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PLANCK CONSTANT

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

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 phonton to the frequency Fp of that photon via the formula:
Ep = h Fp
However, that formula gave no insight as to the underlying mechanism.

This author has shown that a quantum charge spheromak with energy Es and frequency Fs stores energy in accordance with:
Es = hs Fs

If energy Es changes from Esa to Esb and frequency Fs changes from Fsa to Fsb then:
(Esa - Esb) = hs (Fsa - Fsb)

Note that (Esa - Esb) is a change in spheromak potential energy.

The constant hs can be determined theoretically by calculation of:
hs = (Esa - Esb) / (Fsa - Fsb)
 

LINEAR COMBINATIONS OF SPHEROMAKS:
Assume that a stable particle is composed of multiple spheromaks. Each of these component spheromaks behaves in accordance with:
Ei = h Fi
where i indicates a particular spheromak.

Then the total particle energy is given by:
Sum over Ei = h (Sum over Fi)
and
[(Sum over Eia) - (Sum over Eib)] = h [(Sum over Fia) - (Sum over Fib)]
or
[Sum over (Eia - Eib)] = h [Sum over (Fia - Fib)]

For a particle composed of multiple spheromaks that absorb or emit photons at a characteristic frequency this formula constrains the change in distribution of energy between the various particle component spheromaks. Note that if (Fia - Fib) is common to all the component spheromaks then the (Eia - Eib) values are the same for all the component spheromaks but the Eia values are different. Hence the fractional change in energy of the component spheromaks is different. As a result only certain spheromak energy combinations result in stable particles.

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

However, this assumption is not quite correct because the photon has both energy and momentum. When a photon is emitted by an electron there is recoil momentum that becomes electron kinetic energy rather than photon energy. Thus the change in spheromak energy is:
(Esa - Esb) = Ep + dEk
where:
dEk = electron recoil kinetic energy

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

The internationally accepted value for h which as of November 2018 will be used for defining a kilogram is:
h = 6.62607015 X 10^-34 J-s
= 6.62607015 X 10^-34 kg m^2 / s
 

KILOGRAM DEFINITION:
The accepted definition of a second is the time required for photons absorbed by a well defined Cs-133 energy transition to resonate 9,192,631,770 times. This is a 9.19263177 GHz microwave absorption by cesium vapor in a field free enclosure.

The definition of a metre is the distance light travels in a vacuum in a specified time. The speed of light is defined as precisely:
299,792,458 m / s

Hence an assumption of:
h = 6.62607015 X 10^-34 kg m^2 / s
defines the size of a kilogram.

However, the problem with this methodology for defining a kilogram, which will likely cause endless confusion in the future, is that the real physical constant is not the experimentally measured value of h, it is the theoretically calculated value hs. As shown herein below:
hs = 6.625943023 X 10^-34 J-s

The difference between h and hs is primarily due to the electron recoil kinetic energy associated with photon emission. Another source of problems in experimental measurement of h is fields external to the charged particle spheromak which distort the spheromak.
 

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 hs is actually a geometrical constant that is a function of an electron charge, the speed of light and the permiability of free space. The geometrical constant arises from co-incidence of an integer ratio with an irrational number at the minimum energy state of a spheromak in free space.

It is shown that most methods of experimentally measuring the Planck Constant generate small errors due to the momentum of emitted or absorbed photons. Although the Planck Constant is normally defined in terms of photon properties the photon quantization is actually due to energy quantization within the electromagnetic spheromaks that absorb or emit the photons.

Another issue in high precision experimental measurement of the Planck Constant is suppression of electric and magnetic fields that can slightly distort the spheromak geometry. These issues make the measured value of the Planck Constant slightly dependent on the precise definition of the Planck constant and on the method used for its experimental measurement.
 

RADIATION AND MATTER:
Atomic quantum charged particles have associated electromagnetic spheromaks. Electromagnetic spheromaks attempt to reach stable energy states by absorbion or emission of radiation photons. During radiation absorption and emission total energy and total momentum are conserved. Every packet of potential energy, including both charged particles and photons, has a characteristic frequency. During photon emission the emitting particle's characteristic frequency decreases and the number of photons present increases. During photon absorption the absorbing particle's characteristic frequency increases and the number of photons present decreases.

The energy versus frequency behavior of charged particles and photons is determined by the Planck constant, which lies at the heart of quantum mechanics. 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.

In our local universe there is an overall tendency for the photon population to gradually change from a small number of high energy photons into a large number of lower energy photons. This tendancy determines the direction of evolution of the universe.
 

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

It is shown herein that the total energy of a spheromak has a relative minimum at:
So^2 ~ 4.104474517.

Over time natural spheromaks will absorb or emit photons until they reach this stable minimum energy condition. Hence particles such as electrons and protons that have existed for a long time all have So values that satisfy:
So^2 ~ 4.104474517.
At this value of So the value of (dEtt / dFh) for an electromagmetic spheromak is given by:
(dEtt / dFh) = hs = Planck constant, where:
Fh = the characteristic frequency of the circulating quantum charge that forms an electromagnetic spheromak.
The characteristic frequency is the frequency at which a circulating quantum charge retraces its previous path.
 

PLANCK CONSTANT:
The Planck Constant relates a spheromak's energy to its characteristic frequency. The origin of the Planck Constant lies in the manner in which energy is stored in an electromagnetic quantum charge spheromak.

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

The Planck Constant is a fundamental parameter in quantum mechanics. The Planck Constant is a frequently reoccurring composite of other physical and mathematical constants. This web page shows the origin of the Planck Constant.

A spheromak's electric and magnetic field structure allows individual quantized charges to act as stable stores of potential energy. The behavior of these energy stores is governed by electrodynamics. This web page shows the mathematical relationship between spheromaks and quantum mechanics.

It is shown herein that the total field energy Ett of a quantum charge electromagnetic spheromak at steady state in field free space is given by:
Ett = hs Fh
where Fh is the characteristic frequency of the spheromak and hs is a composite constant generally referred to as the Planck constant.

When a charged particle 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 charged particle 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 charged particle at rest in field free space with initial spheromak potential energy Eao absorbs a photon with energy Ep to conserve momentum the charged particle 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 charged particle 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 charged particle at rest with initial potential energy Eao emits a photon with energy Ep to conserve momentum the charged particle 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 charged particle rest mass potential energy after emission of the photon.

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

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

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

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

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 charged particles 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 charged particle potential energy (Ea -Eb) is slightly greater than the photon energy Ep and on photon absorption by a charged particle at rest the change in charged particle potential energy is slightly less than the photon energy Ep. These issues are further complicated by thermal motion of the charged particles.
 

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

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

Using spheromak theory, subject to the accuracy of the spacial energy density assumptions, we can precisely calculate a theoretical values for:
hs = (Eb - Ea) / (Fhb - Fha)
= dEtt / dFh

Note that the spheromak spacial energy density assumptions are only truly valid in field free space, which is often not the case during a practical measurement of the Planck Constant. While the particle spheromak magnetic fields are large compared to an applied laboratory magnetic field, the system is not totally distortion free.
 

ASSUMED VALUE FOR PLANCK CONSTANT:
The published new assumed value for the Planck Constant h is:
h = 6.62607015 X 10^-34 J-s
= 6.62607015 10-34 m^2 kg / s.

The reason for assuming this value is to precisely define a kilogram. However, defining a kilogram in this manner forces new precise definitions of a quantum charge Q and permiability of free space Mu.

The accuracy of measurement of a quantum charge has been limited by the experimental error in determination of h. Hence it is highly desirable to find a precise value for hs.
 

ORIGIN OF PLANCK CONSTANT:
The parameter hs is a function of:
Mu = permiability of free space;
C = speed of light in a vacuum;
Q = proton charge;
Pi = (circumference / diameter) of a circle
= 3.141592653589793
Pi^2 = 9.869604401

To understand the relationship of spheromak parameters to the Planck constant it is necessary to derive a closed form expression for the total field energy of a spheromak and hence an elementary atomic charged particle.
 

PARAMETER DEFINITIONS:
Define for a spheromak in free space:
Rc = minimum radius of inner spheromak wall;
Rs = maximum radius of outer spheromak wall;
Ro = (Rs Rc)^0.5 = radius where the potential energy well is deepest;
So = [Rs / Ro] = [Ro / Rc] = spheromak shape parameter;
So^2 = (Rs / Rc);
Hs = distance of spheromak wall from the equatoral plane;
Hf = maximum value of |Hs|
2 Hf = spheromak overall length;
Lh = charge hose length;
Np = number of poloidal charge motion path turns contained in Lh;
Nt = number of toroidal charge motion path turns contained in Lh;
Nr = Np / Nt;
Rf = spheromak wall radius at H = Hf and H = -Hf;
Lp = Pi (Rs + Rc) = wall tangential poloidal turn length;
Lt = Pi (Rs - Rc) = wall tangential toroidal turn length;
Bpo = poloidal magnetic field strength at the center of the spheromak;
Upo = (Bpo^2 / 2 Mu) = maximum field energy density at the center of the spheromak;
Upo = (Uo) = maximum field energy density at the center of the 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 = [(Mu C Qa) / (4 Pi Ro^2)]

As shown on the web page titled ELECTROMAGNETIC SPHEROMAK the peak magnetic field strength Bpo at the center of a spheromak can also be expressed as:
Bpo = I [(Mu Qs C) / (2 Pi^2 Rc^2)] {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5}
= I (Mu Qs C) / (2 Pi^2 Ro^2)(Ro / Rc)^2 {Nr / {[Nr (So^2 + 1)]^2 + [So^2 - 1]^2}^0.5}
= I (Mu 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.
 

RELATIONSHIPS BETWEEN Efs, Ett and Fh:
An upper limit on the possible spheromak field energy is Efs. As shown on the web page titled: SPHEROMAK ENERGY this upper limit is given by:
Efs = (Bpo^2 / 2 Mu) (Ro)^3 Pi^2

As shown on the web page titled: SPHEROMAK ENERGY the actual field energy trapped by a spheromak in free space at steady state is given by:
Ett = Efs {1 - [(So -1)^2 / (So^2 + 1)]^2}
where:
So ~ 2.026

Thus:
(Ett / Efs) = {1 - [(So -1)^2 / (So^2 + 1)]^2}

The characteristic frequency Fh of an atomic particle spheromak is given by:
Fh = (C / Lh)
= C / {[2 Pi Np (Rs + Rc) / 2]^2 + [2 Pi Nt (Rs - Rc) / 2]^2}^0.5
= C / [Pi {[Np (Rs + Rc)]^2 + [Nt (Rs - Rc)]^2}^0.5]
= C / [Pi Nt {[Nr (Rs + Rc)]^2 + [(Rs - Rc)]^2}^0.5]
= C / [Pi Nt Rc {[Nr (So^2 + 1)]^2 + [(So^2 - 1)]^2}^0.5]
= C Ro / [Pi Nt Rc Ro {[Nr (So^2 + 1)]^2 + [(So^2 - 1)]^2}^0.5]
= [C / (Pi Nt Ro)] [So / {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5]

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

Thus:
Efs = (Bpo^2 / 2 Mu) (Ro)^3 Pi^2
= (Bpo^2 / 2 Mu) (Ro)^4 Pi^2 [1 / Ro]
= (Bpo^2 / 2 Mu) (Ro)^4 Pi^2 Fh Pi Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [C So]

As shown on the web page ELECTROMAGNETIC SPHEROMAK from far field spheromak energy density matching:
Bpo^2 = [Mu^2 C^2][Qa / (4 Pi Ro^2)]^2

Thus:
Efs = (Bpo^2 / 2 Mu) (Ro)^4 Pi^2 Fh Pi Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [C So]
= ([Mu^2 C^2][Qa / (4 Pi Ro^2)]^2 / 2 Mu) (Ro)^4 Pi^2 Fh Pi Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [C So]
= Fh ([Mu C Qa^2] / 32) Pi Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]
= Fh ([Mu C Qa^2] / 4 Pi) [Pi^2 / 8] Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

 

The spheromak field energy Ett is given by:
Ett = (Ett / Efs) Efs
= (Ett / Efs) Fh ([Mu C Qa^2] / 4 Pi) [Pi^2 / 8] Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]
= h Fh
where the Planck constant h is given by:
h = ([Ett / Efs] [Mu C Qa^2] / 4 Pi) [Pi^2 / 8] Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]
= ({1 - [(So -1)^2 / (So^2 + 1)]^2} [(Mu C Qa^2) / (4 Pi)] [Pi^2 / 8] Nt {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

In this formula at steady state So spontaneously adopts the value that minimizes Ett. As shown by the following graph of the So dependent term of Ett vs So the operating value of So is:
So ~ 2.026

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

In the expression for the Planck constant:
So ~ 2.026
Pi = 3.141592653589793
and
Pi^2 = 9.869604401

As shown on the web page titled: SPHEROMAK ENERGY at the spheromak minimum energy operating point the approximate values are:
{1 - [(So -1)^2 / (So^2 + 1)]^2} {[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So] ~ [2.2882]
So ~ 2.026
So^2 ~ 4.104676
Nr^2 ~ 0.5297583716
Nr ~ 0.7278450189
Ett = [Mu C Qa^2 / 4 Pi] [Pi^2 / 8] Fh Nt [2.2882]
Ett = h Fh

Under the proposed new basic constant definitions the value of h is:
h = 6.62607015 10-34 m^2 kg / s
= 6.62607015 X 10^-34 J-s

and the speed of light C in a vacuum is:
C = 2.99792458 X 10^8 m / s
and
[Pi^2 / 8] = 1.23370055
which taken together force a value on the product:
Qa^2 Mu

The nominal values are:
Qa = 1.60217662 X 10^-19 coulombs
and
Mu = 4 Pi X 10^-7 T^2 m^3 / J
but the least significant digits of Qa and Mu must be adjusted to cause h to become:
h = 6.62607015 X 10^-34 J-s

Hence:
Nt = h / {[Mu C Qa^2 / 4 Pi] [Pi^2 / 8] [2.2882]
= 6.62607015 X 10^-34 J-s / {10^-7 X 2.99792458 X 10^8 m / s X (1.60217662 X 10^-19 coulombs)^2 x 1.23370055 X 2.2882}
= 6.62607015 X 10^-34 J-s / {(7.695582224 X 10^-37) X 1.23370055 X 2.2882}
= 305.0077026 turns

Then:
Np = Nr Nt
= 0.0.7278450189 (305.0077026)
= 221.998337 turns

Since Np and Nt must be exact integers with no common factors the only possible exact solution for Np, Nt is:
Nt = 305
Np = 222

which gives the precise value of Nr as:
Nr = (Np / Nt)
= 222 / 305
= 0.7278688525
 

EXACT VALUES:
Since Np and Nt are integers we know that the exact value of Nr is given by:
Nr = (Np / Nt)
= 222 / 305
= 0.7278688525

Hence the exact value of Nr^2 is given by:
Nr^2 = (0.7278688525)^2
= 0.5297930664

Since we now have an accurate value for Nr^2 we can accurately determine So^2 using the common boundary condition. With an accurate value of So^2 we can calculate an accurate value of h in terms of Mu, C and Q.
 

Recall that from the common boundary condition:
Nr^2 = ([8 - {[Pi]^2 [(So^2 - 1) / (So^2 + 1)]^2}] / {[Pi]^2 - [16 / (So^2 - 1)^2]})

This boundary condition is only precisely valid in field free space.

Using the precise value of Nr^2 we can find the corresponding precise value of So^2. In order to do so use the approximate value of So^2 in combination with the slope of the Nr^2 versus So^2 curve at the approximate solution to find the exact solution.

Let X = So^2 - 1
Then:
Nr^2 = {8 - [Pi]^2 [X / (X + 2)]^2} / {[Pi]^2 - [16 / (X^2)]}

Recall that:
Pi^2 = 9.869604401

Try So = 2.0260000
Then:
X = So^2 - 1
= 3.104676
and
X^2 = 9.639013065
giving a trial value of Nr^2 as:
Trial Nr^2 = {8 - [Pi^2] [X / (X + 2)]^2} / {[Pi]^2 - [16 / (X^2)]}
= {8 - [9.869604401] [3.104676 / (5.104676)]^2} / {[9.869604401] - [16 / (9.639013065)]}
= 4.34913352 / {8.209683422}
= 0.5297565444

Recall that the target value of Nr^2 is:
(222/ 305)^2 = 0.5297930664

Thus for a trial value of So = 2.02600000 the target value of Nr^2 exceeds the trial value of Nr^2 by: 0.5297930664 - 0.5297565444 = 3.652198 X 10^-5

We need to make the trial value of Nr^2 slightly larger which implies making the trial value of So slightly smaller.

 

Try So = 2.02590000
Then:
X = So^2 - 1
= 3.10427081
and
X^2 = 9.636497262
giving a trial value of Nr^2 as:
Trial Nr^2 = {8 - [Pi^2] [X / (X + 2)]^2} / {[Pi]^2 - [16 / (X^2)]}
= {8 - [9.869604401] [3.10427081 / (5.10427081)]^2} / {[9.869604401] - [16 / ( 9.636497262)]}
= 4.349506903 / 8.209250066
= 0.5298299928

Recall that the target value of Nr^2 is:
(222 / 305)^2 = 0.5297930664

Thus for a trial value of So = 2.02590000 the trial value of Nr^2 exceeds the target value of Nr^2 by: 0.5298299928 - 0.5297930664 = 3.692635 X 10^-5
 

FINDING EXACT VALUES:
Thus:
d(Trial value of Nr^2) / dSo = (0.5298299928 - 0.5297565444) / (2.02590000 - 2.0260000)
= 7.34484 X 10^-5 / (- 10^-4)
= - 0.734484

Hence: [So|target] = [So|trial] + [dSo / dNr^2] {[Nr^2|target] - [Nr^2|trial]}
= 2.02600000 + [- 1 / 0.734484] [0.5297930664 - 0.5297565444]
= 2.02600000 - [4.972470469 X 10^-5]
= 2.02600000 - .00004972470469
= 2.025950275

The corresponding value of X is:
X = So^2 - 1
= 3.104474517

X^2 = 9.637762026

We can check this calculation by recalculating Nr^2.
Nr^2 = {8 - [Pi^2] [X / (X + 2)]^2} / {[Pi^2] - [16 / (X^2)]}
= {8 - [9.869604401] [3.104474517 / (5.104474517)]^2} / {[9.869604401] - [16 / (9.637762026)]}
= 4.349319182 / 8.209467955
= 0.529793064

Hence:
Nr = 0.7278688508 as compared to target value of:
(222 / 305) = 0.7278688525

Thus the calculated precise So value gives a precise Nr value that is accurate to eight significant figures.

Thus at the spheromak operating point So is:
So = 2.025950275
and
So^2 = 4.104474517

Recall that the theoretical value of hs is given by:
hs = [(Mu C Qs^2) / (4 Pi)] [Pi^2 / 8] Nt
{1 - [(So -1)^2 / (So^2 + 1)]^2}
{[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]

Evaluation of the terms of hs gives:
[(Mu C Qs^2) / (4 Pi)] [Pi^2 / 8] Nt
= {(7.695582224 X 10^-37) X 1.23370055 X 305 J-s
= 2.895683427 X 10^-34 J-s

In evaluating this term it is important to be aware that the published value of the proton charge Qs may not be accurate due to being derived from a value for h which is inaccurate.  

{1 - [(So -1)^2 / (So^2 + 1)]^2}
= {1 - [(1.025950275)^2 / (5.104474517)]^2}
= {1 - 0.0425209706}
= 0.9574790294

{[Nr^2 (So^2 + 1)^2] + [(So^2 - 1)]^2}^0.5 / [So]
= {[0.5297930664 (5.104474517)^2] + [(3.104474517)]^2}^0.5 / [2.025950275]
= {[13.80410806] + [9.637762027]}^0.5 / [2.025950275]
= 2.389831854

Hence the theoretical value of hs using Q, Mu, and C values valid as of May 2018 is:
hs = [2.895683427 X 10^-34 J-s] [0.9574790294] [2.389831854]
= 6.625943023 X 10^-34 J-s

By comparison the value for h chosen for defining a kilogram after November 2018 is:
h = 6.62607015 10-34 m^2 kg / s.
 

RECOIL KINETIC ENERGY ERROR:
Using May 2018 definitions of Qs and Mu these two values for h are in agreement to an accuracy of 1.9 parts in 100,000. The May 2018 discrepency may be due to errors in Qs and Mu or due to conservation of linear momentum on the emission of a photon by an excited charged particle or may be due to spheromak distortion in a high magnetic field. It is necessary to examine exactly how h is experimentally measured to understand these error sources. The fractional discrepency is:
[(6.62607015 - 6.625943023) / 6.62606957] = 1.918588 X 10^-5

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

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

The photon energy experimentally used for determination of h by non-magnetic methods is typically of the order of:
Ep = 20 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) = 20 eV / [2 (51.12 X 10^4 eV)]
= 2 / (102.24 X 10^3)
~ 1.95 X 10^-5

Thus the discrepency between the theoretical value of hs and the experimental value of h may be caused by failure to properly take into account recoil kinetic energy when an electron emits a photon during an atomic energy transition. When an electron emits a 20 eV photon about 2 / 100,000 of the change in potential energy becomes electron linear recoil kinetic energy rather than photon energy. If this error is uncorrected the experimentally measured value of h will be slightly larger than the theoretical value of hs calculated herein. Note that the hs value calculated herein is actually the change in charged particle potential energy with respect to a change in charged particle natural frequency. The energy carried away by the photon will be slightly less than the decrease in charged particle potential energy due to the small increase in charged particle kinetic energy on emission of a photon. This increase in charged particle kinetic energy must occur to satisfy the law of conservation of linear momentum.

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

Note that an experimental measurement of h might be more precise if the charged particle emitting or absorbing the photon is a proton instead of an electron. Then the recoil momentum results in less recoil energy. This issue 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 some disturbance of the applied magnetic fields caused by the circulating electrons in the hydrogen.
 

CONCLUSION:
The spheromak model of a charged particle provides a means of accurately calculating the Planck constant hs in terms of Pi, Mu, Q and C. In highly accurate experimental measurements of h it is necessary to take into account the charged particle recoil kinetic energy. When recoil kinetic energy is properly taken into account there is apparent agreement between the theoretical hs and the experimental determinations of the Planck constant h to about 6 significant figures.

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

This web page last updated May 7, 2018.

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