc - 2 X]^2}^0.5
/ [(Xs - X)(X - Xc )]^0.5

and:
X = R / Ro
Xs = Rs / Ro = (So / A)
Xc = Rc / Ro = (1 / (A So))

Thus from the known value of [Np / Nt] we can calculate the value of:
[A (So^2 + 1) / So]
 

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EXPERIMENTAL RESULTS:
The published CODATA experimentally measured value of (1 / Alpha) = 137.03599915 corresponding to:
h = 6.636070150 X 10^-34 J-s.

In this context (1 / Alpha) 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 potentially provide a means of calculating the theoretical value Alpha.

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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 Planck Constant 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.
 

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REVIEW AND CORRECT

Alpha QUANTIFICATION STRATEGY:
The formula for {1 / Alpha] is:
[1 / Alpha]
= (Lh / Ro)(1 / 4 A^2 B) {[So] [So^2 - So + 1] / [(So^2 + 1)^2]}

Note that [1 / Alpha] has a relative peak with respect to [So / (So^2 + 1)]. {[So] [So^2 - So + 1] / [(So^2 + 1)^2]} = [So / (So^2 + 1)] {1 - [So / (So^2 + 1)]} which peaks at:
[So / (So^2 + 1)] = 1 / 2
or
2 So = So^2 + 1
or
So^2 - 2 So + 1 = 0
or
So = {2 +/- [4 - 4(1)]^0.5} / 2 = 1
This peak does not set the spheromak operating point.

In order to calculate [1 / Alpha] we must quantify the parameters:
(Lh / Ro), (1 / 4 A^2 B) and So.

Alpha QUANTIFICATION CONSTRAINT #1:
On the web page titled: ELECTROMAGNETIC SPHEROMAK it is shown that:
Nt = [Lh / Ro][1 / (2 A)][So / (So^2 + 1)]

Recall from constraint #2 that:
Np / Nt = [Kc^2 (So^2 - 1)^2 / (So^2 + 1)^2] [2]

Alpha QUANTIFICATION CONSTRAINT #5:
The requirement that Np and Nt have no common factors other than one gives rise to the constraint that:
P = 2 Np + Nt
where:
P = a prime number.
 

Alpha QUANTIFICATION CONSTRAINT #7:
Recall that: (Np / Nt) = [2 Kc^2 (So^2 - 1)^2 /(So^2 + 1)^2]

Recall that:
(Np / Nt) = [(P - Nt) / 2 Nt]

Thus:
[(P - Nt) / 2 Nt] = [2 Kc^2 (So^2 - 1)^2 /(So^2 + 1)^2]
or
(P / 2 Nt) - (1 / 2) = [2 Kc^2 (So^2 - 1)^2 /(So^2 + 1)^2]
or
(P / Nt) = 1 + [4 Kc^2 (So^2 - 1)^2 /(So^2 + 1)^2]
Use this expression to explore for consistent P, Nt values.

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We then know all the parameters necessary to calculate [1 / Alpha] using the equation:
[1 / Alpha] = (Lh / Ro)(1 / 4 A^2 B) {[So] [So^2 - So + 1] / [(So^2 + 1)^2]}
where:
(Lh / 2 Pi Ro)
= [1 / 2 A] {[Np] [P / 2]}^0.5 {(So^2 + 1) / So}

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

THE ELECTRON:
To grasp the basic issues involved in the Planck Constant one need look no further than an electron. It has a rest mass Me = 9.109 X 10^-31 kg and a charge Q = 1.602 X 10^-19 coul. Assuming that the rest mass Ee is due to energy contained in a spherical electric field with energy density:
U = Uo {Ro^2 / [Ro^2 + R^2]}^2
then the total energy is given by:
Ee = Integral from R = Ro to R = infinity of:
U 4 Pi R^2 dR
= Integral from R = 0 to R = infinity of:
Uo {Ro^2 / [Ro^2 + R^2]^2} 4 Pi R^2 dR
= Integral from R = 0 to R = infinity of:
Uo Ro^4 4 Pi [R^2 dR / (Ro^2 + R^2)^2]
= Uo Ro^4 4 Pi {-2 Ro^2 R / [(4 Ro^2)(R^2 + Ro^2)]}|R = infinity
+ Uo Ro^4 4 Pi (2 Ro^2 / 4 Ro^2)(2 / 2 Ro) arc tan[2 R / 2 Ro]|R = infinity
- Uo Ro^4 4 Pi {-2 Ro^2 R / [(4 Ro^2)(R^2 + Ro^2)]}|R = 0
- Uo Ro^4 4 Pi (2 Ro^2 / 4 Ro^2)(2 / 2 Ro) arc tan[2 R / 2 Ro]|R = 0
 
= Uo Ro^4 4 Pi (2 Ro^2 / 4 Ro^2)(2 / 2 Ro) (Pi / 2)
 
= Uo Ro^3 Pi^2

For R >> Ro the energy density is:
U = Uo [Ro^2 / (Ro^2 + R^2)]^2
~ Uo Ro^4 / R^4

For a spherical electric field t large distances:
U = (Epsiono / 2) [Q / (4 Pi Epsilono R^2]^2
= [Q^2 / (32 Pi^2 Epsilono)] [ 1 / R^4]

Equating the two energy densities at large distances gives:
Uo Ro^4 = [Q^2 / (32 Pi^2 Epsilono)]

Hence the total electron energy Ee is:
Ee = Uo Ro^3 Pi^2
= [Uo Ro^4] [Pi^2 / Ro]
= [Q^2 / (32 Pi^2 Epsilono)][Pi^2 / Ro]
= [Q^2 / (32 Epsilono Ro)]
= [Q^2 Muo C^2 / 32 Ro]

If one assumes that the electron charge Q is concentrated in a ring of radius Ro that rotates at the speed of light C about the electron center of mass one can calculate a frequency:
Fe = C / (2 Pi Ro)
or
Ee / Fe = [Q^2 Muo C^2 / 32 Ro] / [C / 2 Pi Ro] = Q^2 Muo C Pi / 16
= (1.60217662 X 10^-19 coul)^2 X 1.25663706 10-6 m kg s-2 A-2 X 299 792 458 m / s X Pi / 16
= 18.988088 X 10^-37 kg m^2 / s
= 18.988088 X 10^-37 J-s

However, that calculated value for:
(Ee / Fe) = C Q^2 Muo Pi / 16
is much smaller than the experimentally measured value of the Planck constant:
h = (Ee / Fh).
= 6.62607004 10-34 m2 kg / s

In order for the theoretical value of (Ee / Fe) to match the experimental value of (Ee / Fh) the current path around the electron center of mass, instead of being a simple ring, must be a complex multi-turn closed path of length Lh where:
[Lh / 2 Pi Ro] = (6.62607004 10-34 m2 kg / s) / (18.988088 X 10^-37 J-s)
= 348.9593075

It is also necesary to assume that electromagnetic radiation interacts with an electron's field rather than with individual current path turns. The wavelength of the interacting radiation must be about 349X the apparent physical circumference of the electron.
 

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STRUCTURE OF [Lh / (2 Pi Ro)]:
Recall that:
However, from the web page titled: ELECTROMAGNETIC SPHEROMAK the spheromak boundary condition:
[Nt / Np = [Bto / Bpo][INTEGRAL]
= [Uto / Uo]^0.5 [INTEGRAL]

From the web page titled: THEORETICAL SPHEROMAK
[Uto / Uo] = [1 / A^2] [So^2 /(So^2 + 1)^2]
Hence:
[Nt / Np] = [1 / A] [So /(So^2 + 1)][INTEGRAL]
is a function of only So. Hence for a particular value of So the ratio (Nt / Np) is constant which sets a stable value for:
[Lh / (2 Pi Ro)]
 

CENTRAL BOUNDARY CONDITION:
As shown on the web page titled: ELECTROMAGNETIC SPHEROMAK the value of [Nt / Np] is further confined by the central boundary condition:
[Bpo Nt / Bto Np] = [INTEGRAL]
or
[Nt / Np] = [Bto / Bpo][INTEGRAL]
= [Uto / Uo]^0.5 [INTEGRAL]
= [1 / A] [So /(So^2 + 1)][INTEGRAL]
where [INTEGRAL] is just a function of So.
 

This equation indicates that at a stable value of (1 / Alpha):
Np, Nt, So and A are all constant. Note that (1 / Alpha) has a characteristic value.
 

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FUNDAMENTAL EQUATIONS OF QUANTUM MECHANICS:
Charged atomic particles with rest mass contain spheromaks. A spheromak is an electric current which follows a closed spiral path that traces out the shape of the wall of a toroid with an elliptical cross section. The closed current path has both has both toroidal and poloidal circulation components. The toroidal surface is referred to as the spheromak wall.

The current circulates at the speed of light. At the spheromak geometry the total field energy density just inside the spheromak wall equals the total field energy density just outside the spheromak wall. Hence the electric and magnetic forces are in balance everywhere on the spheromak wall making the spheromak physically stable. This requirement for field energy density balance leads to the spheromak boundary condition equation:
hence non-existence of an atomic particle's rest mass. When two particles interact with each other their extended electric and magnetic fields overlap causing small changes in parameter A for each particle, which leads to small integer changes in Np and Nt (quantum jumps). However, if there is a large change in parameter A, such as in a particle-anti-particle interaction, then parameter A can take a value which causes a spheromak collapse, in which case the particle rest mass energy becomes a propagating photon.

In atoms and in crystals there are multiple interacting particles each of which experience minor changes in parameter A. These changes lead to changes in the solutions for (Np / Nt) and So which in turn change the available energy states. Everything is governed by the aforementioned quadratic equations. If an external magnetic field is applied the quadratic generates generate two solutions close to the original stable solution. That phenomena is known as magnetic resonance.

The existence, mass and other properties of atomic particles is governed in part by the limited set of prime numbers P that simultaneously satisfy all of the aforementioned equations. Since P, Np and Nt must be integers and Np and Nt cannot share common factors the number such prime numbers and hence the number of real atomic particles is distinctly limited.

One way of investigating this entire matter is to identify values of parameter A at which spheromaks collapse and hence particles cannot exist. For a particle to exist in an atom or a crystal it must have some elbow room around its nominal parameter A value in isolation in a vacuum. Thus by identifying parameter A values where particles cannot exist we can identify ranges of the parameter A value where stable particles can exist. That is the direction of my current work.

This thinking suggests that a cause of particle instability is an environment which causes an unfavorable parameter A value. However, the opposite can also be true. For example, the life time of a neutron in a stable atomic nucleus is much longer than the lifetime of a free neutron.

At first glance an A =1.000000000 seems like an obvious solution to the above equations. However, detailed examination shows that there is a spheromak collapse at:BR> A = 1.002989071
Hence a particle which relies on A = 1.00000000 for existence is inherently unstable because its spheromak will collapse if an external field causes its parameter A to increase by only 0.3%. Certain larger parameter A values result in much more stable spheromaks.

One of the practical aspects of this formulation of quantum mechanics is that solving even simple problems requires testing numerous prime numbers in nested quadratic equations. While such tests can readily be done using a computer it is not easy to convey spheromak stability concepts to persons who lack a good understanding of mathematics.

I expect that when Np, Nt and A are replaced by their nominal values and then small deviations of Np, Nt and A from these nominal values are examined what will emerge is the well known Schrodinger representation of quantum mechanics. ie Conventional quantum mechanics is a linear approximation of a spheromak solution. Conventional quantum mechanics relies on common real particles having parameter A values that almost always result in stable spheromaks.
 

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Note that Nr is the ratio of two integers which have no common factors. A change in the physical environment, such as application of a strong external magnetic field, can change the parameter A which in turn might change the integers. On this web page we are primarily concerned about spheromaks in free space. Spheromaks in an atomic, molecular or crystal environment present additional complications.
 

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

PREVENTION OF SPHEROMAK COLLAPSE:
If Np and Nt have a common factor other than unity a spheromak can collapse. To prevent this situation occuring spheromaks operate on a line of constant P.
P = 2 Np + Nt
where P is a prime number. Along this line prime number theory indicates that Np and Nt have no common factors. Additional spheromak stability against external disturbances is realized by makinb both Np and Nt prime numbers.

Experimental data both in the form of measurements of (1 / Alpha) and in the form of plasma spheromak photographs indicate that in reality [A / B]is significantly greater than unity.
 

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

PREFERRED ENERGY STATE:
The value of:
So^2 = 4.115
is almost coincident with the observed geometry of plasma spheromaks. Hence experimental plasma data strongly points to M = (1 / 2). M = (1 / 2) results in significantly larger So^2 values and hence lower spheromak energies than M = 2. Hence in real life M = (1 / 2) is much more energy probable than M = 2. However, M = 2 results in a higher density of states, which in certain circumstances may outweigh simple energy issues.
 

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CONSTRAINTS ON THE RANGES OF Np and Nt:
For M = 2:
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)]}
 

For M = (1 / 2):
Nr = (P - 2 Nt) / Nt and
Nr^2 + (B Kc)^2 = 4 A^4 / Pi^2

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

This inequality has the feature that if the Nt value is approximately known it limits the number of possible P values that need to be investigated to determine the value of P.
 

PRIME NUMBER P QUOTIENT
1229 552.422894
1223 549.7259555
1217 547.0290171**
1213 545.2310581
1201 539.8371812
1193 536.2412632
1187 533.5443248
1181 530.8473863
1171 526.3524889
1163 522.7565709
1153 518.2616735
1151 517.362694
1129 507.4739197
1123 504.7769812
1117 502.0800428*
1109 498.4841248
1103 495.7871864
1097 493.0902479*
1093 491.2922889
1091 490.3933095
1087 488.5953505
1069 480.5045351
1063 477.8075967
1061 476.9086172
1051 472.4137197
1049 471.5147403
1039 467.0198428***
1033 464.3229044
1031 463.4239249
1021 458.9290275
1019 458.030048*
1013 455.3331095
1009 453.5351505
997 448.1412736
991 445.4443352
983 441.8484172
977 439.1514788
971 436.4545403
967 434.6565813
953 428.3637249
947 425.6667865
941 422.969848
937 421.1718891

Best match is at P = 1039, Nt = 467, Np = 105

Nr = (105 / 467) = 0.2248394004

Nr^2 = 0.050552756

B^2 = [(4 A^4 / Pi^2) - (Nr^2)] / Kc^2
= [(4 / Pi^2) - (Nr^2)] / Kc^2
= [0.4052847355 - 0.050552756] / Kc^2
= 0.3547319795 / Kc^2

Kc = [(1 + A)/ 2] Kh
= [(1 + 1.053907365) / 2] Kh
= 1.026953682 Kh

h = [(A-1) / (A + 1)]^2
= [.053907365 / 2.053907365]^2
= 6.888656083 X 10^-4

h^2 = 47.453583 X 10^-8

Kh = [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]

Kh = 1 + (6.888656083 X 10^-4 / 4) + (47.453583 X 10^-8 / 64) + ...

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.

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

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

as in M = 2
.

The key was to recognize that A, So and Nr behave as constants. To solve a real quantum mechanical problem we must find the Nr, P, Np and Nt values corresponding to a particular [A / B] 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 looking for a rational number which equals the calculated (Np / Nt) value.
 

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DETERMINATION OF SPHEROMAK PARAMETERS:
We must determine the value of the spheromak parameters via the following analysis:
 

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

Recall that a spheromak will adopt an A value where:
(1 / Alpha) = (Nt / 2 A)[1 - (So / (So^2 + 1)]
and
[Lh / Ro] = 2 A Nt [(So^2 + 1) / So]

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

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

Rearrange this equation to get:
Nt^2 - Nt [Lh / (Ro 2 A)] + Lh / (Ro Alpha) = 0

This quadratic equation has solutions:
Nt
= {[Lh / (Ro 2 A)] +/- [(Lh / (Ro 2 A))^2 - 4 (Lh / (Ro Alpha))]^0.5} / 2
= [Lh / (Ro 4 A)][1 +/- [1 - ((Ro 2 A) / Lh)^2 4 (Lh / (Ro Alpha))]^0.5}
 
= [Lh / (Ro 4 A)][1 +/- [1 - ((Ro 4 A) / Lh) (4 A / Alpha)]^0.5]

However, since Lh > 2 Pi Ro, only the quadratic solution:
Nt = [Lh / (Ro 4 A)][1 - [1 - ((Ro 4 A) / Lh) (4 A / Alpha)]^0.5]
is real. Note that Nt is a positive integer.

To solve this equation the quantity:
[1 - ((Ro 4 A) / Lh) (4 A / Alpha)]
must be a rational number of the form: [1 - ((Ro 4 A) / Lh) (4 A / Alpha)] = X^2 / Y^2
where X and Y are both integers.

Further consider:
1 - (X / Y = (Y - X) / Y

The product:
[Lh / (Ro 4 A)][(Y - X) / Y]
must be an integer. Hence Y must be an integer factor of [Lh / (Ro 4 A)], which also must be an integer. These constraints severely restrict the possible integer values of X and Y.

Recall that:
[Lh / Ro} = 2 A Nt (So^2 + 1) / So
or
So = 2 A Nt (So^2 + 1) Ro / Lh or
(So^2 + 1)((Ro 2 A Nt) / Lh) - So = 0
or
(So^2 + 1) - (Lh / (Ro 2 A Nt)So = 0
or
So = {(Lh / (Ro 2 A Nt) + / - [(Lh / (Ro 2 A Nt))^2 - 4]^0.5} / 2
 
= [Lh / (Ro 4 A Nt)][1 +/- [1 - 4 (Ro 2 A Nt)^2 / Lh^2]^0.5
 
= [Lh / (Ro 4 A Nt)][1 +/- [1 - (Ro 4 A Nt)^2 / Lh^2]^0.5

However, in real life So > 1 so the positive quadratic solution applies giving:
So = [Lh / (Ro 4 A Nt)][1 + [1 - (Ro 4 A Nt)^2 / Lh^2]^0.5

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

SUMMARY:
We have three relevant equations:
Nt = [Lh / (Ro 4 A)][1 - [1 - ((Ro 4 A) / Lh) (4 A / Alpha)]^0.5]
and
So = [Lh / (Ro 4 A Nt)]{1 + [1 - (Ro 4 A Nt)^2 / Lh^2]^0.5}
and
(1 / Alpha) = (Nt / 4 A)[2 - ((4 A Ro Nt) / Lh)]

The trick is to choose parameters:
[Lh / 4 A Ro] and [4 A / Alpha] such that all the constraints imposed by all three equations are all simultaneously met. In particular:
Nt = positive integer
So ~ 2
and
(1 / Alpha) = 137.03599915

************************************************************ a ***********************************************************

In order to do each itteration step for a specifed value of A the computer must choose M and then calculate:
h = [(A - 1)^2 / (A + 1)^2  
Kh = [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]
 
[dKh / dh] = [(1 / 2^2) + (h / 2^5) + (3 h^2 / 2^8)
+ (5^2 h^3 / 2^12) + (7^2 5 h^4 / 2^16) + (21^2 6 h^5 / 2^20) + ....
 
Kc = [(A + 1) / 2] [Kh]
and
dKc / dA = (Kh / 2) + [(A + 1) / 2][dKh / dA]
= (Kh / 2) + [(A + 1) / 2][dKh / dh][dh / dA]

Recall that:
h = [(A - 1)^2 / (A + 1)^2]
or
dh / dA = [(A + 1)^2 2 (A - 1) - (A -1)^2 2 (A + 1)] / (A + 1)^4
= [(A + 1) 2 (A - 1) - (A -1)^2 2] / (A + 1)^3
= [2 (A - 1)(A + 1 - A + 1) / (A + 1)^3
= [4 (A - 1) / (A + 1)^3]

Hence:
dKc / dA = (Kh / 2) + [(A + 1) / 2][dKh / dh][dh / dA]
= (Kh / 2) + [(A + 1) / 2][4 (A - 1) / (A + 1)^3][dKh / dh]
= (Kh / 2) + [2 (A - 1) / (A + 1)^2][dKh / dh]

 
Kc^2 = [Kc]^2  
Kc^2 B^2 = (1 / 2 M^2) {-1 + [1 + (M^2 16 A^4 / Pi^2)]^0.5}
 
B^2 = [Kc^2 B^2] / Kc^2
and
B = [B^2]^0.5
and
So^2 = (1 + B) / (1 - B)
and
So = [So^2]^0.5
 

DETERMINATION OF A and Kc:
Start at A = 1.000, Kc = 1.0000 Increment Kc, calculate an interim A value. Use that interim A value to calculate a new Kc value. Repeat the process as necessary to converge to the exact A and Kc values.

Use the computed A value to find Nr.

Use the Nr value to find Nt.

Compare the theoretical and experimaental values for (1 / Alpha).

At a particular A value:
h = [(A - 1)]^2 / [A + 1]^2

Then:
Kh = [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]
 
and
Kc = [(A + 1) / 2] [Kh]

Then:
[dKh / dh] = [(1 / 2^2) + (h / 2^5) + (3 h^2 / 2^8)
+ (5^2 h^3 / 2^12) + (7^2 5 h^4 / 2^16) + (21^2 6 h^5 / 2^20) + ....

Then:
dKc / dA = [Kh / 2] + {[2 (A - 1)] / [A + 1]^2} [dKh / dh]
 

Once A is precisely determined calculate the remaining spheromak parameters using the following equations:
Find Nr using:
Nr = {(- 1 / 2 M) +(1 / 2 M) [1 + (16 M^2 A^4 / Pi^2)]^0.5}

Then convert Nr to a rational number using the formula:
Nr = Np / [P - 2 Np]
at
M = 2
or:
Nr = (P - 2 Nt) / Nt
at
M = (1 / 2)

This step requires testing all the candidate prime numbers to find the best P, Np and Nt values. Remember that P = prime number and that:
(P - 2 N)
is an odd integer. Experimental measurements of the Fine Structure constant suggest that P is likely in the range:
700 < P < 1100.

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

Compare the calculated value of (1 / Alpha) to the precise experimentally measured value of 137.03599915 corresponding to:
h = 6.636070150 X 10^-34 J-s

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

Note that to obtain really accurate results more terms will be required in the power series expansions of Kc and dKc.

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

Prime numbers less than 1428 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, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193 , 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777,1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003
 

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

EXPLANATION:
Recall that:
Ett = [Muo C Qs^2 / 32 A^2] [Fh][Lh / Ro]
{4 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 / 32 A^2][Lh / Ro] Fh
which increases with decreasing Ro:
and an energy reducing spheromak shape parameter function:
S(So) = {4 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.

<

Z QUANTIZATION:
The Fine Structure constant stability comes from inherent So versus A 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 balanced by incrementation or decrementation in Nt.

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

Note that at So = 2 this equation simplifies and gives:
S(So) = 8 [3 / 25]
 

Note that:
(1 / Alpha) = [1 / 16 A^2][Lh / Ro] [4 So]{[ (So^2 - 1) + (2 - So)] / [(So^2 + 1)^2]}
= [1 / 16 A^2] [Lh / Ro] S(So)

Recall that:
Z = [Lh / Ro] [So / Pi]
giving:
(1 / Alpha)
= [(Pi / So) Z (1 / 16 A^2)] [4 So]{[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [(Pi / 4 A^2) Z] {[ So^2 - So + 1] / [(So^2 + 1)^2]}
 
= [(Pi / 4 A^2) Z] {[(So^2 - 1) + (2 - So)] / [(So^2 + 1)^2]}

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

At So = (2 + dSo):
S(So) / 8 = (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 crude 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) = [1 / 16 A^2] [Lh / Ro] S(So) = (1 / 16 A^2) (Lh / Ro) 8 (3 / 25)
which suggests that for:
(1 / Alpha) = 137.03599915:
then:
[Lh / Pi Ro] = (16 A^2 / 8) (25 / 3 Pi)(1 / Alpha)
= (50 A^2 / 3 Pi)(1 / Alpha)
= (50 A^2 / 3 Pi)(137.03599915) = 726.9985557 A^2
~ 727 A^2

[Lh / Pi Ro]^2 = (727 A^2)^2
= 727^2 A^4

[Lh / Pi Ro]^2
= [(Rc^2 / Ro^2) {[Np (So^2 + 1)]^2 + [Nt (So^2 - 1) Kc]^2}]

So = Ro / A Rc

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

At So = 2:
[Lh / Pi Ro]^2
= [(1 / A^2 4) {[Np^2 25] + [Nt^2 Kc^2 9]
or
4 [Lh / Pi Ro]^2 = [(1 / A^2) {[Np^2 25] + [Nt^2 Kc^2 9]}
or
4 [727 A^2]^2 = 25 (Np / A)^2 + 9 (Nt Kc / A)^2
or
[1454 A^3]^2 = (5 Np)^2 + (3 Nt Kc)^2 = 2,114,116 A^6

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 of 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 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 November 14, 2020.

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