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**CONFINED PHOTONS:**

This web page develops the relationships between charged particle magnetic moment and charged particle mass.

Confined photons account for almost all the inertial mass Mp and the de Broglie wavelength Lamdap properties of charged particles such as electrons, protons and atomic nuclei. Each charged particle can be thought of as having an associated confined photon within its spheromak walls which oscillates forming a standing wave both radially and circumferentially around the spheromak symmetry axis.

The mathematical model for a spheromak with a quantum charge Q can be applied to electrons and protons. In each case if the spheromak radius Ro is chosen so that the theoretical and experimental magnetic moments agree then the spheromak energy content calculated from the static spheromak electric and magnetic fields is much much smaller than the particle energy calculated from the experimentally measured particle inertial mass. Hence a simple quantum charge spheromak with static electric and magnetic fields does not by itself adequately model a real charged particle.

In order for theoretical calculations match experimental data it is necessary to assume that real quantum charged particles such as electrons, protons and atomic nuclei consist of a static field spheromak with an additional amount of electrically neutral energy contained within a confined photon of frequency Fc that provides the missing rest mass. The confined photon exists within the spheromak walls. Its propagation rate measured at the outer wall on the spheromak equatorial plane is the speed of light C. The spheromak static electric and magnetic fields account for the particle charge and particle magnetic moment but the energy associated with the confined photon accounts for almost all of the particle's rest mass.

Note that the frequency Fc of the confined photon is hundreds of times larger than the frequency Fh of the static field spheromak because the confined photon follows a simple circular path with speed C at the outer wall of the spheromak whereas the spheromak charge circulation follows a long spiral path, also at speed C.

The confined photons account for the particle mass and the wave-particle duality behaviour of observed in the de Broglie two slit experiment.

Note that with respect to wave-particle duality protons and molecular nuclei behave in a manner analogous to an electron. In each case the particle rest mass energy is dominated by the energy contained in the particle's confined photon rather than by energy contained in the static electric and magnetic fields of the particle's spheromak.

**DERIVATION OF DE BROGLIE WAVELENGTH:**

Assume the basic hypothesis of quantum mechanics that for any quantum particle:

**E = h F**

In this relationship:

E = the total particle energy (essentially the confined photon energy)

and

F = C / Lamda.

C = speed of light;

and

Lamda = wavelength in free space at frequency F.

Define:

Eo = energy of confined photon for a charged particle at rest;

Fo = frequency of confined photon for a charged particle at rest;

Lamdao = wavelength of confined photon in free space when the charged particle is at rest;

Lamdo Fo = C;

Mo = equivalent rest mass of confined photon at rest;

Eo = Mo C^2 (special relativity);

Eo = h Fo (Quantum mechanics hypothesis);

V = axial velocity of charged particle

Mc = confined photon energy equivalent mass when particle is in motion at velocity V;

Ec = energy of confined photon for charged particle moving with axial velocity V;

Ec = Mc C^2 (special relativity);

Fc = frequency of confined photon for charged particle moving with velocity V;

Ec = h Fc (Quantum mechanics hypothesis)

Lamdac = wavelength of confined photon in free space for charged particle moving with velocity V;

Fc Lamdac = C;

Fp Lamdap = C

Lamdap = apparent wavelength of particle calculated from the interference pattern;

Fp = apparent particle wave frequency calculated from the particle beam interference pattern;

P = momentum each charged particle

Hence:

**P** = Mc V = (Ec V) / C^2

= (h Fc V) / C^2

= (h V) / C^2)(C / Lamdac)

= (h / Lamdac)(V / C)

= (h / Lamdap)

= **(h Fp / C)**

which is the de Broglie equation.

Hence a particle is characterized by two frequencies, Fc associated with its confined photon and Fp associated with its linear momentum.

Note that as the electron velocity V increases Lamdap, which is the apparent electron wavelength as determined from the dual slit interference pattern, decreases. Lamdao is the wavelength of the standing wave inside a stationary electron. In a practical experimental apparatus:

Lamdap >> Lamdao

**BASIC PHOTON CONFINEMENT THEORY:**

From the web page titled: THEORETICAL SPHEROMAK outside the spheromak wall the field energy density U is given by:

Up = Uo [(Ro^2 / (Ro^2 + R^2 + H^2)]^2

where:

Up = field energy density outside the spheromak wall;

R = cylindrical radius from the spheromak symetry axis;

H = height above the spheromak equatorial plane

Ro = a spheromak radius indicative of the spheromak size.

Hence:

dU / dR = 2 Uo [(Ro^2 / (Ro^2 + R^2 + H^2)][(-Ro^2 2 R) / (Ro^2 + R^2 + H^2)^2]

= 2 Uo [(Ro^4 / (Ro^2 + R^2 + H^2)^2][(- 2 R) / (Ro^2 + R^2 + H^2)]

= 2 U [(- 2 R) / (Ro^2 + R^2 + H^2)]

or

**- dU / dR = 4 U R / (Ro^2 + R^2 + H^2)**

From the web page titled: THEORETICAL SPHEROMAK for points on the spheromak wall:

Hs^2 = [(Rs - R) (R - Rc)]

= Rs R - Rs Rc - R^2 + R Rc

where:

Rc = inside radius of spheromak;

and

Rs = outside radius of spheromak:

and

Rs Rc = Ro^2

For points outside the spheromak wall:

H^2 > Rs R - Rs Rc - R^2 + R Rc

Hence:

(Ro^2 + R^2 + H^2) > Ro^2 + R^2 + Rs R - Rs Rc - R^2 + R Rc

= Ro^2 + Rs R - Rs Rc + R Rc

Recall that:

Ro^2 = Rs Rc

giving:

(Ro^2 + R^2 + H^2) > Rs Rc + Rs R - Rs Rc + R Rc

= R (Rs + Rc)

Hence outside the spheromak wall:

(- dU / dR) < 4 U R /[R (Rs + Rc)]

or
- dU / dR < 4 U / (Rs + Rc)

or
-dU / dR < 2 U / [(Rs + Rc) / 2]

Inside the spheromak wall:

Ut = Uto(Ro / R)^2

or

dUt / dR = 2 Uto (Ro / R)(- Ro / R^2)

or

**-dU / dR = 2 U / R**

In general (- dU / dR) inside the spheromak wall is everywhere greater than (- dU / dR) outside the spheromak wall. Hence the maximum photon confinement due to change in field energy density occurs inside the spheromak wall.

The condition for photon circular orbital confinement with wavelength Lamda is:

d(Lamda) / Lamda = dR / R

or

d(V / Fc) / (V / Fc) = dR / R

or

dV / V = dR / R

Define:

Vs = photon propagation rate at R = Rs;
Vo = photon propagation rate at R = Ro

**Assume that the index of refraction (Vs / V) inside the spheromak wall is given by:
Vs / V = [U / Us]^0.5**

or

(Vs / Vo) (Vo / V) = [Uo / Us]^0.5 [U / Uo]^0.5

where:

(Vs / Vo) = [Uo / Us]^0.5

and

(Vo / V) = [U / Uo]^0.5

Hence:

V / Vo = [Uo / U]^0.5

= [Uto / U]^0.5

= [Uto / {Uto [Ro / R]^2}]^0.5

= (R / Ro)

which meets the confinement condition.

Thus photons orbiting the spheromak axis of symmetry are confined inside the spheromak wall.

At R = Ro:

The confined photons on a circumferential path will satisfy:

N Lamdao = 2 Pi Ro

where Lamdao is the wave length at R = Ro and N is a small positive integer.

Hence:

N (Vo / Fc) = 2 Pi Ro

or

Fc = N Vo / 2 Pi Ro

The time for a photon to move radially from Rc to Rs is given by:

dR / dT = V = Vo (R / Ro)

or

dR / R = (Vo / Ro) dT

or

Ln(Rs / Rc) = (Vo / Ro)(Ts - Tc)

Fc = M / [2 (Ts - Tc)]

= [M / 2] [Vo / Ro Ln(Rs / Rc)]

where M is a small positive integer

For radial photon oscillations:

Fc = [M / 2] [Vo / (Ro Ln(Rs / Rc))]

= [M / 2] [Vo / Vs][Vs / (Ro Ln(Rs / Rc))]

= [M / 2] [Ro / Rs][Vs / Ro Ln(Rs / Rc)]

For Vs ~ C and Ro / Rs = 1 / So:

Fc ~ [M / 2 So] [C / Ro Ln(Rs / Rc)]

For a confined photon oscillation around the 2 Pi Ro circumference:

Fc = Vo / Lamdao

= N Vo / 2 Pi Ro

= N Vs (Vo / Vs) / 2 Pi Ro

= N Vs (Ro / Rs) / 2 Pi Ro

~ N C / So 2 Pi Ro

For a spheromak:

Np = 222

Nt = 305

Fh = spheromak natural frequency

giving:

Fh = C / [(Np 2 Pi (Rs +Rc) / 2)^2 + (Nt 2 Pi(Rs - Rc) / 2)^2]^0.5

or

Fh = C / Pi Nt Rc [(Np / Nt)^2 ((Rs/ Rc) + 1))^2 + ((Rs / Rc) - 1))^2]^0.5

= C / Pi Nt Ro (Rc / Ro) [(Np / Nt)^2 ((Rs/ Rc) + 1))^2 + ((Rs / Rc) - 1))^2]^0.5

= C So / {Pi Nt Ro [(Nr)^2 (So^2 + 1))^2 + (So^2 - 1))^2]^0.5}

= [C / Pi Ro Nt] [So / {[(Nr)^2 (So^2 + 1))^2 + (So^2 - 1))^2]^0.5}]

From GF Spheromak Magnetic Moment:

[So / {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5]
= 0.418439

Hence:

Fh = [C / Pi Ro Nt][.418439]

Recall that for circumferential photon oscillations:

Fc ~ N C / So Ro

and that:

So = 2.025950275

Hence for circumferential oscillations:

Fc / Fh = [N C / So 2 Pi Ro] / {[C / Pi Ro Nt][.418439]}

= [N / 2 So] / {[1 / Nt][.418439]}

= [N Nt] / [2 (.418439) So]

= N [305] / [2 X 0.418439 X 2.025950275]

= N (179.89)

For radial confined oscillations
Fc = [M / 2 So] [C / Ro Ln(Rs / Rc)]

Fh = [C / Pi Ro Nt][.418439]

Fc / Fh = [M / 2 So] [C / Ro Ln(Rs / Rc)] / {[C / Pi Ro Nt][.418439]}

= [M / 2 So] [1 / Ln(Rs / Rc)] / {[1 / Pi Nt][.418439]}

= [M Pi Nt / [2 So (0.418439) Ln(Rs / Rc)]

= M [Pi Nt / [2 X 2.02595 X 0.418439 X 1.4303]

= M [1601]

= M [395.122]

If both oscillation modes simultaneously prevail:

Fc = [M / 2 So] [Vs / Ro Ln(Rs / Rc)] = N Vs / So 2 Pi Ro

or

[M] [ 1 / Ln(Rs / Rc)] = N / Pi

or

N / M = Pi / Ln(Rs / Rc)

= 2.1969

~ 2

From the web page titled: SPHEROMAK MAGNETIC MOMENT for a proton:

Experimental data gives:

Fc / Fh = 100 / 0.09235 = 1082.83

corresponding to N = 6, M = 3.

From the web page titled: SPHEROMAK MAGNETIC MOMENT for an electron:

Experimental data gives:

Fc / Fh = 100 / 0.25814

= 387.38

corresponding to N = 2, M = 1.

**SUMMARY:**

It appears that the proton's confined photon oscillates with N = 6, M = 3. It appears that the electron's confined photon oscillates with N = 2, M = 1.

Thus we have a mathematical model that connects the masses of electrons and protons to their magnetic moments.

It is possible that the proton states N = 2, M = 1 and N = 4, M = 2 correspond to quark states. These exist mathematically but are not found in real life.

**COMMENTS:**

1) Almost all of the charged particle rest mass energy is contained in its confined photon.

2) The photon confinement is a result of the change in index of refraction along the photon wave front due to the radially dependent static fields of the spheromak.

3) At smaller radii within the spheromak wall the static fields are higher than at larger radii. The higher fields reduce the photon propagation rate at small radii. The photon frequency remains unchanged. By this radially dependent index of refraction the electrically neutral photon is confined to orbit the spheromak axis of symmetry inside the spheromak wall.

4) In practical experiments net particle charges only exist in Q increments. Only spheromaks with static Q increment charges emit and absorb photons.

5) A neutron has no net static charge, has a net magnetic moment comparable to a proton and in free space spontaneously decays into a proton, an electron and a neutrino. A neutron might consist of two concentric spheromaks. The larger spheromak dominates the response to an externally applied magnetic field. This presumed neutron structure is only semi-stable in free space.

This web page last updated July 22, 2018.

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