derivation of fundamental particle radii: electron, proton & neutron

1

Derivation of fundamental particle radii

Electron, Proton & Neutron

Riccardo C. Storti1, Todd J. Desiato

Abstract

Experimental predictions are derived from first principles for the root-mean-square (RMS)

charge radii of a free Electron, Proton and Neutron to high computational precision [0.0118(fm),

0.8305±0.0001(fm) and 0.8269(fm) respectively]. This places the derived value of Proton radius to

within “0.38(%)” of the average Simon and Hand predictions [0.8335(fm)], arguably the two most

precise and widely cited references since the 1960's. Most importantly, the SELEX Collaboration has

experimentally verified the Proton radius prediction derived herein to extremely high precision as being

[√[0.69(fm2)] = 0.8307(fm)]. The derived value of Electron radius compares favourably to results

obtained in High-Energy scattering experiments [0.01(fm)] as reported by Milonni et. al. It is also

illustrated that a change in Electron mass of “≈ +0.04(%)” accompanies the High-Energy scattering

measurements. This suggests that the Electron radius depends on the manner in which it's measured and

the energy absorbed by the Electron during the measuring process. The Fine Structure Constant is also

derived, to within “0.026(%)” of its “National Institute of Standards and Technology” (NIST) value,

utilising the Electron and Proton radii construct herein. In addition, it is also illustrated that the

terminating gravitational spectral frequency for each particle, as described previously by Storti et. al.,

may be expressed simply in terms of Compton Frequencies.

ote: within this manuscript, “ħ” (i.e. Dirac’s Constant) is applied to Compton Frequencies, whilst

“h” (i.e. Planck’s Constant) is utilised in Compton Wavelengths.

1 [email protected]

2

1 ITRODUCTIO [1-6]

It is widely believed by proponents of the Polarizable Vacuum (PV) and Zero-Point-Field

(ZPF) models of gravity that the Compton Frequency of an Electron “ωCe” represents some sort of

boundary condition. [5] We may expand this hypothesis by recognising that the Compton Frequency of

a Proton “ωCP” and Neutron “ωCN” are multiples of “ωCe”.

Subsequently, it follows that “ωCP” and “ωCN” may also represent natural boundary conditions.

The construct herein utilises Electro-Gravi-Magnetics2 (EGM) principles [2] to facilitate the derivation

of the root-mean-square (RMS) charge radius of a free Electron, Proton3 and Neutron to high

computational precision [rε = 0.0118(fm), rπ = 0.8305±0.0001(fm) and rν = 0.8269(fm) respectively].

The Fine Structure Constant “α” is also derived in terms of “rε” and “rπ” to within 0.026(%) of

its NIST value4. Moreover, it is conjectured that the High-Energy scattering measurements of “rε”

results in a change in Electron mass “∆me” of “≈ +0.04(%)”. In addition, the harmonic cut-off

frequency “ωΩ”, conjectured in [1-6], for the Electron, Proton and Neutron are simplified5 to being

[ωΩ(rε,me) = 2ωΩ(rπ,mp), ωΩ(rπ,mp) = ωCP2/ωCe and ωΩ(rν,mn) = ωCN

2/ωCe respectively].

2 THEORETICAL MODELLIG6

2.1 SENSE CHECKS & RULES OF THUMB

A series of sense checks and rules of thumb were defined in [5] acting as indicators for order-

of-magnitude relationships and results. Considering “ωCP” and “ωCN” as hypothetical boundaries, it

follows that the Sense Checks (“Stη” and “Stθ”) may be formulated utilising the ratio of “ωΩ” (as

defined in [4]) of the Proton and Neutron to their respective Compton frequencies.

Since |ωΩ| → 0 as r → ∞, we might also expect that [Stη,Stθ] → 0 as r → ∞ according to the

“1x2” matrix block as follows7,

St η r π mp, St θ r ν mn,ω Ω r π mp,

ω CP

ω Ω r ν mn,

ω CN (1)

Where, “mp” and “mn” denote the rest mass of a Proton and Neutron respectively.

2.2 THE PROTON

When “Stη” is forced to consider RMS charge radii predictions for free Protons as reported by

“Stein8” in [7], tempting assumptions may be inferred. Table (1), illustrates the value of “Stη” in

relation to 4 possible radii configurations. Based on the computed values of “Stη” and “Stθ” as stated in

table (1), we may hypothesise that the accuracy of the RMS charge radius of a free Proton may be

numerically and analytically derived.

By equating the value of “Stη” to the Proton to Electron mass ratio “mp/me”, highly precise

predictions for radius may be articulated. The significance of the radii derivation is not in the precise

results predicted, rather, its significance lies in the fact that the predictions returned are very close to

likely experimental results. This is an important realisation given that the EGM model has been derived

utilising mass systems on a planetary scale. Subsequently, the structural validity of EGM is

qualitatively implied.

2 Electro-Gravi-Magnetics (EGM) is based on Buckingham’s Π Theory. [2] 3 The National Institute of Standards and Technology (NIST) states that the RMS charge radius of a

free Proton to be [2002 Value]: rp = 0.8750 ± 0.0068(fm). [8] Where, “fm” represents femtometre

[1(fm) = 10-15 m]. 4 α = 7.297352568x10-3 [2002 Value]. [8] 5 The subscript “e,ε” denotes classical and scattered Electron parameters respectively derived herein. 6 All mathematical modelling and output was formed using “MathCad 8 Professional” and appears in

standard product notation. 7 Also see Appendix (A). 8 Stein is a contributing editor to the American Institute of Physics [“Physics Update”] whom reports

the RMS charge radii to be: 0.805 ± 0.011(fm) and 0.862 ± 0.012(fm). [7,9-11]

3

Stηηηη(rp,mp) Value Description

Stη(0.875(fm),mp) 1783.8 Utilising the NIST 2002 value of “rp” as reported in [8] Stη(0.862(fm),mp) 1798.7 Utilising the value of “rp” as reported by [7,9,11]

Stη(0.845(fm),mp) 1818.7 Utilising the value of “rp” as reported by [12] Stη(0.805(fm),mp) 1868.4 Utilising the value of “rp” as reported by [7,9-11]

Stη(0.834(fm),mp) 1832.6 Utilising the Average value of “rp” as reported by [7,9-11]

Table 1, sense checks.

3 MATHEMATICAL MODELLIG

3.1 DERIVATION OF PROTON & NEUTRON RADII

3.1.1 NUMERICAL

Utilising the results defined in table (1), we shall hypothesise that a numerically exact

relationship exists between the ratio of the Compton wavelength of an Electron “λCe” to the Compton

wavelength of a Proton “λCP” and the Proton to Electron mass. Similarly, we shall hypothesise that a

numerically exact relationship exists between the ratio of “λCe” to the Compton wavelength of a

Neutron “λCN” and the Neutron to Electron mass ratio according to the “2x2” matrix block as follows,

St η r π mp,

St θ r ν mn,

St η r π mp,

St θ r ν mn,

λ Ce

λ CP

λ Ce

λ CN

mp

me

mn

me (2)

Where, “rπ” and “rν” denote values satisfying equation (2) utilising the “Given” function within the

“MathCad 8 Professional” environment. Hence,

[rπ rν] = [0.8306 0.8269] (fm) (3)

Comparing the results for “rπ” to the values illustrated in table (1), it is apparent that “rπ”

compares favourably, within 1.8(%), to the prediction [0.845(fm)] determined by Andrews et. al. [12]

Moreover, considering “rπ” in relation to the predictions derived by Simon [9] and Hand [10], arguably

the two most precise and cited relevant works referenced by science since the 1960’s, [13] we find that

“rπ” is within 0.38(%) of the average Simon and Hand predictions [0.8335(fm)].

3.1.2 ANALYTICAL

Performing the appropriate substitutions from9 [4] into the mass ratio relationships for “Stη”

and “Stθ” in equation (2), useful analytical representations for “rπ” and “rν” may be formed in terms of

Compton wavelengths and particle mass as follows10,

r π

r ν

c me.

8 π2.

5

27 me4.

128 G. π. h3.

.

λ CP

K PV r π mp, mp.

5

λ CP4

K PV r π mp, 4mp

2..

λ CN

K PV r ν mn, mn.

5

λ CN4

K PV r ν mn, 4mn

2..

.

(4)

Where, the Refractive Index “KPV” [14] may be usefully approximated to unity according to,

9 See Appendix A: Equation (A2-A8). 10 “c” represents the speed of light in a vacuum. “G” and “h” denote the Gravitational and Planck

constants respectively.

4

K PV r M,( ) e

2G M.

r c2.

.

≈ 1 (5)

Utilising the approximations and exact expressions described by equation set (A1) in Appendix A,

equation (4) may be simplified in terms of Compton, Planck (λh,ωh,mh) and particle mass

characteristics. Hence, 3 highly precise analytical approximation forms of “rπ” and “rν” may be written

as follows,

r πλ CP

3

16 π2. λ Ce

.

5

27

4 π. λ h2.

λ CP

λ Ce4

..c ω Ce

.

4 ω CP3.

5

27 ω h2.

32 π4.

ω Ce4

ω CP

..h me

.

16 c. π2. mp

3.

5

27 mh2.

4 π.

me4

mp

..

(6)

r νλ CN

3

16 π2. λ Ce

.

5

27

4 π. λ h2.

λ CN

λ Ce4

..c ω Ce

.

4 ω CN3.

5

27 ω h2.

32 π4.

ω Ce4

ω CN

..h me

.

16 c. π2. mn

3.

5

27 mh2.

4 π.

me4

mn

..

(7)

Subsequently, the analytical approximation error relative to the numerically precise result for “rπ” and

“rν” returned by “MathCad 8 Professional” may be shown11 to be trivial.

3.2 DERIVATION OF ELECTRON RADIUS

3.2.1 NUMERICAL

Storti et. al. conjectured in [6] that the mass-energy threshold of a Photon “mγ” based on the

classical Electron radius “re” may be deduced by the summation of a finite reciprocal harmonic series.

Since there are half as many odd harmonics as there are “odd + even” harmonics in a broad Fourier

distribution, as the harmonic cut-off mode “nΩ” tends to infinity, the following relationship was derived

in [6],

mg

m γ

1

2ln 2 n Ω r e me,. γ.>

(8)

Where, “mg” represents the odd harmonic spectral mass-energy contribution and “γ” denotes Euler’s Constant.

Applying Buckingham Π Theory (BPT) in terms of dynamic, kinematic and geometric

similarity to the preceding equation and recognising that the mass-energy terms may be replaced by

“ωΩ”, leads to an expression where “rε” may be numerically determined as follows,

ω Ω r ε me,

ω Ω r e me,

1

2ln 2 n Ω r e me,. γ.

(9)

It is conjectured in [4-6] that the gravitational spectrum of a solid spherical mass may be

characterised by a frequency distribution terminating at “ωΩ”. Subsequently, it follows that IFF “re”

represents a conditional experimental observation dimension, we may conjecture that the radius of an

Electron occupies a range of values dependent on how it is measured as suggested by recent scattering

experiments. [15]

Therefore, the preceding equation represents a robust mathematical condition defining the

lower boundary of the Electron radius that preserves the gravitational nature of the works covered in

[1-6]. Utilising the “Given” function within the “MathCad 8 Professional” environment, a highly

precise numerical approximation for “rε” is determined to be,

r ε 0.0118 fm( ). (10)

11 The magnitude of error for any line element is less than 10-6(%).

5

3.2.2 ANALYTICAL

An analytical representation of equation (9) may be formulated by performing the appropriate

substitutions for “ωΩ” as stated in Appendix A and reference [4] leading to the following relationship

with trivial error,

r ε r e1


9

5

.

(11)

3.3 DERIVATION OF THE FINE STRUCTURE CONSTANT

An analytical approximation of “α” incorporating “rε” and “rπ” may be obtained utilising

exploratory factor analysis. Applying the radii approximations above, a useful exponential

relationship12 between the Proton and Electron may be defined as follows,

αr ε

r πe

2

3.

(12)

We shall conjecture that the “2/3” index is a qualitative indicator by considering the derivations in [1]

where it was illustrated that “2/3” relates the experimental relationship function “K0” to “KPV”. This

assumption shall be further developed in the proceeding section.

3.4 ELECTRON CUT-OFF FREQUENCY

The calculated results imply that EGM may be a useful tool by which to enhance nuclear

understanding in the fields of Quantum-Electro-Dynamics (QED) and Quantum-Chromo-Dynamics

(QCD). Subsequently, exploratory factor analysis in conjunction with the preceding formulations

suggest that “ωΩ” for a free Electron may be usefully approximated13 as follows,

ωΩ(rε,me) = 2ωΩ(rπ,mp) (13)

3.5 REFINEMENT OF ELECTRON RADIUS

Assuming equation (12,13) to be exact representations may provide an opportunity for greater

computational precision of “rε”. This may be achieved by utilising the “Given” function satisfying the

following “1x3” matrix block within the “MathCad 8 Professional” environment,

αr ε

r e

ω Ω r ε me,

ω Ω r π mp,

r ε

r πe

2

3. 1


9

5

2

(14)

Returns the result14,

rε = 0.011802(fm) (15)

3.6 DERIVATION OF ELECTRON SCATTERING MASS

The mass of the Electron based on its classical radius has been established and scientifically

accepted for many years. However, considering that scattering experiments have cast doubt on its

radius, we may conjecture that the introduction of energy to the state of the Electron during radius

measurements by scattering techniques affects its mass.

It is also well accepted that the mass of a particle increases as its energy state increases leading

to a reduction in its physical dimensions. Subsequently, we may conjecture that “rε” derived herein is

accompanied by “∆me” when “rε” is measured utilising High-Energy techniques as conducted by “Los

Alamos National Laboratories” (LANL) [15] and the Stanford Linear Accelerator (SLAC). [16]

12 To within 0.026(%) of its NIST value based on approximations of “rε” and “rπ” derived herein. 13 To within 0.018(%). 14 “ωΩ(0.011802(fm),me) = 2ωΩ(rπ,mp)” to within 10

-6(%).

6

Therefore, the Electron scattering mass “mε” may be determined utilising the “Given” function

satisfying the following “1x2” matrix block within the “MathCad 8 Professional” environment,

ω Ω r ε m ε,

ω Ω r e me,

ω Ω r ε m ε,

ω Ω r π mp,

1

2ln 2 n Ω r e me,. γ. 2

(16)

Yields,

∆me ≈ +0.04(%) (17)

3.7 HARMONIC CUT-OFF FREQUENCIES

Utilising the preceding construct in conjunction with exploratory factor analysis, a simple family

of equations may be formulated expressing the terminating gravitational spectral frequency for a free

Electron, Proton and Neutron explicitly in terms of Compton Frequencies in the form a “1x3” matrix as

follows,

ω Ω r ε me, ω Ω r π mp, ω Ω r ν mn, 2 ω Ω r π mp,.ω CP

2

ω Ce

ω CN2

ω Ce (18)

4 PHYSICAL MODELLIG

4.1 ELECTRON

Equation (10,15) agrees favourably with the results of High-Energy scattering experiments

reported in [15]. It states that, if the Electron is not a point particle, its physical dimensions are

approximately no larger than “0.01(fm)” and it seems improbable that the Electron has any “structure”.

Theses results strongly support EGM because “nΩ” is a function of radius and mass. Hence, it

may be stated that EGM also implies that structure does not exist beyond “rε” due to the lack of

harmonic modes beyond that dimension. Therefore, we may conjecture that the free Electron radius and

mass varies according to its energy level and may be physically modelled over the following set,

(r,M): rε ≤ r ≤ re ∩ me ≤ M ≤ mε (19)

4.2 PROTON

Relating equation (12) to the standard calculation form “α = 2πre / λCe” yields a set of highly

precise physical modelling boundaries for “rπ” in terms of Compton, Planck and exponential

characteristics as follows,

r ε

r e

c

ω Ce

. e

2

3. r πc ω Ce

.

4 ω CP3.

5

27 ω h2.

32 π4.

ω Ce4

ω CP

..

(20)

Therefore, “rπ” may be approximately written as: rπ = 0.8305±0.0001(fm)

4.3 NEUTRON

In addition to “rν” predicted above, a set of useful physical modelling approximations15 to

assist with experimental design considerations may be defined based on the proceeding equations as

follows,

r ν

r π

λ CN

λ CP

ω CP

ω CN

mp

mn (21)

Notably, a convenient shorthand physical modelling tool is the ratio of “rε” to the difference in

radii between a free Proton and Neutron as follows,

15 To within 0.3(%).

7

r ε

r π r ν

≈ π

(22)

If we assume that equation (22) represents an exact analytical boundary solution where “rε” from

equation (15) is utilised in conjunction with “rπ” from equation (6), the result returned for “rν” may be

expressed in terms of Compton, Planck and trigonometric characteristics as follows16,

r πr ε

πr ν

c ω Ce.

4 ω CN3.

5

27 ω h2.

32 π4.

ω Ce4

ω CN

..

(23) Therefore, “rν” may be approximately written as: rν = 0.8269(fm)

5 EXPERIMETATIO

The SELEX Collaboration is an international effort pursuing experimental verification of

particle properties such as the radius of a Proton. The Proton radius prediction derived herein has been

experimentally verified to extremely high precision as illustrated in [17].

6 COCLUSIOS

It has been illustrated that the EGM model of gravity predicts experimentally supported RMS

charge radii values of a free Electron, Proton and Neutron from an almost entirely mathematical

foundation. Experimental predictions have been derived from first principles for the RMS charge radii

of a free Electron, Proton and Neutron to high computational precision. This places the derived value of

Proton radius to within “0.38(%)” of the average Simon and Hand predictions, arguably the two most

precise and widely cited references since the 1960's.

Most importantly, the SELEX Collaboration has experimentally verified the Proton radius prediction

derived herein to extremely high precision [√[0.69(fm2)] = 0.8307(fm)].

The derived value of Electron radius compares favourably to results obtained in High-Energy

scattering experiments conducted at LANL. It has also been illustrated that a change in Electron mass

of “≈ +0.04(%)” accompanies the High-Energy scattering measurements. This suggests that the

Electron radius depends on the manner in which it's measured and the energy absorbed by the Electron

during the measuring process.

The Fine Structure Constant has also been derived, to within “0.026(%)” of its NIST value,

utilising the Electron and Proton radii construct herein. In addition, it is predicted that the terminating

gravitational spectral frequency for each particle, as described previously by Storti et. al., may be

expressed simply in terms of Compton Frequencies.

16 The error on the Left Hand Side (LHS), with respect to the Right Hand Side (RHS) is less than

0.013(%).

8

References

[1] R. C. Storti, T. J. Desiato, “Electro-Gravi-Magnetics (EGM) - Practical modelling methods of the

polarizable vacuum – I”, http://www.deltagroupengineering.com/Docs/EGM_1.pdf


polarizable vacuum – II”, http://www.deltagroupengineering.com/Docs/EGM_2.pdf


polarizable vacuum – III”, http://www.deltagroupengineering.com/Docs/EGM_3.pdf


polarizable vacuum – IV”, http://www.deltagroupengineering.com/Docs/EGM_4.pdf


polarizable vacuum – V”, http://www.deltagroupengineering.com/Docs/EGM_5.pdf

[6] R. C. Storti, T. J. Desiato, “Derivation of the Photon mass-energy threshold”,

http://www.deltagroupengineering.com/Docs/Photon_Mass-Energy_Threshold.pdf

[7] Stein, B. P. "Physics Update." Physics Today 48, 9, Oct. 1995.

[8] NIST: http://physics.nist.gov/cgi-bin/cuu/Value?rp|search_for=rms+charge+radius+of+proton

[9] G.G. Simon et al., Nucl. Phys. A333, 381 (1980).

[10] L.N. Hand, D.G. Miller, and R. Wilson, Rev. Mod. Phys. 35, 335 (1963).

[11] Wolfram Research: http://scienceworld.wolfram.com/physics/Proton.html

[12] D. A. Andrews et al., 1977 J. Phys. G: ucl. Phys. 3 L91-L92.

[13] A Proposal to the MIT-Bates PAC. Precise Determination of the Proton Charge Radius, August 19

(2003) – Spokespersons: H. Gao, J.R. Calarco [e-mail: [email protected], phone: (617) 258-0256, fax:

(617) 258-5440].

[14] H. E. Puthoff, et. al., “Engineering the Zero-Point Field and Polarizable Vacuum for Interstellar

Flight”, JBIS, Vol. 55, pp.137-144, http://xxx.lanl.gov/abs/astro-ph/0107316, v1, Jul. 2001.

[15] P. W. Milonni, The Quantum Vacuum – An Introduction to Quantum Electrodynamics, Academic

Press, Inc. 1994. Page 403.

[16] Stanford Linear Accelerator: http://www.slac.stanford.edu/

[17] I. Eschrich et. al.; http://arxiv.org/hep-ex/0106053

9

APPEDIX A

me

h ω Ce.

2 π. c2.

mp

λ Ce me.

λ CP

ω CP me.

ω Ce

h ω CP.

2 π. c2.

, mn

λ Ce me.

λ CN

ω CN me.

ω Ce

h ω CN.

2 π. c2.

,

λ h

ω h

mh

G h.

c3

c5

G h.

h c.

G

,

(A1)

ω Ω r M,( ) n Ω r M,( ) ω PV 1 r, M,( ). (A2)

n Ω r M,( )Ω r M,( )

12

4

Ω r M,( )1

(A3)

Ω r M,( )

3

108Um r M,( )

U ω r M,( )

. 12 768 81Um r M,( )

U ω r M,( )

2

..

(A4)

Um r M,( )3 M. c

2.

4 π. r3. (A5)

U ω n PV r, M, U ω r M,( ) n PV 24

n PV4. (A6)

U ω r M,( )h

2 c3.

ω PV 1 r, M,( )4.

(A7)

ω PV n PV r, M,n PV

r

32 c. G. M.

π r.. e

G M.

r c2..

(A8)

Um r M,( )

U ω r M,( )

3 r2. c

4.

4 h. G.

3π r.

2 c. G. M..

(A9)

Ω r M,( ) 3 c.3

6 r2. c.

h G.

3π r.

2 c. G. M...

(A10)

n Ω r M,( )Ω r M,( )

12

c

4

3

6 r2. c.

h G.

3π r.

2 c. G. M...

(A11)

ω Ω r ε me,

ω Ω r e me,

n Ω r ε me, ω PV 1 r ε, me,.

n Ω r e me, ω PV 1 r e, me,.

n Ω r ε me,

r ε

32 c. G. me

.

π r ε.

.

n Ω r e m e,

r e

32 c. G. me

.

π r e.

.

n Ω r ε me,

n Ω r e me,

r e

r ε

.3r e

r ε

.

(A12)

Ω r ε me,

Ω r e me,

r e

r ε

.3r e

r ε

.

c

4

3

6 r ε2. c.

h G.

3π r ε

.

2 c. G. me.

..

c

4

3

6 r e2. c.

h G.

3π r e

.

2 c. G. me.

..

r e

r ε

3r e

r ε

..

3

r ε

r e

2 3r ε

r e

.r e

r ε

3r e

r ε

..r e

r ε

3

r ε

r e

3r ε

r e

..r e

r ε

5

9

(A13)

derivation of fundamental particle radii: electron, proton & neutron

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