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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.
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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 massfrequency equivalence. A geometricelectromagnetic interpretation of mass, relativistic mass, De
Broglie wavelength, Proca, KleinGordon, Dirac and AharonovBohm 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 m1eV
AElectromagnetic vector
potential
V s m1eV
AtTime component of electro-
magnetic four potential
V s m1eV
AElectromagnetic vector
potential module
V s m1eV
mMass kg eV
FElectromagnetic
eld bivector
V s m2eV2
BFlux density eld V s m2=TeV2
EElectric eld V m1eV2
VPotential energy J=kg m2s2eV
JFour current density eld A m2eV3
JCurrent density eld A m2eV3
ρCharge density A s m3=C m3eV3
x, y, z Space coordinates meV1
tTime variable seV1
cLight speed in vacuum 2.997 924 58 ×108m s11
~Reduced Planck constant 1.054 571 726 ×1034 J s 1
µ0Permeability of vacuum 4π×107V s A1m14π
ǫ0Dielectric constant
of vacuum
8.854 187 817×1012 A s V1m11
4π
eElectron charge 1.602 176 565 ×1019 A s 0.085 424 546
αFine structure constant 7.2973525664 ×1037.2973525664 ×103
meElectron rest mass 9.10938356 ×1031kg 0.5109989461 ×106eV
λcElectron Compton wavelength 2.426 310 2389 ×1012 m1.229 588 259 ×105eV1
KJJosephson constant 0.4835978525 ×1015 Hz V12.71914766 ×102
reReduced Compton electron
wavelength (Compton radius)
re=λc
2π
rcElectron charge radius rc=αre
TeZitterbewegung period Te=2πre
c
1.9732705×107m1eV1.
6.5821220 ×1016 s1eV1.
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 eld generated by a point-like charged particle should have an innite
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 ux 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 scientic 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 simplied 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 sufcient 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 eV11.9732705 ×107m0.2µm,
T(1 eV)= 1 eV16.582122 ×1016 s0.66 fs.
Consequently, an angular frequency can be measured in electron volts:
1 eV 1.519268 ×1015 rad s1.
Following these concepts, it is possible to dene 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, dened as re=λc
2π0.38616 ×1012 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 rc2.8179 ×1015 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 ux 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 eld 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
=α0.0854245NU
.(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 veried by experiment, since the frequency of the oscillatory motion is so high and
its amplitude is so small”. In the scientic 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 Aalong 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 Aand 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 fth 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
eANU
.
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+m2A= 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
ψmψ= 0 (23)
should have the form
FmF = 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
FωruF= 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+ωruA= 0,
that can be written as
ruA+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
eA+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 cγt(see Eq. (60) of [1] and Eq. (12))
e
ωA=cγt,(29)
that left multiplying by rubecomes
e
ωruA=rucruγt.(30)
Now, by applying (30) to (28) and remembering that A=F, (28) becomes
eF=ω2(rucruγt).(31)
Applying the identity F= (E+IB)γt(see Eq. (73) of [1], Eq. (31) becomes
e(E+IB)γt=ω2(rucruγt).(32)
This last equation can be split in two equations. The rst one deals with the electric eld 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 ux density eld B:
eIBγt=ω2ruc,
eIB=ω2ruc
that right multiplying for cbecomes:
eIBc =ω2ru.
As Band care orthogonal vectors in the Zitterbewegung model, it is possible to write also:
eIBc=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 conrm 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 ux and electric charge quantization. In this paragraph we call
“wave amplitude” the module Aof vector potential Ain 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 coefcient 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 ux ΦM, a
value two times the magnetic ux 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= 2K1
JNU
,
where
ΦM= 2Φo= 4.13566766 ×1015 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=r1. 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=re1v2
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 ux 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
1v2
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 conned 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=eAcan 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 pis 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=c2v2
z
r=c2v2
z
re1v2
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, nally,
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 T8.1×1021 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
eld, 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 ux density eld 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 eld vector: θπ
3,2π
3:
~2
k+~2
=~2,~k=±1
2~.
The torque exerted by the external ux density eld 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=EHEL= 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 ux density eld 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,µ1.645 ×1026 and BE= 1.5 T, the NMR frequency is νNMR
24.8 MHz, whereas for isotope 11
5B we have s= 3/2,µ1.36 ×1026 J T1and NMR frequency is νNMR
20.5 MHz. Another example deals with isotope 87
38Sr with s= 9/2and µ5.52 ×1027 J T1. 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 eld, is subjected to Larmor precession
and its spin value ±~/2is interpreted as the intrinsic angular momentum component parallel to the magnetic eld.
It is interesting to note that a hypothetical technology, able to align the intrinsic angular momentum of a sufcient
number of electrons, could favor the formation of a coherent superconducting and super-uid 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 lamentation or
clustering of electronic charge suggests that under certain conditions strong coulomb repulsion can be overcome by
cohesive forces as yet imprecisely dened”.
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, nds a possible deep energetic level of E03.7 keV (see Eqs. (16) and (17)) at a
distance r0from the nucleus. In particular Naudts demonstrates that
E0mec2α3.7 keV
at a distance from nucleus equal to
r0~
mec0.39 ×1012 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×
1012 mhas been computed, a value much smaller than the distance of about 74 ×1012 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 3.3×1010 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 =λc2.42 ×1012 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
cλc
π2
2.3×1012 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 αmec23.7 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 dening 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+αmec2514.728 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
re1
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 eld B, to a force fproportional to the gradient of
B
f=(B·µ).
Therefore, the magnetic eld 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 eld 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 rst 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 nd, 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 ow 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
nd 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 difcult
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 ux 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.
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View publication statsView publication stats
... According to another Zitterbewegung electron model [9,16], the electron can be modeled by a current loop, with radius r e , generated by a charge distribution that rotates at the speed of light. This current loop is proposed as the origin of the electron's mass, inertia, angular momentum, spin and magnetic momentum. ...
... The very restrictive conditions under which eq. 6 can be applied may be created only in very peculiar environments. A possible solution has been suggested in [16] where the spin value ± /2 is interpreted as the component of the electron's angular momentum parallel to an external magnetic eld while the electron, like a tiny gyroscope, is subjected to Larmor precession. This particular, semi-classical, interpretation of spin does not exclude the possibility that the electron's angular momentum may be aligned, in particular conditions, to the external magnetic eld, so that electrons behave as elementary particles with whole spin . ...
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High-energy particles are detected from spontaneous processes in an ultra-dense deuterium D(0) layer. Intense distributions of such penetrating particles are observed using energy spectroscopy and glass converters. Laser-induced emission of neutral particles with time-of-flight energies of 1–30 MeV u−1 was previously reported in the same system. Both spontaneous line-spectra and a spontaneous broad energy distribution similar to a beta-decay distribution are observed. The broad distribution is concluded to be due to nuclear particles, giving straight-line Kurie-like plots. It is observed even at a distance of 3 m in air and has a total rate of 107–1010 s−1. If spontaneous nuclear fusion or other nuclear processes take place in D(0), it may give rise to the high-energy particle signal. Low energy nuclear reactions (LENR) and so called cold fusion may also give rise to such particles.
Article
Laser-induced Coulomb explosions in ultra-dense hydrogen clusters prove that the interatomic distance in such clusters is a few picometers, since the well-defined kinetic energy release (Coulomb repulsion energy) is many hundred eV. In the best characterized case which is ultra-dense deuterium D(-1) or d(-1), the kinetic energy given to the fragments in a cluster is normally 630 eV. This implies a D-D distance of 2.3 +/- 0.1 pm (2.15 +/- 0.02 pm measured in D-4 clusters). A description based on theoretical work by J.E. Hirsch using only the electron spins predicts that several excitation levels of spin quantum numbers exist in these quantum materials. The agreement with the experimental D-D distance is excellent at 2.23 pm for s=2. Theory is now verified by experimentally detecting states with spin values s=1 and s=3. The D-D distance is only 0.56 pm in the lowest excitation level s=1. The previously suggested "inverted" structure of D(-1) is obsolete since the new theory gives a better explanation of the accumulated experimental results.