Derivation Of Electromagnetic Radii

Derivation of the ElectroMagnetic radii of fundamental particles, the emission / absorption spectrum of the Hydrogen atom & an experimentally implicit definition of the Bohr Radius Riccardo C. Storti1

Abstract This manuscript demonstrates that the Electro-Gravi-Magnetics (EGM) construct accurately derives: (i) the Neutron Mean Square Charge Radius [to within 0.296(%)], (ii) the Neutron Magnetic Radius [to within 0.003(%)], (iii) the Proton Electric Radius [to within 0.062(%)], (iv) the Proton Magnetic Radius [to within 0.825(%)], (v) the Classical Proton RMS Charge Radius [to within 0.046(%)], (vi) a precise mathematical representation of the Neutron Charge Distribution and all its key features and (vii) the “1st” term of the emission / absorption spectrum of the Hydrogen atom (Balmer Series) [to within 0.131(%)]. Subsequently, the derivation of the complete spectrum is inferred. Additionally, an experimentally implicit definition of the Bohr Radius [to within 0.353(%) of the classical representation] is also derived.

1

[email protected]

1

1

I#TRODUCTIO#

The construct herein covers two principle areas in particle and quantum physics: (i) the experimentally verified Mean-Square charge radius of the Neutron “KX = -0.113(fm2)” {“fm” denotes “femtometre” [x10-15(m)]} [1] and (ii), the first term2 in the emission / absorption spectrum of the Hydrogen atom3 “λB = 656.46962(nm)”. [2] It shall be demonstrated that “KX” and “λB” may be derived precisely [to within 0.296(%) and 0.131(%) respectively] utilising the principles of ElectroGravi-Magnetics (EGM). [3-15] An experimentally implicit definition “rX” of the Bohr Radius “rBohr” is also derived utilising the fundamental harmonic wavelength of the EGM spectrum “λPV(1,r,M)”, by application of the experimentally implicit definition of the Planck Scale “Kλλh” as derived in [10]. Consequently, it is demonstrated that the first wavelength term of the Balmer Series may be derived in accordance with EGM methodology, denoted by “λA”. Storti et. Al. demonstrated in [14,15] that the mass-energy and radii of all fundamental particles may be easily described by the ratio of two spectra resulting in a simple harmonic sequence. This was formulated (in part) by the derivation of the Root-Mean-Square (RMS) charge radius of a free Proton and Neutron (“rπ” and “rν” respectively). [12] The RMS charge radius of the Neutron gained its name by dimensional analogy to “KX”. Since “KX” is a mean squared value and has a dimensional representation of “fm2”, the mathematical square-root operation on this dimension results in the terminology “RMS charge radius of a free eutron”. Although it is not an accurate portrayal of the nature of an RMS value, it is “quasimeaningful” terminology given that the conventional representation for “KX” (a physical displacement) takes form of a negative squared term (i.e. “-fm2”). 2

THEORETICAL MODELLI#G

2.1

THE NEUTRON

Assuming spherical Neutron geometry, the physical meaning of “rν” denotes the radial displacement value where the charge density is zero. Thus, “rν” represents the boundary between the positively charged core and negatively charged shell; denoting the radial ordinate at which the Neutron is in equilibrium with the surrounding Zero-Point-Field (ZPF). Graphically, this may be represented as follows, Neutron Charge Distribution

Charge Density

rν

ρ ch( r ) ρ ch r 0 ρ ch r dr

r Radius

Charge Density Maximum Charge Density Minimum Charge Density

Figure 1, 2 3

i.e. the experimentally verified wavelength. i.e. the Balmer Series.

2

r dr

Where, “rdr” denotes the radial displacement of zero charge density gradient (local minima). The first hypothesis to be tested is that “rν” may be converted to the conventional representation of “KX” as a “-fm2” quantity. This may be achieved by utilising the Neutron Charge Distribution “ρch” curve as follows, r

2 ρ ch ( r )

bi

1 . π

3

2

1 . b1. e 3 3 a 1 π

ai

.e

2

r

3

i= 1 a i

2

r

b2

a1

.e

a2

3

a2

(1)

Where, “a1, a2, b1 and b2” are mathematical constants physically satisfying the preceding equation, “r” denotes the magnitude of the radial position vector. Testing the hypothesis that “rν” is a highly precise alternate representation of the experimentally verified value of “KX” {to within 0.296(%), as presented by [1]}, also leads to the derivation of: i. The Neutron Magnetic Radius {to within 0.003(%), as presented by [16]]. ii. The Proton Electric Radius {to within 0.062(%), as presented by [16]]. iii. The Proton Magnetic Radius {to within 0.825(%), as presented by [16]}. iv. The classical RMS charge radius of the Proton {to within 0.046(%), as defined by [17]}. 2.2

THE HYDROGEN ATOM

The second hypothesis to be tested proposes a relationship exists between the first term of the Balmer Series, the fundamental harmonic wavelength of the EGM spectrum and the harmonic cut-off mode “nΩ”. This is shown to produce a “1st” term wavelength to within “0.131(%)” of the classical representation, as presented by [2]. Subsequently, it is inferred that all Balmer Series terms naturally proceed. An experimentally implicit definition of the Bohr Radius is also derived, to within “0.353(%)” of the classical representation as presented by [2]. 3

MATHEMATICAL MODELLI#G

3.1

CONVERSION OF NEUTRON POSITIVE CORE RADIUS Recognising that, ∞ 2 r . ρ ch ( r ) d r b 1

ρ ch ( r ) d r d r d r 4 . π .

Q

b2 0

0 ∞ 4.π . 0

(2)

3 4 2 r . ρ ch ( r ) d r . a 1 2

2 a 2 .b 1

(3) -3

where, “Q” denotes the charge density per unit Coulomb and takes the units “fm ”. Subsequently, equation (2) yields the relationship “b2 = -b1” such that, r

ρ ch ( r )

b1 π

.

1 . e

2

a1

1 . e

3

2

a2

3

a1

3

r

a2

(4)

“rν” represents the Zero-Point-Field (ZPF) equilibrium radius and intersects the radial axis at “r = rν” in accordance with equation (1). Hence, an expression for “rν” may be defined in terms of “a1, a2 and b1” as follows, rν

b1 π

3

.

1 . e

a1

3

2

rν

1 . e

a2

2

0

3

a1

a2

3

(5)

rν

2

a2

3 . ln

2

a 1.a 2

.

a1

2

2

a2

a1

(6)

The maximum value of “ρch” occurs at “r = 0” and may be determined (assuming spherical Neutron geometry) according to, 4 3 V( r ) . π . r 3 (7) Hence, the Charge Density per unit Coulomb “Q(r)” is expressed by [Q(r)
C/m3 * 1/C = 1/m3]: 1

Q( r )

V( r )

(8)

Therefore, the Charge Density per unit Coulomb per unit Quark “Qch(r)” may be written as, Q( r )

Q ch ( r )

3

(9)

Evaluating yields, 1

Q ch r ν = 0.1408

3

fm

(10)

This result may be expressed analytically by relating equation (4) to (7-9) when “r = 0” as follows, b1

1 4.π .r ν

3

π

.

1

1 3

3

a1

3

a2

(11)

Hence, 3

rν

3 . π . a1a2 4.π .b 1 a 3 a 3 2 1

3

(12)

The radial position “rdr” (as a function of “rν”) for which the gradient of the Charge Density “dρch(r)/dr” is zero may be determined as follows, r

d b1. 1 . e dr 3 a 3 1 π

a1

2

r

1 . e

2

r

2.b 1.r

a2

3

a2

. a 5 .e 1

3

π . a 1.a 2

a2

2

r 5 a 2 .e

a1

2

0

5

(13)

Simplifying yields, r dr

2

a2

5 a 1 .e

r dr

2

a1

5 a 2 .e

(14)

Therefore, r dr

rν

2

2

5 . ln

a1 a2

a 1.a 2

.

ln a 2 2 3 r dr . . 5 ln a 1 5.

r dr

3

4

rν

2

a2

2 2

a1

(15)

ln a 1 ln a 2

(16)

(17)

Evaluating yields, r dr = 1.0674 ( fm)

(18)

Exploratory factor analysis, with respect to equation (4), indicates that an infinite family of solutions for “a1, a2 and b1” exists to satisfy “ρch”. Therefore, we shall assume that “a2 = xa1” and “a1 = rν”. Subsequently, the values of “a2, b1 and x” may be determined as follows, substitute , a 2 x. a 1 3.

2

2 a 2 .b 1 K S

a1

2

KS

2.

substitute , a 1 r ν

3

solve , b 1

2 rν . 1

2

x

(19)

where, “KS” denotes the MS charge radius of a Neutron as derived utilising EGM methodology. Hence, 2 . KS

b1

3.r ν

2

2

1

x

(20)

Substituting “b1” into equation (11) yields an expression for “KS” in terms of “rν and x” as follows,

b1

1 4.π .r ν

3

π

.

3

1

1 3

3 a 2 2

2

a1

1

3.

substitute , a 2 x. a 1

3

a1

KS

2.

substitute , b 1

a2

8

2.

rν π

substitute , a 1 r ν

2

( x 1)

. x3 .

2

x

x 1

solve , K S

(21)

Hence, 2 3 . π . r ν ( 1 x) . x3 . 2 8 1 x x

KS

(22)

A solution for “x” may be found by performing the appropriate substitutions into equation (6) and solving numerically utilising the “Given” and “Find” commands with the “MathCad 8 Professional” environment as follows,

rν

2

3 . ln

a2 a1

.

a 1.a 2 2

a2

2 2

a1

substitute , a 2 x. a 1 rν

substitute , a 1 r ν

2

2 3 . ln( x) . r ν .

factor

2

x ( ( x 1) .( x 1) )

(23)

Given 2

1 3

2

x x

1

x

ln( x) .

(24)

Find( x)

(25)

Evaluating yields, x = 0.6829 a1 a2

=

0.8268 0.5647

(26) ( fm)

(26)

b 1 = 0.2071

(28) 2

K S = 0.1133 fm

5

(29)

The error produced by “KS” in relation to its experimental value “KX” [1] may be calculated according to “1 - KS / KX” as follows, 2 K X 0.113. fm (30) KX

1

= 0.295 ( % )

KS

(31)

ote: The experimental uncertainty of “KX” is “±0.005(fm2)” (as defined by [1]). Consequently, “KS” matches experimental measurement precisely, with zero error. The error described by equation (31) assumes an exact experimental value as defined by equation (30). We may graphically reinforce the preceding derivation by substituting the results for “KS, rν and x” into equation (4) and working in dimensionless form as follows, 2

r

KS

2.

ρ ch ( r )

3

3

π .r ν

5.

rν

. e

1.

e

r x .r ν

2

3

2

x

x

1

(32)

Evaluating “ρch” at specific conditions yields the appropriate results, ρ ch r 0

0.1408

ρ ch r ν

= 5.768.10

12 ρ ch 10 .( fm)

1

9

3

fm

0

(33)

Utilising the “Given” and “Find” commands, we may determine graphical inflections at “r1” and “r2” according to, Given r1

d

2

KS

2.

d r 12 3

. e

rν

d

5 2 π .r ν . x

d r 23 3

3

π .r ν

5.

1.

e

2

x .r ν

0

x

1

KS

2.

r1

3

3

(34) r2

3

2

. e

2

rν

r2

1.

e

x .r ν

2

0

3

2

x

x

1

r1

(35) Find r 1 , r 2

r2

(36)

Evaluating yields [r1 = 0.3766(fm), r2 = 0.6624(fm)], r1 r2

=

6

0.3766 0.6624

(37)

r1

d dr

ρ ch( r )

d dr 1 d

r2

ρ ch r 1

2

d r2 d

ρ ch( r )

2

d r 22 d

ρ ch r 2

2

d r 02

ρ ch r 0

r Radius

Figure 2, Evaluating specific conditions yields the appropriate results, d ρ ch r 1 dr 1 d

0.2539

2

d r 22 d

ρ ch r 2

= 0.5447 1.1032

2

d r 02

ρ ch r 0

(38)

rν 4 r . ρ ch ( r ) d r

0 ∞ 4 r . ρ ch ( r ) d r

4.π .

rν

0.0166 =

rν 2.

r ρ ch ( r ) d r

0.13 0.0705 0.0705

0 ∞ 2 r . ρ ch ( r ) d r

rν

(39)

7

We shall perform an additional test to ensure that no obvious algebraic errors have been inadvertently performed. To achieve this, we shall employ the exact analytical representation of the integrand, in this case “ρch”, as defined by standard mathematics tables as follows, [18] ∞ 2

r

b1 π

3

.

a1

1 . e

2

r a2

1 . e

3

dr

3

a1

a2

b1

. 1 2.π a 2 1

0

1 2

a2

(40)

Substituting appropriately produces, 2 . KS b1

.

1

2.π a 2 1

3.r ν

1 2

a2

2

2

x

1

1

.

2.π

rν

KS

1 2

x. r ν

2

4 2 3.π .r ν .x

(41)

Evaluating yields, KS

= 0.0552

4 2 3.π .r ν .x

(42)

Whereas the result computed by numerical approximation is, ∞ ρ ch ( r ) d r = 0.0552 0

(43)

Since the results of the two preceding equations are identical, no obvious algebraic or numerical errors have been performed. Assuming “KX” has zero uncertainty, equation (20) may be transposed and utilised to convert “KX” to an equivalent RMS charge radius form “rX” as follows, 2

6.b 1.K X . x

rX

2

3.b 1. x

1

1

Evaluating and converting dimensionally produces: 0.8071 ≤ rX(fm) ≤ 0.8437 3.2

(44) (45)

NEUTRON MAGNETIC RADIUS

Continuing in dimensionless form, the Neutron Magnetic Radius “rνM” may be usefully approximated to high computational precision {to within 3.2 x10-3(%) of the experimental result [0.879(fm)]} [16] utilising “ρch”, “rν” and “rdr”. Firstly, recognising that “d2ρch(rdr/rν)/dr2 = 0” and graphing over the domain “rν ≤ r ≤ 1.8(fm)”,

8

r dr

r dr

d

rν

2

d r2

ρ ch( r )

r Radius

Figure 3, Provokes the solution, Given r dr rν r ν . ρ ch r νM

ρ ch ( r ) d r rν

(46)

Find r νM

(47)

rνM = 0.87897(fm)

(48)

r νM

Evaluating and converting dimensionally produces,

Visualising graphically over the domain “rν ≤ r ≤ 1.8(fm)” yields, Neutron Charge Distribution r νM

r dr

ρ ch r νM ρ ch( r )

r Radius

Figure 4, 3.3

PROTON ELECTRIC RADIUS

Similarly, in dimensionless form, the Proton Electric Radius “rπE” may be usefully approximated to high computational precision {to within 6.2 x10-2(%) of the experimental result [0.848(fm)]} [16] utilising “ρch”, “rν” and “rdr” as follows,

9

Given r dr r ν . ρ ch r πE

ρ ch ( r ) d r rν

(49)

Find r πE

r πE

(50)

Evaluating and converting dimensionally produces, rπE = 0.84853(fm) 3.4

(51)

PROTON MAGNETIC RADIUS

Again, in dimensionless form, the Proton Magnetic Radius “rπM” may be usefully approximated to high computational precision {to within 0.82(%) of the experimental result [0.857(fm)]} [16] utilising “ρch”, “rν” and “rdr” as follows, Given ∞ r ν . ρ ch r πM

ρ ch ( r ) d r r dr rν

(52)

Find r πM

(53)

rπM = 0.84993(fm)

(54)

r πM

Evaluating and converting dimensionally produces,

3.5

CLASSICAL PROTON RMS CHARGE RADIUS

Finally, in dimensionless form, the Proton RMS charge radius “rp” may be usefully approximated to high computational precision {to within 0.05(%) of the National Institute of Standards and Technology (NIST) result [0.8750(fm)]} [17]2005 as follows, r P r πE

1. 2

r νM

rν

(55)

Evaluating and converting dimensionally produces, rp = 0.87459(fm) 3.6

(56)

DERIVATION OF THE HYDROGEN ATOM SPECTRUM (BALMER SERIES)

It is possible to utilise EGM to derive the first term in the Balmer Series of the Hydrogen atom spectrum. Subsequently by inference, the remaining terms may also be produced. Moreover, an experimentally implicit definition of the Bohr Radius “rBhor” may also be derived. 3.6.1

CLASSICAL DERIVATION OF THE ATOMIC EMISSION / ABSORPTION SPECTRUM [2]

1. Calculate the reduced mass of Hydrogen “µ”, µ

m e .m p me

mp

2. Calculate the Rydberg Constant “R∞” (Qe denotes Electric charge),

10

(57)

2 4 2.π .µ .Q e

R∞

2

h 3. Calculate the Electronic energy level “E” at an arbitrary quantum number “nq”,

E nq

(58)

R∞ nq

2

(59)

4. Calculate the transition energy “∆E”, ∆E n q

E nq

E( 2 )

(60)

5. Calculate the Balmer Series wavelength “λB”, λB nq

h .c ∆E n q

(61)

6. Specify the quantum range variable “nq = 3, 4…12” and plot the spectrum, The Hydrogen Spectrum (Balmer Series)

1

350

400

450

500

550

600

650

700

λB nq nm Wavelength

Figure 5, 7. Evaluate the first term, λB(3) = 656.46962(nm) 3.6.2

(62)

GENERAL FORMULATION OF ATOMIC EMISSION / ABSORPTION SPECTRA BY EGM

Assumptions: 1. “rBohr” defines a usefully approximate position of the Zero-Point-Field (ZPF) equilibrium radius. 2. The fundamental wavelength of the Polarisable Vacuum (PV) spectrum of the Hydrogen atom coincides with the longest wavelength in the Balmer Series. 3. The Hydrogen atom may be usefully represented by an “imaginary particle” (spherical) of radius “rBohr” with approximately the mass of a Proton. 4. The ZPF is in equilibrium with an “imaginary field” surrounding the atom at approximately “rBohr”. EGM utilises Fourier Series to develop a spectral representation of the PV model of gravity to describe the field energy induced by mass. The EGM spectrum is a discrete version of the continuous ZPF spectrum based on a Fourier distribution. The PV spectrum is a subset of the EGM spectrum with a non-zero fundamental frequency. It occupies a bandwidth of the EGM spectrum and is system or particle specific, based upon the distribution of energy density. It assumes that the ZPF energy outside a region of certain radius is in equilibrium with the field energy inside that region. Since EGM is based upon a Fourier distribution, the amplitude spectrum within it is usefully represented by a decay function (asymptotically tending to zero). Subsequently, we would expect that the ratio of the fundamental PV wavelength to the longest wavelength in the Balmer Series might relate to the total number of modes by an index value.

11

Since the PV spectrum as described by EGM is double sided, symmetrical about the “0th” mode, the wavelength of the PV spectrum “λPV” for a spherical mass may be applied to determine the first term in the Balmer Series of the Hydrogen atom as follows, c

λ PV n PV, r , M

ω PV n PV, r , M

(63)

where, “λPV(1,r,M)” denotes the fundamental (starting) wavelength of the PV spectrum of arbitrary mass and radius. If “λA” approximates the first term of the Balmer series [i.e. the longest wavelength such that λA ≅ λB(3)] in the Hydrogen atom emission / absorption spectrum, then a relationship to the EGM method may be assumed and tested as follows, Let: ψ

λ PV( 1 , r , M )

2 .n Ω ( r , M )

λ A( r, M )

(64)

where, “2nΩ” denotes the total number of modes (odd + even) on both sides of the EGM spectrum, symmetrical about the “0th” mode defined by, [9] n Ω ( r, M )

Ω ( r, M )

4

12

Ω ( r, M )

1

(65)

Hence, testing an obvious value of “ψ = 2” yields the general formulation, λ A( r, M )

3.6.3

λ PV( 1 , r , M ) 2 .n Ω ( r , M )

(66)

APPLICATION OF THE GENERAL FORMULATION BY EGM

3.6.3.1 METHOD 1 The Bohr Radius is a non-physical quantum average property incorporating Planck’s Constant “h” and may be defined as follows, 2 ε 0 .h r Bohr 2 π .m e .Q e (67) Evaluating yields, [2] rBohr = 5.291772108 x10-11(m) (68) It was demonstrated in [10] that the historical representation of the Planck scale was approximately “16(%)” too small. Since “h” is a function of “rBohr” and represents a non-physical quantum average property, it follows that “rBohr” is approximately “16(%)” too large and must also be re-scaled for application to equation (66) under the EGM method by a factor of “Kω”. Hence, 3

Kω

2 π

λ A K ω .r Bohr , m p = 657.32901 ( nm )

(69) (70)

Evaluating “λA” and comparing to “λB” yields the EGM error associated with the first term in the Balmer series as follows, λ A K ω .r Bohr , m p 1 = 0.13091 ( % ) λB (71)

12

3.6.3.2 METHOD 2 If we assume “rBohr” to be correct and constrain the EGM predicted Balmer Series wavelength to be exactly equal to the classical representation, then we may calculate the required “imaginary particle mass” (mx) utilising the “Given” and “Find” commands within the “MathCad 8 Professional” environment as follows, Given λ A K ω .r Bohr , m x

mx

1

λB

(72)

Find m x

(73)

m x = 1.68052 10

27 .

kg

(74)

Notably, “mx” is very close to the Proton mass and the Atomic Mass Constant “mµ”. Determining EGM mass errors yields, mx 1 = 0.47208 ( % ) mp (75) mx

1 = 1.20316 ( % )

mµ

(76)

3.6.3.3 METHOD 3 If we assume the Atomic mass constant to be correct, we may determine the correct value of ZPF equilibrium radius based upon the experimentally implicit definition of the Planck scale derived in [10] as follows, Given λ A K ω .r x, m µ

1

λB

rx

(77)

Find r x

. r x = 5.2731910

(78) 11

( m)

(79)

Comparing “rx” to “rBohr” yields the difference between them, r Bohr

1 = 0.35238 ( % )

rx

(80)

Hence, the ZPF equilibrium radius coincides with the Bohr Radius to within “0.353(%)” and suggests an experimentally implicit4 definition of “rBohr”. Therefore, a useful approximation to the first term in the Hydrogen atom spectrum (Balmer series) may be given by, λA

4

λ PV 1 , K ω .r Bohr , m p 2 .n Ω K ω .r Bohr , m p

Refer to [10] for factors of experimental implication.

13

(81)

4

CO#CLUSIO#

It has been illustrated that EGM accurately derives the following physical properties of fundamental particles, i. The Neutron Mean Square Charge Radius. ii. The Neutron Magnetic Radius {to within 0.003(%), as presented by [16]]. iii. The Proton Electric Radius {to within 0.062(%), as presented by [16]]. iv. The Proton Magnetic Radius {to within 0.825(%), as presented by [16]}. v. The Classical Proton RMS Charge Radius {to within 0.046(%), as defined by [17]}. vi. A precise mathematical representation of the Neutron Charge Distribution and all its key features. vii. The “1st” term of the emission / absorption spectrum for the Hydrogen atom (Balmer Series) {to within “0.131(%)” of the classical representation, as presented by [2]}. Subsequently, the derivation of the complete spectrum is inferred. viii. An experimentally implicit definition of the Bohr Radius, to within “0.353(%)” of the classical representation as presented by [2]. References [1] Karmanov et. Al., http://arxiv.org/abs/hep-ph/0106349 [2] University of Tel Aviv, http://www.tau.ac.il/~phchlab/experiments/hydrogen/balmer.htm [3] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum – I, Physics Essays: Vol. 19, No. 1: March 2006. [4] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum – II, Physics Essays: Vol. 19, No. 2: June 2006. [5] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum – III, Physics Essays: Vol. 19. No.3: September 2006. [6] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum – IV, Physics Essays: Vol. 19. No.4: December 2006. [7] “Riccardo C. Storti, Todd J. Desiato”, Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum – V, Physics Essays: Vol. 20. No.1: March 2007. [8] “Riccardo C. Storti, Todd J. Desiato”, “Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum – VI”, Physics Essays: Vol. 20. No.2: June 2007. [9] “Riccardo C. Storti, Todd J. Desiato”, “Electro-Gravi-Magnetics (EGM), Practical modelling methods of the polarizable vacuum – VII”, Physics Essays: Vol. 20. No.3: September 2007. [10] Quinta Essentia: A Practical Guide to Space-Time Engineering – Part 3; Riccardo C. Storti, ISBN13: 978-1847539427, LuLu Press, Ch. 3.13. [11] “Riccardo C. Storti, Todd J. Desiato”, Derivation of the Photon mass-energy threshold, The Nature of Light: What Is a Photon?, edited by C. Roychoudhuri, K. Creath, A. Kracklauer, Proceedings of SPIE Vol. 5866 (SPIE, Bellingham, WA, 2005) [pg. 207 - 213]. [12] “Riccardo C. Storti, Todd J. Desiato”, “Derivation of fundamental particle radii (Electron, Proton & Neutron)”, Physics Essays: Vol. 22. No.1: March 2009. [13] “Riccardo C. Storti, Todd J. Desiato”, Derivation of the Photon & Graviton mass-energies & radii, The Nature of Light: What Is a Photon?, edited by C. Roychoudhuri, K. Creath, A. Kracklauer, Proceedings of SPIE Vol. 5866 (SPIE, Bellingham, WA, 2005) [pg. 214 - 217]. [14] Quinta Essentia: A Practical Guide to Space-Time Engineering – Part 3; Riccardo C. Storti, ISBN13: 978-1847539427, LuLu Press, Ch. 3.11. [15] Quinta Essentia: A Practical Guide to Space-Time Engineering – Part 3; Riccardo C. Storti, ISBN13: 978-1847539427, LuLu Press, Ch. 3.12. [16] Hammer and Meißner et. Al., http://arxiv.org/abs/hep-ph/0312081 [17] National Institute of Standards and Technology (NIST), http://physics.nist.gov/cuu/ [18] Lennart Rade, Bertil Westergren, “Beta Mathematics Handbook Second Edition”, Chartwell-Bratt Ltd, 1990, Page 470.

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