# XYLENE POWER LTD.

## PLANCK CONSTANT AND FINE STRUCTURE CONSTANT

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

DEFINITION:
The Planck constant h relates the quantum of energy dE transferred between rest mass and electromagnetic radiation to the radiation frequency dF via the relationship:
h = dE / dF
The quantum of electromagnetic radiation is known as a photon.

When an electromagnetic wave passes a particle with rest mass such as an electron, a proton, atom or a molecule one of three things happen:
a) Energy is transferred from the particle, atom or molecule to the electromagnetic wave;
b) Energy is transferred from the electromagnetic wave to the particle, atom or molecule;
c) No energy transfer occurs.

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

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

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

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

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

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

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

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

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

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

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

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

SPHEROMAK GEOMETRY:
The toroidal shape of a spheromak can be characterized by its inner radius Rc, its outer radius Rs and its height 2 Zf parallel to its 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 close to unity.

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.

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

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

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

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.

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

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

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

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

Issues in high precision experimental measurement of the Planck Constant include suppression of external electric and magnetic fields that can distort the spheromak geometry and allowance for recoil energy. These two issues make the experimentally measured value of the Planck Constant 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 STRINGS OR CHARGE HOSE:
Our universe is composed of a large number of closed charge strings. Each closed charge string contains one quantum of net electric charge, approximately 1.602 X 10^-19 coulombs. The net charge is uniformly distributed along the charge string. The net charge current circulates around the closed charge string path (current path) at the speed of light C, approximately 3 X 10^8 m / s. 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.

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 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);
Ro = A (Rs Rc)^0.5 = radius at which the spheromak potential energy well is deepest;
So = [A Rs / Ro] = [Ro / A Rc] = spheromak shape parameter;
So^2 = (Rs / Rc);
Lh = spheromak charge hose path length;
Np = integer number of poloidal charge hose turns contained in Lh;
Nt = integer number of toroidal charge hose turns contained in Lh;
Nr = Np / Nt;
Lp = Pi (Rs + Rc) = charge hose poloidal turn length;
Lt = Kc Pi (Rs - Rc) = charge hose toroidal turn length;
Kc = (ellipse perimeter length) / (circle perimeter length);
Bpo = poloidal magnetic field strength at the center of the spheromak;
Uo = (Bpo^2 / 2 Mu) = maximum static field energy density at the center of the spheromak;

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

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

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

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

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

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

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

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

= [Pi / A][Z / So][Fh / C]
where:
Ro = (A Rs / So)
= (A Rc So)

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

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

The ratio (Lh / Ro) is the ratio of the spheromak charge hose length Lh to the spheromak geometric radius Ro.

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

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

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

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

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

= [Muo C^2 Qs^2 / 32 A^2] [Pi Fh / A C][Z / So]
{4 So [ So^2 - So + 1] / [(So^2 + 1)^2]}

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

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

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

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

The parameter h is known as the Planck Constant.

In a stable spheromak the parameters So, [Lh / Ro] and A are all constant.

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

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

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

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

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

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

= [Pi / A So]^2 [Np^2 (So^2 + 1)^2][1 + [(1 - Fx) / Fx]]

= [Pi / A So]^2 [Np^2 (So^2 + 1)^2][1 / Fx]

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

= [1 / So]^2 [Nt^2 (So^2 + 1)^2][4 A^2]

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

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

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

P>The published CODATA experimentally measured value of (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 provides a means of calculating the theoretical value Alpha.

(1 / Alpha) = 2 h / (Muo C Q^2)

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

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

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

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

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

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

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

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

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

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

or
dNp Np [(So^2 + 1) / So]^2 + dNt Nt [(So^2 - 1) Kc / So]^2
+ Np^2 [(So^2 + 1) / So] [(So 2 So - (So^2 + 1)) dSo / So^2]
+ Nt^2 (So^2 - 1) Kc / So] [(2 So^2 Kc dSo - (So^2 - 1) Kc dSo) / So^2]
- [Lh / Ro]^2 [2 dA / A] = 0
or
dNp Np [(So^2 + 1) / So]^2 + dNt Nt [(So^2 - 1) Kc / So]^2
+ Np^2 [(So^2 + 1) / So] [(So^2 - 1) dSo / So^2]
+ Nt^2 (So^2 - 1) Kc / So] [(So^2 + 1) Kc dSo / So^2]
- [Lh / Ro]^2 [2 dA / A] = 0
or
dNp Np [(So^2 + 1) / So]^2 + dNt Nt [(So^2 - 1) Kc / So]^2
+ Np^2 [(So^4 - 1) / So^3] dSo
+ Nt^2 [(So^4 - 1) Kc^2 / So^3] dSo
- [Lh / Ro]^2 [2 dA / A] = 0
or
dNp Np [(So^2 + 1) / So]^2 + dNt Nt [(So^2 - 1) Kc / So]^2
+ [Np^2 + Nt^2 Kc^2][(So^4 - 1) / So^3] dSo
- [Lh / Ro]^2 [2 dA / A] = 0

NO COMMON FACTORS:
A spheromak will collapse if Np and Nt share a common factor other than one. A family of potential solutions for (Np / Nt) should have the characteristic that Np and Nt share no common factors and:
dSo / dN = 0
amd
dA / dN = 0

Then the equation becomes:
(dNp / dN) Np [(So^2 + 1) / So]^2 + (dNt / dN) Nt [(So^2 - 1) Kc / So]^2
+ [Np^2 + Nt^2 Kc^2][(So^4 - 1) / So^3] (dSo / dN)
- [Lh / Ro]^2 [(2 / A)] (dA / dN) = 0

However:
(dSo / dN) = 0
and
(dA / dN) = 0

Hence this equation becomes:
(dNp / dN) Np [(So^2 + 1) / So]^2 + (dNt / dN) Nt [(So^2 - 1) Kc / So]^2 = 0

TWO RATIONAL NUMBER SOLUTION FAMILIES:
There are two families of rational numbers with no common factors.

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

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

Thus at the N value corresponding to spheromak stability:
dNt = - M dNp
where M = 2 for Rational Number Family #1 and M = (1 / 2) for Rational Number Family #2.

Recall that:
(dNp/dN) Np [(So^2 + 1) / So]^2 + (dNt /dN) Nt [(So^2 - 1) Kc / So]^2 = 0.
However:
(dNt / dN) = - M (dNp / dN)
which gives:
(dNp/dN) Np [(So^2 + 1) / So]^2 + (- M dNp /dN) Nt [(So^2 - 1) Kc / So]^2 = 0
or
Np [(So^2 + 1) / So]^2 + (- M) Nt [(So^2 - 1) Kc / So]^2 = 0
or
Np [(So^2 + 1) / So]^2 = M Nt [(So^2 - 1) Kc / So]^2
or
Nr / M = (Np / Nt) / M
= [(So^2 - 1) Kc]^2 / [(So^2 + 1)]^2

or
Nr = M (Kc B)^2

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

In this equation:
Nr = (Np / Nt)
Np = integer number of poloidal current turns
Nt = integer number of toroidal current turns
Kc = (ellipse perimeter length) / (contained circle perimeter length)
B = (So^2 - 1) / (So^2 + 1)
So^2 = Rs / Rc
Rs = maximum spheromak wall radius from the axis of symmetry
Rc = minimum spheromak wall radius from the axis of symmetry
A = 2 Zf / (Rs - Rc)
2 Zf = maximum spheromak dimension parallel to the spheromak axis of symmetry

There are two families of spheromak mathematical solutions. The sources of these families are the sets of rational numbers in which the numerator and denominator have no common factors. Both families of solutions are subject to the constraint that:
Nr = M (Kc B)^2
where for one family M = 2 and for the other family M = (1 / 2). This constraint arises from the requirement that for stability the spheromak's shape must remain the same independent of the spheromak's radius.

Substitution of:
Nr = M (Kc B)^2
into the spheromak boundary condition leads to two sets of quadratic equations, one set for M = 2 and one set for M = (1 / 2):
M^2 (Kc B)^4 + Kc^2 B^2 = 4 A^4 / Pi^2
and
Nr^2 + (Nr / M) = 4 A^4 / Pi^2

Each of these four quadratic equations has 2 solutions of which only one represents physical reality. The parameter:
Nr = (Np / Nt)
must be a rational number (ratio of integers) which introduces an element of quantization. The net circulating charge is also quantized which introduces another element of energy quantization.

In these equations if M = 2 then:
Nr = (Np / Nt) = Np / (P - 2 Np)
but if M = (1 / 2) then:
Nr = (Np / Nt) = (P - 2 Nt) / Nt

The parameter P is a situation dependent prime number which introduces yet another element of quantization. The parameter A, which is the ratio of the axes of an ellipse, is near unity but must meet the quantization constraints set by the aforementioned equations.

There is an additional important constraint on parameter A relating to real particles such as electrons and protons. To meet the Planck constant:
~ 450 < Nt < ~ 520.
If parameter A adopts a value which leads to Np / Nt becoming a simple fraction, that will cause spheromak collapse and 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.

I suspect that real particles, such as electrons, actually have two solutions, one for M = 2 and one for M = (1 / 2), which share a common A value. It appears that the M = 2 solution points to a trapped photon which contains most of the particle's rest mass. The M = (1 / 2) solution points to a spheromak which gives the particle physical stability and electro-magnetic properties that are reflected in the Planck constant.

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.

VALUE OF Nr:
The spheromak boundary condition gave:
Nr^2 + [(So^2 - 1) Kc]^2 / [(So^2 + 1)]^2 = [4 A^4 / Pi^2]

Hence:
Nr^2 + (Nr / M) = 4 A^4 / Pi^2
or
Nr = {(- 1 / M) + [(1 / M^2) + (16 A^4 / Pi^2)]^0.5} / 2
= (1 / 2 M){-1 + [1 + (16 M^2 A^4 / Pi^2)]^0.5}

Recall that if M = 2 then:
Nr = Np / (P - 2 Np)
and if M = (1 / 2) then:
Nr = (P - 2 Nt) / Nt

Thus there are two families of spheromaks. If M = 2 then:
Nr = (Np / Nt)
= [Np / (P - 2 Np)]
= (1 / 4){-1 + [1 + (64 A^4 / Pi^2)]^0.5}

and if M = (1 / 2) then:
Nr = Np / Nt
= (P - 2 Nt) / Nt
= {-1 + [1 + (4 A^4 / Pi^2)]^0.5}

These two equations together with the spheromak boundary condition which is:
Nr^2 + [Kc (So^2 - 1) / (So^2 + 1)]^2 = 4 A^4 / Pi^2

For M = 2:
Nr = (-1 / 4) + (1 / 4) [1 +(64 A^4 / Pi^2)]^0.5
which at A = 1.0000 gives:
Nr = 0.4339479041

For M = (1 / 2):
Nr = -1 + [1 + (4 A^4 / Pi^2]^0.5
which at A = 1.0000 gives:
Nr = 0.185447061

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 will 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.

PARAMETER A VALUES WHICH CAUSE SPHEROMAK COLLAPSE:
Certain values of parameter A will cause a spheromak collapse due to Nr becoming a simple fraction. For example:
If M = (1 / 2):
and if:
[1 + (4 A^4 / Pi^2)]^0.5 = 5 / 4
or
[1 + (4 A^4 / Pi^2)] = 25 / 16
or
(4 A^4 / Pi^2) = 9 / 16
or
A^4 = Pi^2 (9 / 64)
or
A^2 = Pi (3 / 8)
or
A = 1.085401881

In this case:
Nr = -1 + [1 + (4 A^4 / Pi^2]^0.5
= -1 + (5 / 4)
= (1 / 4)
which is a form of spheromak collapse. To enable the Planck constant:
Nr = Np / Nt
must be the ratio of two integers representing several hundred spheromak turns. Hence spheromak collapse points indicate parameter A values at which the Planck constant cannot exist for the rational number family in question.

We need to tabulate all the spheromak collapse A values for both M = 2 and M = (1 / 2) spheromaks for
0.8 < A^4 < 1.2

COLLAPSE POINTS FOR M = 2 SPHEROMAKS:
Recall that in general:
Nr = (1 / 2 M){-1 + [1 + (16 M^2 A^4 / Pi^2)]^0.5}

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

This expression will collapse when:
[1 + (64 A^4 / Pi^2)] = X^2 / Y^2
where X and Y are integers. Our concern relates to integer combinations which are such that the ratio of their squares is close to
[1 + (64 / Pi^2)] = 7.4845
where A = 1.00000

eg X^2 / Y^2 = (9 / 1) = 9
(25 / 4) = 6.25,
(36 / 4) = 9,
(49 / 9) = 5.44,
(64 / 9) = 7.111,
(81 / 9) = 9
(81 / 16) = 5.0625,
(100 / 16) = 6.25,
(121 / 16) = 7.5625, (144 / 25) = 5.76,
(169 / 25) = 6.76,
(196 / 25) = 7.84,
(225 / 25) = 9.00,
(256 / 36) = 7.111,
(289 / 36) = 8.027

The following table for M = 2 spheromaks shows spheromak collapse points,the corresponding Nr values, the corresponding (4 A^4 / Pi^2) values, the corresponding [1 + (4 A^4 / Pi^2)] values and the corresponding A^4 values. This table shows that M = (1 / 2) spheromaks can exist at numerous A values where M = 2 spheromaks collapse. A frequent M = 2 spheromak collapse on this table occurs at:
[1 + (64 A^4 / Pi^2)] = 9
or
(64 A^4 / Pi^2) = 8
or
A^4 = (Pi^2 / 8)

[1 + (64 A^4 / Pi^2)]Nr(4 A^4 / Pi^2) [1 + (4 A^4 / Pi^2)]A^4
(25 / 4)(3 / 8)(21 / 4(16))(85 / 64)Pi^2 (21 / 256) = 0.8096
(36 / 4)(2 / 4)(32 / 4(16))(96 / 64)Pi^2 (32 / 256) = 1.2337
(49 / 9)(4 / 12)(40 / 9(16))(184 / 16)Pi^2 (10 / 144) = 0.6854
(64 / 9)(5 / 12)(55 / 9(16))(199 / 16)Pi^2 (55 /4(144)) = 0.94241
(81 / 9)(2 / 4)(72 / 9(16))(216 / 144)Pi^2 / 8 = 1.2337
(121 / 16)(7 / 16)(105 / 256)(361 / 256) = (19 / 16)^2Pi^2 (105 / 1024) = 1.01201
(144 / 16)(2 / 4)(128 / 256)384 / 256Pi^2 / 8 = 1.2337
(169 / 25)(8 / 20)(144 / 25(16))(544 / 400)Pi^2 (36 / 400) = 0.88826
(196 / 25)(9 / 20)(171 / 25(16))(571 / 400)Pi^2 (171 / 1600) = 1.054813960
(225 / 25)(10 / 20)(200 / 25 (16))(600 / 400)Pi^2 / 8 = 1.2337
(256 / 36)(10 / 24)(220 / 36(16))(796 / 36(16))Pi^2 (55 / 576) = 0.94241
(289 / 36)(11 / 24)(253 / 36(16))(829 / 36(16))Pi^2 (253 / 36 (64)) = 1.083771661
(324 / 49)(11 / 28)(275 / 49 (16))(1059 / 49 (16))Pi^2 (275 /49(64))= 0.86547
(361 / 49)(12 / 28)(312 / 49(16)1096 /49(16))Pi^2 (312 / 64(49)) = 0.98192
(400 / 49)(13 / 28)(351/ 49 (16)(1135 / 49 (16))Pi^2 (351 / 49(64)) = 1.104665541

The ratio of the squares closest to 7.4845 is (121 / 16). It gives:
[1 + (64 A^4 / Pi^2)] = 121 / 16
or
(64 A^4 / Pi^2) = 105 / 16
or
A^4 = (Pi^2 / 64)(105 / 16)
= Pi^2 (105 / 1024)
= 1.01201998
or
A = 1.002991544<
or
(A - 1) = 0.002991544
This is the value of A closest to unity which causes a M = 2 spheromak to collapse. This A value also causes collapse of M = (1 / 2) spheromaks.

However, nature has found a way to keep
A^4 > Pi^2 (105 / 1024)
which avoids this potential problem of a collapse of both possible rational number solutions.

At the M = 2 spheromak collapse at A = 1.002991544:
Nr = (1 / 4){- 1 + [1 +(64 A^4 / Pi^2)]^0.5}
= (1 / 4){- 1 + [121 / 16]^0.5}
= (1 / 4){- 1 + (11/ 4)}
= (1 / 4){7 / 4}
= 7 / 16

Based on measurements of So^2 plasma spheromaks appear to be M = (1 / 2) spheromaks.

Other M = 2 collapse ratios near 7.4845 and hence worthy of further examination are:
64 / 9 = 7.1111
and
196 / 25 = 7.84000

CONSIDER THE RATIO 64 / 9:
In a M = 2 spheromak:
[1 + (64 A^4 / Pi^2)] = 64 / 9
or
(64 A^4 / Pi^2) = 55 / 9
or
A^4 = (Pi^2 / 64)(55 / 9) = 0.9424101403
or
A = 0.9852807268

Apply this A value to a M = (1 / 2) spheromak:
64 A^4 / Pi^2 = 55 / 9
or
4 A^4 / Pi^2 = [55 / 9(16)]
or
[1 + (4 A^4 / Pi^2)] = 199 / 9 (16)
which will not collapse in the M = (1 / 2) spheromak.

CONSIDER THE RATIO 196 / 25:
[1 + (64 A^4 / Pi^2)] = 196 / 25
or
(64 A^4 / Pi^2) = 171 / 25
or
A^4 = (Pi^2 / 64)(171 / 25)
= 1.054813968
or
A^2 = (Pi / 8)[(171)^0.5 / 5]
= 1.027041366
or
A = 1.013430494
It appears that this A value may enable M = (1 / 2) spheromaks at this value of A.

For the ratio of squares = 196 / 25:
64 A^4 / Pi^2 = 171 / 25
or
4 A^4 / Pi^2 = [171 / 25 (16)]
or
[1 + (4 A^4 / Pi^2)] = 571 / 400
which will not collapse in the M = (1 / 2) spheromak.
The corresponding A value is given by:
A^4 = (Pi^2 / 4)[171 / 400]
= 1.054813968

The corresponding collapsed Nr value in the M = 2 spheromak is given by:
Nr = (1 / 4)[1 + (64 A^4 / Pi^2)]^0.5
= (1 / 4){-1 + [196 / 25]^0.5}
= (1 / 4){-1 + [14 / 5]}
= (1 / 4){9 / 5}
= 9 / 20

COLLAPSE POINTS OF M = (1 / 2) SPHEROMAKS:
M = (1 / 2) spheromaks cannot exist at A values which cause their collapse, regardless of the corresponding status of M = 2 spheromaks.
Recall that:
Nr = (1 / 2 M){-1 + [1 + (16 M^2 A^4 / Pi^2)]^0.5}

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

This M = (1 / 2) spheromak solution will potentially collapse at:
[1 + (4 A^4 / Pi^2)] = X^2 / Y^2
where X and Y are integers with a quotient of squares near
[1 + (4 / Pi^2)] = 1.4042

Potential values of (X^2 / Y^2) at M = (1 / 2) spheromak collapse are:
16 / 9 = 1.77777
25 / 16 = 1.5625
36 / 25 = 1.4400
49 / 36 = 1.36111
64 / 49 = 1.30612
81 / 49 = 1.6530
81 / 64 = 1.2656
100 / 64 = 1.5625
121 / 81 = 1.4938
144 / 100 = 1.4400
169 / 100 =1.6900
169 / 121 = 1.39669
196 / 144 = 1.3611
225 / 169 = 1.3313
256 / 196 = 1.3061
289 / 196 = 1.4744
289 / 225 = 1.2844
324 / 225 = 1.4400
324 / 256 = 1.2656
361 / 256 = 1.4101
400 / 256 = 1.5625
400 / 289 = 1.3840
441 / 324 = 1.3611

Of the above ratios the one closest to 1.4042 is 361 / 256 = 1.4101

[1 + (4 A^4 / Pi^2)] = 361 /256
or
4 A^4 / Pi^2 = 105 / 256
or
A^4 = Pi^2 (105 / 1024)
= 1.0120199833

Other M = (1 / 2) ratios worth of further examination are: 1.36111 = 49 / 36, 1.3840 = 400 / 289, 1.39669 = 169 / 121, and 1.44 = 36 / 25. Spheromak collapse at these ratios will impose further constraints on parameter A values.

[1 + (4 A^4 / Pi^2)] = 49 / 36
or
(4 A^4 / Pi^2) = 13 / 36
or
A^4 = (Pi^2 / 4)(13 / 36)
= 0.8910059508

[1 + (4 A^4 / Pi^2)] = 400 / 289
or
(4 A^4 / Pi^2) = 111 / 289
or
A^4 = (Pi^2 / 4)(111 / 289) = 0.9476869256

[1 + (4 A^4 / Pi^2)] = 169 / 121
or
(4 A^4 / Pi^2) = 48 / 121
or
A^4 = Pi^2 (12 / 121)
= 0.97880374

[1 + (4 A^4 / Pi^2)] = 36 / 25
or
(4 A^4 / Pi^2) = 11 / 25
or
A^4 = Pi^2 (11 / 100) = 1.085656482

This M = (1 / 2) spheromak collapse will prevent the existence of a M = (1 / 2) spheromak at
A^4 = Pi^2 (11 / 100)
even though such spheromaks might potentially be enabled by the M = 2 spheromak collapse at
A^4 = _______.

None of these values interferes with the contemplated M = (1 / 2) spheromak operation at:
A^4 = 1.1
which is enabled by a M = 2 spheromak collapse at that A value.

SUMMARY OF CONSEQUENCES OF SPHEROMAK COLLAPSE NEAR A = 1.000000:
Both the M = 2 and M = (1 / 2) spheromaks collapse at:
A^4 = Pi^2 (105 / 1024)
or at
A = 1.002991545

Hence to prevent spheromak collapse close to A = 1.0000, A^4 must satisfy:
1.00000000 < A^4 < 1.0120199833

This is a very tight specification which is difficut to meet with integer changes in Np and Nt.

For M = (1 / 2) spheromaks find the range of Nr that is free from spheromak collapse for:
1.00000 < A < 1.002991545

At A^4 = Pi^2 (105 / 1024):
Nr = -1 + [1 + 4 A^4 / Pi^2]^0.5
= -1 + [1 + 4 (105 / 1024)]^0.5
= 0.1875

At A^4 = 1.000000:
Nr = -1 + [1 + 4 A^4 / Pi^2]^0.5
= -1 + [1 + (4 / Pi^2)]^0.5
= 0.1854470612

Hence for M = (1 / 2) spheromaks while theoretically there is no spheromak collapse problem provided that Nr remains in the range:
0.1854470612 < Nr < 0.1875
meeting that specification is extremely difficult.

This range is so close to A = 1.000000 that in practise stable spheromaks cannot exist at A = 1.00000. However, a stable spheromak can theoretically exist at A^4 = 1.054813968
which is potentially stable for the M = (1 / 2) spheromak because the M = 2 spheromak has a collapse to Nr = 9 / 20 at that A value.

There is another potentially stable point at A^4 = 1.1 again enabled by collapse of the M = 2 spheromak.

M = (1 / 2) spheromaks can exist in the range:
1.0120199833 < A^4 < 1.085656482
and will likely reside at
A^4 = 1.054813968

Recall that for M = (1 / 2) spheromaks:
Nr = (P - 2 Nt) / Nt
or
P = Nt (2 + Nr)
= Nt (2.1854470612)

Thus we can find Nt for M = (1 / 2) spheromaks at A = 1.00000 by dividing prime numbers by:
[2 + Nr] = (2.1854470612)
and looking for integer quotients.

PRIME NUMBER P QUOTIENT
1117 511.1082395
1109 507.4476613
1103 504.7022276
1097 501.9567939
1093 500.1265047
1091 499.2113602
1087 497.381071
1069 489.14477
1063 486.3993363
1061 485.4841917
1051 480.9084689
1049 479.9933243***
1039 475.4176015
1033 472.6721678
1031 471.7570232
1021 467.1813004
1019 466.2661558
1013 463.5207221
1009 461.690433
997 456.1995656
991 453.4541319
983 449.7935537
977 447.04812**
971 444.3026863
967 442.4723971
953 436.0663852*
947 433.3209515
941 430.5755178
937 428.7452287

For M = (1 / 2) at A = 1.0000 this table gives a best value of:
Nt = 480 at P = 1049, Np = 89
where:
Nr = 89 / 480 = 0.1854166667

Recall that at A = 1.0000:
Nr = 0.1854470612
or
Nr^2 = 0.0343906125

For M = (1 / 2) at A = 1.00000
B^2 = [(4 A^4 / Pi^2) - (Nr^2)] / Kc^2
= [(4 / Pi^2) - (Nr^2)]
= 0.4052847355 - 0.0343906125
= 0.370894123
giving:
(1 - B^2)^0.5 = 0.7931619488
{1 - [So / (So^2 + 1)]}
= {1 - [(1 - B^2)^0.5 / 2]
= [1 - [(0.7931619488 / 2)]
= 0.6034190256

Hence for M = (1 / 2):
(1 / Alpha) = [Nt / 2 A][1 - [So / (So^2 + 1)]]
= [480 / 2 A] [0.6034190256]
= 144.8205661 / A

For M = 2 spheromaks find the range of Nr that is free from spheromak collapse corresponding to the parameter A range:
1.00000 < A < 1.002991545

At A^4 = Pi^2 (105 / 1024):
Nr = (1 / 4){-1 + [1 + 64 A^4 / Pi^2]^0.5
= (1 / 4){-1 + [1 + 64 (105 / 1024)]^0.5
= 0.4375

At A^4 = 1.000000:
Nr = (1 / 4){-1 + [1 + 64 A^4 / Pi^2]^0.5}
= (1 / 4){-1 + [1 + (64 / Pi^2)]^0.5}
= 0.4339479041

Hence for M = 2 spheromaks there is no spheromak collapse problem provided that Nr remains in the range:
0.4339479041 < Nr < 0.4375

Recall that for M = 2 spheromaks:
Nr = Np / (P - 2 Np)
or
(P - 2 Np) = Np / Nr
or
P = Np [(1 / Nr) + 2]

Thus we can find Np for M = 2 spheromaks at A = 1.00000 by dividing prime numbers by:
[2 + (1 / Nr)] = 4.304424081
and looking for integer quotients.

Prime Number P Quotient
997 231.6221593
991 230.2282446
983 228.3696916
977 226.9757769
971 225.5818622
967 224.6525858
953 221.4001181
947 220.0062034***
941 218.6122887
937 217.6830123
929 215.8244593
919 213.5012682
911 211.6427152

Note that at P = 947 the quotient 220.0062034 resulting from:
(P / 4.304424081)
has a near zero remainder. Hence:
Np = 220
P = 947
Nt = 947 - 2(220) = 507

Hence:
Nr = 220 / 507 = 0.4339250493
or
Nr^2 = (220 / 507)^2 = 0.1882909484

Recall that:
So / (So^2 + 1) = {[1 - B^2]^0.5} / 2 For M = 2 at A = 1.00000
B^2 = [(4 A^4 / Pi^2) - (Nr^2)] / Kc^2
= [(4 / Pi^2) - (Nr^2)]
= [0.4052847355 - 0.1882909484]
= 0.2169937871
giving:
(1 - B^2)^0.5 = 0.8848763828

{1 - [So / (So^2 + 1)]}
= {1 - [(1 - B^2)^0.5 / 2]
= [1 - [(0.8848763828 / 2)]
= 0.557561809

Hence for M = 2:
(1 / Alpha) = [Nt / 2 A][1 - [So / (So^2 + 1)]]
= [507 / 2 A] [0.5575618086]
= 141.3419185 / A

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

ISSUES RELATED TO PARAMETER A:
There are numerous A values that can potentially cause spheromak collapse. Depending on the value of parameter A there can be collapses of M = (1 / 2) spheromaks, collapses of M = 2 spheromaks or collapses of both M = (1 / 2) and M = 2 spheromaks.

HYPOTHESIS:
Our present hypothesis is that spheromaks do not operate at A = 1.0000 due to the close proximity of spheromak collapse points. Our present hypothesis is that spheromaks actually operate at:
A^4 = 1.1
which is stable for the M = (1 / 2) spheromak because the M = 2 spheromak has a collapse to Nr = ________ at that A value.

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

The nearest M = (1 / 2) spheromak collapses are at A^4 = 1.012019983 and A^4 = 1.085656482 which are both far from the contemplated spheromak operating point at
4 A^4 / Pi^2 = 171 / 400 or
A^4 = (Pi^2 / 4)(171 / 400)
A^4 = 1.054813968

For M = (1 / 2) spheromaks the corresponding Nr values are given by: Nr = - 1 + [1 + (4 A^4 / Pi^2)]^0.5
= -1 + [1 + (171 / 400)]^0.5
= 0.1947803145

Recall that for M = (1 / 2) spheromaks:
Nr = (P - 2 Nt) / Nt
or
P = Nt (Nr + 2)
= Nt (2.1947803145)

If this hypothesis is correct we should be able to use the
A^4 = (Pi^2 / 4)(171 / 400)
= 1.054813968
value at the M = 2 spheromak collapse to determine the corresponding value of (1 / Alpha) for the M = (1 / 2) spheromak.

Thus if this theory is correct dividing primes by 2.1947803145 corresponding to A^4 = 1.054813968 should give an integer result for Nt.

PRIME NUMBER P QUOTIENT
1229 559.9649276
1223 557.2311688
1217 554.49741
1213 552.6749042
1201 547.2073866
1193 543.5623748
1187 540.828616
1181 538.0948572
1171 533.5385926
1163 529.8935808
1153 525.3373162
1151 524.4260632
1129 514.402281
1123 511.6685222
1117 508.9347634
1109 505.2897516
1103 502.5559928
1097 499.822234
1093 497.9997282***
1091 497.0884752
1087 495.2659694
1069 487.064693
1063 484.3309342
1061 483.4196812
1051 478.8634166
1049 477.9521636
1039 473.395899
1033 470.6621402
1031 469.7508872
1021 465.1946226
1019 464.2833696
1013 461.5496108
1009 459.727105
997 454.2595874
991 451.5258286
983 447.8808168
977 445.147058
971 442.4132992
967 440.5907934
953 434.2120228
947 431.478264
941 428.7445052
937 426.9219993

Best match is at P = 1093, Nt = 498, Np = 97

Nr = (97 / 498) = 0.1947791165

Nr^2 = 0.0379389042

B^2 = [(4 A^4 / Pi^2) - (Nr^2)] / Kc^2
= [(171 / 400) - (0.0379389042)] / Kc^2
= 0.3895610958 / Kc^2

A^4 = 1.054813968
or
A = 1.013430494

Kc = [(1 + A)/ 2] Kh
= [(1 + 1.013430494) / 2] Kh
= 1.006715247 Kh

h = [(A-1) / (A + 1)]^2
= [.013430494 / 2.013430494]^2
= 4.449494695 X 10^-5

h^2 = 1.979800304 X 10^-9

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 + (4.449494695 X 10^-5 / 4) + ...
= 1 + (1.112373674 X 10^-5) + ...
= 1.00001112373674 + ....
= 1.00001112373674

Kc = [(1 + A)/ 2] Kh
= 1.006715247 Kh
= 1.006715247(1.00001112373674)
= 1.006726445

Kc^2 = 1.013498136

B^2 = 0.3895610958 / Kc^2 = 0.3895610958 /1.013498136
= 0.3843727797

Hence:
(1 - B^2)^0.5 = 0.7846191562
{1 - [So / (So^2 + 1)]}
= {1 - [(1 - B^2)^0.5 / 2]
= [1 - [(0.7846191562 / 2)]
= 0.6076904219

Hence for M = (1 / 2):
(1 / Alpha) = [Nt / 2 A][1 - [So / (So^2 + 1)]]
= [498 / 2 (1.013430494)] [0.6076904219]
= 149.3096132

The conclusion is that the true value of parameter A is significantly larger than used in this calculation. There are two possibilities. One is that A is set at about 1.03 by an unrecognized external factor. Another possibility is that A is trapped at about 1.03 by a different M = 2 spheromak collapse. We need to further examine M = 2 spheromak collapses.

VALUE OF (Kc B)^2:
The spheromak boundary condition gave:
Nr^2 + [(So^2 - 1) Kc]^2 / [(So^2 + 1)]^2 = [4 A^4 / Pi^2]
or
Nr^2 + Kc^2 B^2 = [4 A^4 / Pi^2]

Recall that:
Nr = M [(So^2 - 1) Kc]^2 / [(So^2 + 1)]^2
= M Kc^2 B^2

Substitution into the boundary condition gives:
M^2 Kc^4 B^4 + Kc^2 B^2 = [4 A^4 / Pi^2]
or
(Kc B)^2 = [1 / 2 M^2]{- 1 + [1 + (M^2 16 A^4 / Pi^2)]^0.5}

Consistency Check:
Nr^2 + Kc^2 B^2 = (1 / 2 M)^2{-1 + [1 + (16 M^2 A^4 / Pi^2)]^0.5}^2
+ (1 / 2 M^2){- 1 + [1 + (M^2 16 A^4 / Pi^2)]^0.5}

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

= 2 (1 / 2 M)^2 + [4 A^4 / Pi^2] - 2 (1 / 2 M)^2[1 + (16 M^2 A^4 / Pi^2)]^0.5} - (1 / 2 M^2) + (1 / 2 M^2)[1 + (M^2 16 A^4 / Pi^2)]^0.5
= [4 A^4 / Pi^2]
as expected.

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

Hence:
B (So^2 + 1) = (So^2 - 1)
or
So^2 ( 1 - B) = (1 + B)
or
So^2 = (1 + B) / (1 - B)

For the special case of A = 1.0000 and hence Kc = 1.0000:
B^2 = (1 / 2 M^2 Kc^2){- 1 + [1 + (M^2 16 A^4 / Pi^2)]^0.5}
= (1 / 2 M^2){- 1 + [1 + (M^2 16 / Pi^2)]^0.5}

For M = 2:
B^2 = (1 / 2 M^2){- 1 + [1 + (M^2 16 / Pi^2)]^0.5}
= (1 / 8){- 1 + [1 + (64 / Pi^2)]^0.5}
= 0.216973952

The corresponding value of B is:
B = 0.4658046286

For M = (1 / 2):
Nr = (P - 2 Nt) / Nt
and
Nr = (1 / 2 M){-1 + [1 + (16 M^2 A^4 / Pi^2)]^0.5}

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

Hence:
(P - 2 Nt) = Nt {-1 + [1 + (4 A^4 / Pi^2)]^0.5}
or
P = Nt {1 + [1 + (4 A^4 / Pi^2)]^0.5}
This equation is of fundamental importance in spheromak analysis.

Continuing with M = (1 / 2):
B^2 = (1 / 2 M^2){- 1 + [1 + (M^2 16 A^4 / Pi^2)]^0.5}
= (2){- 1 + [1 + (4 A^4 / Pi^2)]^0.5}
= 0.3708941225

At A = 1.0000 The corresponding value of B is:
B = 0.6090107737

The corresponding value of So^2 is:
So^2 = (1 + B) / (1 - B)
= 1.6090107737 / 0.3909892263
= 4.11523046
which closely matches the experimentally observed So value for plasma spheromaks.

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.

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.

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

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

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

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

Hence:
1 < So^2 < 4.508

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

and for M = (1 / 2):
(Np / Nt) = [P - 2 Nt] / Nt

If A = 1.00000 the computed value of (Np / Nt) for M = 2 is:
Nr = (Np / Nt) = 0.4339479041
and the computed value of (Np / Nt) for M = (1 / 2) is:
Nr = (Np / Nt) = 0.185447061

One of the challenges is precise determination of the A value. THE VALUE OF (Np / Nt) AND ITS DEPENDENCE ON THE VALUE OF A IS KEY TO UNDERSTANDING SPHEROMAKS. Initial calculations are done with A = 1.0000. However, in general A is not precisely equal to one.

Note that as shown later if A = 1.000 then for M = 2:
P = 947
and
(Np / Nt) = (220 / 507)
= 0.4339250493

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

For M = 2:
(Np / Nt) = Np / [P - 2 Np]

For M = (1 / 2):
(Np / Nt) = [P - 2 Nt] / Nt = - 1 + [1 + (4 A^4 / Pi^2)]^0.5
and
P = Nt [1 + (4 A^4 / Pi^2)]^0.5

FUNCTIONS OF Nr:
Recall that:
Nr = (Np / Nt)
= (- 1 / 2 M) + (1 / 2 M)[1 + (16 M^2 A^4 / Pi^2)]^0.5

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

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

For M = 2:
At A = 1.00000, Np = 220, Nt = 507, P = 947:
(Np / Nt) = 0.4339250493
and
(Np / Nt)^2 = 0.1882909484

For M = (1 / 2):
At A = 1.00000, Np = 83, Nt = 447, P = 977:
Nr = (Np / Nt) = 0.185447061
and
Nr^2 = (Np / Nt)^2
= 0.0343906124

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

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

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

For M = 2 at A = 1.000000, Kc = 1.00000, Np = 220, Nt = 507
B^2 = [(4 A^4 / Pi^2) - (Nr^2)] / Kc^2
= [0.4052847355 - 0.1882909484]
= 0.2169937871
and
B = 0.4658259193

For M = (1 / 2) at A = 1.000000, Kc = 1.00000
B^2 = [(4 A^4 / Pi^2) - (Nr^2)] / Kc^2
= [0.4052847355 - 0.0343906124]
= 0.3708941211
and
B = 0.6090107726

FUNCTIONS OF So:

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

At A = 1.00000 for Solution Family #1:
So^2 = [(1 + B) / (1 - B)]
= 1.4658259193 / 0.5341740807
= 2.744097799

At A = 1.00000 for M = (1 / 2):
So^2 = [(1 + B) / (1 - B)]
= 1 + 0.6090107726 / ( 1- 0.6090107726)
= 1.6090107726 / (0.3909892274)
= 4.11523o446

Note that for M = 2:
So^2 ~ 2.744
and for M = (1 / 2):
So^2 ~ 4.115
whereas experimentally photographed plasma spheromaks exhibit:
So^2 ~ 4.1

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

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

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

FOR M = (1 / 2)
Np = P - 2 N
and
Nt = N
where:
P = prime number
and N = integer in the range:
0 < 2 N < P

Note that for M = (1 / 2) for each quantum number (Np / Nt) pair Np and Nt have no common factors and between adjacent quantum number pairs the relationship:
dNp = - 2 dNt
applies.

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

Thus for M = (1 / 2):
Np / Nt = [(P - 2 N) / N]
= {(- 1 / 2 M) +(1 / 2 M) [1 + (16 M^2 A^4 / Pi^2)]^0.5}
= {(- 1) + [1 + (4 A^4 / Pi^2)]^0.5}

For M = (1 / 2) there is a spheromak collapse point at:
[1 + (4 A^4 / Pi^2)] = (25 / 16)
or
[1 + (4 A^4 / Pi^2)]^0.5 = (5 / 4)
at which point Np / Nt = (1 / 4)
and
(4 A^4 / Pi^2) = 9 / 16
or
A^4 = (Pi^2 / 4)(9 / 16)
or
A^4 = Pi^2 (9 / 64)

Note that the A value
A^4 = Pi^2 (9 / 64)
which triggers a spheromak collapse for M = (1 / 2) is larger than the A value
A = Pi^2 / 8
which triggers a spheromak collapse for M = 2. To avoid two conflicting real solutions we suspect that the M = 2 solution jams its A value at:
A^4 = Pi^2 / 8
or
A = 1.053907365
which allows the M = (1 / 2) solution to exist at that A value.

Hence for M = (1 / 2) and A^4 = Pi^2 / 8:
Np / Nt = {(- 1) + [1 + (4 A^4 / Pi^2)]^0.5}
= Np / Nt = {(- 1) + [1 + (4 / 8)]^0.5}
= {(- 1) + [3 / 2]^0.5}
= Nr = 0.224744871

(P - 2 Nt) / Nt = Nr
or
P = Nr Nt + 2 Nt
= (Nr + 2) Nt
= 2.224744871 Nt

Thus if this theory is correct dividing primes by 2.224744871 should give an integer result for Nt.

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) + ...
= 1 + (1.722164021 X 10^-4) + (0.7414622344 X 10^-8) + ...
= 1.0001722164021 + .000000007414622344 + ....
= 1.000172224

Kc = [(1 + A)/ 2] Kh
= 1.026953682 Kh
= 1.026953682 (1.000172224)
= 1.027130548

Kc^2 = 1.054997163

B^2 = 0.3547319795 / Kc^2
= 0.3547319795 / 1.054997163
= 0.3362397473

Hence:
(1 - B^2)^0.5 = 0.8147148291
{1 - [So / (So^2 + 1)]}
= {1 - [(1 - B^2)^0.5 / 2]
= [1 - [(0.8147148291 / 2)]
= 0.5926425855

Hence for M = (1 / 2):
(1 / Alpha) = [Nt / 2 A][1 - [So / (So^2 + 1)]]
= [467 / 2 (1.053907365] [0.5926425855]
= 131.3038017

REDO THIS CALCULATION FOR A SPHEROMAK COLLAPSE AT ABOUT A ~ 1.03

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
or
Nr = (Np / Nt) = [P - 2 Nt] / Nt
as in M = (1 / 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 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.

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

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

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

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

The efficient means of solving this problem is to make a reasonable guess and then check for compliance. We will start with the equation for So and guess that:
[1 - (Ro 4 A Nt)^2 / Lh^2] = 9 / 25
Making this guess required some trial and error.

Then:
(Ro 4 A Nt)^2 / Lh^2 = 16 / 25
or
(Ro 4 A Nt) / Lh = 4 / 5

Then:
So = [Lh / (Ro 4 A Nt)]{1 + [1 - (Ro 4 A Nt)^2 / Lh^2]^0.5}
= [5 / 4]{1 + [1 - (16 / 25)]^0.5}
= [5 / 4]{1 + [9 / 25]^0.5}
= [5 / 4]{1 + [3 / 5]}
= 2

Then:
Nt = [Lh / (Ro 4 A)][1 - [1 - ((Ro 4 A) / Lh) (4 A / Alpha)]^0.5]
= [5 Nt / 4] [1 - [1 - (4 / 5 Nt)(4 A / Alpha)]^0.5]
which implies that:
[1 - (4 / 5 Nt)(4 A / Alpha)]^0.5 = (1 / 5)
or
[1 - (4 / 5 Nt)(4 A / Alpha)] = (1 / 25)
or
(4 / 5 Nt)(4 A / Alpha) = 24 / 25 or
(4 A / Alpha) = (24 / 25)(5 Nt / 4)
= (6 Nt / 5)

Hence:
Nt = [5 Nt / 4] [1 - [1 - (4 / 5 Nt)(4 A / Alpha)]^0.5]
= [5 Nt / 4] [1 - [1 - (4 / 5 Nt)(6 Nt / 5)]^0.5]
= [5 Nt / 4] [1 - [1 - (24 / 25)]^0.5]
= [5 Nt / 4] [1 - [1 / 25]^0.5]
= [5 Nt / 4] [1 - [1 / 5]]
= [5 Nt / 4] [4 / 5]
= Nt

and
(4 A / Alpha) = (6 Nt / 5)

Hence:
(1 / Alpha) = (6 Nt / 5)(1 / 4 A)
= (3 / 10)(Nt / A)
= (6 / 10)(Nt / 2 A)

Consistency check:
{1 - [So / (So^2 + 1)]}
= {1 - [2 / 5]}
= (3 / 5)
= (6 / 10)

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

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

Relevant conditions are:
dB = 0
dA = 0
dSo = 0
So = 2
B = (So^2 - 1) / (So^2 + 1)
= 3 / 5

Hence:
Nr^2 + Kc^2 (9 / 25) = [4 A^4 / Pi^2]
or
Nr^2 = [4 A^4 / Pi^2] - Kc^2 (9 / 25)
where Kc is a function of A.

We have another equation for Nr of the form:
Nr = (1 / 2 M)(-1 + [1 + 16 M^2 A^4 / Pi^2]^0.5
or
Nr^2 = (1 / 4 M^2)[1 - 2 [1 + 16 M^2 A^4 / Pi^2]^0.5 + [1 + 16 M^2 A^4 / Pi^2]

= (1 / 4 M^2)[2 - 2 [1 + 16 M^2 A^4 / Pi^2]^0.5 + 16 M^2 A^4 / Pi^2]

For the case of M = 0.5:
Nr^2 = (1 / 4 M^2)[2 - 2 [1 + 16 M^2 A^4 / Pi^2]^0.5 + 16 M^2 A^4 / Pi^2]
Nr^2 = [2 - 2 [1 + 4 A^4 / Pi^2]^0.5 + 4 A^4 / Pi^2]

For the case of M = 2:
Nr^2 = (1 / 4 M^2)[2 - 2 [1 + 16 M^2 A^4 / Pi^2]^0.5 + 16 M^2 A^4 / Pi^2]
= (1 / 16)[2 - 2 [1 + 64 A^4 / Pi^2]^0.5 + 64 A^4 / Pi^2]

FIX

For the case of M = (1 / 2)
equate the two expressions for Nr^2 to get:
[4 A^4 / Pi^2] - Kc^2 (9 / 25)
= [2 - 2 [1 + 4 A^4 / Pi^2]^0.5 + 4 A^4 / Pi^2]
or
- Kc^2 (9 / 25) = [2 - 2 [1 + 4 A^4 / Pi^2]^0.5]
or
2 [1 + 4 A^4 / Pi^2]^0.5 = 2 + Kc^2 (9 / 25)
or
[1 + 4 A^4 / Pi^2]^0.5 = 1 + Kc^2 (9 / 50)
or
[1 + 4 A^4 / Pi^2] = 1 + Kc^2 (9 / 25) + Kc^4 (81 / 2500)
or
4 A^4 / Pi^2 = Kc^2 (9 / 25) + Kc^4 (81 / 2500)
or
A^4 = Pi^2 Kc^2 (9 / 100) + Pi^2 Kc^4 (81 / 10,000)

This expression gives A as a function of Kc for M = (1 / 2).

For the case of M = 2
equate the two expressions for Nr^2 to get:
[4 A^4 / Pi^2] - Kc^2 (9 / 25) = (1 / 16)[2 - 2 [1 + 64 A^4 / Pi^2]^0.5 + 64 A^4 / Pi^2]
or
- Kc^2 (9 / 25) = (1 / 16)[2 - 2 [1 + 64 A^4 / Pi^2]^0.5]
or
- Kc^2 (72 / 25) = 1 - [1 + 64 A^4 / Pi^2]^0.5 or
[1 + 64 A^4 / Pi^2]^0.5 = 1 + Kc^2 (72 / 25)
or
[1 + 64 A^4 / Pi^2] = 1 + 2 Kc^2 (72 / 25) + Kc^4 (72 / 25)^2
or
64 A^4 / Pi^2 = 2 Kc^2 (72 / 25) + Kc^4 (72 / 25)^2
or
A^4 = [Pi^2]{Kc^2 (72 / (32 X 25) + Kc^4 (9 / 25)^2}
= [Pi^2]{Kc^2 (9 / (4 X 25) + Kc^4 (9 / 25)^2}
= [Pi^2]{Kc^2 (9 / 100) + Kc^4 (81 / 625)}

We also have an expression for Kc as a function of A from ellipse geometry.
Define:
h = [(A - 1) / (A + 1)]^2
and
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) + ....]

Then:
Kc = [(1 + A) / 2] Kh

The spheromak operating point should lie at the intersection point of those two functions where Kc > 1.

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

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 ~ ^2 + Pi^2

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

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 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.

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

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.

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

Z QUANTIZATION:
The Fine Structure constant stability comes from inherent So 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

Try Np = 83, Nt = 465, P = 1013
= 172,225 + 1,946,025 = 2,118,250

P = Np + 2 Nt = 1013

Try Np = 83, Nt = 463, P = 1009
= 172,225 + 1,929,321 Kc^2 = 2,101,546 + (Kc^2 - 1)1,929,321

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

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

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

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

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

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

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

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^2 = (Np / Nt)^2 that correspond to a precise balance between the electric and magnetic forces along the entire length of the charge string path. 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.
applies.

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

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

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

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

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

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

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

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

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

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

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

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

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

Note that the 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.

CONFINED PHOTONS:

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

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

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

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

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

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

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

Hence:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Note that in plotting this graph Nr^2 is itself a complex function of So. In the expression for the Planck constant:
Pi = 3.141592653589793
and
Pi^2 = 9.869604401

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

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

At steady state Fh is constant so:
dFh = 0
and
Nr = (Np / Nt)
(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 June 1, 2019.