Content uploaded by Andrea Rossi

Author content

All content in this area was uploaded by Andrea Rossi on Oct 22, 2022

Content may be subject to copyright.

E-Cat SK and long-range particle interactions

Andrea Rossi

Premise

This is a major revision made in December 24 2020 of the original pre-print

Last update: October 21st 2022

Abstract

Some theoretical frameworks that explore the possible formation of dense exotic

electron clusters in the E-Cat SK are presented. Some considerations on the proba-

ble role of Casimir, Aharonov-Bohm, and collective eects in the formation of such

structures are proposed. A relativistic interaction Lagrangian, based on a pure elec-

tromagnetic electron model, that suggests the possible existence of very low entropy

charge aggregates and that highlights the primary role of the electromagnetic potentials

in these clusters, is presented. The formation of these cluster may be associated to a

localized Vacuum polarization generated by a rapid radial charge displacement. The

formation of these dense electron clusters are introduced as a probable precursor for

the formation of proton-electron aggregates at pico-metric scale, stressing the impor-

tance of evaluating the plausibility of special electron-nucleon interactions, as already

suggested in [21]. An observed isotopic dependence of a particular spectral line in the

visible range of E-Cat plasma spectrum seems to conrm the presence of a specic

proton-electron interaction at electron Compton wavelength scale.

keywords:

Aharonov-Bohm eect, Anomalous Heat Eect, Bose-Einstein Condensate,

Casimir eect, charge clusters, collective eects, Darwin Lagrangian, electron model, Elec-

trum Validum, geometric phase coherence, long range interactions, low entropy aggregates,

pico-metric structures, Electron Energy Distribution Function (EEDF), relativistic interac-

tion Lagrangian, vector potential, Zitterbewegung electron model

Introduction

The E-Cat technology poses a serious and interesting challenge to the conceptual founda-

tions of modern physics. Particularly promising, for understanding this technology, is the

exploration of long-range particle interactions. In paragraph

Nuclear Reactions in Distant

Collisions

[49], E. P. Wigner highlights their importance in nuclear transfer reactions:

The

fact that nuclear reactions of the type

Au197 +N14 →Au198 +N13

take place at energies

at which colliding nuclei do not come in contact is an interesting though little-advertised

discovery

. More recently a possible double role of electrons in long range interactions has

been suggested in

Nucleon polarizability and long range strong force from

σI = 2

meson

exchange potential

" [21]:

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.

1

In this paper we propose that, at a relatively long distance, intermediate between the

atomic and nuclear scale, in the same order of magnitude of electron Compton wavelength,

the eects of magnetic force, the Casimir force and quantum vacuum/virtual particles should

not be dismissed. In particular, in section 1 we show that Coulomb repulsion between elec-

trons at a distance of four reduced Compton wavelengths can be balanced by the Casimir

force in specic geometric congurations. The possible role of Casimir forces in the E-Cat

technology has been rstly proposed by Professor Sven Kullander during our discussions in

2013. In section 2, extending to leptons the N.D. Cook, V. Dallacasa and P. Di Sia nu-

clear force model [13, 14], based on the magnetic attraction between nucleons, and applying

the condition that the four-distance between charges in Minkowski space-time is a light-like

vector, a possible balance of magnetic and Coulomb force is proposed. A relativistic inter-

action Lagrangian that suggests the possibility of these coherent low entropy aggregates is

presented. In section 3, it is hypothesized that a relatively narrow Electron Energy Distri-

bution Function (EEDF) is a pre-condition that may favor the formation of these coherent

aggregates. A mechanism that may allow Zero Point Energy within the E-Cat technology

will be presented in section 4. In section 5 dense electron clusters are introduced as a prob-

able precursor for the formation of proton-electron aggregates at pico-metric scale. In this

last section one spectroscopic signature of these structures is discussed. Section 6 contains

a brief description of the experimental setup, while in section 7 the E-Cat SK performance

is computed.

1 Charge clusters and the Casimir force

Putho and Piestrup in their paper

Charge connement by Casimir force

[41] propose,

as a possible cause of the high-density charge clustering seen by K. Shoulders [44] and

other researchers, the vacuum pressure hypothesized in 1948 by H. B. G. Casimir and

experimentally veried by S. K. Lamoreaux [32] in 1996. To compensate electron Coulomb

repulsion with vacuum pressure in a spherical shell distribution of

N

electrons, Putho found

a critical value for the sphere radius

RN

:

RN≈~√N

2mec=c√N

2ωe

=re√N

2,

(1)

where

re=c

ωe=λe

2π

is the reduced electron Compton wavelength. This value is derived by

applying the Compton angular frequency

ωe=mec2

/~

as the cuto frequency for electron-

vacuum interactions and assuming a vacuum spectral energy density

ρ(ω) :

ρ(ω) = ~ω3

2π2c3dω.

For a charge cluster of

N= 1011

electrons, the computed cluster size

D

is approximately

D= 2RN≈0.12 µm

, a value not too far from the typical charge cluster size seen by Shoul-

ders. The electron distance

dE

in the spherical shell that minimizes electrostatic potential

can be roughly approximated as

dE≈r4πR2

N

N=√πre≈1.78re≈0.68 ·10−12 m.

(2)

It's interesting to note that this distance is not a function of

N

but a constant value of

the same order of the reduced electron Compton wavelength

re=λe

/2π≈0.38 ·10−12 m.

At this scale the electron should not be modeled as a point-like particle, not even as a rst

2

approximation. Consequently, a more detailed and realistic electron model is preferable to

evaluate the Casimir eect in free electron clusters.

An interesting approach along this direction is proposed by J. Maruani in his paper

The

Dirac Electron and Elementary Interactions

[33]. To compute the Casimir force between

electrons, Maruani suggests applying the Casimir force

FC

formula per unit area

A

for

the

ideal case of perfect plates in perfect vacuum at 0 Kelvin

:

FC(d)

A=π2~c

240d4.

(3)

where

d

is the distance between plates and

c

is the light speed in vacuum. Maruani

considers a Zitterbewegung [26, 25, 23, 8] electron model where the reduced Compton wave-

length is the electron

diameter

. In this case the plate area in (3) becomes

A=π(λe

/4π)2

and the attractive Casimir Force

FC(d)

between electrons can be computed and compared

with the Coulomb repulsion force

Fe(d) :

FC(d) = π~cλ2

e

3840d4,

(4)

Fe(d) = 1

4πε0

e2

d2.

(5)

According to this approach, the Casimir force balances Coulomb repulsion approximately

at a distance

db≈2λe

/2π≈0.77 ·10−12m

, a value close to that of two reduced Compton

wavelengths (see Fig. 1 in [33]).

According to another Zitterbewegung electron model [8, 15, 31], the electron can be

modeled by a current loop, with radius

re

, 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. In this case the area enclosed

by the zbw current is

A=π(λe

/2π)2=πr2

e

, a value four times larger than that used by

Maruani, and consequently the Casimir force may reach a value four times greater than

the one indicated in (4). With this larger area, Coulomb repulsion is balanced at a distance

db≈4λe

/2π≈1.54·10−12 m

, as shown in Fig. 1, where in a logarithmic scale the hypothesized

Casimir force between two electrons is plotted together with Coulomb and a magnetic force

computed considering the electrons as two parallel aligned current loops. We can nd the

idea of an internal rapid motion (Zitterbewegung) at light-speed in electrons in the P.A.M.

Dirac Nobel lecture [16].

2 Charge clusters and magnetic interactions

2.1 Space-charge, vacuum polarization and virtual particles

An important eect in vacuum tubes is the so-called space-charge. This name is related

to the spontaneous formation of an electron cloud around a cathode heated in vacuum. Al-

though well known and exploited since the early years of vacuum tube technology, this eect

lacks a well-dened theory. This statement is supported by the observation that the forma-

tion of a stable space-charge should be prevented by the Coulomb repulsion between free

electrons. L. Nelson in US patent 6465965 proposes, as a rationale for this long-range elec-

trostatic screening, a possible vacuum polarization, generated by the creation-annihilation

of virtual charges pairs as a consequence of the quantum vacuum uctuations predicted by

the Heisenberg uncertainty principle. The lifetime of such particle-antiparticle couples is

3

Figure 1: Trends of Casimir, Coulomb and magnetic forces as a function of distance.

inversely proportional to their mass-energy, but, during their short existence, these may act

as the charges in the solid dielectric of a capacitor that, screening the electric eld, lower the

voltage required to accumulate a charge in capacitor plates. The creation of these virtual

particles is favored by the high density of allowable energy states in vacuum and is hindered

by the relatively low number of permitted states in an ordinary metallic conductor. Ac-

cording to Nelson, this dierence may be exploited to generate a macroscopic voltage and

an energy gain. Alternative hypotheses, based on self-organizing Zitterbewegung electron

phases in vacuum and Lorentz force, are however possible as will be shown in the next

sub-sections.

In any case, the long-range interaction between the electrons in the space charge is a

phenomenon that deserves to be seriously studied and investigated [47].

2.2 Lorentz force and Zitterbewegung phase coherence

According to [8, 10, 31], the electron is associated with a magnetic ux

ΦM=h

/e

equal to the

ratio of the Planck constant

h

and the elementary charge

e

. Consequently, the possible role

of a magnetic attraction in charge connement cannot be dismissed

a priori

. As shown in

Fig. 1, the magnetic force between two electrons, if naively modeled as two parallel aligned

current loops, cannot compensate for the Coulomb repulsion. However, at this point, it

is important to remember that the Zitterbewegung current is generated by an elementary

charge

e

that rotates at light-speed

c

along a circumference equal to the electron Compton

wavelength [8, 31] and, consequently, that a rotation phase coherence between charges in the

same light cone may greatly enhance the magnetic attraction.

In this case, the force can be computed as the Lorentz force

FL

acting on an elementary

charge moving at the speed of light. Its value can balance the Coulomb repulsion:

FL(d) = ecB (d) = µo

4π·e2c2

d2=1

4π0·e2

d2,

(6)

where

B(d) = µoec

4πd2

(7)

4

is the magnetic ux density generated by another elementary charge that moves in parallel

at light-speed

c

at a distance vector

~

d

orthogonal to the charge velocity vector.

A similar approach has been suggested by Norman Cook, Paolo Di Sia and Valerio

Dallacasa [13, 14, 11], as the possible magnetic origin of the strong nuclear force. The

condition that the charges must be in the same light-cone [34] can be satised if the electron

distance

d

is an integer multiple of Compton wavelength while the rotating charges have the

same Zitterbewegung phase:

d=nλe

(8)

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 [15] 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

~

. In this case electron clusters may form Bose-Einstein condensates where electron

Zitterbewegung phases are synchronized and electron distances respect equation (8). In this

highly ordered, low entropy, hypothetical structure the Coulomb repulsion is balanced by

the magnetic force

FL

in agreement with (6). In section 2.3 we will propose a Lagrangian for

N interacting charged particles that suggests the possible existence of these coherent states.

In [15]

a fundamental connection

between Aharonov-Bohm equations and an electron

model is proposed, starting from a geometric interpretation of the electron wave-function

complex phase [24, 23, 25]. This approach suggests the possibility of eciently creating

electron condensates exploiting the Aharonov-Bohm eect, a phenomenon that shows the

dependence of the electron wave-function phase from electromagnetic potentials [1]. In [15] it

is hypothesized that a voltage pulse with a very short, critical rise time may favor the creation

of coherent and dense electron clusters:

The conjecture is based on the possibility that, as a

consequence of Aharonov-Bohm eect, a rapid, collective and simultaneous variation of the

Zitterbewegung phase catalyzes the creation of coherent systems

.

2.3 Darwin Lagrangian

In his work

Magnetic energy, superconductivity, and dark matter

[17] Prof. Essén empha-

sizes the importance of long-range magnetic interactions in systems where a large number

of charged particles are involved. He proposes, as a possible useful tool in modeling such

interactions, a Darwin Lagrangian

LD

, that relates the electromagnetic potentials with the

kinetic energy:

LD=

N

X

a=1 hma

2v2

a−ea

2φa(ra) + ea

2cva·Aa(ra)i

(9)

Aa(ra) =

N

X

b6=a

eb[vb+ (vb·ruab)ruab ]

2crab

φa(ra) =

N

X

b6=a

eb

rab

5

rab =|ra−rb|

ruab =ra−rb

|ra−rb|

In these equations the letters

a

and

b

are used as indexes of the massive charged particles,

ra

are their spatial coordinates,

ea

their charge value,

va

their velocity,

ma

their mass,

Aa(ra)

and

φ(ra)

are respectively the vector and electric potential at

ra

and

N

is the total number

of the interacting particles. Gaussian unit system has been used.

The Darwin Lagrangian can be conceptually simplied recognizing that the mechanical

momentum

pa

of a massive charged elementary particle has a pure electromagnetic origin:

pa=mava=ea

cAazp

In this last equation

Aazp

is the component of the vector potential

Aaz

, generated by

the Zitterbewegung current, parallel to the particle's velocity vector

va

. This means that we

can write a kinetic energy term that is only a function of the magnetic vector potential:

ma

2v2

a=p2

a

2ma

p2

a

2ma

=e2

aA2

azp

2c2ma

ma=eaAaz

c2

p2

a

2ma

=eaA2

azp

2Aaz

For non-relativistic speed we can write:

Aazp

Aaz 'va

c

(10)

Being that

va

and

Aazp

are parallel vectors it's possible to substitute the product of their

modules with the dot product:

ma

2v2

a=eavaAazp

2c=ea

2cva·Aazp

(11)

consequently we can encapsulate the kinetic energy terms inside the vector potential

ones:

Aat (ra) = Aazp +Aa(ra)

(12)

LD=

N

X

a=1 h−ea

2φa(ra) + ea

2cva·Aat (ra)i

(13)

Equation 13 is a rewriting of eq. 9 that clearly shows a more fundamental role of the

electromagnetic potentials, considering that all the kinetic energy terms can be expressed as

a function of the magnetic vector potential.

6

2.4 Zitterbewegung Lagrangian

The component

Aazp

in eq. 12, for non-relativistic speeds, is a tiny fraction of the Zitter-

bewegung generated vector potential

Aaz

as shown in eq. 10. This observation suggests the

possibility to write a new Lagrangian that does not exclude the role of

Aaz.

Accepting an appropriate, pure electromagnetic, Zitterbewegung model for the electrons

[15, 31], the rst step along this path starts with substituting the concept of massive charged

particles with the more fundamental idea of mass-less elementary charges

e

moving at

the speed of light with a mechanical momentum proportional to the dot product of their

velocity and the vector potential value [31]. This choice implies a possible active role of the

vector potential, associated with the rest-mass energy, in the magnetic interactions. The

interactions occur only between charges that are in the same light-cone. This means that

their distance in Minkowski space-time must be a light-like (nilpotent) vector (eq. 23). Using

natural units, where

~=c= 1,

this relativistic interaction Lagrangian has a very simple

form:

Lz=

N

X

a=1

[eaca·Aa(ra)−eaφa(ra)]

(14)

To conrm the validity of 14 we must demonstrate that it satises the classical Lagrangian

denition:

L=T−U

(15)

T=

N

X

a=1

eaca·Aa(ra)

U=

N

X

a=1

eaφa(ra)

Now, according to the Ehrenberg-Siday-Aharonov-Bohm equations, the Zitterbewegung

geometric phase is ruled by the vector potential (eq. 16) and by the electric potential (eq.

17):

dϕaM =eaAa(ra)·dl

dl=cadt

dϕaM =eaAa(ra)·cadt

(16)

dϕaE =eaφa(ra)dt

(17)

Dividing eq. 16 for

dt

we obtain the value of the Zitterbewegung frequency of the charge

ea,

a value equal to the relativistic mass-energy of the particle

a

:

ωzbw =dϕaM

dt =ma

7

T=

N

X

a=1

ma

These observations conrm that eq. 14 respects the classical Lagrangian denition, con-

sidering that the kinetic energy of the electron's mass-less charge is exactly equal to its

relativistic mass.

The vector potential

Aa(ra)

is the sum of the self-interaction term

eaca

αrea

with an interac-

tion term:

Aa(ra) = eaca

αrea

+X

b6=a

eb[cb+ (cb·ruab)ruab ]

rab

(18)

φa(ra) = ea

αrea

+X

b6=a

eb

rab

(19)

Lz=

N

X

a=1 (1

rea

+X

b6=a

αca·cb+α(cb·ruab) (ca·ruab )

rab −"1

rea

+X

b6=a

α

rab #)

(20)

Lz=

N

X

a=1 X

b6=a

α[ca·cb+ (cb·ruab) (ca·ruab )−1]

rab

(21)

rab =ra−rb

rab =|ra−rb|

ruab =rab

rab

ca·cb= cos (ϑab1)

ca·ruab = cos (ϑab2)

cb·ruab = cos (ϑab3)

ϑab1=ϑab2−ϑab3

Lz=

N

X

a=1 X

b6=a

α[cos (ϑab2−ϑab3) + cos (ϑab2) cos (ϑab3)−1]

rab

Lzab =α[cos (ϑab2−ϑab3) + cos (ϑab2) cos (ϑab3)−1]

rab

8

In these latter equations

ra

is the generic spatial position of the mass-less charge

ea

,

ca

its unit velocity vector

(c2

a= 1)

,

α=e2

a

the ne structure constant

(α−1≈137.036)

,

rea

the

Zitterbewegung radius,

rab

is the Euclidean distance between the mass-less charges and

tab

their time distance.

ruab

is a unit vector that has the same direction of

rab

. The inverse of

the Zitterbewegung radius in natural units is equal to the value of the relativistic mass of

the charged particle

(mea =r−1

ea )

. The product

αrea

is the charge radius.

The phase space trajectory of the N charges is determined by the stationary Action

condition

δ(S)=0

S=ˆ4T

Lzdt.

According to eq. 21 the Action has the following simple form

S=

N

X

a=1 X

b6=aˆ4T

α[cos (ϑab2−ϑab3) + cos (ϑab2) cos (ϑab3)−1]

rab

dt

(22)

S=

N

X

a=1 X

b6=aˆ4T

Lzabdt

δ(Lzab) = 0 =⇒δ(S) = 0

δ(Lzab (rab, ϑab2, ϑab3)) = ∂Lz

∂rab

δrab +∂Lz

∂ϑab2

δϑab2+∂Lz

∂ϑab3

δϑab3= 0

∂Lz

∂rab

=−α[cos (ϑab2−ϑab3) + cos (ϑab2) cos (ϑab3)−1]

r2

ab

∂Lz

∂ϑab2

=α[−sin (ϑab2−ϑab3)−sin (ϑab2) cos (ϑab3)]

rab

∂Lz

∂ϑab3

=α[sin (ϑab2−ϑab3)−cos (ϑab2) sin (ϑab3)]

rab

r2

ab −c2t2

ab = 0

(23)

From these equations we can see that the

coherence condition

(eq. 24) satises the

principle of stationary Action

(ϑab1= 2πn)∩ϑab2=π

2+πm=⇒δ(S) = 0 (n, m ∈Z).

(24)

When these coherence conditions are satised the Coulomb repulsion is balanced by the

Lorentz force as already shown in par. 2.2. This may explain the high density of the electron

clusters studied by Kenneth Shoulders [44][46][45].

9

2.5 Entropy of the coherent clusters

Although the formation of charge cluster coherent states is compatible with the condition of

stationary Action (eq. 22), its probability is heavily hindered by the extremely low entropy of

such states. The order of magnitude of the entropy ratio between non-coherent and coherent

congurations is approximately equal to the number N of interacting particles, considering

that the coherent state can be described by a single wave-function, as in BEC condensate.

The Darwin Lagrangian may be used for non coherent states of N interacting electrons

observed at time scales larger than the Zitterbewegung period

(tz'8.1·10−21s)

. In this case

the average value of the component of the vector potential orthogonal to electron velocity

vanishes and does not play a role in magnetic interactions, but is hidden in the kinetic energy

term, being the value of its module multiplied by the elementary charge equal to the electron

rest mass in natural units.

3 Coherent clusters and EEDF

The Zitterbewegung angular frequency

ωzbw

is exactly equal the electron relativistic mass-

energy

m

in natural units:

ωzbw =m

The relativistic mass-energy

m

is the sum of the rest mass

m0

and the kinetic energy

Ek

m=m0+Ek

The collective phase-lock in the charge clusters requires monochromatic electrons [3] or

a very narrow distribution of Zitterbewegung frequency and consequently an environment

with a narrow Electron Energy Distribution Function (EEDF) may favor the creation of

these coherent structures. The EEDF in a gas mixture plasma discharge is a function of the

pressure and gas composition [6] [19], consequently an appropriate choice of these parameters

[37], narrowing the EEDF, may favor the formation of these aggregates.

4 Energy from the Vacuum

In his book An Introduction To A Realistic Quantum Physics [38, 39], Giuliano Preparata

denes the Vacuum as

the template of physical reality

that

does not precede creation but

is, actually, a fundamental piece of it

.

Following this point of view some authors [2] claim that the keys to understand the emer-

gence of matter-energy from Vacuum are the magnetic vector potential and the Aharonov-

Bohm eect, completely reversing the widely accepted idea that considers the vector potential

only as an useful math tool. In their work

Aharonov-Bohm eect as the basis of electromag-

netic energy inherent in the vacuum

[2] the authors, starting from this concept, deduce that

devices can be manufactured in principle to take an unlimited amount of electromagnetic

energy from the vacuum as dened by the Aharonov-Bohm eect, without violating Noether's

Theorem

.

Within this conceptual framework Putho has explored [9][40] the idea that, in principle,

it's possible to extract energy and heat from electromagnetic zero-point radiation via the use

of Vacuum pressure. A device that may reach this goal has been proposed in the US patent

US7379286 [22], where the authors consider the possibility of a local energy extraction that

is

replenished globally from and by the electromagnetic quantum vacuum

.

10

The idea that Vacuum is structured and that can be exploited to localize energy extracted

from the environment needs a clear denition of this structure and its relation with energy

and matter. The rst step starts with recognizing the electromagnetic potentials as the

Vacuum structure and consequently as the fundamental entities of the physical reality.

The second one requires an encoding of their relations with both the energy density and the

energy ux in the space-time continuum. Calling

the four-gradient and

γt

the unit vector

along the time axis of the Minkowski space-time, we can dene a scalar eld

S

that is the

four-divergence of the electromagnetic four-potential

A

:

A=A+γtφ

·A=S

Now, the derivatives of the four-potential can be viewed as an operator that rotates in

the four dimensions of space-time the unit vector

γt,

giving raise to a four-vector that has

the time component equal to the Vacuum energy density

U

and the three space components

equal to the energy density ux vector

P

[31]:

1

8πAγtg

A=Uγt+P

P=−1

4π(E×B− SE)

(25)

As we can see the vector

P

is the sum of two vectors: the rst one is the Poynting vector

while the second one is a vector equal to the product of the scalar eld

S

and the electric

eld

E.

In mainstream literature, as a consequence of the widespread application of the

Lorenz gauge

(·A= 0)

, the scalar eld S and the vector

SE

are generally ignored, but

nevertheless their signicance has been highlighted by many authors [42, 48, 7, 36, 28, 29,

35, 50, 51, 43].

Combining the Gauss law (eq. 26) with the component

SE

/4π

of the generalized Poynting

vector

P

, there emerges a non-null divergence of an energy ux density that clearly implies

the presence of a power source or a power sink where both charge density

ρ

and scalar eld

S

are not vanishing. In this case the time derivative of the energy density U is equal to the

product of the charge density

ρ

and the scalar eld

S

4πρ =∇ · E

(26)

ρS=1

4π∇ · ES

dU

dt =ρS

(27)

This time derivative of the energy density can be interpreted as a power ux that is a

consequence of a non-null derivative of the electric potential

φ

:

dU

dt =ρdφ

dt

Integrating over a volume that contains a single electron the eq. 27 becomes:

11

eS =edφ

dt

(28)

Now, combining eq. 28 with the dierential form of the electric Aharonov-Bohm equation

(eq. 29), we can see that the presence of a scalar eld implies a variation

δωzbw

of the

electron's Zitterbewegung frequency

ωzbw

and the appearance of a force

fS

:

eφ =dϕ

dt

(29)

δωzbw =dϕ

dt

eS=deφ

dt

eS=d2ϕ

dt2

eS=dωzbw

dt

(30)

φ=ˆSdt

(31)

fS=−e∇φ

Experimental data suggest that an intense impulsive current with a radial/cylindrical

symmetry and a critical rise time creates a Scalar eld that generates radial forces that,

conning the charges, create the conditions for the formation of the coherent aggregates

discussed in the previous sections. This radial charge displacement can be generated by an

appropriate electrode geometry [4, 20, 12] or by the pinch eect generated by short and

intense current impulses [46]. Eq. 30 describes an energy-mass change, the sign of which

depends on the sign of the product

eS.

This process is a consequence of a Vacuum polarization

caused by the presence of the Scalar eld

S.

This implies the possibility of a long range

interaction that consists in a mass-energy transfer from positive charged particles to negative

ones or vice-versa. This mass-transfer obviously does not violate the principle of energy

conservation and apparently does not lead to the instability of the nuclei of the positive ions

present in the plasma, even if some authors claim the possibility that Scalar elds may alter

the nucleus dynamic [27]. The tiny energy lost by the nuclei can be replenished by their

interaction with the active Vacuum.

The hypothesis that a mass-transfer mechanism may be the cause of the anomalous heat

seen in the E-Cat QX has been presented in a previous paper [21], inside however a dierent

theoretical framework.

To evaluate the power generated in a device as the E-Cat-QX, assuming that the thermal

energy is generated by the electron transition from a coherent to an incoherent state and

assuming an electron distance in the coherent state that is equal to the electron Compton

wavelength

(λe≈2.43 ·10−12m)

, we get a power output in the order of several tens of watts:

I= 0.25A

12

dne

dt =I

e= 1.56 ·1018ne

/s

Ee=1

4πε0

e2

λe

wout =Ee

dne

dt '150w

5 Neutral pico-metric aggregates

Coherent charge clusters may form, in presence of protons, compact neutral aggregates at

a pico-metric

(10−12 m)

scale, intermediate between the atomic

(10−10 m)

and nuclear size

(10−15 m)

, formed by a coherent chain of bosonic electrons with protons located in the center

of their Zitterbewegung orbits [15]. A critical, cathode-temperature-dependent, threshold of

electron density is an important precondition for the creation of such structures.

The existence of electron-proton and electron-deuteron structures at this scale has been

already experimentally veried and studied [5, 52, 18]. In [30] Holmlid recognizes the electron

Zitterbewegung as the underlying rationale for such aggregates:

This electron spin motion

may be interpreted as a motion of the charge with orbit radius

rq=~

/2mec≈0.192

pm and

with the velocity of light c (`zitterbewegung')

. It's important to note that this radius value,

as proposed by Holmlid, Maruani and Hestenes [23], is one half the zbw radius value

re

in [8,

31], and that the choice of such value

(rq=re

/2)

implies that no distinction is made between

electron intrinsic angular momentum and spin, excluding consequently the possibility of

existence of bosonic electrons with spin=

~

.

An interesting aspect of the electron-proton interactions proposed in [15] is given by

the possibility to experimentally verify the existence of some specic spectral signatures.

According to [15] the electron's charge can orbit around a proton at a distance of about

re= 0.38 pm

. The intense magnetic ux density

Bzbw

generated by the rotating charge at

the center of the Zitterbewegung current loop is [8]

Bzbw = 32.21 ·106T.

Now, the proton magnetogyric ratio

gH

is

gH= 267.52 ·106rad ·s−1·T−1

and consequently the Nuclear Magnetic Resonance frequency is

νNMR =gHBzbw

2π= 1.3714 ·1015 Hz

and the relative precession frequency

νp

is

νp=νNMR

/2= 6.8571 ·1014 Hz.

This frequency corresponds to a wavelength in the visible spectrum

λp=c

νp

= 4.372 ·10−7m

13

The presence of this line in the E-Cat plasma spectrum is a possible indication of the

existence of this type of pico-metric aggregate. A stronger and reliable clue in this direction

comes from observing that the amplitude of this spectral line is a clear function of the hydro-

gen isotope present in the plasma: the line is strongly reduced when deuterium is used in the

charge instead of protium. This consideration is supported by the observation that a deuteron

has a much smaller magnetogyric ratio than proton

(gD= 41.066 ·106rad ·s−1·T−1)

. Con-

sequently, considering the strong chemical similarity of deuterium and protium, this large

macroscopic dierence in spectral emission under the same conditions reveals its nuclear

origin.

6 Experimental Setup

The plausibility of these hypotheses is supported by a series of experiments made with the

E-cat SK. The E-cat SK has been put in a position that allows the lens of a spectrometer

to exactly view the plasma in a dark room: an ohmmeter measures the resistance across

the circuit that gives energy to the E-Cat; the control panel is connected to a

220 V

outlet,

while the two cables connected with the plasma electrodes start from the control panel.

A frequency meter, a laser, and a tesla-meter have been connected with the plasma for

auxiliary measurements and a Van de Graa electron accelerator

(200 kV )

has been used for

the examination of the plasma electric charge. Other instruments used in the experimental

setup are: a voltage generator/modulator; two oscilloscopes, one for the power source and

one for monitoring the energy consumed by the E-Cat; Omega thermocouples to measure

the delta T of the cooling air; IR thermometer; a frequency generator; a Geiger counter and

bubbles columns to measure emissions of ionizing radiations and neutrons.

7 Evaluation of E-Cat SK performance

The performance of the E-Cat SK is summarized in the following calculations. The plasma

temperature can be calculated applying the Wien equation. Calling

b

the Wien displacement

law constant and

λmax

the observed peak wavelength of the radiation we have

Tk=b

λmax

Tk=2.898 ·10−3

0.3575 ·10−6= 8106 K.

Power emission and the average energy produced in one hour can be computed applying

the Stefan-Boltzmann law

Wout =σεT 4

kA≈22 kW

Eout = 22 k W h

where

σ= 5.67 ·10−8W m−2K−4

,

ε= 0.9

(assuming a non-perfect black body) and

A≈10−4m2

(the length of the cylindrical shaped plasma core is

l≈1cm

, while its diameter

is

d≈0.3cm

).

This value must be compared with the calorimetric measurements, considering that the

spectrum of the radiations has not a full Maxwellian curve. The E-Cat has been installed

14

in a laboratory of an industry in the State of Tennessee, in the USA, to keep warm a room

that has a surface of 3000 sq.ft (about 300 sq.m.) and a height of 15 ft (about 5 m).

The temperature outside when we made the measurements was about 32

°

F (0

°

C) and the

temperature in the room was about 61

°

F (16

°

C). To keep this temperature it was used before

a heater of about 20-22 kW.

In detail:

Fan ow rate: 5500

m3

/h'

6700

kg

/h

delta T = 16

°

C

Cp air = 0.17

W = 6700 x 0.17 x 16 = 18224

Kcal

/h

= 20.5

kW h

/h

We also made a test with an air ow of 330

m3

/h

and obtained a deltaT of 312

°

C.

Every 60 days of continued operation the E-Cat SK produces- as we can nd with a simple

extrapolation- 30000 kWh of heat, approximately the equivalent of 2600 kg of heating oil,

therefore avoiding, at the same time, the emission of more than 8000 kg of

CO2

. Now, calling

Einp

the energy consumed by the control panel in one hour

Einp = 380 W h

we can compute the average coecient of performance (COP), as the ratio of output and

input energies

COP =Eout

Einp ≈54

Conclusions

In this paper, three dierent, not mutually exclusive Ansätze, for long-range particle inter-

actions in E-Cat SK have been proposed. The rst one is based on the possible role of the

Casimir force in dense electron aggregates: two dierent approaches, one of which is based

on Zitterbewegung electron models, both indicate that Coulomb repulsion between electrons

may be balanced at a pico-metric scale. The second one, in analogy with the Norman Cook

idea of magnetic origin of strong force [13, 11], deals with the Lorentz forces in coherent

systems, where electron Zitterbewegung phases are synchronized and electron charges are in

the same light cone. A relativistic interaction Lagrangian for a set of elementary charged

particles that suggests the possible existence of these coherent states has been proposed. An

hypothesis that peculiar discharge geometries and dynamics create a Vacuum polarization

that favors the formation of these low entropy structure, has been presented. The third one

is based on the possible electrostatic screening eect of virtual particle pairs created by the

uctuations of quantum vacuum.

As a consequence of these relatively long-range interactions, the possible formation of

dense aggregates at pico-metric scale has been proposed. An E-Cat plasma spectral signa-

ture, isotopic dependent, in the visible range of a proton-electron pico-metric structure has

been reported.

R.P.Y∞X

Acknowledgments

I acknowledge, for interesting discussions and collaborations on the subject, Carl Oscar

Gullström and Giorgio Vassallo.

15

Bibliography

References

[1] Y. Aharonov and D. Bohm. Signicance of Electromagnetic Potentials in the Quantum

Theory. In:

Physical Review

115 (Aug. 1959), pp. 485491.

doi

:

10.1103/PhysRev.

115.485

.

[2] P. Anastasovski et al. Aharonov-Bohm eect as the basis of electromagnetic energy

inherent in the vacuum. In:

Foundations of Physics Letters

15 (Dec. 2002), pp. 561

568.

[3] M.J. Arman and C. Chase.

System and methods for generating coherent matterwave

beams

. US Patent US9502202. 2016.

[4] H. Aspden.

Power from space: The Correa invention

. 1996.

[5] S. Badiei, P.U. Andersson, and L. Holmlid. High-energy Coulomb explosions in ultra-

dense deuterium: Time-of-ight-mass spectrometry with variable energy and ight

length. In:

International Journal of Mass Spectrometry

282.12 (2009), pp. 7076.

issn

: 1387-3806.

[6] N. L. Bassett and D. J. Economou. Eect of Cl2 additions to an argon glow discharge.

In:

Journal of Applied Physics

75.4 (1994), pp. 19311939.

doi

:

10.1063/1.356340

.

[7] G. Bettini. Cliord Algebra, 3 and 4-Dimensional Analytic Functions with Appli-

cations. Manuscripts of the Last Century. In:

viXra.org

Quantum Physics (2011).

http://vixra.org/abs/1107.0060, pp. 163.

url

:

http://vixra.org/abs/1107.0060

.

[8] F. Celani, A.O. Di Tommaso, and G. Vassallo. The electron and Occam's razor. In:

Journal of Condensed matter nuclear science

25 (2017), pp. 7699.

[9] D. Cole and H. Putho. Extracting energy and heat from the vacuum. In:

Physical

review. E, Statistical physics, plasmas, uids, and related interdisciplinary topics

48

(Sept. 1993), pp. 15621565.

[10] O. Consa. Helical Model of the Electron. In:

The General Science Journal

(2014),

pp. 114.

[11] Norman D. Cook and Andrea Rossi.

On the Nuclear Mechanisms Underlying the Heat

Production by the E-Cat

. 2015. arXiv:

1504.01261 [physics.gen-ph]

.

[12] P.N. Correa and A.N. Correa.

Direct current energized pulse generator utilizing auto-

genous cyclical pulsed abnormal glow discharge

. US Patent US5502354. 1986.

[13] V. Dallacasa and N. D. Cook.

Models of the Atomic Nucleus

. Springer, 2010.

isbn

:

3540285695.

[14] P. Di Sia.

A solution to the 80 years old problem of the nuclear force

. doi = 10.5281/zen-

odo.1472981. Oct. 2018.

[15] A.O. Di Tommaso and G. Vassallo. Electron structure, Ultra-Dense Hydrogen and

Low Energy Nuclear Reactions. In:

Journal of Condensed Matter Nuclear Science

29

(2019), pp. 525547.

[16] P.A.M. Dirac.

Theory of Electrons and Positrons

. www.nobelprize.org, Nobel Founda-

tion. 1933.

[17] H. Essén. Magnetic energy, superconductivity, and dark matter. In:

Progress in

Physics

16 (Apr. 2020), pp. 2932.

16

[18] J.M. Frederick and J.R. Reitz. Electromagnetic Composites at the Compton Scale.

In:

International Journal of Theoretical Physics

51.1 (2012), pp. 322330.

issn

: 1572-

9575.

[19] V. A. Godyak, R.B. Piejak, and B.M. Alexandrovich. Measurement of electron energy

distribution in low-pressure RF discharges. In:

Plasma Sources Science and Technology

1.1 (1992), pp. 3658.

[20] E.V. Gray.

Ecient power supply suitable for inductive loads

. US Patent US4595975A.

1986.

[21] C.O. Gullström and A. Rossi.

Nucleon polarizability and long range strong force from

σI=2

meson exchange potential

. 2017. arXiv:

1703.05249 [physics.gen-ph]

.

[22] B. Haisch and G. Moddell.

Quantum Vacuum Energy Extraction

. US Patent US7379286.

2008.

[23] D. Hestenes. Hunting for Snarks in Quantum Mechanics. In:

American Institute of

Physics Conference Series

. Ed. by P. M. Goggans and C.-Y. Chan. Vol. 1193. American

Institute of Physics Conference Series. Dec. 2009, pp. 115131.

[24] D. Hestenes. The zitterbewegung interpretation of quantum mechanics. In:

Founda-

tions of Physics

20.10 (1990), pp. 12131232.

[25] D. Hestenes. Zitterbewegung in quantum mechanics. In:

Foundations of Physics

40.1

(2010), pp. 154.

[26] D. Hestenes. Zitterbewegung Modeling. In:

Foundations of Physics

23.3 (1993),

pp. 365387.

issn

: 1572-9516.

[27] L. Hively.

Systems, apparatus, and methods for generating and/or utilizing scalar-

longitudinal waves

. US Patent US9306527. 2016.

[28] L. Hively and G. Giakos. Toward a More Complete Electrodynamic Theory. In:

International Journal of Signal and Imaging Systems Engineering

5 (May 2012), pp. 3

10.

[29] L. Hively and A. Loebl. Classical and extended electrodynamics. In:

Physics Essays

32 (Mar. 2019), pp. 112126.

[30] L. Holmlid and S. Olafsson. Spontaneous Ejection of High-energy Particles from Ultra-

dense Deuterium D(0). In:

International Journal of Hydrogen Energy

40.33 (2015),

pp. 10559 10567.

issn

: 0360-3199.

[31] A. Kovacs et al.

Unied Field Theory and Occam's Razor

. World Scientic, June 2022.

isbn

: 978-1-80061-129-0.

[32] S. K. Lamoreaux. Demonstration of the Casimir force in the 0.6 to 6 micrometers

range. In:

Phys. Rev. Lett.

78 (1997), pp. 58.

[33] J. Maruani. The Dirac Electron and Elementary Interactions: The Gyromagnetic Fac-

tor, Fine-Structure Constant, and Gravitational Invariant: Derivations from Whole

Numbers. In: Jan. 2018, pp. 361380.

[34] C. Mead. The nature of light: what are photons? In:

Proc. SPIE

8832 (2013).

[35] K. Meyl.

Scalar Wave Eects according to Tesla

. Jan. 2006.

[36] G. Modanese. Generalized Maxwell equations and charge conservation censorship.

In:

Modern Physics Letters B

31 (Aug. 2016).

17

[37] J. Papp.

Method and means of converting atomic energy into utilizable kinetic energy

.

US Patent US3670494. 1972.

[38] G. Preparata.

An Introduction to a Realistic Quantum Physics

. World Scientic, 2002.

isbn

: 9789812381767.

[39] G Preparata.

QED Coherence in Matter

. World Scientic, 1995.

[40] H.E. Putho and E.W. Davis. On Extracting Energy from the Quantum Vacuum. In:

Frontiers of Propulsion Science

. American Institute of Aeronautics and Astronautics,

Inc., 2009. Chap. 19, pp. 569603.

[41] H.E. Putho and M.A. Piestrup.

Charge connement by Casimir forces

. 2004. arXiv:

physics/0408114 [physics.gen-ph]

.

[42] D. Reed. Unravelling the potentials puzzle and corresponding case for the scalar lon-

gitudinal electrodynamic wave. In:

Journal of Physics: Conference Series

1251 (2019).

[43] D. Reed and L. Hively. Implications of Gauge-Free Extended Electrodynamics. In:

Symmetry

12 (Dec. 2020).

[44] K. Shoulders.

EV, A Tale of Discovery

. Austin, TX, 1987.

[45] K. Shoulders.

Permittivity transitions

. Bodega, CA 94922, 2000.

[46] K. Shoulders and J. Sarfatti.

Energy Conversion From The Exotic Vacuum

. 2004.

[47] C.P. Tinsley. An interview with Martin Fleischmann. In:

Innite Energy Magazine

(11 1996).

[48] K.J. Van Vlaenderen.

A generalisation of classical electrodynamics for the prediction

of scalar eld eects

. 2003. arXiv:

physics/0305098 [physics.class-ph]

.

[49] Eugene Paul Wigner, Alvin M. Weinberg, and Arthur Wightman.

The Collected Works

of Eugene Paul Wigner: the Scientic Papers

. Berlin: Springer, 1993.

url

:

https:

//cds.cern.ch/record/247324

.

[50] D. A. Woodside. Three-vector and scalar eld identities and uniqueness theorems

in Euclidean and Minkowski spaces. In:

American Journal of Physics

77.5 (2009),

pp. 438446.

[51] O. Zaimidoroga. An Electroscalar Energy of the Sun: Observation and Research. In:

Journal of Modern Physics

07 (Jan. 2016), pp. 806818.

[52] S. Zeiner-Gundersen and S. Olafsson. Hydrogen reactor for Rydberg Matter and Ul-

tra Dense Hydrogen, a replication of Leif Holmlid. In: International Conference on

Condensed Matter Nuclear Science, ICCF-21. Fort Collins, USA, 2018.

18