Content uploaded by Giorgio Vassallo

Author content

All content in this area was uploaded by Giorgio Vassallo on Oct 08, 2019

Content may be subject to copyright.

J. Condensed Matter Nucl. Sci. 29 (2019) 525–547

Research Article

Electron Structure, Ultra-dense Hydrogen and Low Energy Nuclear

Reactions

Antonino Oscar Di Tommaso and Giorgio Vassallo∗,†

Università degli Studi di Palermo, Dipartimento di Ingegneria (DI), Viale delle Scienze, 90128 Palermo, Italy

Abstract

In this paper, a simple Zitterbewegung electron model, proposed in a previous work, is presented from a different perspective

based on the principle of mass−frequency equivalence. A geometric−electromagnetic interpretation of mass, relativistic mass, De

Broglie wavelength, Proca, Klein−Gordon, Dirac and Aharonov−Bohm equations in agreement with the model is proposed. A

non-relativistic, Zitterbewegung interpretation of the 3.7 keV deep hydrogen level found by J. Naudts is presented. According to

this perspective, ultra-dense hydrogen can be conceived as a coherent chain of bosonic electrons with protons or deuterons located

in the center of their Zitterbewegung orbits. This approach suggests a possible role of ultra-dense hydrogen in some aneutronic and

many-body low energy nuclear reactions.

c

2019 ISCMNS. All rights reserved. ISSN 2227-3123

Keywords: Aharonov–Bohm equations, Aneutronic and many-body low energy nuclear reactions, Compact structures, De Broglie

wavelength, Electron structure, Dirac equation, ESR, Heisenberg’s uncertainty principle, Klein–Gordon equation, Josephson con-

stant, LENR, natural units, Proca equation, relativistic mass, Ultra-dense hydrogen, Zitterbewegung

Nomenclature

γ2

x=γ2

y=γ2

z=−γ2

t=1were {γx,γy, γz, γt}are the four basis vectors of Cl3,1(R)Clifford algebra,

isomorphic to Majorana matrices algebra [1]

γiγj=−γjγiwith i6=jand i, j ∈{x, y, z, t};

∂=γx

∂

∂x+γy

∂

∂y+γz

∂

∂z+γt1

c

∂

∂t

I=γxγyγzγt

I△=γxγyγz

1. Introduction

According to Carver Mead, mainstream physics literature has a long history of hindering fundamental conceptual

reasoning, often “involving assumptions that are not clearly stated” [2]. One of these is the unrealistic assumption of

∗Corresponding author. E-mail: giorgio.vassallo@unipa.it.

†Also at: International Society for Condensed Matter Nuclear Science (ISCMNS).

c

�2019 ISCMNS. All rights reserved. ISSN 2227-3123

526 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

Nomenclature

Symbol Name SI units Natural units (NU)

AElectromagnetic four-

potential

V s m−1eV

A△Electromagnetic vector

potential

V s m−1eV

AtTime component of electro-

magnetic four potential

V s m−1eV

AElectromagnetic vector

potential module

V s m−1eV

mMass kg eV

FElectromagnetic

ﬁeld bivector

V s m−2eV2

BFlux density ﬁeld V s m−2=TeV2

EElectric ﬁeld V m−1eV2

VPotential energy J=kg m2s−2eV

JFour current density ﬁeld A m−2eV3

J△Current density ﬁeld A m−2eV3

ρCharge density A s m−3=C m−3eV3

x, y, z Space coordinates m∗eV−1

tTime variable s†eV−1

cLight speed in vacuum 2.997 924 58 ×108m s−11

~Reduced Planck constant 1.054 571 726 ×10−34 J s 1

µ0Permeability of vacuum 4π×10−7V s A−1m−14π

ǫ0Dielectric constant

of vacuum

8.854 187 817×10−12 A s V−1m−11

4π

eElectron charge 1.602 176 565 ×10−19 A s 0.085 424 546

αFine structure constant 7.2973525664 ×10−37.2973525664 ×10−3

meElectron rest mass 9.10938356 ×10−31kg 0.5109989461 ×106eV

λcElectron Compton wavelength 2.426 310 2389 ×10−12 m1.229 588 259 ×10−5eV−1

KJJosephson constant 0.4835978525 ×1015 Hz V−12.71914766 ×10−2

reReduced Compton electron

wavelength (Compton radius)

re=λc

2π

rcElectron charge radius rc=αre

TeZitterbewegung period Te=2πre

c

∗1.9732705×10−7m≃1eV−1.

†6.5821220 ×10−16 s≃1eV−1.

point-like shaped elementary particles with intrinsic properties as mass, charge, angular momentum, magnetic moment

and spin. According to the laws of mechanics and electromagnetism, a point-like particle cannot have an “intrinsic

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 527

angular momentum”. Moreover, a magnetic moment must necessarily be generated by a current loop, that cannot exist

in a point-like particle. Furthermore, the electric ﬁeld generated by a point-like charged particle should have an inﬁnite

energy. Therefore, an alternative realistic approach that fully addresses these very basic problems is indispensable. A

possibility is given by a Zitterbewegung interpretation of quantum mechanics, according to which charged elementary

particles can be modeled by a current ring generated by a massless charge distribution rotating at light speed along

a circumference whose length is equal to particle Compton wavelength [3,4]. As a consequence, every elementary

charge is always associated with a magnetic ﬂux quantum and every charge is coupled to all other charges on its

light cone by time-symmetric interactions [2]. The aim of this paper is to present a gentle introduction to an electron

Zitterbewegung model together with some observations that deems to reinforce its plausibility.

The present paper is structured in the following way. In Section 2 the deep connection between some basic concept

as space, time, energy, mass, frequency, and information is exposed. In Section 3 an introduction to a Zitterbewegung

electron model is presented, together with a geometric-electromagnetic interpretation of Proca, Klein–Gordon, Dirac

and Aharonov–Bohm equations. In Section 4 a simple geometric interpretation of relativistic mass and De Broglie

wavelength is proposed. In Section 5 the relation of Electronic Spin Resonance (ESR) frequency with Larmor pre-

cession frequency of the Zitterbewegung orbit is presented. Finally, in Section 6 some hypotheses on the structure of

ultra-dense hydrogen are formulated, whereas Section 7 deals with the possible role of ultra-dense hydrogen in low

energy nuclear reactions.

N.B. In this paper all equations enclosed in square brackets with subscript “NU” have dimensions expressed in nat-

ural units. The mathematical notation used in Sections 3.3–3.5, based on real Clifford algebra Cl3,1(R), is introduced

in [1].

2. Energy, Mass, Frequency and Information

The concept of measurement plays a fundamental role in all scientiﬁc disciplines based on experimental evidence.

The most used measurement units (such as the international system, SI) are based mainly on human conventions not

directly related to fundamental constants. To simplify the conceptual understanding of certain physical quantities it is

convenient to adopt in some cases a measurement system based on universal constants, such as the speed of light cand

the Planck’s quantum ~.

Considering that a measure is an event localized in space and time, the quantum of action can be seen, in some

cases, as an objective entity in some respects analogous to a bit of information located in the space-time continuum.

In accordance with Heisenberg’s uncertainty principle, the result of the measurement of some values (such as angular

momentum) cannot have an accuracy less than half a single Planck’s quantum. Therefore, to simplify the interpretation

of physical quantities, it may be useful to adopt a system in which both the speed of light and the quantum of action

are dimensionless quantities (pure numbers) having a unit value, i.e.: c= 1 and ~= 1. In this system, the constancy

of light speed makes possible to use a single measurement unit for space and time, simplifying, in many cases, the

conceptual interpretation of physical quantities. The energy of a photon, a “particle of light”, is equal to Planck’s

quantum multiplied by the photon angular frequency. By using the symbol Tto indicate the period of a single complete

oscillation and λthe relative wavelength, it is, therefore, possible to write

E=~ω=2π~

T=2π~c

λ.(1)

By using natural units, period and wavelength coincide and the above expression is simpliﬁed in

E=ω=2π

T=2π

λNU

.(2)

528 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

The subscript NU highlights the use of natural units for expressions contained within square brackets. This equation

indissolubly links some fundamental concepts, as space, time, energy and mass, giving the possibility to express an

energy value simply as a frequency or as the inverse of a time, or even as the inverse of a length. Vice versa, it allows

to use as a measurement unit of both space and time a value equal to the inverse of a particular energy value as the

electron-volt. Therefore, to compute photon wavelength in vacuum with natural units it is sufﬁcient to divide the

constant 2πby its energy. This value will correspond exactly to the period of a complete oscillation. Hence, in natural

units the inverse of an eV can be used as a measurement unit for space and time:

L(1 eV)= 1 eV−1≈1.9732705 ×10−7m≈0.2µm,

T(1 eV)= 1 eV−1≈6.582122 ×10−16 s≈0.66 fs.

Consequently, an angular frequency can be measured in electron volts:

1 eV ≈1.519268 ×1015 rad s−1.

Following these concepts, it is possible to deﬁne a link between fundamental concepts of information, space, time,

frequency and energy. A “quantum of information” carried by a single photon will have a “necessary reading time”

and a “spatial dimension” inversely proportional to its energy. A simple example is given by radio antennas (dipoles),

whose length is proportional to the received (or transmitted) “radio photons” wavelength and inversely proportional

to their frequency and to the number of bits that can be received in a unit of time. In this perspective, the concept of

energy is closely linked to the “density” of information in space and in time.

3. Electron Structure

The famous Einstein’s formula E=mc2becomes particularly explanatory if expressed in natural units:

[E=m]NU .

Mass is energy and it is, therefore, possible to associate a precise amount of energy to a particle having a given

mass. Taking up the considerations made on the deep bond existing between the concepts of space, time, frequency

and energy, it is interesting trying to associate the electron rest mass meto an angular frequency ωe, a length reand a

time Te. In fact Einstein’s formula can be expressed as

Ee=mec2=~ωe=~c

re

=h

Te

(3)

or adopting natural units

Ee=me=ωe=1

re

=2π

TeNU

.(4)

These constants have a simple and clear interpretation if one accepts a particular electron model consisting of a

current ring generated by a massless charge rotating at the speed of light along a circumference whose radius is equal

to the electron reduced Compton wavelength, deﬁned as re=λc

2π≈0.38616 ×10−12 m[3–6]. According to the

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 529

model described in [4] the charge is not a point-like entity, but it is distributed on a spherical surface whose radius is

equal to the electron classical radius rc≈2.8179 ×10−15 m. In Eq. (4) ωeis the angular frequency of the rotating

charge, reis its orbit radius and Teits period. The current loop is associated with a quantized magnetic ﬂux ΦMequal

to Planck’s constant (h= 2π~) divided by the elementary charge e(see Eq. (34) p. 84 [4])

ΦM=h/e

or in natural units

[ΦM= 2π/e]NU .

The rotation is caused by the centripetal Lorentz force due to the magnetic ﬁeld associated with the current loop

generated by the elementary rotating charge (Eq. (36)).The value of this elementary charge, in natural units, is a pure

number and is equal to the square root of the ratio between the charge radius rcand the the orbit radius re(see Eqs.

(39) and (40) p. 85 [4]:

e=rc

re

=√α≈0.0854245NU

.(5)

Similar models, based on the concept of “current loop”, have been proposed by many authors, but have often

been ignored for their incompatibility with the most widespread interpretations of Quantum Mechanics [3,5–10]. It is

interesting to remember how, already in his Nobel lecture of 1933, P.A.M. Dirac referred to an internal high-frequency

oscillation of the electron: “It is found that an electron which seems to us to be moving slowly, must actually have

a very high frequency oscillatory motion of small amplitude superposed on the regular motion which appears to us.

As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light.This is a

prediction which cannot be directly veriﬁed by experiment, since the frequency of the oscillatory motion is so high and

its amplitude is so small”. In the scientiﬁc literature, the German word Zitterbewegung (ZBW) is often used to indicate

this rapid oscillation/rotation of the electron charge. The rotating charge is characterized by a momentum pcof purely

electromagnetic nature:

pc=eA =eΦM

2πre

=~ωe

c=~

re

=mec.

In this formula the variable A=~/ereindicates the vector potential seen by the rotating charge (see Eq. (25), p.

82 [4]. Multiplying the charge momentum pcby the radius rewe obtain the “intrinsic” angular momentum ~of the

electron:

pcre=~.(6)

Using natural units the momentum pchas the dimension of energy and it is exactly equal to the electron mass–

energy at rest me:

pc=eA =Ee=1

re

=me=ωeNU

.

530 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

3.1. Aharonov–Bohm equations and Zitterbewegung model

The magnetic Aharonov–Bohm effect is described by a quantum law that gives the phase variation ϕof the “electron

wave function” starting from the integral of the vector potential A△along a path [11], i.e.

ϕ=e

~A△·dl.(7)

In the proposed Zitterbewegung model, the electron “wave function phase” has a precise geometric meaning: the

charge rotation phase. By using (7), a possible counter-test consists in verifying that the phase shift ϕalong the

circumference of the Zitterbewegung orbit is equal exactly to 2πradians. In fact

ϕ=e

~A△·dl=e

~2πre

0

Adl=e

~2πre

0

~

ere

dl=e

~

~

ere

2πre= 2π,(8)

because vectors A△and dlhave the same direction tangent to the elementary charge trajectory. This result is also

consistent with the prediction of the electric Aharonov–Bohm effect, a quantum phenomenon that establishes the

variation of phase ϕas a function of the integral of electric potential Vin a time interval T, i.e.:

ϕ=e

~T

Vdt. (9)

Applying the electric Aharonov–Bohm effect formula to compute the phase shift ϕwithin a time interval Te=2π

ωe

equal to a Zitterbewegung period we obtain the expected result, i.e. ϕ= 2π. In fact, the electric potential of the

electron rotating charge can be expressed as

V=e

4πε0rc

=e

rcNU

and its period as

Te=2πre

c= [2πre]NU .

A simple calculation, applying (9) and (5), yields the same results:

ϕ=e

~

Te

0

Vdt=e

~V Te=e

~V2πre

c=e2

rc

2πreNU

= 2π.(10)

Now, by equating the ﬁfth term of (8) and the fourth term of (10) it is possible to demonstrate that

At=V

c=A=|A△|,

[At=V=A=|A△|]NU ,

A2

= (A△+γtAt)2=A2

△−A2

t= 0.(11)

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 531

By introducing the differential form of (9) we obtain

dϕ=e

~Vdt

and this yields the phase speed

dϕ

dt=ωe=e

~V=e2

4πε0~rc

=cα

rc

=c

re

=mec2

~=ce

~A,

dϕ

dt=ωe=me=eV =eANU

.(12)

3.2. Proca equation and Zitterbewegung electron model

A deep connection of Maxwell’s equations (see Eq. (97), p. 121 [1])

∂∂∧A+µ0J= 0 (13)

with Proca equation for a particle of mass m

∂∂∧A+mc

~2

A= 0,(14)

∂∂∧A+m2A= 0NU (15)

emerges if we prove that equation µ0J=m2ANU can be applied to the electron Zitterbewegung model intro-

duced in [4]. In this model the electron’s charge orbit delimits a disc-shaped volume with radius reand height 2rc.

Inside this volume the average Zitterbewegung current density ¯

Jecan be computed dividing the Zitterbewegung cur-

rent by one half the disc vertical section A:

¯

Je=Ie

A,

where

A= 2rerc= 2αr2

e,

¯

Je=Ie

A=Ie

2αr2

e

.(16)

From [4], p. 82, we have that

Ie=αA△

2πNU

532 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

and substituting it in (16) we get

¯

Je=A△

4πr2

eNU

,

µ0¯

Je= 4π¯

Je=A△

r2

e

=ω2

eA△=m2

eA△NU

.

Remembering that the electron’s electromagnetic four potential A=A△+γtAtassociated to the rotating charge is

a light-like vector (i.e. A2

= 0, see Eq. (11)) we can write the following relations:

µ0Jet =At

r2

e

=ω2

eAt=m2

eAtNU

,

µ0

¯

Je=µ0¯

Je+γtJet=m2

e(A△+γtAt) = m2

eANU ,

and consequently (QED):

µ0

¯

Je=m2

eANU .(17)

3.3. Proca and electromagnetic Klein–Gordon equations

In this paragraph and in the next one we will use only natural units, omitting the subscript NU. The aim is to show the

connection of Proca equation with an “electromagnetic version” of Klein–Gordon equation. By applying the operator

∂∧to Proca equation

∂∂∧A+m2A= 0,(18)

∂F+m2A= 0,

we get

∂∧∂F+m2∂∧A= 0,

∂∧∂F+m2F= 0.

Now, by writing Maxwell’s equations considering an averaged four-current vector density

∂F=−4π

¯

J,(19)

and by applying to both members the operator ∂·we obtain the following expression

∂·∂F=−4π∂·

¯

J= 0,

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 533

that is equal to zero as a consequence of the charge–current conservation law. For this reason, the term ∂∧∂Fcan be

safely substituted by the term ∂2F:

∂∧∂F=∂2F−∂·∂F=∂2F.

As a result we obtain a Klein–Gordon-like equation where the electromagnetic bivector Fsubstitutes the “wavefunc-

tion” ψ:

∂2F+m2F= 0.(20)

A similar equation for the electromagnetic four potential can be obtained simply by applying the Lorenz gauge

condition ∂A=∂∧Ato Proca equation:

∂2A+m2A= 0 (21)

or

∂2A+ω2A= 0.(22)

It is important to note that the Lorenz gauge condition can been applied to Maxwell’s equations (19) only when an

averaged four current density vector value is used. In this case the electromagnetic four potential is also an averaged

value and no more an harmonic function of space–time [1].

3.4. The electromagnetic Dirac equation

By following the same conceptual pattern of the previous paragraph, an electromagnetic–geometric version of the

Dirac equation (23),

i✁

∂ψ−mψ= 0 (23)

should have the form

∂F−mF = 0.(24)

Here mcannot be a scalar, being ∂Fa vector and Fa bivector, respectively, but rather a space-like vector with

module m. A possible candidate for mis a vector that has the same direction of the Zitterbewegung radius rand a

module m=1

r=ω. Calling rua unit vector in the same direction of r,Eq. (24) becomes

∂F−ωruF= 0,(25)

where the operator ∂of Cl3,1(R)substitutes the Dirac operator i✁

∂,the Zitterbewegung angular frequency ωthe

electron mass and the electromagnetic bivector Fthe wave function ψ.The unit vector ruis always orthogonal to the

vector potential and therefore:

r2

u= 1,

534 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

ωr=ru,

r·A= 0.

By applying (19) to (25) we can write

4πJ+ωruF= 0 (26)

whereas, by applying (17) to (26) and remembering that F=∂A,we obtain:

ω2A+ωru∂A= 0,

that can be written as

ru∂A+ωA= 0.

Now, by left multiplying the last equation for the unit vector ruwe obtain a Dirac-like equation for the electromagnetic

four potential

∂A+ωruA= 0.(27)

Multiplying for the elementary charge e(27) becomes

e∂A+eωruA= 0.(28)

Moreover, by multiplying the electromagnetic four-potential for the ratio e

ω, we obtain a light-like vector that can

be interpreted as the charge four-velocity c−γt(see Eq. (60) of [1] and Eq. (12))

e

ωA=c−γt,(29)

that left multiplying by rubecomes

e

ωruA=ruc−ruγt.(30)

Now, by applying (30) to (28) and remembering that ∂A=F, (28) becomes

eF=−ω2(ruc−ruγt).(31)

Applying the identity F= (E+IB)γt(see Eq. (73) of [1], Eq. (31) becomes

e(E+IB)γt=−ω2(ruc−ruγt).(32)

This last equation can be split in two equations. The ﬁrst one deals with the electric ﬁeld E:

eEγt=ω2ruγt.

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 535

Applying the identity eA =ω, the square ω2can be written as eAω,namely a term that is equal to the module of the

force generated on an elementary electric charge by the time derivative of a rotating vector potential:

eE=eAωru=−edA△

dt.(33)

This electric force has the same value of the centrifugal force acting on a mass mrotating with angular frequency

ωat distance rfrom its orbit center:

ru=ωr=mr,

eE=mω2r.

The second part of (32) deals with the magnetic ﬂux density ﬁeld B:

eIBγt=−ω2ruc,

eI△B=−ω2ruc

that right multiplying for cbecomes:

eI△Bc =−ω2ru.

As Band care orthogonal vectors in the Zitterbewegung model, it is possible to write also:

eI△B∧c=−ω2ru

that, using ordinary vector algebra, becomes:

ec×B=−ω2ru.(34)

Finally, merging (33) with (34) we obtain an equation that tell us that the mass-less rotating charge, with momentum

p=eA△,is subjected to a centripetal magnetic force −ω2ru:

ec×B=edA△

dt=dp

dt,(35)

ec×B=−mω2r=−ω2ru.(36)

These easy to interpret equations conﬁrm the correctness of the original choice of ωrufor the vector min the electro-

magnetic version of Dirac equation (24).

536 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

3.5. Proca equation, electric charge quantization and Josephson constant

An interesting consequence of Eq. (22) is the magnetic ﬂux and electric charge quantization. In this paragraph we call

“wave amplitude” the module Aof vector potential A△in Eq. (22)

A=A△+γtAt, A =|A△|=At

Substituting ωwith eA in Eq. (22) we obtain a non-linear wave equation for the electromagnetic four potential, where

the wave angular frequency is proportional to the wave amplitude and the proportionality coefﬁcient is the “electric

charge quantum”, i.e. the elementary charge e.

∂2A+e2A2A= 0NU ,(37)

∂2A+αA2A= 0NU .(38)

In this equation the ratio frequency/amplitude, ν/A, expressed in natural units is a pure number equal to half the

value of Josephson constant KJ:

v

A=1

2KJNU

.

The product of wave amplitude and wave period Tis equal to another constant exactly equal to a magnetic ﬂux ΦM, a

value two times the magnetic ﬂux quantum Φo(see Fig. 1). It is a reasonable conjecture to consider (37) also valid for

other charged elementary particles. In natural units we have

AT =ωT

e=h

e=ΦM= 2K−1

JNU

,

where

ΦM= 2Φo= 4.13566766 ×10−15 V s,

[ΦM= 73.55246018]NU .

4. Geometric Interpretation of Relativistic Electron Mass and De Broglie Wavelength

If an electron moves along an axis zorthogonal to its charge rotation plane, it will describe an helical trajectory whose

length is L=cΔtand whose z-axis length is l=vzΔt. The electron mass is exactly equal to the inverse of the

helix radius rif expressed in NU, i.e. m=r−1. An acceleration along z, implies a smaller radius and, hence, a mass

increase. Using the Pythagorean theorem it is possible to write the value of the radius ras a function of vz[4,5]:

r=re1−v2

z

c2

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 537

Figure 1. A possible explanation of magnetic ﬂux and electric charge quantization: in electromagnetic Klein–Gordon/Proca equation vector

potential amplitude time wave period is a constant Φm=h/e.

and the related mass variation

m=~ω

c2=me

1−v2

z

c2

.

The charge momentum is proportional to the angular frequency and it has a direction tangent to the helical path. The

relativistic momentum of charge is, then,

pc=eA =~ω

c=~

r(39)

or, using natural units,

pc=ω=1

r=mNU

.

Equation (39) suggests a particular interpretation of the Heisenberg uncertainty principle: an electron, whose charge

has a momentum pc, cannot be conﬁned within a spherical space of radius Rless than r. This means that it must be

R > r =~

pc

.

538 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

Now, the charge momentum vector pc=eA△can be decomposed into two components: p⊥, that is orthogonal to

electron velocity and another one, pk, that is parallel, i.e. in the z-direction. Therefore the charge momentum can be

expressed as

pc=p⊥+pk.

The magnitude of component p⊥is a constant, independent from velocity vz, and is proportional to the charge angular

speed ωein the xy-plane [12]. Therefore,

p⊥=~ωe

c=mec

or in natural units

[p⊥=ωe=me]NU ,

whereas the component pkis the momentum of the electron and is proportional to the instantaneous angular frequency

ωz=vz/r

pk=~ωz

c=~vz

cr =~ω

c2vz=mvz

or in natural units

pk=ωz=vz

r=mvzNU

.

Using again the Pythagorean theorem it is possible to write the following equations

ωe=v⊥

r=c2−v2

z

r=c2−v2

z

re1−v2

z

c2

=c

re

.(40)

and, as a consequence of (40), also

ω=c

r.

But

ωz=vz

r

and, therefore, the sum of squares of the angular frequencies yields the following relations

ω2=ω2

e+ω2

z, p2

c=p2

⊥+p2

k,

and, ﬁnally,

m2c2=m2

ec2+m2v2

z.(41)

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 539

For the sake of simplicity we will use the symbol pto indicate the electron momentum pk

p=pk=mvz.

According to De Broglie hypothesis, ωzis the instantaneous angular frequency associated to a particle with rest mass

me, relativistic mass mand velocity vz=ωzr. As a consequence

p=mvz=~ω

c2vz=~

cr vz=~ωz

c=~2π

λ=~k(42)

or

p=mvz=ωvz=vz

r=ωz=2π

λ=kNU

.

Equation (42) yields

p

k=pλ

2π=~.(43)

where the term k= 2π/λis the wave number of the electron and λthe related De Broglie wavelength. Of course, if we

observe the electron at a spatial scale much larger than its Compton wavelength and at a time scale much higher than

the very short period T≈8.1×10−21 sof the Zitterbewegung rotation period, for a constant speed vz, the electron

can be approximated to a point particle, provided with “mass” and charge, which moves with a uniform motion along

the z-axis of the helix. Particularly, Fig. 2 represents the helical trajectories of electrons moving at different speeds.

5. ESR, NMR, Spin and Intrinsic Angular Momentum

As shown in the previous paragraph, in the proposed model, the electron has an angular momentum ~

~

~and a magnetic

moment µB, equal to Bohr magneton. It is, therefore, reasonable to assume that, in presence of an external magnetic

ﬁeld, the electron is subjected, as a small gyroscope, to a torque τand to a Larmor precession with frequency ωp. The

only difference with a classical gyroscope is the quantization of the ~kcomponent of the angular momentum ~

~

~along

the external ﬂux density ﬁeld BE. This component can take only two possible spin values, namely ~k=±1

2~(see

[4], p. 83). The two spin values will correspond to two possible values for the angle θformed between the angular

momentum vector and the external magnetic ﬁeld vector: θ∈π

3,2π

3:

~2

k+~2

⊥=~2,~k=±1

2~.

The torque exerted by the external ﬂux density ﬁeld BEis

τ=|µB×BE|=µBBEsin (θ)

and the related Larmor precession angular frequency is

ωp=BEµB

~.(44)

The precession angular frequency will correspond to two possible energy levels:

540 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

Figure 2. Zitterbewegung trajectories for different speeds.

EH=~ωpif θ=2π

3

and

EL=−~ωpif θ=π

3.

The difference of energy levels corresponds to the Spin Electronic Resonance (ESR) frequency νESR:

ΔE=EH−EL= 2~ωp=~ωESR =hνESR.(45)

From (44) and (45) it is possible to determine the ESR frequency as

νESR = 2BEµB

h.(46)

For instance, an external magnetic ﬂux density ﬁeld equal to BE= 1.5 T yields a frequency νESR ≈42 GHz. By

calling sthe spin value and µthe nuclear magnetic moment we can also generalize (46) for particles other than the

electron. In this case the term used is Nuclear Magnetic Resonance (NMR) frequency, which is equal to

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 541

νNMR ≈BEµ

hs .(47)

For instance, for isotope 7

3Li, with s= 3/2,µ≈1.645 ×10−26 and BE= 1.5 T, the NMR frequency is νNMR ≈

24.8 MHz, whereas for isotope 11

5B we have s= 3/2,µ≈1.36 ×10−26 J T−1and NMR frequency is νNMR ≈

20.5 MHz. Another example deals with isotope 87

38Sr with s= 9/2and µ≈5.52 ×10−27 J T−1. In this case NMR

frequency is νNMR ≈278 kHz for BE= 0.15 T with a Larmor frequency ωp

2π=1

2νNMR ≈139 kHz.

5.1. Electron spin and coherent systems

In the proposed model, the electron, in presence of an external magnetic ﬁeld, is subjected to Larmor precession

and its spin value ±~/2is interpreted as the intrinsic angular momentum component parallel to the magnetic ﬁeld.

It is interesting to note that a hypothetical technology, able to align the intrinsic angular momentum of a sufﬁcient

number of electrons, could favor the formation of a coherent superconducting and super-ﬂuid condensate state. In this

state, the electrons would behave as particles with whole spin ~and would no longer be subject to the Fermi–Dirac

statistic. The compression effect (pinch) of an electrical discharge, accurately localized in a very small “capillary”

volume, inside which a very rapid and uniform variation of the electric potential occurs, could favor the formation of a

superconducting plasma. The conjecture is based on the possibility that, as a consequence of Aharonov–Bohm effect,

a rapid, collective and simultaneous variation of the Zitterbewegung phase catalyzes the creation of coherent systems

like those described by K. Shoulders and H. Puthoff [13]: “Laboratory observation of high-density ﬁlamentation or

clustering of electronic charge suggests that under certain conditions strong coulomb repulsion can be overcome by

cohesive forces as yet imprecisely deﬁned”.

6. Hypotheses on the Structure Of Ultra-dense Hydrogen

In relativistic quantum mechanics, the Klein-Gordon equation describes a charge density distribution in space and

time. In this equation a term m2c2/~2appears, whose interpretation becomes simple and intuitive if one uses natural

units and the principle of mass–energy–frequency equivalence. In particular, it is possible to recognize this term as the

square of the Zitterbewegung angular frequency ω:

m2c2

~2=m2=ω2NU

.

In the paper “On the hydrino state of the relativistic hydrogen atom” [14], the author, by applying the Klein–Gordon

equation to the hydrogen atom, ﬁnds a possible deep energetic level of E0≈3.7 keV (see Eqs. (16) and (17)) at a

distance r0from the nucleus. In particular Naudts demonstrates that

E0≈mec2α≈3.7 keV

at a distance from nucleus equal to

r0≈~

mec≈0.39 ×10−12 m.

According to the author, the E0level corresponds to the hypothetical state of a relativistic electron: “The other set of

solutions contains one eigenstate which describes a very relativistic particle with a binding energy which is a large

542 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

fraction of the rest mass energy”. It is possible to formulate an alternative hypothesis according to which the radius

r0is simply the radius reof the Zitterbewegung orbit, in the center of which the proton is located. Consequently the

energy, E0, can be interpreted as the electrostatic potential energy between the electron charge and the proton:

E0=1

4πǫ0

e2

re

=~

re

αc=mec2α,E0=e2

re

=α

re

=ωee2=meαNU

.

A series of numerous experiments conducted by Leif Holmlid of the University of Gothenburg, recently replicated

by Sindre Zeiner–Gundersen [15], seems to demonstrate the existence of a very compact form of deuterium [16–18].

Starting from the kinetic energy (about 630 eV) of the nuclei emitted in some experiments, achieved by irradiating

this particular form of ultra-dense deuterium with a small laser, a distance between deuterium nuclei of about 2.3×

10−12 mhas been computed, a value much smaller than the distance of about 74 ×10−12 mthat separates the

nuclei of a normal deuterium molecule. Therefore, it is possible to advance an hypothesis on the structure of ultra-

dense hydrogen (UDH) starting from the electron Zitterbewegung model. The proton is considerably smaller than

Zitterbewegung orbit radius re, consequently an hypothetical structure formed by an electron with a proton (or a

deuterium nucleus) in its center would have a potential energy of

−e2

re≈ −3.7 keVNU

,

a value corresponding to the energy in the X-ray range with a wavelength of about 3.3×10−10 m. The distance

between the deuterium nuclei in the Holmlid experiment could be explained by an ordered linear sequence of ultra-

dense particles in which the rotation planes of the electron charges are parallel and equidistant. In these hypothetical

aggregates, the Zitterbewegung phases of two neighboring electrons differ by πradians and the distance dcbetween

the charges of the two electrons is equal to the distance traveled by light in a time equal to a rotation period T. This

distance amounts to dc=cT =λc≈2.42 ×10−12 m. In this case, the distance between the nuclei dican be obtained

by applying the Pythagorean theorem, as shown in Fig. 3, yielding the value

di=λ2

c−λc

π2

≈2.3×10−12 m.

This UDH model is in agreement with the third assumption of Carver Mead “Alternate World View”: “every element

of matter is coupled to all other charges on its light cone by time-symmetric interactions” [2].

6.1. Ultra-dense hydrogen and anomalous heat generation in metal–hydrogen systems

The combustion of a mole of hydrogen (about two grams) generates an energy of 286 kJ (or 240 kJ if we do not take

into account the latent heat of vaporization of water), a value that corresponds to an energy of 1.48 eV per atom. The

formation of an ultra-dense hydrogen atom would release an energy of 3.7 keV per atom, a value 2500 times higher.

The conversion of only two grams of hydrogen into ultra-dense hydrogen would then be able to generate an energy

of 715 MJ ≈198 kWh. Consequently, the hypothesis, according to which in some experiments the development of

anomalous heat is partially or totally due to the formation of ultra-dense hydrogen, cannot be excluded. Following an

alternative hypothesis, the αmec2≈3.7 keV energy is not emitted as an X-ray photon but is stored in the electron

mass–frequency–energy, with a consequent small Zitterbewegung orbit radius reduction. By deﬁning meu and reu the

mass and the radius, respectively, in this new state we have:

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 543

Figure 3. Ultra-dense hydrogen model..

meuc2=mec2+αmec2≈514.728 keV.(48)

The mass increase implies a Zitterbewegung radius reduction. In fact

mec2=~ωe=~c

re

,

meuc2=me(1+α)c2=~ωeu =~c

reu

,

and therefore

reu =~

me(1+α)c=re

1 + α.

This radius reduction generates a potential energy decrease:

△Ep=e2

4πε01

re−1

reu =e2α

4πε0re

=α2

reNU ≈27.2 eV.

544 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

Following the Carver Mead “transactional” interpretation of photons, the eventual (or necessary?) emission of the

ultraviolet 27.2 eV photon may be favored by a “Mills catalyst” [19,2].

Another Zitterbewegung model for deep electron states has been recently presented by A. Kovacs et al., aimed at

explaining their impressive experimental results [20].

7. Ultra-dense Hydrogen and Low-energy Nuclear Reactions

In the proposed model the particles of hydrogen or ultra-dense deuterium are electrically neutral but have a magnetic

moment almost equal to electron’s one. This is a value 960 times higher than the neutron magnetic moment. A particle

with magnetic moment µis subjected, in presence of a magnetic ﬁeld B, to a force fproportional to the gradient of

B

f=∇(B·µ).

Therefore, the magnetic ﬁeld Bgenerated by a nucleus could exert a considerable “remote action” on the particles of

ultra-dense hydrogen. This force could be the source of the “long range potential” mentioned in a theoretical work of

Gullström and Rossi, “Nucleon polarizability and long range strong force from σI= 2 meson exchange potential”

[21]:

“A less probable alternative to the long range potential is if the e-N coupling in the special EM ﬁeld environment

would create a strong enough binding to compare an electron with a full nuclide. In this hypothesis, no constraints

on the target nuclide are set, and nucleon transition to excited states in the target nuclide should be possible. In other

words these two views deals with the electrons role, one is as a carrier of the nucleon and the other is as a trigger for

a long range potential of the nucleon”.

Hence, it is possible that, according to this scenario, electrons would have a fundamental dual role as catalysts

of low-energy nuclear reactions (LENR): the ﬁrst as neutralization-masking effect of the positive charge of hydrogen

or deuterium nuclei, a necessary condition to overcome the Coulomb barrier, the second as the source of a relatively

long-range magnetic force.

By using the Holmlid notation “H(0)” to indicate ultra-dense hydrogen particles, it is possible to hypothesize a

LENR reaction involving the 7

3Li, an isotope that constitutes more than 92% of the natural Lithium

7

3Li +H(0) →24

2He +e.(49)

This reaction would produce an energy of about 17.34 MeV mainly in the form of kinetic energy of helium nuclei,

without emission of neutrons or penetrating gamma rays. A similar reaction, able to release about 8.67 MeV, could

be hypothesized for the isotope 11

5B

11

5B+H(0) →34

2He +e.(50)

Emissions in the X-ray range would still be present in the form of braking radiation (Bremsstrahlung) generated by the

deceleration caused by impacts of helium nuclei with other atomic nuclei.

The three “miracles” required by the low-energy nuclear reactions could therefore ﬁnd, for example, in the reaction

(49) a possible explanation:

(1) Overcoming the Coulomb barrier: the ultra-dense hydrogen particles are electrically neutral.

(2) No neutrons are emitted: the reactions products of (49) and (50) consist exclusively of helium nuclei and an

electron.

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 545

(3) Absence of penetrating gamma radiation: the energy produced is mainly manifested as kinetic energy of the

reaction products and as X-ray emission from bremsstrahlung. However a probability for gamma radiation

from excited intermediate products and from secondary interaction of high energy alpha particles could not be

completely dismissed.

The mechanical energy of the alpha particles produced by the reactions could be converted with a reasonable yield

directly into electrical energy or into usable mechanical energy [22], avoiding the need for an intermediate conversion

into thermal energy. None of the three miracles is required to justify the production of abnormal heat due to ultra-dense

hydrogen formation.

In the Iwamura experiment the low-energy nuclear transmutation of elements deposited on a system formed by

alternating thin layers of palladium (Pd) and calcium oxide (CaO) was observed. The transmutation occurs when

the system is crossed by a ﬂow of deuterium. The CaO layer, essential for the transmutation, is hundreds of atomic

layers far from the surface where the atoms to transmute are deposited or implanted. It is, therefore, necessary to

ﬁnd a mechanism that explains the remote action, the role of the CaO and the overcoming of the Coulomb barrier

by deuterium nuclei. An interesting hypothesis could derive from considering the formation of ultra-dense deuterium

(UDD) at the interface between calcium oxide and palladium, an area in which the high difference in the work function

between Pd and CaO favors the formation of a layer with high electron density (Swimming Electron Layer or SEL)

[23]. The ultra-dense deuterium could subsequently migrate to the area where the atoms to transmute are present.

Therefore, aggregates of neutral charged ultra-dense deuterium would be, according to this hypothesis, the probable

responsible for the transmutation of Cs into Pr and Sr into Mo. It is possible that strontium oxide, with its very low

work function, substitutes the calcium oxide role in Celani’s experiments [24]. By using again the Holmlid notation

“D(0)” to indicate “atoms” of ultra-dense deuterium, the hypothesized

many-body reactions

in Iwamura experiments

[25] would be very simple:

133

55 Cs + 4D(0) →141

59 Pr + 4e,

88

38Sr + 4D(0) →96

42Mo + 4e,

138

56 Ba + 6D(0) →150

62 Sm + 6e.

In the above equations the symbols 4D(0) and 6D(0) represent picometric, coherent chains of ultra-dense deuterium

particles. The short distance between deuterons in such hypothetical structures may favor these otherwise difﬁcult

to explain many-body nuclear transmutation. In this context, the electrons would have the precise role of deuterium

nucleus vectors within the nucleus to be transmuted.

8. Conclusions

In this paper a simple Zitterbewegung electron model has been introduced, where the concepts of mass-energy, mo-

mentum, magnetic momentum and spin naturally emerge from its geometric and electromagnetic parameters, thus

avoiding the obscure concept of “intrinsic property” of a “point-like” particle. An intuitive geometric interpretation

of relativistic mass and De Broglie wavelength has been presented. Using only electromagnetic and geometric con-

cepts an interpretation of Proca, Dirac, Klein–Gordon and Aharonov–Bohm equations based on this particular electron

model has been presented. A non linear equation for electromagnetic four potential has been introduced that directly

implies electric charge and magnetic ﬂux quantization.

Electronic Spin Resonance (ESR)frequency has been computed starting from a spin model based on the Larmor

precession frequency of Zitterbewegung rotation plane. A very simple model for ultra-dense hydrogen, where electron

546 A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547

has only spin angular momentum, has been proposed, highlighting its possible role in many-body and aneutronic low

energy nuclear reactions.

Acknowledgements

Authors wish to thank Francesco Celani and Giuliano Bettini for helpful discussions and suggestions.

References

[1] F. Celani and A.O. Di Tommaso and G. Vassallo, Maxwell’s equations and Occam’s razor, J. Condensed Matter Nucl. Sci.

25 (2017) 100–128.

[2] Carver Mead, The nature of light: what are photons? Proc. SPIE, 8832: 8832 – 8832 – 7, 2013.

[3] David Hestenes, Quantum mechanics from self-interaction, Found. Phys. 15(1) (1985) 63–87.

[4] F. Celani, A.O. Di Tommaso and G. Vassallo, The electron and Occam’s razor, J. Condensed Matter Nucl. Sci. 25 (2017)

76–99.

[5] Oliver Consa, Helical model of the electron, General Sci. J. (2014) 1–14.

https://www.gsjournal.net/Science-Journals/Research

[6] J. Paul Wesley and David L. Bergman, Spinning charged ring model of electron yielding anomalous magnetic moment,

Galilean Electrodynamics 1(1990) 63–67.

[7] A.L. Parson and Smithsonian Institution, A Magneton Theory of the Structure of the Atom (with Two Plates), Vol. 65, in

Publication (Smithsonian Institution), Smithsonian Institution, 1915.

[8] D.L. Bergman and C.W. Lucas, Credibility of common sense science, Found. Sci. (2003) 1–17. Reprinted by permission of

Galilean Electrodynamics 1(1990) 63—67.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.490.4984&rep=rep1&type=pdf.

[9] David Hestenes, The zitterbewegung interpretation of quantum mechanics, Found. Phys. 20(10) (1990) 1213–1232.

[10] Richard Gauthier, The electron is a charged photon, Vol. 60, APS April Meeting 2015, 2015.

http://meetings.aps.org/link/BAPS.2015.APR.Y16.4.

[11] Y. Aharonov and D. Bohm, Signiﬁcance of electromagnetic potentials in the quantum theory, Phys. Rev. 115 (1959

485–491.

[12] G. Bettini, The Moebius Strip: a Biology of Elementary Particles, viXra.org,Quantum Physics, 2010.

http://vixra.org/pdf/1004.0035v4.pdf.

[13] H.E. Puthoff and M.A. Piestrup, Charge conﬁnement by casimir forces, arXiv:physics/0408114, 2004.

[14] J. Naudts, On the hydrino state of the relativistic hydrogen atom, ArXiv Physics e-prints, July 2005.

[15] S. Zeiner-Gundersen and S. Olafsson, Hydrogen reactor for Rydberg matter and ultra dense hydrogen, a replication of Leif

Holmlid, Int. Conf. Condensed Matter Nucl. Sci., ICCF-21, Fort Collins, USA, 2018.

[16] Shahriar Badiei, Patrik U. Andersson and Leif Holmlid, High-energy Coulomb explosions in ultra-dense deuterium: Time-

of-ﬂight-mass spectrometry with variable energy and ﬂight length. Int. J. Mass Spectrometry 282(1–2) (2009) 70–76.

[17] Leif Holmlid. Excitation levels in ultra-dense hydrogen p(-1) and d(-1) clusters: structure of spin-based Rydberg matter, Int.

J. Mass Spectrometry 352 (2013) 1–8.

[18] Leif Holmlid and Sveinn Olafsson, Spontaneous ejection of high-energy particles from ultra-dense deuterium D(0), Int. J.

Hydrogen Energy 40(33) (2015) 10559–10567.

[19] R.L. Mills, J.J. Farrell and W.R. Good, Uniﬁcation of Spacetime, the Forces, Matter, and Energy, Science Press, Ephrata, PA

17522, 1992.

[20] Andras Kovacs, Dawei Wang and Pavel N. Ivanov, Investigation of electron mediated nuclear reactions, J. Condensed Matter

Nucl. Sci. 29 (2019).

[21] Carl-Oscar Gullström and Andrea Rossi, Nucleon polarizability and long range strong force from σI=2 meson exchange

potential, arXiv 1703.05249, 2017.

[22] Alfonso Tarditi, Aneutronic fusion spacecraft architecture, 2012, NASA-NIAC 2001 PHASE I RESEARCH GRANT, Final

A.O. Di Tommaso and G. Vassallo / Journal of Condensed Matter Nuclear Science 29 (2019) 525–547 547

Research Activity Report (SEPTEMBER 2012), pp. 1–33,

https://www.nasa.gov/sites/default/ﬁles/atoms/ﬁles/niac_2011_phasei_tardittianeutronicfusionspacecraftarchitecture_tagged.pdf.

[23] H. Hora, G.H. Miley, J.C. Kelly and F. Osman, Shrinking of hydrogen atoms in host metals by dielectric effects and inglis-

teller depression of ionization potentials, Proc. 9th Int. Conf. on Cold Fusion, (ICCF9), 2002, pp. 1–6.

[24] F. Celani, C. Lorenzetti, G. Vassallo, E. Purchi, S. Fiorilla, S. Cupellini, M. Nakamura, P. Boccanera, R. Burri, B. Ortenzi,

L. Notargiacomo and A. Spallone, Steps to identify main parameters for ahe generation in sub-micrometric materials: mea-

surements by isoperibolic and air-ﬂow calorimetry, ICCF21, Int. Conf. on Cold Fusion, 3–8 June 2018, pp. 1–19, DOI:

10.13140/RG.2.2.16425.29287.

[25] Y. Iwamura, T. Itoh and N. Yamazaki, H. Yonemura, K. Fukutani and D. Sekiba, Recent advances in deuterium permeation

transmutation experiments. J. Condensed Matter Nucl. Sci. 10 (2013) 76–99.

View publication statsView publication stats